Prepared for submission to JHEP
Introduction
Topological defect lines are by now a standard tool in two-dimensional conformal field theory: they encode the symmetries of a theory, including the non-invertible ones, they act on bulk and boundary fields, and they constrain renormalisation-group flows [1 , 2 , 3 , 4 , 5 ] . In a diagonal minimal model M ( p , p + 1 ) M(p,p+1) M ( p , p + 1 ) D ( r , s ) D_{(r,s)} D ( r , s ) [6 ] asked what happens when such a line is perturbed by one of its own relevant defect fields: the defect is no longer topological, and it flows, in the manner of a boundary RG flow, to a new conformal interface, which may be topological again or may be a genuinely transmitting, non-topological defect. Perturbed topological defects of minimal models have been studied from several angles—chiral perturbations, which preserve pure transmission while generally breaking conformal invariance away from fixed points [7 , 8 ] ; RG interfaces between the UV and IR theories of a bulk flow [9 , 10 ] ; the defect flows of the tricritical Ising model [11 ] —but the non-topological endpoints of the KRW flows are known only through perturbation theory in 1 / p 1/p 1/ p [6 , 12 , 13 ] and through level-truncated string field theory [14 ] . A first-principles, non-perturbative lattice determination of these interfaces has been missing.
In this paper, we study the flow of the ( 2 , 1 ) (2,1) ( 2 , 1 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) \phi_{(1,3)} \otimes\bar{\phi}_{(3,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) p = 4 p=4 p = 4 c = 7 / 10 c=7/10 c = 7/10 p = 5 p=5 p = 5 M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) c = 4 / 5 c=4/5 c = 4/5 p p p D ( 1 , 2 ) D_{(1,2)} D ( 1 , 2 ) λ l ϕ ( 1 , 3 ) ⊗ 1 + λ r 1 ⊗ ϕ ˉ ( 1 , 3 ) \lambda_l\phi_{(1,3)}\otimes1+\lambda_r 1\otimes\bar{\phi}_{(1,3)} λ l ϕ ( 1 , 3 ) ⊗ 1 + λ r 1 ⊗ ϕ ˉ ( 1 , 3 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) \phi_{(1,3)} \otimes\bar{\phi}_{(3,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) λ l = λ r > 0 \lambda_l=\lambda_r>0 λ l = λ r > 0 D ( 1 , 2 ) D_{(1,2)} D ( 1 , 2 ) p = 5 p=5 p = 5 C 5 C_5 C 5 g g g p = 4 p=4 p = 4 η = D ( 3 , 1 ) \eta=D_{(3,1)} η = D ( 3 , 1 ) C p C_p C p p p p C p C_p C p C 5 C_5 C 5 η \eta η p → 4 p\to4 p → 4 η = D ( 1 , 4 ) \eta=D_{(1,4)} η = D ( 1 , 4 ) p = 4 p=4 p = 4 D ( 1 , 4 ) D_{(1,4)} D ( 1 , 4 ) p p p p → 4 p\to4 p → 4 C p C_p C p D ( 1 , 4 ) D_{(1,4)} D ( 1 , 4 )
Two facts make a lattice construction possible, and they are the two themes of the paper. The first is a property of the flow itself: it preserves the ( r , 1 ) (r,1) ( r , 1 ) M ( p , p + 1 ) M(p,p+1) M ( p , p + 1 ) C p \mathcal{C}_p C p S U ( 2 ) p − 2 SU(2)_{p-2} S U ( 2 ) p − 2 [4 , 5 ] .1 { 1 , σ ′ = D ( 2 , 1 ) , η = D ( 3 , 1 ) } \{\mathbf{1},\sigma'=D_{(2,1)},\eta=D_{(3,1)}\} { 1 , σ ′ = D ( 2 , 1 ) , η = D ( 3 , 1 ) } p = 4 p=4 p = 4 ⊠ Z 2 \boxtimes\mathbb{Z}_2 ⊠ Z 2 D ( 2 , 1 ) = w = τ F i b ⊗ η D_{(2,1)}=\mathbf{w}=\tau_{\mathrm{Fib}}\otimes\eta D ( 2 , 1 ) = w = τ Fib ⊗ η φ = 1 + 5 2 \varphi= \frac{1+\sqrt{5}}{2} φ = 2 1 + 5 p = 5 p = 5 p = 5 § 2 : a line x x x x ⊗ D ( 2 , 1 ) x \otimes D_{(2,1)} x ⊗ D ( 2 , 1 ) ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) \phi_{(1,3)} \otimes\bar{\phi}_{(3,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) Σ × [ − 1 , 1 ] \Sigma\times[-1,1] Σ × [ − 1 , 1 ] D ( 1 , 2 ) D_{(1,2)} D ( 1 , 2 ) A p − 1 A_{p-1} A p − 1 [15 , 16 ] is precisely an anyon chain [17 , 18 ] of C p C_p C p p p p S U ( 2 ) p − 2 SU(2)_{p-2} S U ( 2 ) p − 2 w w w 2 ( r , 1 ) (r,1) ( r , 1 ) F F F R R R [19 , 20 ] and of the Temperley–Lieb constructions of [21 , 22 , 23 ] . (For the special case p = 4 p = 4 p = 4 C 4 C_4 C 4 [24 ] within the fusion-category lattice models of [25 ] .) Because the symmetry preserved along the flow is the one the lattice realises exactly, the whole flow can be followed with the symmetry imposed, sector by sector; and the perturbing field has a transparent meaning in this language, as a quantum-dimension-weighted combination of the projectors onto the two fusion channels of the defect anyon and its neighbouring domain wall.
The paper is organised as follows. § 2 reviews the KRW flows and establishes, in the continuum, which symmetry the flow D ( 2 , 1 ) → C D_{(2,1)} \to C D ( 2 , 1 ) → C § 3 builds the lattice model: the dilute height chain, the topological seam, the lattice symmetry operators, the clean deformation operator, the symmetry-resolved numerics and the observables used to characterise an interface. § 4 and § 5 report the results at p = 4 p = 4 p = 4 p = 5 p = 5 p = 5 § 6 reformulates the construction as a loop model valid at continuous p p p § 7 follows the family C p C_p C p 4 ≤ p ≤ 5 4 \leq p \leq5 4 ≤ p ≤ 5 p = 4 p = 4 p = 4 § 8 summarises and compares with conformal perturbation theory, integrable defect flows and string field theory.
The KRW flow and the symmetry it preserves
Perturbed topological defects: the work of Kormos, Runkel and Watts
A topological defect D D D [ L n , D ] = [ L ˉ n , D ] = 0 [L_n,D] = [\bar{L}_n,D] = 0 [ L n , D ] = [ L ˉ n , D ] = 0 D ( r , s ) D_{(r,s)} D ( r , s ) [1 , 2 ] , one for each Kac label, which act on the primary field ϕ ( r ′ , s ′ ) \phi_{(r',s')} ϕ ( r ′ , s ′ ) S ( r , s ) , ( r ′ , s ′ ) / S ( 1 , 1 ) , ( r ′ , s ′ ) S_{(r,s),(r',s')}/S_{(1,1),(r',s')} S ( r , s ) , ( r ′ , s ′ ) / S ( 1 , 1 ) , ( r ′ , s ′ ) d ( r , s ) = S ( r , s ) , ( 1 , 1 ) / S ( 1 , 1 ) , ( 1 , 1 ) d_{(r,s)} = S_{(r,s),(1,1)}/S_{(1,1),(1,1)} d ( r , s ) = S ( r , s ) , ( 1 , 1 ) / S ( 1 , 1 ) , ( 1 , 1 ) g g g [26 ] , and only the ( 1 , 1 ) (1,1) ( 1 , 1 ) η = D ( p − 1 , 1 ) ≅ D ( 1 , p ) \eta= D_{(p-1,1)} \cong D_{(1,p)} η = D ( p − 1 , 1 ) ≅ D ( 1 , p ) g = 1 g = 1 g = 1 D D D D ⊗ D ˉ D \otimes\bar{D} D ⊗ D ˉ § 2.2 ), and it contains in general chiral fields of weight ( h , 0 ) (h,0) ( h , 0 ) ( 0 , h ) (0,h) ( 0 , h )
Kormos, Runkel and Watts [6 ] considered the perturbation of a topological defect by its relevant defect fields and asked where the resulting flow ends. Their starting point is the defect D ( 1 , 2 ) D_{(1,2)} D ( 1 , 2 ) M ( p , p + 1 ) M(p,p+1) M ( p , p + 1 ) ϕ = ϕ ( 1 , 3 ) ⊗ 1 \phi= \phi_{(1,3)} \otimes1 ϕ = ϕ ( 1 , 3 ) ⊗ 1 ϕ ˉ = 1 ⊗ ϕ ˉ ( 1 , 3 ) \bar{\phi} = 1 \otimes\bar{\phi}_{(1,3)} ϕ ˉ = 1 ⊗ ϕ ˉ ( 1 , 3 ) ( h 1 , 3 , 0 ) (h_{1,3},0) ( h 1 , 3 , 0 ) ( 0 , h 1 , 3 ) (0,h_{1,3}) ( 0 , h 1 , 3 ) h 1 , 3 = ( p − 1 ) / ( p + 1 ) h_{1,3} = (p-1)/(p+1) h 1 , 3 = ( p − 1 ) / ( p + 1 )
D ( 1 , 2 ) ( λ l , λ r ) = D ( 1 , 2 ) exp ( λ l ∫ ϕ + λ r ∫ ϕ ˉ ) (1) D_{(1,2)}(\lambda_l,\lambda_r) = D_{(1,2)} \exp\left(\lambda_l \int\phi+ \lambda_r \int\bar{\phi}\right) \tag*{(1)} D ( 1 , 2 ) ( λ l , λ r ) = D ( 1 , 2 ) exp ( λ l ∫ ϕ + λ r ∫ ϕ ˉ ) (1) with y = 1 − h 1 , 3 = 2 / ( p + 1 ) y = 1 - h_{1,3} = 2/(p+1) y = 1 − h 1 , 3 = 2/ ( p + 1 ) y y y Figure 1 , and three of its features organise everything that follows.
Figure 1. The space of flows of the (1, 2) defect perturbed by λ l ϕ + λ r ϕ ˉ \lambda_l\phi+\lambda_r\bar{\phi} λ l ϕ + λ r ϕ ˉ [6 ] (schematic; p > 3 p > 3 p > 3 D ′ = D ( 2 , 1 ) D' = D_{(2,1)} D ′ = D ( 2 , 1 ) I I I C C C I I I F F F D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) C C C ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) \phi_{(1,3)} \otimes\bar{\phi}_{(3,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) I I I D ( 2 , 1 ) → C D_{(2,1)} \to C D ( 2 , 1 ) → C
First, a purely chiral perturbation (λ r = 0 \lambda_r = 0 λ r = 0 λ l = 0 \lambda_l = 0 λ l = 0 [7 , 8 ] , but need not preserve conformal invariance along the flow. A conformal endpoint of such a flow is topological; matching the action on the boundary state ∥ 1 , 1 ⟩ ⟩ \|1,1\rangle\rangle ∥1 , 1 ⟩⟩ [6 ] , §3.1): the positive chiral flows of D ( 1 , 2 ) D_{(1,2)} D ( 1 , 2 ) D ′ = D ( 2 , 1 ) D' = D_{(2,1)} D ′ = D ( 2 , 1 ) I I I [27 , 28 ] . In the present paper the perturbation is non-chiral and the endpoint is a genuinely conformal defect (or, at p = 4 p = 4 p = 4 λ l λ r ≠ 0 \lambda_l\lambda_r \ne0 λ l λ r = 0 I I I λ l = λ r < 0 \lambda_l = \lambda_r < 0 λ l = λ r < 0 F F F ( r , 1 ) (r,1) ( r , 1 ) C C C c = 1 c = 1 c = 1 C C C [6 ] , abstract and §6). This picture is established for large p p p M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) M ( 4 , 5 ) M(4,5) M ( 4 , 5 ) [6 ] , §6.2.4). For C C C g g g y y y p p p ln g C − ln g D ( 2 , 1 ) = − π 2 4 y 3 + O ( y 4 ) \ln g_C - \ln g_{D_{(2,1)}} = -\frac{\pi^2}{4}y^3 + O(y^4) ln g C − ln g D ( 2 , 1 ) = − 4 π 2 y 3 + O ( y 4 ) [6 , 14 ] and R C = 1 − T = 9 8 π 2 y 4 + O ( y 5 ) R_C = 1 - \mathcal{T} = \frac{9}{8}\pi^2 y^4 + O(y^5) R C = 1 − T = 8 9 π 2 y 4 + O ( y 5 ) [12 ] . Third, and most relevant here, the boundaries of these basins of attraction are themselves flows out of D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) D ′ D' D ′ C C C ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) \phi_{(1,3)} \otimes\bar{\phi}_{(3,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) λ r > λ l \lambda_r > \lambda_l λ r > λ l ϕ ( 3 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) \phi_{(3,3)} \otimes\bar{\phi}_{(1,3)} ϕ ( 3 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) λ l > λ r \lambda_l > \lambda_r λ l > λ r D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) I I I D ( 1 , 2 ) D_{(1,2)} D ( 1 , 2 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) \phi_{(1,3)} \otimes\bar{\phi}_{(3,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) Δ = h 1 , 3 + h 3 , 3 = 1 − 2 ( p − 1 ) / ( p ( p + 1 ) ) \Delta= h_{1,3} + h_{3,3} = 1 - 2(p - 1)/(p(p + 1)) Δ = h 1 , 3 + h 3 , 3 = 1 − 2 ( p − 1 ) / ( p ( p + 1 )) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) C C C I I I D ( 2 , 1 ) → C D_{(2,1)} \to C D ( 2 , 1 ) → C
KRW also observed that the perturbed defects (1) commute with the topological lines D ( r , 1 ) D_{(r,1)} D ( r , 1 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) D ( r , 1 ) D_{(r,1)} D ( r , 1 )
Defect operators of D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 )
In radial quantisation about a point of the ring, the states of a ring crossed by a line D D D D D D defect operators of D D D D D D [2 , 4 ]
H D × D = ⨁ k N D D k H k , H k = ⨁ i , j N k i j V i ⊗ V ‾ j , (2) \mathcal{H}_{D\times D}=\bigoplus_k N^k_{DD}\mathcal{H}_k,\qquad\mathcal{H}_k=\bigoplus_{i,j}N^j_{ki}V_i\otimes\overline{V}_j, \tag*{(2)} H D × D = k ⨁ N D D k H k , H k = i , j ⨁ N k i j V i ⊗ V j , (2) where H k \mathcal{H}_k H k k k k H 1 \mathcal{H}_1 H 1 i = ( r ′ , s ) i=(r',s) i = ( r ′ , s ) j = ( r ′ ′ , s ) j=(r'',s) j = ( r ′′ , s ) L L L D = D ( 2 , 1 ) D=D_{(2,1)} D = D ( 2 , 1 ) D × D = D ( 1 , 1 ) + D ( 3 , 1 ) D\times D=D_{(1,1)}+D_{(3,1)} D × D = D ( 1 , 1 ) + D ( 3 , 1 ) H 1 ⊕ H ( 3 , 1 ) \mathcal{H}_1\oplus\mathcal{H}_{(3,1)} H 1 ⊕ H ( 3 , 1 ) V i ⊗ V ‾ i V_i\otimes\overline{V}_i V i ⊗ V i V ( r ′ , s ) ⊗ V ‾ ( r ′ ′ , s ) V_{(r',s)}\otimes\overline{V}_{(r'',s)} V ( r ′ , s ) ⊗ V ( r ′′ , s ) r ′ ′ ∈ r ′ ⊗ 3 r''\in r'\otimes3 r ′′ ∈ r ′ ⊗ 3 D D D H D = ⨁ i , j N D i j V i ⊗ V ‾ j \mathcal{H}_D=\bigoplus_{i,j}N^j_{Di}V_i\otimes\overline{V}_j H D = ⨁ i , j N D i j V i ⊗ V j § 4 uses.
Lassos: a crossing is a choice
A wrapped ( r , 1 ) (r,1) ( r , 1 ) x x x u ∈ Hom ( x ⊗ D , D ⊗ x ) u\in\operatorname{Hom}(x\otimes D,D\otimes x) u ∈ Hom ( x ⊗ D , D ⊗ x ) x ⊗ D x\otimes D x ⊗ D u D ′ u_{D'} u D ′ x x x D D D D ′ ∈ x ⊗ D D'\in x\otimes D D ′ ∈ x ⊗ D D ( 1 , 2 ) D_{(1,2)} D ( 1 , 2 ) x = ( r , 1 ) x=(r,1) x = ( r , 1 ) ( r , 1 ) ⊗ ( 1 , 2 ) = ( r , 2 ) (r,1)\otimes(1,2)=(r,2) ( r , 1 ) ⊗ ( 1 , 2 ) = ( r , 2 ) D ( 2 , 1 ) = w D_{(2,1)}=\mathbf{w} D ( 2 , 1 ) = w x = w x=\mathbf{w} x = w w ⊗ w = 1 ⊕ adj \mathbf{w}\otimes\mathbf{w}=\mathbf{1}\oplus\operatorname{adj} w ⊗ w = 1 ⊕ adj x = η x=\eta x = η x x x lasso , an element of the tube algebra of D D D [5 , 29 ] . As long as D D D x x x D D D x x x D D D x x x ( i , j ˉ ) (i,\bar{j}) ( i , j ˉ ) C ⊠ C ‾ \mathcal{C}\boxtimes\overline{\mathcal{C}} C ⊠ C [29 , 30 ] , whereas a generic junction need not act diagonally within an irreducible tube-algebra representation or on the corresponding defect multiplicity spaces. It does not mix distinct irreducible representation blocks. We denote the two half-braidings by B B B B − 1 B^{-1} B − 1 B = F R F − 1 B=FRF^{-1} B = F R F − 1 § 3 and its inverse, and their explicit expansion in the channel basis is given in § 3.3 , eq. (19) . On a defect field V i ⊗ V ‾ j V_i\otimes\overline{V}_j V i ⊗ V j S x i / S 0 i S_{xi}/S_{0i} S x i / S 0 i S x j / S 0 j S_{xj}/S_{0j} S x j / S 0 j
The reason a crossing reads one chiral label is most transparent in the three-dimensional description of the diagonal theory [3 , 4 ] , Figure 2 . There the CFT on a surface Σ \Sigma Σ Σ × [ − 1 , 1 ] \Sigma\times[-1,1] Σ × [ − 1 , 1 ] Σ × { + 1 } \Sigma\times\{+1\} Σ × { + 1 } Σ × { − 1 } \Sigma\times\{-1\} Σ × { − 1 } x x x D D D Σ × { 0 } \Sigma\times\{0\} Σ × { 0 } ( i , j ˉ ) (i,\bar{j}) ( i , j ˉ ) i i i j j j i = j i=j i = j D D D D D D x x x D D D x x x D D D i i i S x i / S 0 i S_{xi}/S_{0i} S x i / S 0 i j j j S x j / S 0 j S_{xj}/S_{0j} S x j / S 0 j i = j i=j i = j i ≠ j i \ne j i = j F F F θ a \theta_a θ a § 3.3 ); which of the two reads the holomorphic label is a matter of orientation conventions, fixed here empirically by the undeformed-ring check of Table 1.
Figure 2. Why a lasso reads one chiral label. In the Reshetikhin–Turaev description on Σ × [ − 1 , 1 ] \Sigma\times[-1,1] Σ × [ − 1 , 1 ] ( i , j ˉ ) (i,\bar{j}) ( i , j ˉ ) P P P i i i j ˉ \bar{j} j ˉ x x x D D D i i i S x i / S 0 i S_{xi}/S_{0i} S x i / S 0 i j ˉ \bar{j} j ˉ S x j ˉ / S 0 j ˉ S_{x\bar{j}}/S_{0\bar{j}} S x j ˉ / S 0 j ˉ
The deformation fixes the lasso
Once D D D ϕ \phi ϕ x x x ϕ \phi ϕ D D D
u ∘ ( i d x ⊗ ϕ ) = ( ϕ ⊗ i d x ) ∘ u , (3) u \circ(\mathrm{id}_x \otimes\phi) = (\phi\otimes\mathrm{id}_x) \circ u, \tag*{(3)} u ∘ ( id x ⊗ ϕ ) = ( ϕ ⊗ id x ) ∘ u , (3) read with time running along D D D ϕ \phi ϕ D D D ϕ : D → D \phi: D \to D ϕ : D → D i d x ⊗ ϕ \mathrm{id}_x \otimes\phi id x ⊗ ϕ ϕ ⊗ i d x \phi\otimes\mathrm{id}_x ϕ ⊗ id x x x x D D D u : x ⊗ D → D ⊗ x u : x \otimes D \to D \otimes x u : x ⊗ D → D ⊗ x x ⊗ D → D ⊗ x x \otimes D \to D \otimes x x ⊗ D → D ⊗ x D D D ϕ \phi ϕ x x x ϕ \phi ϕ Y x ϕ = d x ϕ Y_x\phi= d_x\phi Y x ϕ = d x ϕ d x d_x d x Figure 3 ). Since the over-crossing reads the left label of ϕ \phi ϕ ϕ = ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) \phi= \phi_{(1,3)} \otimes\bar{\phi}_{(3,3)} ϕ = ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) ( 1 , 3 ) (1,3) ( 1 , 3 ) r ′ = 1 r' = 1 r ′ = 1 d x d_x d x ( r , 1 ) (r,1) ( r , 1 ) ϕ \phi ϕ ( 3 , 3 ) (3,3) ( 3 , 3 ) Y ( 2 , 1 ) = − ϕ − 1 ≠ ϕ Y_{(2,1)} = -\phi^{-1} \ne\phi Y ( 2 , 1 ) = − ϕ − 1 = ϕ p = 5 p=5 p = 5 ( 3 , 3 ) = ( 2 , 3 ) (3,3) = (2,3) ( 3 , 3 ) = ( 2 , 3 ) Y ( 2 , 1 ) = − 2 ≠ 2 Y_{(2,1)} = -\sqrt{2} \ne\sqrt{2} Y ( 2 , 1 ) = − 2 = 2 p = 4 p=4 p = 4 ( 3 , 3 ) = ( 1 , 2 ) (3,3) = (1,2) ( 3 , 3 ) = ( 1 , 2 ) D ( 2 , 1 ) → C D_{(2,1)} \to C D ( 2 , 1 ) → C x x x ϕ ( 3 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) \phi_{(3,3)} \otimes\bar{\phi}_{(1,3)} ϕ ( 3 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) D ( 2 , 1 ) → C D_{(2,1)} \to C D ( 2 , 1 ) → C Δ = 11 / 15 \Delta= 11/15 Δ = 11/15 a + a+ a + c + c+ c +
Figure 3. Why the deformation selects a lasso. (a) The junction u ∈ Hom ( x ⊗ D , D ⊗ x ) u\in\operatorname{Hom}(x\otimes D,D\otimes x) u ∈ Hom ( x ⊗ D , D ⊗ x ) D ( 1 , 2 ) D_{(1,2)} D ( 1 , 2 ) ( r , 1 ) ⊗ ( 1 , 2 ) = ( r , 2 ) (r,1)\otimes(1,2)=(r,2) ( r , 1 ) ⊗ ( 1 , 2 ) = ( r , 2 ) D ( 2 , 1 ) = w D_{(2,1)}=w D ( 2 , 1 ) = w w ⊗ w = 1 ⊕ adj w\otimes w=1\oplus\operatorname{adj} w ⊗ w = 1 ⊕ adj B B B B − 1 B^{-1} B − 1 x x x ϕ \phi ϕ (3) , i.e. only if ϕ \phi ϕ ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) \phi_{(1,3)}\otimes\bar{\phi}_{(3,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 )
With two strands deformed by the same field, the preserved operator is the lasso with the over-crossing at both strands, and with this choice (2) is diagonalised by the left labels:
Y ( r , 1 ) ∣ V ( r ′ , s ) ⊗ V ‾ ( r ′ ′ , s ) = S ( r , 1 ) , ( r ′ , s ) S ( 1 , 1 ) , ( r ′ , s ) = ( − 1 ) ( r − 1 ) s sin ( π r r ′ p + 1 p ) sin ( π r ′ r p + 1 p ) (4) Y_{(r,1)}\big|_{V_{(r',s)}\otimes\overline{V}_{(r'',s)}}=\frac{S_{(r,1),(r',s)}}{S_{(1,1),(r',s)}}=(-1)^{(r-1)s}\frac{\sin\left(\pi rr'\frac{p+1}{p}\right)}{\sin\left(\pi r'r\frac{p+1}{p}\right)} \tag*{(4)} Y ( r , 1 ) V ( r ′ , s ) ⊗ V ( r ′′ , s ) = S ( 1 , 1 ) , ( r ′ , s ) S ( r , 1 ) , ( r ′ , s ) = ( − 1 ) ( r − 1 ) s sin ( π r ′ r p p + 1 ) sin ( π r r ′ p p + 1 ) (4) independently of r ′ ′ r'' r ′′ H k \mathcal{H}_k H k S U ( 2 ) p − 2 SU(2)_{p-2} S U ( 2 ) p − 2 p = 4 p=4 p = 4 Y ( 2 , 1 ) ∈ { + 2 , 0 , − 2 } Y_{(2,1)}\in\{+\sqrt{2},0,-\sqrt{2}\} Y ( 2 , 1 ) ∈ { + 2 , 0 , − 2 } η = Y ( 3 , 1 ) = ± 1 \eta=Y_{(3,1)}=\pm1 η = Y ( 3 , 1 ) = ± 1 p = 5 p=5 p = 5 Y ( 2 , 1 ) = ± φ , ± φ − 1 Y_{(2,1)}=\pm\varphi,\pm\varphi^{-1} Y ( 2 , 1 ) = ± φ , ± φ − 1 η = Y ( 4 , 1 ) = ± 1 \eta=Y_{(4,1)}=\pm1 η = Y ( 4 , 1 ) = ± 1 Y ( 2 , 1 ) = Y τ F i b η Y_{(2,1)}=Y_{\mathrm{\tau Fib}}\eta Y ( 2 , 1 ) = Y τ Fib η
Since the preserved lasso commutes with the deformed Hamiltonian, these eigenvalues are conserved along the whole flow: every operator of the infrared interface inherits a ( Y ( 2 , 1 ) , η ) (Y_{(2,1)},\eta) ( Y ( 2 , 1 ) , η ) C C C § 4 , § 5 ); at the endpoint η \eta η x ⊗ η x\otimes\eta x ⊗ η
Kormos, Runkel and Watts prove, by a matrix-element argument ([6 ] , §2.4.2), that the perturbed defect (1) commutes with all the lines D ( r , 1 ) D_{(r,1)} D ( r , 1 ) [6 ] , §3.2). In their setting the statement is unambiguous for two reasons that the preceding analysis makes explicit: the junction space of x = ( r , 1 ) x=(r,1) x = ( r , 1 ) D ( 1 , 2 ) D_{(1,2)} D ( 1 , 2 ) ( 1 , 3 ) ⊗ ( 1 , 1 ) (1,3)\otimes(1,1) ( 1 , 3 ) ⊗ ( 1 , 1 ) ( 1 , 1 ) ⊗ ( 1 , 3 ) (1,1)\otimes(1,3) ( 1 , 1 ) ⊗ ( 1 , 3 ) ( 1 , s ) (1,s) ( 1 , s ) D ( r , 1 ) D_{(r,1)} D ( r , 1 ) (3) holds automatically. The junction never has to be discussed. Our flow starts from D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) ( r , 1 ) (r,1) ( r , 1 ) D ( 2 , 1 ) → C D_{(2,1)}\to C D ( 2 , 1 ) → C
This is also what makes a lattice realisation possible. The preserved symmetry is a line of the fusion category C p \mathcal{C}_p C p F F F A p − 1 A_{p-1} A p − 1 (4) are then imposed exactly, sector by sector, along the whole flow.
The lattice model
The dilute RSOS chain
Degrees of freedom. We use the following convention throughout the paper: the height variables live on the bonds of the chain, and the anyons (the domain walls, or their absence) live on the sites between consecutive bonds. Consider a ring of L L L j j j
a j ∈ { 1 , … , p − 1 } (three values for tricritical Ising, four for M ( 5 , 6 ) ) , (5) a_j \in\{1,\ldots,p-1\} \qquad\text{(three values for tricritical Ising, four for }M(5,6)\text{)}, \tag*{(5)} a j ∈ { 1 , … , p − 1 } (three values for tricritical Ising, four for M ( 5 , 6 ) ) , (5) subject to the admissibility rule that neighbouring heights differ by at most one,
∣ a j − a j + 1 ∣ ≤ 1 at every site . (6) \lvert a_j-a_{j+1}\rvert\le1 \qquad\text{at every site}. \tag*{(6)} ∣ a j − a j + 1 ∣ ≤ 1 at every site . (6) Between two consecutive bonds there is a site . A site carries no variable of its own; we call it empty if the two heights around it are equal and occupied if they differ,
site between bonds a , b : a = b ⇒ empty , a ≠ b ⇒ occupied . (7) \text{site between bonds }a,b:\qquad a=b\Rightarrow\text{ empty},\qquad a\ne b\Rightarrow\text{ occupied}. \tag*{(7)} site between bonds a , b : a = b ⇒ empty , a = b ⇒ occupied . (7) An occupied site is a domain wall between two height domains. The word “dilute” refers to the fact that empty sites are allowed (in the ordinary, dense A p − 1 A_{p-1} A p − 1 Figure 4 . The dense A p − 1 A_{p-1} A p − 1 [31 ] , whose critical points realise the unitary minimal models [32 , 33 ] ; the dilute A p − 1 A_{p-1} A p − 1 [15 ] from the O ( n ) O(n) O ( n ) [34 , 35 ] .
Figure 4. The dilute A p − 1 A_{p-1} A p − 1 p = 5 p=5 p = 5 j j j a j ∈ { 1 , … , p − 1 } a_j \in\{1,\ldots,p-1\} a j ∈ { 1 , … , p − 1 } (11) hops these walls and creates/annihilates them in pairs.
Local Hilbert space. The heights are the only degrees of freedom, so the local Hilbert space of a bond is
H j = span { ∣ a j ⟩ : a j = 1 , … , p − 1 } , dim H j = p − 1 , (8) \mathcal{H}_j = \operatorname{span}\{ \lvert a_j \rangle: a_j = 1,\ldots,p-1 \}, \qquad\dim\mathcal{H}_j = p-1, \tag*{(8)} H j = span {∣ a j ⟩ : a j = 1 , … , p − 1 } , dim H j = p − 1 , (8) a three-state space at p = 4 p=4 p = 4 p = 5 p=5 p = 5 ⨂ j H j \bigotimes_j \mathcal{H}_j ⨂ j H j admissible subspace, spanned by the height configurations obeying (6) at every site. Every operator introduced below (the Hamiltonian, the seam, the wrapped lines, the deformation) preserves this subspace. In exact diagonalisation we enumerate the admissible configurations and work in that basis; in DMRG we keep the full ( p − 1 ) (p-1) ( p − 1 )
Anyon-chain reading. The same data have a second, equivalent description that will be essential for the defect. Identify a height a a a C p C_p C p S U ( 2 ) p − 2 SU(2)_{p-2} S U ( 2 ) p − 2 j = a − 1 2 j=\frac{a-1}{2} j = 2 a − 1 1 2 \frac{1}{2} 2 1
d w = 2 cos π p = S 2 S 1 , S a ≡ sin π a p (9) d_{\mathrm{w}} = 2\cos\frac{\pi}{p} = \frac{S_2}{S_1}, \qquad S_a \equiv\sin\frac{\pi a}{p} \tag*{(9)} d w = 2 cos p π = S 1 S 2 , S a ≡ sin p π a (9) (d w = 2 d_{\mathrm{w}}=\sqrt{2} d w = 2 p = 4 p=4 p = 4 d w = φ = 1.618 d_{\mathrm{w}}=\varphi=1.618 d w = φ = 1.618 p = 5 p=5 p = 5 S a S_a S a (6) says exactly that a j + 1 ∈ a j ⊗ ( 1 ⊕ w ) a_{j+1}\in a_j\otimes(1\oplus\mathrm{w}) a j + 1 ∈ a j ⊗ ( 1 ⊕ w ) { a j } \{a_j\} { a j } fusion-tree basis of a chain of w anyons sitting on the occupied sites, with the heights recording the intermediate total charge as one moves along the ring. “A w lives on every domain wall” is the precise statement, not just a picture. The two kinds of labels play different categorical roles, and it is worth stating them precisely, since this is the structure the defect construction will exploit. The site labels 1 and w w w x x x C = C p \mathcal{C} = \mathcal{C}_p C = C p module category M \mathcal{M} M C \mathcal{C} C C \mathcal{C} C a ↦ a ⊗ w a \mapsto a \otimes w a ↦ a ⊗ w A p − 1 A_{p-1} A p − 1 M \mathcal{M} M C \mathcal{C} C S U ( 2 ) p − 2 SU(2)_{p-2} S U ( 2 ) p − 2 D D D E E E [33 ] , the heights instead label vertices of other ADE graphs, illustrating why the action on heights must be specified separately from the fusion rules, and the module-category structure is the general statement that an anyon chain is defined by a pair ( C , M ) (\mathcal{C},\mathcal{M}) ( C , M ) [17 , 18 ] . This is the lattice counterpart of the correspondence between module categories and conformal boundary conditions, or equivalently between module categories and the ways a chiral algebra can be completed to a full CFT [36 , 37 ] ; on the lattice it is the organising principle of the fusion-category lattice models of [20 , 25 ] and of the general framework of dualities and non-invertible symmetries in one-dimensional quantum lattice models developed by Lootens, Delcamp, Ortiz and Verstraete [38 , 39 ] , which is reviewed in Delcamp’s IHES lectures [40 ] .
The Hamiltonian. The critical dynamics is a nearest-neighbour Hamiltonian
H 0 = − ∑ j = 1 L h j , (10) H_0 = -\sum_{j=1}^{L} h_j, \tag*{(10)} H 0 = − j = 1 ∑ L h j , (10) where h j h_j h j l = a j − 1 l = a_{j-1} l = a j − 1 a = a j a = a_j a = a j r = a j + 1 r = a_{j+1} r = a j + 1 a → a ′ a \to a' a → a ′ P l = ∣ l ⟩ ⟨ l ∣ P_l = |l\rangle\langle l| P l = ∣ l ⟩ ⟨ l ∣
h j = ∑ l , r ∑ a , a ′ W a → a ′ ( l , r ) P l ∣ a ′ ⟩ ⟨ a ∣ P r , (11) h_j = \sum_{l,r}\sum_{a,a'} W_{a\to a'}^{(l,r)} P_l \lvert a'\rangle\langle a\rvert P_r, \tag*{(11)} h j = l , r ∑ a , a ′ ∑ W a → a ′ ( l , r ) P l ∣ a ′ ⟩ ⟨ a ∣ P r , (11) so h j h_j h j ( p − 1 ) 3 × ( p − 1 ) 3 (p-1)^3 \times(p-1)^3 ( p − 1 ) 3 × ( p − 1 ) 3 W a → a ′ ( l , r ) W_{a\to a'}^{(l,r)} W a → a ′ ( l , r ) A p − 1 A_{p-1} A p − 1 [34 ] and of Warnaar, Nienhuis and Seaton [15 , 16 ] , taken at the critical branch with crossing parameter
λ = π ( p − 1 ) 4 p ( λ = 3 π 16 at p = 4 , λ = π 5 at p = 5 ) . (12) \lambda= \frac{\pi(p-1)}{4p} \qquad\left(\lambda= \frac{3\pi}{16}\text{ at }p=4,\quad\lambda= \frac{\pi}{5}\text{ at }p=5\right). \tag*{(12)} λ = 4 p π ( p − 1 ) ( λ = 16 3 π at p = 4 , λ = 5 π at p = 5 ) . (12) They depend only on the shape of the move, i.e. on which of the two internal sites are domain walls before (o L = [ l ≠ a ] o_L = [l \ne a] o L = [ l = a ] o R = [ a ≠ r ] o_R = [a \ne r] o R = [ a = r ] o L ′ = [ l ≠ a ′ ] o'_L = [l \ne a'] o L ′ = [ l = a ′ ] o R ′ = [ a ′ ≠ r ] o'_R = [a' \ne r] o R ′ = [ a ′ = r ]
empty ( a = a ′ , o L = o R = 0 ) : w ∅ = sin 3 λ N 0 , one wall ( a = a ′ , o L ≠ o R ) : w ∥ = − sin 2 λ cos 3 λ N 0 , two walls ( a = a ′ , o L = o R = 1 ) : w = = − sin 5 λ N 0 , hop ( o L o R = 0 , wall moves side ) : w ↷ = sin 3 λ N 0 , create/annihilate ( l = r , one bump ) : w ∪ ∩ = sin 2 λ N 0 S m S l , TL ( l = r , both walls, any a ′ ) : w e = − sin λ N 0 S a S a ′ S l . (13) \begin{aligned}
\text{empty }(a=a',\ o_L=o_R=0):\quad&w_{\varnothing}=\frac{\sin3\lambda}{\mathcal{N}_0},\\
\text{one wall }(a=a',\ o_L\ne o_R):\quad&w_{\parallel}=-\frac{\sin2\lambda\cos3\lambda}{\mathcal{N}_0},\\
\text{two walls }(a=a',\ o_L=o_R=1):\quad&w_{=}=-\frac{\sin5\lambda}{\mathcal{N}_0},\\
\text{hop }(o_Lo_R=0,\ \text{wall moves side}):\quad&w_{\curvearrowright}=\frac{\sin3\lambda}{\mathcal{N}_0},\\
\text{create/annihilate }(l=r,\ \text{one bump}):\quad&w_{\cup\cap}=\frac{\sin2\lambda}{\mathcal{N}_0}\sqrt{\frac{S_m}{S_l}},\\
\text{TL }(l=r,\ \text{both walls, any }a'):\quad&w_e=-\frac{\sin\lambda}{\mathcal{N}_0}\frac{\sqrt{S_aS_{a'}}}{S_l}. \tag*{(13)}
\end{aligned} empty ( a = a ′ , o L = o R = 0 ) : one wall ( a = a ′ , o L = o R ) : two walls ( a = a ′ , o L = o R = 1 ) : hop ( o L o R = 0 , wall moves side ) : create/annihilate ( l = r , one bump ) : TL ( l = r , both walls, any a ′ ) : w ∅ = N 0 sin 3 λ , w ∥ = − N 0 sin 2 λ cos 3 λ , w = = − N 0 sin 5 λ , w ↷ = N 0 sin 3 λ , w ∪∩ = N 0 sin 2 λ S l S m , w e = − N 0 sin λ S l S a S a ′ . (13) with N 0 = sin 2 λ sin 3 λ \mathcal{N}_0=\sin2\lambda\sin3\lambda N 0 = sin 2 λ sin 3 λ Figure 5 . Two points of reading are worth making explicit, since they decide the matrix. In the creation/annihilation weight m m m bump height, the middle height of the configuration in which both sites are walls, whichever side of the move it is on: W l → m ( l , l ) = W m → l ( l , l ) = sin 2 λ N 0 S m / S l W^{(l,l)}_{l\to m}=W^{(l,l)}_{m\to l}=\frac{\sin2\lambda}{\mathcal{N}_0}\sqrt{S_m/S_l} W l → m ( l , l ) = W m → l ( l , l ) = N 0 s i n 2 λ S m / S l ∣ m − l ∣ = 1 |m-l|=1 ∣ m − l ∣ = 1 l = r l=r l = r including a ′ = a a'=a a ′ = a W a → a ( l , l ) = w e = − sin λ N 0 S a S l W^{(l,l)}_{a\to a}=w_e=-\frac{\sin\lambda}{\mathcal{N}_0}\frac{S_a}{S_l} W a → a ( l , l ) = w e = − N 0 s i n λ S l S a h j h_j h j H = − ∑ j P j , j + 1 1 H=-\sum_j P^1_{j,j+1} H = − ∑ j P j , j + 1 1 [17 ] , in which every site carries a w w w § 3.4 , where they are needed to build the deformation. The first three are diagonal (the middle height does not change): they are the energies of an empty site, of an isolated wall, and of two adjacent walls. The last three are off-diagonal: a wall hops across the middle bond; a pair of walls is created from, or annihilated into, an empty stretch (this needs l = r l=r l = r Temperley–Lieb move (with the diagonal part just mentioned), in which two adjacent walls with l = r l=r l = r h j h_j h j S a S_a S a d w d_w d w H 0 H_0 H 0 M ( p , p + 1 ) M(p,p+1) M ( p , p + 1 ) c = 7 / 10 c=7/10 c = 7/10 v = 1.778 v=1.778 v = 1.778 p = 4 p=4 p = 4 c = 4 / 5 c=4/5 c = 4/5 v = 5 / 3 v=5/3 v = 5/3 p = 5 p=5 p = 5 [16 , 41 ] ; we use these sound velocities to convert all gaps into scaling dimensions, Δ = ( E − E 0 ) L / ( 2 π v ) \Delta=(E-E_0)L/(2\pi v) Δ = ( E − E 0 ) L / ( 2 π v ) [42 , 43 , 44 ] .
Figure 5. The six face-weight moves of (13) , each drawn as a “before → after” configuration of the three bonds ( l , a , r ) → ( l , a ′ , r ) (l,a,r) \to(l,a',r) ( l , a , r ) → ( l , a ′ , r ) l = r l=r l = r e e e l = r l=r l = r
The topological ( r , 1 ) (r,1) ( r , 1 )
What we want. A defect line is topological if it can be deformed freely without changing any correlation function [2 , 3 , 4 ] . In radial quantisation on the ring, the defect is a static worldline that pierces the spatial circle at one point, between two sites; on the lattice
“topological” means that this insertion commutes with the Hamiltonian in the sense that the strand can be moved from one site to the next without changing the spectrum, and more strongly that the wrapped version of the same line (the operator that transports it once around the ring) commutes with H H H ( r , 1 ) (r,1) ( r , 1 )
The line as an anyon. In the anyon-chain reading of § 3.1 , the ( r , 1 ) (r,1) ( r , 1 ) C p \mathcal{C}_p C p x = r − 1 x=r-1 x = r − 1 x = 1 x=1 x = 1 w = D ( 2 , 1 ) \mathrm{w}=D_{(2,1)} w = D ( 2 , 1 ) x = p − 2 x=p-2 x = p − 2 η = D ( p − 1 , 1 ) \eta=D_{(p-1,1)} η = D ( p − 1 , 1 ) x x x one extra bond c j c_j c j a j a_j a j a j + 1 a_{j+1} a j + 1 x x x
c j ∈ a j ⊗ x ( for x = w : ∣ c j − a j ∣ = 1 ) . (14) c_j \in a_j \otimes x \qquad(\text{for }x=\mathrm{w}: \lvert c_j-a_j\rvert=1). \tag*{(14)} c j ∈ a j ⊗ x ( for x = w : ∣ c j − a j ∣ = 1 ) . (14) So the strand of anyon x x x x = w x=\mathrm{w} x = w a j a_j a j c j c_j c j c j c_j c j a j + 1 a_{j+1} a j + 1 (6) . Away from the strand nothing changes.
Face operators across the strand: braiding. The only question is how the Hamiltonian acts near the strand (Figure 6 ). The face operator h j h_j h j (11) needs the two physical neighbours of the middle bond a j a_j a j a j − 1 a_{j-1} a j − 1 a j + 1 a_{j+1} a j + 1 a j a_j a j c j c_j c j h j h_j h j x x x c j c_j c j a j + 1 a_{j+1} a j + 1 before acting and to put it back afterwards.
Figure 6. How the Hamiltonian acts across the strand. (1) The face operator h j h_j h j ( a j − 1 , a j , a j + 1 ) (a_{j-1},a_j,a_{j+1}) ( a j − 1 , a j , a j + 1 ) a j a_j a j c j c_j c j a j a_j a j a j + 1 a_{j+1} a j + 1 x x x y y y j + 1 j+1 j + 1 B B B (15) ; the label between a j a_j a j b ′ ∈ a j ⊗ y b' \in a_j \otimes y b ′ ∈ a j ⊗ y ( a j − 1 , a j , b ′ ) (a_{j-1},a_j,b') ( a j − 1 , a j , b ′ ) B − 1 B^{-1} B − 1 h j s e a m = B − 1 h j f a c e B h_j^{\mathrm{seam}}=B^{-1}h_j^{\mathrm{face}}B h j seam = B − 1 h j face B a j a_j a j c j c_j c j a j ± 1 a_{j\pm1} a j ± 1 h j h_j h j Figure 21 (b), time upwards: the thick line is the ( 2 , 1 ) (2,1) ( 2 , 1 ) j j j j + 1 j+1 j + 1 x x x y y y B B B j j j j + 1 j+1 j + 1 y y y B − 1 B^{-1} B − 1 § 2.3 .
Moving an anyon x x x y y y y = 1 y=1 y = 1 j + 1 j+1 j + 1 y = w y=w y = w
B = F a y x R y x ( F a x y ) − 1 , (15) B=F^{a y x}R^{y x}(F^{a x y})^{-1}, \tag*{(15)} B = F a y x R y x ( F a x y ) − 1 , (15) where F F F 6 j 6j 6 j R y x R^{y x} R y x [30 , 45 ] , and the matrices act on the intermediate label—i.e. on the strand height c j c_j c j b ′ ∈ a j ⊗ y b'\in a_j\otimes y b ′ ∈ a j ⊗ y
h j s e a m = B − 1 ( h j f a c e ) B , (16) h_j^{\mathrm{seam}}=B^{-1}\left(h_j^{\mathrm{face}}\right)B, \tag*{(16)} h j seam = B − 1 ( h j face ) B , (16) where h j f a c e h_j^{\mathrm{face}} h j face (11) acting on the triple ( a j − 1 , a j , b ′ ) (a_{j-1},a_j,b') ( a j − 1 , a j , b ′ ) B B B h j s e a m h_j^{\mathrm{seam}} h j seam h j f a c e h_j^{\mathrm{face}} h j face L L L p = 4 p=4 p = 4 x = x ′ = w = σ ′ x=x'=w=\sigma' x = x ′ = w = σ ′ η \eta η
F w w w w = 1 2 ( 1 1 1 − 1 ) , R 1 w w = e − 7 i π / 8 , R η w w = e + 5 i π / 8 , F^{www}_{w}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix},\qquad R^{ww}_{1}=e^{-7i\pi/8},\qquad R^{ww}_{\eta}=e^{+5i\pi/8}, F w w w w = 2 1 ( 1 1 1 − 1 ) , R 1 w w = e − 7 iπ /8 , R η w w = e + 5 iπ /8 , so (15) is a 2 × 2 2 \times2 2 × 2 w × w = 1 ⊕ η w \times w = 1 \oplus\eta w × w = 1 ⊕ η p = 5 p = 5 p = 5 d w = φ d_w = \varphi d w = φ R R R R 1 w w = e − i 4 π / 5 R^{ww}_{1} = e^{-i4\pi/5} R 1 w w = e − i 4 π /5 R a d j w w = e + 3 i π / 5 R^{ww}_{\mathrm{adj}} = e^{+3i\pi/5} R adj w w = e + 3 iπ /5
Which category. Fusion rules do not determine a category, and it is worth stating precisely which data the construction uses, since the seam operator is sensitive to them. The ( r , 1 ) (r,1) ( r , 1 ) S U ( 2 ) p − 2 SU(2)_{p-2} S U ( 2 ) p − 2 a = r − 1 = 2 j a = r - 1 = 2j a = r − 1 = 2 j M ( p , p + 1 ) M(p,p+1) M ( p , p + 1 ) ( r , 1 ) (r,1) ( r , 1 ) θ ( r , 1 ) = e 2 π i h ( r , 1 ) \theta_{(r,1)} = e^{2\pi i h_{(r,1)}} θ ( r , 1 ) = e 2 π i h ( r , 1 ) e 2 π i j ( j + 1 ) / p e^{2\pi i j(j+1)/p} e 2 π ij ( j + 1 ) / p i [ a ] i^{[a]} i [ a ] [ a ] = a m o d 2 [a] = a \bmod2 [ a ] = a mod 2 h ( 2 , 1 ) = 3 4 p + 1 4 h_{(2,1)} = \frac{3}{4p} + \frac{1}{4} h ( 2 , 1 ) = 4 p 3 + 4 1 F F F 6 j 6j 6 j S U ( 2 ) p − 2 SU(2)_{p-2} S U ( 2 ) p − 2 Z 2 \mathbb{Z}_2 Z 2 ( − 1 ) [ a ] [ b ] [ c ] [ d ] (-1)^{[a][b][c][d]} ( − 1 ) [ a ] [ b ] [ c ] [ d ] p = 4 p = 4 p = 4 F w w w w = 1 2 ( 1 1 1 − 1 ) F^{www}_{w} = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1\\1 & -1\end{pmatrix} F w w w w = 2 1 ( 1 1 1 − 1 ) S U ( 2 ) 2 SU(2)_2 S U ( 2 ) 2 (24) ; commutation of the wrapped lines (21) with H 0 H_0 H 0 R R R
R c a b = i [ a ] [ b ] ( − 1 ) ( a + b − c ) / 2 q j c ( j c + 1 ) − j a ( j a + 1 ) − j b ( j b + 1 ) , q = e i π / p , (17) R^{ab}_{c} = i^{[a][b]}(-1)^{(a+b-c)/2}q^{j_c(j_c+1)-j_a(j_a+1)-j_b(j_b+1)}, \qquad q = e^{i\pi/p}, \tag*{(17)} R c ab = i [ a ] [ b ] ( − 1 ) ( a + b − c ) /2 q j c ( j c + 1 ) − j a ( j a + 1 ) − j b ( j b + 1 ) , q = e iπ / p , (17) which solve the hexagon equations with these F F F θ w = 1 d w ∑ c d c R c w w = e 2 π i h ( 2 , 1 ) \theta_w = \frac{1}{d_w}\sum_c d_c R^{ww}_c = e^{2\pi i h_{(2,1)}} θ w = d w 1 ∑ c d c R c w w = e 2 π i h ( 2 , 1 ) i [ a ] [ b ] i^{[a][b]} i [ a ] [ b ] (15) is 1 1 1 R c w w R^{ww}_c R c w w h s e a m = B − 1 h B h^{\mathrm{seam}} = B^{-1}hB h seam = B − 1 h B S U ( 2 ) p − 2 SU(2)_{p-2} S U ( 2 ) p − 2 − q − 3 / 2 -q^{-3/2} − q − 3/2 q 1 / 2 q^{1/2} q 1/2 (17) the one-strand ring at p = 4 p = 4 p = 4 L = 6 L = 6 L = 6 0.298 0.298 0.298 0.354 0.354 0.354 0.757 0.757 0.757 D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) 0.1375 0.1375 0.1375 0.4375 0.4375 0.4375 0.6375 0.6375 0.6375 (17) reproduces to 10 − 3 10^{-3} 1 0 − 3 A 1 + A − 1 e A1 + A^{-1}e A 1 + A − 1 e A = i q 1 / 2 A = iq^{1/2} A = i q 1/2 (47) , which agrees with (17) channel by channel (A = i q 1 / 2 = R a d j w w A = iq^{1/2} = R^{ww}_{\mathrm{adj}} A = i q 1/2 = R adj w w A + A − 1 d w = − i q − 3 / 2 = R 1 w w A + A^{-1}d_w = -iq^{-3/2} = R^{ww}_{1} A + A − 1 d w = − i q − 3/2 = R 1 w w (24) fixes the height-basis normalization of the identity-channel projector.
Why this is topological: the wrapped lines. The construction inherits the consistency of the category. The operator Y x Y_x Y x x x x F F F
( Y x ) { b } { a } = ∏ k F b k b k − 1 a k x a k − 1 ℓ k , ℓ k = anyon on slot k ( 1 , w , or x at the strand ) , (18) (Y_x)_{\{b\}\{a\}} = \prod_k F^{x a_{k-1} \ell_k}_{b_k b_{k-1} a_k}, \qquad\ell_k = \text{anyon on slot } k \ (1,\ w,\ \text{or } x \text{ at the strand}), \tag*{(18)} ( Y x ) { b } { a } = k ∏ F b k b k − 1 a k x a k − 1 ℓ k , ℓ k = anyon on slot k ( 1 , w , or x at the strand ) , (18) and it commutes with H 0 H_0 H 0 (16) by the hexagon identity [30 , 45 ] ; this is the same mechanism that makes the wrapped lines of [19 , 20 , 46 ] exact lattice symmetries. There is a duality here with the dense RSOS models: there it is the ( 1 , s ) (1,s) ( 1 , s ) [21 , 23 ] , while the ( r , 1 ) (r,1) ( r , 1 ) [23 ] , and the ( r , 1 ) (r,1) ( r , 1 ) ( r , 1 ) (r,1) ( r , 1 ) ( 2 , 1 ) (2,1) ( 2 , 1 ) Figure 7 ):
Figure 7. Two-interface ring. Heights live on the bonds (black dots mark the sites between them); two ( 2 , 1 ) (2,1) ( 2 , 1 ) Y ( r , 1 ) Y_{(r,1)} Y ( r , 1 ) Z 2 \mathbb{Z}_2 Z 2
∥ [ Y ( 2 , 1 ) , H ] ∥ ∞ , ∥ [ Y ( p − 1 , 1 ) , H ] ∥ ∞ < 10 − 14 . \lVert[Y_{(2,1)},H]\rVert_{\infty},\ \lVert[Y_{(p-1,1)},H]\rVert_{\infty}<10^{-14}. ∥[ Y ( 2 , 1 ) , H ] ∥ ∞ , ∥[ Y ( p − 1 , 1 ) , H ] ∥ ∞ < 1 0 − 14 . The commuting lines are also conserved charges : the simultaneous eigenvalues of Y ( 2 , 1 ) Y_{(2,1)} Y ( 2 , 1 ) Y ( p − 1 , 1 ) = η Y_{(p-1,1)}=\eta Y ( p − 1 , 1 ) = η Z 2 \mathbb{Z}_2 Z 2 § 3.4 , which commutes with the same lines.
Check against the defect spectrum. At zero deformation the ring with two strands is the lattice version of M ( p , p + 1 ) M(p,p+1) M ( p , p + 1 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) [2 ] , H D × D = H 1 ⊕ H ( 3 , 1 ) \mathcal{H}_{D\times D}=\mathcal{H}_1\oplus\mathcal{H}_{(3,1)} H D × D = H 1 ⊕ H ( 3 , 1 ) ( 3 , 1 ) (3,1) ( 3 , 1 ) D ( 2 , 1 ) × D ( 2 , 1 ) = D ( 1 , 1 ) + D ( 3 , 1 ) D_{(2,1)}\times D_{(2,1)}=D_{(1,1)}+D_{(3,1)} D ( 2 , 1 ) × D ( 2 , 1 ) = D ( 1 , 1 ) + D ( 3 , 1 ) § 2.3 (Table 1). At p = 4 p=4 p = 4 g g g g D ( 2 , 1 ) = 2 g_{D_{(2,1)}}=\sqrt{2} g D ( 2 , 1 ) = 2 § 4 ).
The two-strand ring. All computations use a ring of L L L D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) Figure 7 ), both built with the same braiding B B B D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) D ( 2 , 1 ) × D ( 2 , 1 ) ∋ 1 D_{(2,1)}\times D_{(2,1)}\ni1 D ( 2 , 1 ) × D ( 2 , 1 ) ∋ 1 § 5.2 ).
Antiunitary reflection symmetry. The reflection P P P B B B P H P − 1 = H ∗ PHP^{-1}=H^* P H P − 1 = H ∗ K K K P K PK P K H H H ± \pm ± ( Y ( 2 , 1 ) , η ) (Y_{(2,1)},\eta) ( Y ( 2 , 1 ) , η ) F F F Y x Y_x Y x [ P K , Y x ] = [ P , Y x ] K [PK,Y_x]=[P,Y_x]K [ P K , Y x ] = [ P , Y x ] K P P P Y x Y_x Y x B ∗ B^* B ∗ P P P
The wrapped lines on the lattice
The junction compatible with the seam. Section 2.3 showed that a line x x x u ∈ Hom ( x ⊗ x ′ , x ′ ⊗ x ) u\in\operatorname{Hom}(x\otimes x',x'\otimes x) u ∈ Hom ( x ⊗ x ′ , x ′ ⊗ x ) x ′ x' x ′ x = x ′ = w x=x'=\mathrm{w} x = x ′ = w B B B B − 1 B^{-1} B − 1 § 2.4 , Figure 3 ). We now explain which junction the lattice realises and why it is the right one. The wrapped line Y x Y_x Y x Figure 8 ). The ring is a fusion tree: a chain of trivalent vertices, one per slot, at which the site anyon ℓ k \ell_k ℓ k w \mathrm{w} w x ′ x' x ′ a k a_k a k x x x b ∈ a ⊗ x b\in a\otimes x b ∈ a ⊗ x Figure 8 (a), top: x x x a 5 a_5 a 5 b 5 b_5 b 5 k k k Figure 8 (b): the fusion tree in which x x x a k a_k a k x x x a k − 1 a_{k-1} a k − 1 F b k − 1 a k − 1 ℓ k x a k b k F^{x a_k b_k}_{b_{k-1}a_{k-1}\ell_k} F b k − 1 a k − 1 ℓ k x a k b k k k k b k b_k b k k − 1 k-1 k − 1 a k − 1 a_{k-1} a k − 1 b k − 1 ∈ a k − 1 ⊗ x b_{k-1}\in a_{k-1}\otimes x b k − 1 ∈ a k − 1 ⊗ x Figure 8 (a), bottom). In the real gauge used here the F F F b k b_k b k x x x Y x Y_x Y x { a } \{a\} { a } { b } \{b\} { b } F F F (21) below includes the label of the starting segment. The strand is simply one more slot of the tree, with site anyon ℓ k = x ′ \ell_k=x' ℓ k = x ′ c j c_j c j x x x F F F c j → c j ′ ∈ c j ⊗ x c_j\to c'_j\in c_j\otimes x c j → c j ′ ∈ c j ⊗ x R R R Y x Y_x Y x F F F x ⊗ x ′ x \otimes x' x ⊗ x ′ P D ′ P_{D'} P D ′ § 3.4 are the junctions u D ′ u_{D'} u D ′ § 2.3 ) the two half-braidings expand with the R R R B = ∑ D ′ R D ′ x x ′ u D ′ B = \sum_{D'} R^{xx'}_{D'} u_{D'} B = ∑ D ′ R D ′ x x ′ u D ′ B − 1 = ∑ D ′ ( R D ′ x x ′ ) − 1 u D ′ B^{-1} = \sum_{D'} (R^{xx'}_{D'})^{-1}u_{D'} B − 1 = ∑ D ′ ( R D ′ x x ′ ) − 1 u D ′ x = x ′ = w x = x' = w x = x ′ = w R R R (17) and q = e i π / p q = e^{i\pi/p} q = e iπ / p
Figure 8. The wrapped line on the lattice. (a) The ring as a fusion tree (eight slots, the strand at slot 3 with its anyon x ′ x' x ′ c c c x x x x x x a 5 a_5 a 5 b 5 ∈ a 5 ⊗ x b_5 \in a_5 \otimes x b 5 ∈ a 5 ⊗ x x x x b k − 1 b_{k-1} b k − 1 x x x F b k b k − 1 a k x a k − 1 ℓ k F^{x a_{k-1} \ell_k}_{b_k b_{k-1} a_k} F b k b k − 1 a k x a k − 1 ℓ k ℓ k = x ′ \ell_k = x' ℓ k = x ′ a k → b k a_k \to b_k a k → b k c → c ′ c \to c' c → c ′ F F F (21) . (c) The crossing of x x x x ⊗ x ′ x \otimes x' x ⊗ x ′ (19) : the pure-F F F B B B B − 1 B^{-1} B − 1 (20) .
B = − i q − 3 / 2 u 1 + i q 1 / 2 u a d j , B − 1 = i q 3 / 2 u 1 − i q − 1 / 2 u a d j , (19) B = -i q^{-3/2} u_1 + i q^{1/2} u_{\mathrm{adj}}, \qquad B^{-1} = i q^{3/2} u_1 - i q^{-1/2} u_{\mathrm{adj}}, \tag*{(19)} B = − i q − 3/2 u 1 + i q 1/2 u adj , B − 1 = i q 3/2 u 1 − i q − 1/2 u adj , (19) so the two junctions differ by the relative phase q ± 2 = e ± 2 π i / p q^{\pm2} = e^{\pm2\pi i/p} q ± 2 = e ± 2 π i / p Figure 8 (c), which is exactly the phase that distinguishes passing in front of the strand from passing behind it. Why does a line built from F F F (16) , and which of the two junctions is it? Both questions are answered by noticing what the planar F F F x x x Figure 8 ; the strand leaf x ′ x' x ′ V V V F F F x x x V V V x x x x ′ x' x ′ x x x x ′ x' x ′ x ′ x' x ′ x ′ x' x ′ x x x Figure 9 shows the two situations, before and after the move, flattened into the plane: to compare the move with a junction in Hom ( x ⊗ x ′ , x ′ ⊗ x ) \operatorname{Hom}(x \otimes x',x' \otimes x) Hom ( x ⊗ x ′ , x ′ ⊗ x ) x x x a k − 1 a_{k-1} a k − 1 x ′ x' x ′ x ′ x' x ′ V V V U U U θ c / ( θ a θ x ′ ) \theta_c/(\theta_a\theta_{x'}) θ c / ( θ a θ x ′ ) ( a , x ′ → c ) (a,x' \to c) ( a , x ′ → c ) θ i = e 2 π i h i \theta_i = e^{2\pi i h_i} θ i = e 2 π i h i ( r , 1 ) (r,1) ( r , 1 ) i i i Y x Y_x Y x Y x ′ = U Y x U − 1 Y'_x = UY_xU^{-1} Y x ′ = U Y x U − 1 Figure 8 (b), in which x x x V V V x x x V V V V V V ( a k − 1 , x ′ → a k ) (a_{k-1},x' \to a_k) ( a k − 1 , x ′ → a k ) U − 1 U^{-1} U − 1 θ a k / ( θ a k − 1 θ x ′ ) \theta_{a_k}/(\theta_{a_{k-1}}\theta_{x'}) θ a k / ( θ a k − 1 θ x ′ ) x x x V V V V V V ( b k − 1 , x ′ → b k ) (b_{k-1},x' \to b_k) ( b k − 1 , x ′ → b k ) b k − 1 ∈ a k − 1 ⊗ x b_{k-1} \in a_{k-1} \otimes x b k − 1 ∈ a k − 1 ⊗ x U U U θ b k / ( θ b k − 1 θ x ′ ) \theta_{b_k}/(\theta_{b_{k-1}}\theta_{x'}) θ b k / ( θ b k − 1 θ x ′ ) θ x ′ \theta_{x'} θ x ′ θ a k − 1 θ b k / ( θ a k θ b k − 1 ) \theta_{a_{k-1}}\theta_{b_k}/(\theta_{a_k}\theta_{b_{k-1}}) θ a k − 1 θ b k / ( θ a k θ b k − 1 ) x ′ x' x ′ a k − 1 ⊗ x a_{k-1}\otimes x a k − 1 ⊗ x a k − 1 a_{k-1} a k − 1 x ′ x' x ′ x x x B B B B − 1 = B − 2 B B^{-1}=B^{-2}B B − 1 = B − 2 B (19) . ( U U U H H H U Y x U − 1 UY_xU^{-1} U Y x U − 1 § 2.3 are realised on the lattice by the pure-F F F Y x Y_x Y x (21) and by the same line with the F F F
Figure 9. The two junctions as one planar move. The trunk and the line x x x x ′ x^{\prime} x ′ V V V Figure 8 (b), with x x x V V V V V V a k − 1 a_{k-1} a k − 1 a k a_k a k x x x V V V V V V b k − 1 ∈ a k − 1 ⊗ x b_{k-1} \in a_{k-1} \otimes x b k − 1 ∈ a k − 1 ⊗ x b k b_k b k F F F x ′ x^{\prime} x ′ x x x x x x Hom ( x ⊗ x ′ , x ′ ⊗ x ) \operatorname{Hom}(x \otimes x^{\prime},x^{\prime} \otimes x) Hom ( x ⊗ x ′ , x ′ ⊗ x ) x ′ x^{\prime} x ′ V V V x ′ x^{\prime} x ′ θ a k / ( θ a k − 1 θ x ′ ) \theta_{a_k}/(\theta_{a_{k-1}}\theta_{x^{\prime}}) θ a k / ( θ a k − 1 θ x ′ ) θ b k / ( θ b k − 1 θ x ′ ) \theta_{b_k}/(\theta_{b_{k-1}}\theta_{x^{\prime}}) θ b k / ( θ b k − 1 θ x ′ ) F F F Y x Y_x Y x Y x ′ = U Y x U − 1 Y_x^{\prime}=U Y_x U^{-1} Y x ′ = U Y x U − 1 (20) , whose matrix element carries the ratio of the two factors.
Y x ′ = U Y x U − 1 : F b k b k − 1 a k x a k − 1 x ′ ⟶ F b k b k − 1 a k x a k − 1 x ′ θ a k − 1 θ b k θ a k θ b k − 1 at the strand slot (20) Y'_x = UY_xU^{-1}: \qquad F^{x\,a_{k-1}\,x'}_{b_k\,b_{k-1}\,a_k} \longrightarrow F^{x\,a_{k-1}\,x'}_{b_k\,b_{k-1}\,a_k}\frac{\theta_{a_{k-1}}\theta_{b_k}}{\theta_{a_k}\theta_{b_{k-1}}}\quad\text{at the strand slot} \tag*{(20)} Y x ′ = U Y x U − 1 : F b k b k − 1 a k x a k − 1 x ′ ⟶ F b k b k − 1 a k x a k − 1 x ′ θ a k θ b k − 1 θ a k − 1 θ b k at the strand slot (20) (with the twists θ \theta θ R R R (15) ; for a seam built from B ∗ B^* B ∗ F F F R R R p = 4 p=4 p = 4 p = 5 p=5 p = 5 L = 6 L=6 L = 6 Y x Y_x Y x B B B B ∗ B^* B ∗ Y x ′ Y'_x Y x ′ (20) commutes with H 0 H_0 H 0 Y x Y_x Y x { ± φ , ± φ − 1 } \{\pm\varphi,\pm\varphi^{-1}\} { ± φ , ± φ − 1 } p = 5 p=5 p = 5 Y x Y_x Y x ( 1 , 2 ) ⊗ ( 2 , 2 ) (1,2)\otimes(2,2) ( 1 , 2 ) ⊗ ( 2 , 2 ) ( 2 , 2 ) ⊗ ( 1 , 2 ) (2,2)\otimes(1,2) ( 2 , 2 ) ⊗ ( 1 , 2 ) Δ = 3 20 \Delta=\frac{3}{20} Δ = 20 3 Y ( 2 , 1 ) Y_{(2,1)} Y ( 2 , 1 ) ( − φ , + φ − 1 ) (-\varphi,+\varphi^{-1}) ( − φ , + φ − 1 ) Y ( 2 , 1 ) ′ Y'_{(2,1)} Y ( 2 , 1 ) ′ ( + φ − 1 , − φ ) (+\varphi^{-1},-\varphi) ( + φ − 1 , − φ ) η \eta η p = 5 p=5 p = 5 p = 4 p=4 p = 4 σ ′ \sigma' σ ′ θ η ⊗ σ ′ / ( θ η θ σ ′ ) = − 1 \theta_{\eta\otimes\sigma'}/(\theta_\eta\theta_{\sigma'})=-1 θ η ⊗ σ ′ / ( θ η θ σ ′ ) = − 1 Y η ′ = − Y η Y'_\eta=-Y_\eta Y η ′ = − Y η Y η ′ = Y η Y'_\eta=Y_\eta Y η ′ = Y η § 2.4 literally: on the undeformed ring both lassos are exact symmetries, because the strand is topological, and the deformation decides between them.
Which lasso the deformation preserves, and why. The clean field Φ c l \Phi_{\mathrm{cl}} Φ cl § 3.4 is built from the channel projectors P 1 P^1 P 1 P a d j P^{\mathrm{adj}} P adj (24) . These projectors are planar objects of the fusion tree: they are written in the same basis as Y x Y_x Y x [ Y x , Φ c l ] = 0 [Y_x,\Phi_{\mathrm{cl}}]=0 [ Y x , Φ cl ] = 0 [ Y x , H 0 ] = 0 [Y_x,H_0]=0 [ Y x , H 0 ] = 0 Y x ′ = U Y x U − 1 Y'_x=UY_xU^{-1} Y x ′ = U Y x U − 1 Φ c l \Phi_{\mathrm{cl}} Φ cl [ Y x ′ , Φ c l ] = 0 [Y'_x,\Phi_{\mathrm{cl}}]=0 [ Y x ′ , Φ cl ] = 0 [ Y x , U − 1 Φ c l U ] = 0 [Y_x,U^{-1}\Phi_{\mathrm{cl}}U]=0 [ Y x , U − 1 Φ cl U ] = 0 U U U U − 1 Φ c l U U^{-1}\Phi_{\mathrm{cl}}U U − 1 Φ cl U ∥ [ Y x ′ , Φ c l ] ∥ = O ( 1 ) \lVert[Y'_x,\Phi_{\mathrm{cl}}]\rVert=O(1) ∥[ Y x ′ , Φ cl ]∥ = O ( 1 ) ∥ [ Y x , Φ c l ] ∥ < 10 − 14 \lVert[Y_x,\Phi_{\mathrm{cl}}]\rVert<10^{-14} ∥[ Y x , Φ cl ]∥ < 1 0 − 14 § 2.4 this is the statement that the field created by Φ c l \Phi_{\mathrm{cl}} Φ cl p = 5 p=5 p = 5 L = 6 L=6 L = 6 8 8 8 Φ c l ∣ v a c ⟩ \Phi_{\mathrm{cl}}|\mathrm{vac}\rangle Φ cl ∣ vac ⟩ ∥ ( Y ( 2 , 1 ) − φ ) Φ c l ∣ v a c ⟩ ∥ < 10 − 12 \lVert(Y_{(2,1)}-\varphi)\Phi_{\mathrm{cl}}|\mathrm{vac}\rangle\rVert<10^{-12} ∥( Y ( 2 , 1 ) − φ ) Φ cl ∣ vac ⟩∥ < 1 0 − 12 ∥ ( Y ( 2 , 1 ) ′ + φ − 1 ) Φ c l ∣ v a c ⟩ ∥ < 10 − 12 \lVert(Y'_{(2,1)}+\varphi^{-1})\Phi_{\mathrm{cl}}|\mathrm{vac}\rangle\rVert<10^{-12} ∥( Y ( 2 , 1 ) ′ + φ − 1 ) Φ cl ∣ vac ⟩∥ < 1 0 − 12 + φ +\varphi + φ Y ( 2 , 1 ) Y_{(2,1)} Y ( 2 , 1 ) − φ − 1 -\varphi^{-1} − φ − 1 Y ( 2 , 1 ) ′ Y'_{(2,1)} Y ( 2 , 1 ) ′ ( 3 , 3 ) ≡ ( 2 , 3 ) (3,3)\equiv(2,3) ( 3 , 3 ) ≡ ( 2 , 3 ) ( 1 , 3 ) ⊗ ( 3 , 3 ) (1,3)\otimes(3,3) ( 1 , 3 ) ⊗ ( 3 , 3 ) Y x Y_x Y x Y x ′ Y'_x Y x ′ Y x Y_x Y x Y x ′ Y'_x Y x ′ (3) requires of the junction that reads the deformation-neutral label. Which of the two sides of the plane is called the holomorphic one is a convention, the same convention that names the lattice deformation ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) \phi_{(1,3)}\otimes\bar{\phi}_{(3,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) F F F § 2.4 , the one that survives the deformation, which is why [ Y x , H ] = 0 [Y_x,H]=0 [ Y x , H ] = 0 10 − 14 10^{-14} 1 0 − 14 Φ c l \Phi_{\mathrm{cl}} Φ cl H 0 H_0 H 0
The lattice symmetry operators. Explicitly, the wrapped line x x x
Y x ∣ a 0 , a 1 , … ⟩ = ∑ { b } ∏ k F b k b k − 1 a k x a k − 1 ℓ k ∣ b 0 , b 1 , … ⟩ , b k ∈ a k ⊗ x , ℓ k = { 1 site k empty w site k occupied x ′ strand slot (21) Y_x |a_0,a_1,\ldots\rangle= \sum_{\{b\}} \prod_k F^{x\ a_{k-1}\ \ell_k}_{b_k\ b_{k-1}\ a_k} |b_0,b_1,\ldots\rangle,\qquad b_k \in a_k \otimes x,\qquad\ell_k =
\begin{cases}
1 & \text{site } k \text{ empty}\\
w & \text{site } k \text{ occupied}\\
x^{\prime} & \text{strand slot}
\end{cases} \tag*{(21)} Y x ∣ a 0 , a 1 , … ⟩ = { b } ∑ k ∏ F b k b k − 1 a k x a k − 1 ℓ k ∣ b 0 , b 1 , … ⟩ , b k ∈ a k ⊗ x , ℓ k = ⎩ ⎨ ⎧ 1 w x ′ site k empty site k occupied strand slot (21) the product running over all slots of the ring, strand slots included; it is obtained by placing the x x x F F F η = Y ( p − 1 , 1 ) \eta=Y_{(p-1,1)} η = Y ( p − 1 , 1 ) a ⊗ η a \otimes\eta a ⊗ η b = p − a b=p-a b = p − a Y η Y_\eta Y η a k ↦ p − a k a_k \mapsto p-a_k a k ↦ p − a k F F F (21) multiply to + 1 +1 + 1 Y η Y_\eta Y η η \eta η § 3.5 use (21) directly, and the DMRG uses its matrix-product form. For the exact diagonalisation these are sparse matrices in the admissible basis; the charges of the eigenstates are read off from ⟨ ψ ∣ Y ( 2 , 1 ) ∣ ψ ⟩ \langle\psi|Y_{(2,1)}|\psi\rangle ⟨ ψ ∣ Y ( 2 , 1 ) ∣ ψ ⟩ ⟨ ψ ∣ η ∣ ψ ⟩ \langle\psi|\eta|\psi\rangle ⟨ ψ ∣ η ∣ ψ ⟩ Y Y Y H H H η \eta η H H H § 3.5 ). For the DMRG the same operators are written as matrix-product operators: η \eta η Y ( 2 , 1 ) Y_{(2,1)} Y ( 2 , 1 ) ( p − 1 ) 2 (p-1)^2 ( p − 1 ) 2 ( b k − 1 , a k − 1 ) (b_{k-1},a_{k-1}) ( b k − 1 , a k − 1 ) F F F H H H
Check on the undeformed ring. Table 1 tests (4) on the undeformed two-strand ring at p = 5 p=5 p = 5 H 1 ⊕ H ( 3 , 1 ) \mathcal{H}_1\oplus\mathcal{H}_{(3,1)} H 1 ⊕ H ( 3 , 1 ) ( 1 , 3 ) ⊗ ( 3 , 3 ) (1,3)\otimes(3,3) ( 1 , 3 ) ⊗ ( 3 , 3 ) a + a+ a + ( 3 , 3 ) ⊗ ( 1 , 3 ) (3,3)\otimes(1,3) ( 3 , 3 ) ⊗ ( 1 , 3 ) c + c+ c +
Table 1. The undeformed two-strand ring at p = 5 p=5 p = 5 K = λ = 0 K=\lambda=0 K = λ = 0 L = 10 L=10 L = 10 H 1 ⊕ H ( 3 , 1 ) \mathcal{H}_1\oplus\mathcal{H}_{(3,1)} H 1 ⊕ H ( 3 , 1 ) (4) from the left label, and the sector in which the level is found numerically (a + = ( φ , + ) a+=(\varphi,+) a + = ( φ , + ) b − = ( φ − 1 , − ) b-=(\varphi^{-1},-) b − = ( φ − 1 , − ) c + = ( − φ − 1 , + ) c+=(-\varphi^{-1},+) c + = ( − φ − 1 , + ) d − = ( − φ , − ) d-=(-\varphi,-) d − = ( − φ , − ) § 5.3 ). Weights of M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) h 1 , 2 = 1 8 h_{1,2}=\frac{1}{8} h 1 , 2 = 8 1 h 1 , 3 = 2 3 h_{1,3}=\frac{2}{3} h 1 , 3 = 3 2 h 2 , 1 = 1 5 h_{2,1}=\frac{1}{5} h 2 , 1 = 5 1 h 2 , 2 = 1 40 h_{2,2}=\frac{1}{40} h 2 , 2 = 40 1 h 2 , 3 = 1 15 h_{2,3}=\frac{1}{15} h 2 , 3 = 15 1 h 3 , 2 = 21 40 h_{3,2}=\frac{21}{40} h 3 , 2 = 40 21 h 3 , 3 = 1 15 h_{3,3}=\frac{1}{15} h 3 , 3 = 15 1 h 4 , 2 = 13 8 h_{4,2}=\frac{13}{8} h 4 , 2 = 8 13
A single crossing. On a ring crossed once by D D D Y x Y_x Y x H D \mathcal{H}_D H D p = 5 p=5 p = 5 H D ( 2 , 1 ) \mathcal{H}_{D(2,1)} H D ( 2 , 1 ) Δ = 2 15 , 3 20 , 2 5 , 11 20 , 11 15 , … \Delta=\frac{2}{15},\frac{3}{20},\frac{2}{5},\frac{11}{20},\frac{11}{15},\ldots Δ = 15 2 , 20 3 , 5 2 , 20 11 , 15 11 , … ( 1 , 2 ) ⊗ ( 2 , 2 ) (1,2)\otimes(2,2) ( 1 , 2 ) ⊗ ( 2 , 2 ) ( 2 , 2 ) ⊗ ( 1 , 2 ) (2,2)\otimes(1,2) ( 2 , 2 ) ⊗ ( 1 , 2 ) 3 20 \frac{3}{20} 20 3 (4) of their left label; at p = 4 p=4 p = 4 C 4 C_4 C 4 F F F ( 2 , 1 ) (2,1) ( 2 , 1 ) Y ( 2 , 1 ) Y_{(2,1)} Y ( 2 , 1 ) C 4 C_4 C 4 S U ( 2 ) 2 SU(2)_2 S U ( 2 ) 2 § 4 are thus labelled by the Verlinde values of the left label: the Y ( 2 , 1 ) = − 2 Y_{(2,1)}=-\sqrt{2} Y ( 2 , 1 ) = − 2 ( 1 , 2 ) (1,2) ( 1 , 2 ) + 2 +\sqrt{2} + 2 ( 1 , 1 ) (1,1) ( 1 , 1 ) ( 1 , 3 ) (1,3) ( 1 , 3 ) Y ( 2 , 1 ) = 0 Y_{(2,1)}=0 Y ( 2 , 1 ) = 0 ( 2 , s ) (2,s) ( 2 , s )
The clean deformation operator
What the continuum asks for. The KRW flow [6 ] is triggered by a local perturbation on the defect. The field that drives it is the non-chiral defect field ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) \phi_{(1,3)}\otimes\bar{\phi}_{(3,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) ( h , h ˉ ) = ( h 1 , 3 , h 3 , 3 ) (h,\bar{h})=(h_{1,3},h_{3,3}) ( h , h ˉ ) = ( h 1 , 3 , h 3 , 3 )
Δ ϕ = 1 − ε , ε = 2 ( p − 1 ) p ( p + 1 ) ( Δ ϕ = 0.7 at p = 4 , 11 15 at p = 5 ) , (22) \Delta_{\phi}=1-\varepsilon,\qquad\varepsilon=\frac{2(p-1)}{p(p+1)}\quad(\Delta_{\phi}=0.7\text{ at }p=4,\ \frac{11}{15}\text{ at }p=5), \tag*{(22)} Δ ϕ = 1 − ε , ε = p ( p + 1 ) 2 ( p − 1 ) ( Δ ϕ = 0.7 at p = 4 , 15 11 at p = 5 ) , (22) so it is relevant; it carries spin h 1 , 3 − h 3 , 3 h_{1,3}-h_{3,3} h 1 , 3 − h 3 , 3 1 2 \frac{1}{2} 2 1 p = 4 p=4 p = 4 3 5 \frac{3}{5} 5 3 p = 5 p=5 p = 5 ϕ ( 3 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) \phi_{(3,3)}\otimes\bar{\phi}_{(1,3)} ϕ ( 3 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) ( 2 , 1 ) (2,1) ( 2 , 1 ) § 3.2 ): on the undeformed ring the two appear as one level at Δ = 11 15 \Delta=\frac{11}{15} Δ = 15 11 α + \alpha+ α + c + c+ c + § 5.3 ). Among the nonidentity defect primaries in the symmetry sector of the identity, ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) \phi_{(1,3)}\otimes\bar{\phi}_{(3,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) only relevant one. A lattice operator that commutes with the wrapped lines, as ours does, connects the vacuum only to that sector, so the lattice flow is KRW’s single-field perturbation of D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) C C C λ l = λ r \lambda_l=\lambda_r λ l = λ r D ( 1 , 2 ) D_{(1,2)} D ( 1 , 2 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) C C C ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) \phi_{(1,3)}\otimes\bar{\phi}_{(1,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) 2 h 1 , 3 2h_{1,3} 2 h 1 , 3
Two w’s next to each other. Fix a strand x = w x=w x = w j j j Figure 11 )
Figure 11. The heights around a strand. The strand is an extra bond c j c_j c j a j a_j a j a j + 1 a_{j+1} a j + 1 ∣ c j − a j ∣ = 1 |c_j-a_j|=1 ∣ c j − a j ∣ = 1 a j a_j a j c j c_j c j w w w e L , n L e_{\mathrm{L}},n_{\mathrm{L}} e L , n L j j j ( a j − 1 , c j ) (a_{j-1},c_j) ( a j − 1 , c j ) e R , n R e_{\mathrm{R}},n_{\mathrm{R}} e R , n R j + 1 j+1 j + 1 ( a j , a j + 1 ) (a_j,a_{j+1}) ( a j , a j + 1 )
a j − 1 → site j a j → strand c j → site j + 1 a j + 1 . a_{j-1}\xrightarrow{\text{site }j}a_j\xrightarrow{\text{strand}}c_j\xrightarrow{\text{site }j+1}a_{j+1}. a j − 1 site j a j strand c j site j + 1 a j + 1 . The strand bond a j a_j a j c j c_j c j j j j a j − 1 ≠ a j a_{j-1}\ne a_j a j − 1 = a j j + 1 j+1 j + 1 c j ≠ a j + 1 c_j\ne a_{j+1} c j = a j + 1
w ⊗ w = 1 ⊕ adj , (23) w\otimes w=1\oplus\operatorname{adj}, \tag*{(23)} w ⊗ w = 1 ⊕ adj , (23) where “adj” is the object of C p C_p C p ε \varepsilon ε η = ( 3 , 1 ) \eta=(3,1) η = ( 3 , 1 ) p = 4 p=4 p = 4 τ F i b = ( 3 , 1 ) \tau_{\mathrm{Fib}}=(3,1) τ Fib = ( 3 , 1 ) p = 5 p=5 p = 5 j j j ( a j − 1 , c j ) (a_{j-1},c_j) ( a j − 1 , c j ) j + 1 j+1 j + 1 ( a j , a j + 1 ) (a_j,a_{j+1}) ( a j , a j + 1 ) l → m → r l\to m\to r l → m → r ∣ m − l ∣ = ∣ r − m ∣ = 1 \lvert m-l\rvert=\lvert r-m\rvert=1 ∣ m − l ∣ = ∣ r − m ∣ = 1 r ∈ l ⊗ w ⊗ w r\in l\otimes w\otimes w r ∈ l ⊗ w ⊗ w
r = l ± 2 ⇒ channel adj only , r = l ⇒ channels 1 and adj both possible . r=l\pm2\ \Rightarrow\ \text{channel adj only},\qquad r=l\ \Rightarrow\ \text{channels 1 and adj both possible}. r = l ± 2 ⇒ channel adj only , r = l ⇒ channels 1 and adj both possible . On the r = l r=l r = l channel projectors P 1 P^1 P 1 ( l , m , r ) (l,m,r) ( l , m , r )
P 1 ∣ l , m , r ⟩ = δ l , r ∑ m ′ S m S m ′ d w S l ∣ l , m ′ , r ⟩ , P a d j = [ l ≠ m and m ≠ r ] − P 1 (24) P^{1}|l,m,r\rangle=\delta_{l,r}\sum_{m'}\frac{\sqrt{S_mS_{m'}}}{d_wS_l}|l,m',r\rangle,\qquad P^{\mathrm{adj}}=[l\ne m\ \text{and}\ m\ne r]-P^{1} \tag*{(24)} P 1 ∣ l , m , r ⟩ = δ l , r m ′ ∑ d w S l S m S m ′ ∣ l , m ′ , r ⟩ , P adj = [ l = m and m = r ] − P 1 (24) where the sum runs over heights m ′ m' m ′ l l l r r r S a S_a S a d w d_w d w (9) , and [ ⋯ ] [\cdots] [ ⋯ ] ∑ m a d j l S m = d w S l \sum_{m\ \mathrm{adj}\ l}S_m=d_wS_l ∑ m adj l S m = d w S l ( P 1 ) 2 = P 1 (P^{1})^2=P^{1} ( P 1 ) 2 = P 1 P 1 P a d j = 0 P^{1}P^{\mathrm{adj}}=0 P 1 P adj = 0 P 1 + P a d j P^{1}+P^{\mathrm{adj}} P 1 + P adj ∣ w w ; 1 ⟩ ⟨ w w ; 1 ∣ |\mathrm{ww};1\rangle\langle\mathrm{ww};1| ∣ ww ; 1 ⟩ ⟨ ww ; 1∣ ∣ w w ; a d j ⟩ ⟨ w w ; a d j ∣ |\mathrm{ww};\mathrm{adj}\rangle\langle\mathrm{ww};\mathrm{adj}| ∣ ww ; adj ⟩ ⟨ ww ; adj ∣ Figure 10 . In the diagrammatic calculus the projector on the channel c c c c c c d c / θ ( w , w , c ) d_c/\theta(\mathrm{w},\mathrm{w},c) d c / θ ( w , w , c ) θ \theta θ c = 1 c=1 c = 1 P 1 = e / d w P^{1}=e/d_w P 1 = e / d w F F F F l ; m 1 l w w = S m / ( d w S l ) F^{l\ \mathrm{w}\ \mathrm{w}}_{l;m\ 1}=\sqrt{S_m/(d_wS_l)} F l ; m 1 l w w = S m / ( d w S l ) (24) ; for c = a d j c=\mathrm{adj} c = adj n n n (25) below, which is how P a d j P^{\mathrm{adj}} P adj Y ( r , 1 ) Y_{(r,1)} Y ( r , 1 )
Figure 10. The fusion-channel projectors of (24) as fusion diagrams (w lines in red). The projector on the channel c c c c c c d c / θ ( w , w , c ) d_c/\theta(\mathrm{w},\mathrm{w},c) d c / θ ( w , w , c ) c = 1 c=1 c = 1 P 1 P^{1} P 1 d w d_w d w e e e F F F F l ; m 1 l w w = S m / ( d w S l ) F^{l\ \mathrm{w}\ \mathrm{w}}_{l;m\ 1}=\sqrt{S_m/(d_wS_l)} F l ; m 1 l w w = S m / ( d w S l ) S m S m ′ / ( d w S l ) \sqrt{S_mS_{m'}}/(d_wS_l) S m S m ′ / ( d w S l ) (24) . For c = a d j c=\mathrm{adj} c = adj n n n (25) .
The two natural building blocks. Two familiar local operators are linear combinations of these projectors: the Temperley–Lieb generator e e e [47 ] , which is d w d_w d w n n n
e = d w P 1 , n = P 1 + P a d j . (25) e=d_wP^{1},\qquad n=P^{1}+P^{\mathrm{adj}}. \tag*{(25)} e = d w P 1 , n = P 1 + P adj . (25) Each can be placed on either side of the strand. Written out on the four heights ( a j − 1 , a j , c j , a j + 1 ) (a_{j-1},a_j,c_j,a_{j+1}) ( a j − 1 , a j , c j , a j + 1 )
n L = [ a j − 1 ≠ a j ] , n R = [ c j ≠ a j + 1 ] , n_{\mathrm{L}}=\left[a_{j-1}\ne a_j\right],\qquad n_{\mathrm{R}}=\left[c_j\ne a_{j+1}\right], n L = [ a j − 1 = a j ] , n R = [ c j = a j + 1 ] , e L ∣ a j − 1 , a j , c j , a j + 1 ⟩ = δ a j − 1 , c j [ a j − 1 ≠ a j ] ∑ m ′ ≠ a j − 1 S a j S m ′ S a j − 1 ∣ a j − 1 , m ′ , c j , a j + 1 ⟩ , e_{\mathrm{L}}\lvert a_{j-1},a_j,c_j,a_{j+1}\rangle=\delta_{a_{j-1},c_j}\left[a_{j-1}\ne a_j\right]\sum_{m'\ne a_{j-1}}\frac{\sqrt{S_{a_j}S_{m'}}}{S_{a_{j-1}}}\lvert a_{j-1},m',c_j,a_{j+1}\rangle, e L ∣ a j − 1 , a j , c j , a j + 1 ⟩ = δ a j − 1 , c j [ a j − 1 = a j ] m ′ = a j − 1 ∑ S a j − 1 S a j S m ′ ∣ a j − 1 , m ′ , c j , a j + 1 ⟩ , e R ∣ a j − 1 , a j , c j , a j + 1 ⟩ = δ a j , a j + 1 [ c j ≠ a j + 1 ] ∑ m ′ ≠ a j S c j S m ′ S a j ∣ a j − 1 , a j , m ′ , a j + 1 ⟩ (26) e_{\mathrm{R}}\lvert a_{j-1},a_j,c_j,a_{j+1}\rangle=\delta_{a_j,a_{j+1}}\left[c_j\ne a_{j+1}\right]\sum_{m'\ne a_j}\frac{\sqrt{S_{c_j}S_{m'}}}{S_{a_j}}\lvert a_{j-1},a_j,m',a_{j+1}\rangle \tag*{(26)} e R ∣ a j − 1 , a j , c j , a j + 1 ⟩ = δ a j , a j + 1 [ c j = a j + 1 ] m ′ = a j ∑ S a j S c j S m ′ ∣ a j − 1 , a j , m ′ , a j + 1 ⟩ (26) with m ′ m' m ′ e L e_{\mathrm{L}} e L j j j a j a_j a j e R e_{\mathrm{R}} e R ( j + 1 ) (j+1) ( j + 1 ) c j c_j c j n L , n R n_{\mathrm{L}},n_{\mathrm{R}} n L , n R e L , e R e_{\mathrm{L}},e_{\mathrm{R}} e L , e R w e w_e w e (13) and Figure 5 , normalised so that e 2 = d w e e^2=d_w e e 2 = d w e c j c_j c j B B B
Measuring the field content. Which continuum fields does a given lattice operator O O O ∣ Ψ ⟩ \lvert\Psi\rangle ∣ Ψ ⟩ D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) ⟨ Ψ ∣ O ∣ v a c ⟩ \langle\Psi|O|\mathrm{vac}\rangle ⟨ Ψ∣ O ∣ vac ⟩ O O O Table 2 gives the absolute values of these overlaps for the natural candidates at p = 4 p=4 p = 4 Table 3 repeats the exercise at p = 5 p=5 p = 5 n L + n R n_{\mathrm{L}}+n_{\mathrm{R}} n L + n R ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) \phi_{(1,3)} \otimes\bar{\phi}_{(1,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) e L + e R e_{\mathrm{L}}+e_{\mathrm{R}} e L + e R
Defect-field content of the candidate lattice operators on the p = 4 p=4 p = 4 D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) ∣ ⟨ Ψ ∣ O ∣ v a c ⟩ ∣ |\langle\Psi|\mathcal{O}|\mathrm{vac}\rangle| ∣ ⟨ Ψ∣ O ∣ vac ⟩ ∣ L = 10 L=10 L = 10 Φ c l \Phi_{\mathrm{cl}} Φ cl ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) \phi_{(1,3)}\otimes\bar{\phi}_{(1,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 1 , 3 )
Table 2.
Absolute overlap magnitudes, as in Table 2 , at p = 5 p=5 p = 5 D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) L = 10 L=10 L = 10 e L + e R e_{\mathrm{L}}+e_{\mathrm{R}} e L + e R n L + n R n_{\mathrm{L}}+n_{\mathrm{R}} n L + n R 1.62 = φ = d w 1.62=\varphi=d_{\mathrm{w}} 1.62 = φ = d w p = 4 p=4 p = 4 Table 2 , the ratio is 1.41 = 2 = d w 1.41=\sqrt{2}=d_{\mathrm{w}} 1.41 = 2 = d w d w ( e L + e R ) − ( n L + n R ) d_{\mathrm{w}}(e_{\mathrm{L}}+e_{\mathrm{R}})-(n_{\mathrm{L}}+n_{\mathrm{R}}) d w ( e L + e R ) − ( n L + n R ) d w = φ d_{\mathrm{w}}=\varphi d w = φ 2 d w ( e L + e R ) − ( n L + n R ) \frac{2}{d_{\mathrm{w}}}(e_{\mathrm{L}}+e_{\mathrm{R}})-(n_{\mathrm{L}}+n_{\mathrm{R}}) d w 2 ( e L + e R ) − ( n L + n R )
Table 3.
The clean combination. Since both contaminations sit in the same two operators, and in the same ratio (Table 2 and Table 3 ), a single combination of the two channel projectors removes both. It is the traceless combination, the projectors weighted by their inverse quantum dimensions,
P 1 d 1 − P a d j d a d j ∝ d a d j P 1 − P a d j = d w e − n , (27) \frac{P^{1}}{d_{1}}-\frac{P^{\mathrm{adj}}}{d_{\mathrm{adj}}}\propto d_{\mathrm{adj}}P^{1}-P^{\mathrm{adj}}=d_{\mathrm{w}}e-n, \tag*{(27)} d 1 P 1 − d adj P adj ∝ d adj P 1 − P adj = d w e − n , (27) where we used (25) and d w 2 = 1 + d a d j d_{\mathrm{w}}^{2}=1+d_{\mathrm{adj}} d w 2 = 1 + d adj w ⊗ w = 1 ⊕ a d j \mathbf{w}\otimes\mathbf{w}=\mathbf{1}\oplus\mathrm{adj} w ⊗ w = 1 ⊕ adj
Φ c l = ( d a d j P 1 − P a d j ) L + ( d a d j P 1 − P a d j ) R = d w ( e L + e R ) − ( n L + n R ) (28) \Phi_{\mathrm{cl}}=\left(d_{\mathrm{adj}}P^{1}-P^{\mathrm{adj}}\right)_{\mathrm{L}}+\left(d_{\mathrm{adj}}P^{1}-P^{\mathrm{adj}}\right)_{\mathrm{R}}=d_{\mathrm{w}}(e_{\mathrm{L}}+e_{\mathrm{R}})-(n_{\mathrm{L}}+n_{\mathrm{R}}) \tag*{(28)} Φ cl = ( d adj P 1 − P adj ) L + ( d adj P 1 − P adj ) R = d w ( e L + e R ) − ( n L + n R ) (28) At p = 4 p=4 p = 4 η = ( 3 , 1 ) \eta=(3,1) η = ( 3 , 1 ) d a d j = 1 d_{\mathrm{adj}}=1 d adj = 1 d w = 2 = 2 / d w d_{\mathrm{w}}=\sqrt{2}=2/d_{\mathrm{w}} d w = 2 = 2/ d w (28) is simply the projector difference P 1 − P a d j = 2 d w ( e L + e R ) − ( n L + n R ) P^{1}-P^{\mathrm{adj}}=\frac{2}{d_{\mathrm{w}}}(e_{\mathrm{L}}+e_{\mathrm{R}})-(n_{\mathrm{L}}+n_{\mathrm{R}}) P 1 − P adj = d w 2 ( e L + e R ) − ( n L + n R ) p = 5 p=5 p = 5 τ F i b = ( 3 , 1 ) \tau_{\mathrm{Fib}}=(3,1) τ Fib = ( 3 , 1 ) d a d j = φ d_{\mathrm{adj}}=\varphi d adj = φ φ P 1 − P a d j = φ ( e L + e R ) − ( n L + n R ) \varphi P^{1}-P^{\mathrm{adj}}=\varphi(e_{\mathrm{L}}+e_{\mathrm{R}})-(n_{\mathrm{L}}+n_{\mathrm{R}}) φ P 1 − P adj = φ ( e L + e R ) − ( n L + n R ) not the projector difference. The bottom rows of Table 2 and Table 3 show what this buys on the undeformed D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) p p p Φ c l \Phi_{\mathrm{cl}} Φ cl ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) \phi_{(1,3)}\otimes\bar{\phi}_{(1,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) p = 5 p=5 p = 5 P 1 − P a d j P^{1}-P^{\mathrm{adj}} P 1 − P adj Table 3 ). The vanishing of the identity component is not an accident of the table: in the topological ground state the two channels of the strand and its neighbouring wall are populated in the ratio of their quantum dimensions, ⟨ n ⟩ = d w ⟨ e ⟩ \langle n\rangle=d_{\mathrm{w}}\langle e\rangle ⟨ n ⟩ = d w ⟨ e ⟩ ⟨ v a c ∣ Φ c l ∣ v a c ⟩ = 0 \langle\mathrm{vac}|\Phi_{\mathrm{cl}}|\mathrm{vac}\rangle=0 ⟨ vac ∣ Φ cl ∣ vac ⟩ = 0 10 − 15 10^{-15} 1 0 − 15 L ≤ 14 L\leq14 L ≤ 14 p p p d a d j P 1 − P a d j d_{\mathrm{adj}}P^{1}-P^{\mathrm{adj}} d adj P 1 − P adj d a d j d 1 − d a d j = 0 d_{\mathrm{adj}}d_{1}-d_{\mathrm{adj}}=0 d adj d 1 − d adj = 0 Φ c l \Phi_{\mathrm{cl}} Φ cl Φ c l ∣ v a c ⟩ \Phi_{\mathrm{cl}}|\mathrm{vac}\rangle Φ cl ∣ vac ⟩ § 3.3 , which fixes the sector but not a single scaling field); these are irrelevant corrections, and what the construction needs is only that the relevant content of the perturbation be the driving field alone. Second, the symmetry argument of § 2.4 says which fields a symmetric lattice operator may contain, not that every symmetric operator contains the driving field; that Φ c l \Phi_{\mathrm{cl}} Φ cl (28) is transparent. On each side of the strand it asks whether the defect anyon and its neighbouring domain wall fuse to the identity or to the adjoint, weighting the answer so that the trivial (channel-blind) part cancels: it is a Kondo-type coupling [7 , 48 ] that distinguishes the two screening channels of the impurity anyon and realises the driving perturbation up to irrelevant corrections. The main structural result is that this channel asymmetry realises the KRW perturbation in the scaling limit, with the unwanted identity and leading irrelevant-primary overlaps vanishing to numerical precision at the tested sizes.
The deformed Hamiltonian. The model studied in the rest of the paper is
H = H 0 + ∑ s t r a n d s [ K Φ c l + λ ( n L + n R ) ] . (29) H=H_0+\sum_{\mathrm{strands}}\left[K\Phi_{\mathrm{cl}}+\lambda(n_{\mathrm{L}}+n_{\mathrm{R}})\right]. \tag*{(29)} H = H 0 + strands ∑ [ K Φ cl + λ ( n L + n R ) ] . (29) with the two seams built as in § 3.2 . The coupling K K K λ \lambda λ § 4 , § 5 ). Both terms commute with the wrapped lines, so every eigenstate of (29) keeps its ( Y ( 2 , 1 ) , η ) (Y_{(2,1)},\eta) ( Y ( 2 , 1 ) , η )
Symmetry-resolved numerics
The admissible Hilbert space grows quickly with L L L 2.7 × 10 6 2.7\times10^6 2.7 × 1 0 6 p = 4 p=4 p = 4 L = 16 L=16 L = 16 1.9 × 10 6 1.9\times10^6 1.9 × 1 0 6 p = 5 p=5 p = 5 L = 14 L=14 L = 14 Y ( p − 1 , 1 ) = η Y_{(p-1,1)}=\eta Y ( p − 1 , 1 ) = η a ↦ p − a a\mapsto p-a a ↦ p − a η = ± 1 \eta=\pm1 η = ± 1 p = 4 p=4 p = 4 η = + 1 \eta=+1 η = + 1 Y ( 2 , 1 ) Y_{(2,1)} Y ( 2 , 1 ) Y ( 2 , 1 ) 2 = 1 + η = 2 Y_{(2,1)}^2=1+\eta=2 Y ( 2 , 1 ) 2 = 1 + η = 2 Y ( 2 , 1 ) / 2 Y_{(2,1)}/\sqrt{2} Y ( 2 , 1 ) / 2 ( Y ( 2 , 1 ) , η ) (Y_{(2,1)},\eta) ( Y ( 2 , 1 ) , η ) L = 24 , 32 L = 24, 32 L = 24 , 32 p = 5 p = 5 p = 5 [49 , 50 ] on the same Hamiltonian, written as an MPO on the ring, isolating the charge sectors by adding to H H H
Observables
Before turning to the two flows we define the quantities that will be used to characterise an interface, and how each is measured on the ring of § 3 . Throughout, L L L
Scaling dimensions. In radial quantisation about one strand, the eigenstates of H H H [42 , 43 , 44 ]
Δ = ( E − E 0 ) L 2 π v , (30) \Delta= \frac{(E-E_0)L}{2\pi v}, \tag*{(30)} Δ = 2 π v ( E − E 0 ) L , (30) with E 0 E_0 E 0 v v v § 3.1 . For a topological line D D D H D × D = ⨁ k N D D k H k \mathcal{H}_{D\times D} = \bigoplus_k N^k_{DD}\mathcal{H}_k H D × D = ⨁ k N D D k H k § 2.3 ), whose content is fixed by the fusion rules; for a non-topological conformal interface it is not known a priori and is one of the outputs of the computation. At a topological endpoint D × D D\times D D × D (30) measures dimensions from it.
Charge sectors. The wrapped lines Y ( 2 , 1 ) Y_{(2,1)} Y ( 2 , 1 ) η = Y ( p − 1 , 1 ) \eta= Y_{(p-1,1)} η = Y ( p − 1 , 1 ) H H H § 3.2 ), so every level carries a pair of quantum numbers. Their eigenvalues on a state of the ( r ′ , s ′ ) (r',s') ( r ′ , s ′ ) S ( r , 1 ) , ( r ′ , s ′ ) / S ( 1 , 1 ) , ( r ′ , s ′ ) S_{(r,1),(r',s')}/S_{(1,1),(r',s')} S ( r , 1 ) , ( r ′ , s ′ ) / S ( 1 , 1 ) , ( r ′ , s ′ ) ϕ ( r ′ , s ) ⊗ ϕ ˉ ( r ′ ′ , s ) \phi_{(r',s)} \otimes\bar{\phi}_{(r'',s)} ϕ ( r ′ , s ) ⊗ ϕ ˉ ( r ′′ , s ) ( r ′ , s ) (r',s) ( r ′ , s ) § 4 and § 5 . At p = 4 p = 4 p = 4 Y ( 2 , 1 ) ∈ { + 2 , 0 , − 2 } Y_{(2,1)} \in\{+\sqrt{2}, 0, -\sqrt{2}\} Y ( 2 , 1 ) ∈ { + 2 , 0 , − 2 } η = ± 1 \eta= \pm1 η = ± 1 p = 5 p = 5 p = 5 Y ( 2 , 1 ) = Y τ F i b η Y_{(2,1)} = Y_{\mathrm{\tau Fib}}\eta Y ( 2 , 1 ) = Y τ Fib η Y τ F i b ∈ { ϕ , − ϕ − 1 } Y_{\mathrm{\tau Fib}} \in\{\phi, -\phi^{-1}\} Y τ Fib ∈ { ϕ , − ϕ − 1 } η = ± 1 \eta= \pm1 η = ± 1
The g g g The g g g g = ⟨ 0 ∣ D ∣ 0 ⟩ g = \langle0|D|0\rangle g = ⟨ 0∣ D ∣0 ⟩ [26 ] , the overlap of the interface with the vacuum on the cylinder; it equals the quantum dimension for a topological line (g η = 1 g_\eta= 1 g η = 1 g D ( 2 , 1 ) = 2 g_{D_{(2,1)}} = \sqrt{2} g D ( 2 , 1 ) = 2 p = 4 p = 4 p = 4 ϕ \phi ϕ p = 5 p = 5 p = 5 [26 , 51 , 52 ] . We extract it from the entanglement entropy, using the fact that the boundary or impurity entropy appears as an additive constant in the entanglement entropy of an interval that contains the defect [53 , 54 , 55 , 56 ] . Let S D ( ℓ ) S_D(\ell) S D ( ℓ ) ℓ = 2 k + 1 \ell= 2k + 1 ℓ = 2 k + 1 S b u l k ( ℓ ) S_{\mathrm{bulk}}(\ell) S bulk ( ℓ ) ℓ \ell ℓ ℓ \ell ℓ
ln g = S D ( ℓ ) − S b u l k ( ℓ ) (31) \ln g = S_D(\ell) - S_{\mathrm{bulk}}(\ell) \tag*{(31)} ln g = S D ( ℓ ) − S bulk ( ℓ ) (31) up to corrections that vanish as ℓ , L → ∞ \ell,L \to\infty ℓ , L → ∞ p = 4 p = 4 p = 4 § 4 ).
The transmission coefficient. A conformal interface is characterised, beyond g g g [57 ] (see also [58 , 59 ] ), the transmission coefficient T T T ⟨ T ( z ) T ( w ) ⟩ \langle T(z)T(w)\rangle ⟨ T ( z ) T ( w )⟩ z z z w w w T = 1 T=1 T = 1 T = 0 T=0 T = 0
J j = i [ h j , h j + 1 ] , (32) J_j = i [h_j,h_{j+1}], \tag*{(32)} J j = i [ h j , h j + 1 ] , (32) the time derivative of the local energy; being the divergence of a conserved density, its scaling limit is proportional to T − T ˉ T-\bar{T} T − T ˉ 1 / x 4 1/x^4 1/ x 4 J j J_j J j ⟨ J i J j ⟩ c = ⟨ J i J j ⟩ − ⟨ J i ⟩ ⟨ J j ⟩ \langle J_iJ_j\rangle_c=\langle J_iJ_j\rangle-\langle J_i\rangle\langle J_j\rangle ⟨ J i J j ⟩ c = ⟨ J i J j ⟩ − ⟨ J i ⟩ ⟨ J j ⟩ i i i j j j L − 1 L-1 L − 1 d d d
T ( d ) = ⟨ J i J j ⟩ r i n g w i t h s t r a n d s ⟨ J i J j ⟩ p u r e r i n g , d = separation of the blocks i , j , (33) T(d) = \frac{\langle J_iJ_j\rangle^{\mathrm{ring\ with\ strands}}}{\langle J_iJ_j\rangle^{\mathrm{pure\ ring}}}, \qquad d=\text{separation of the blocks }i,j, \tag*{(33)} T ( d ) = ⟨ J i J j ⟩ pure ring ⟨ J i J j ⟩ ring with strands , d = separation of the blocks i , j , (33) which divides out the non-universal normalisation of J J J 1 / d 4 1/d^4 1/ d 4 d d d d d d d d d σ / n p a i r s \sigma/\sqrt{n_{\mathrm{pairs}}} σ / n pairs d d d T ( d ) T(d) T ( d ) d d d L L L d = L / 2 d=L/2 d = L /2 χ = 200 \chi=200 χ = 200 1 / L 1/L 1/ L
Lattice β \beta β The KRW flow can be read off directly on the lattice, by the finite-size scaling method introduced by Lüscher, Weisz and Wolff for asymptotically free theories [60 ] : a running coupling is defined by a finite-volume observable (there the mass gap of the O ( 3 ) O(3) O ( 3 ) σ \sigma σ L L L [61 ] ), and its change from L L L s L sL s L β \beta β
Developed into the Schrödinger-functional scheme [62 , 63 ] , this is how the running coupling of QCD is computed non-perturbatively [64 ] . In the statistical-mechanics literature the same idea, applied to an exponent-like finite-size observable, is the phenomenological renormalisation group of Nightingale and of Roomany and Wyld [65 , 66 ] . One needs a dimensionless running coupling, defined by a finite-volume quantity, that vanishes at the UV defect, is monotonic along the flow and takes a definite value at each fixed point. We take it to be a finite-volume Casimir energy of the single-strand ring (one D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) H D ( 2 , 1 ) \mathcal{H}_{D_{(2,1)}} H D ( 2 , 1 ) § 2.3 ), which is precisely the type of coupling used in [60 ] : the L L L H D ( 2 , 1 ) \mathcal{H}_{D_{(2,1)}} H D ( 2 , 1 ) ( 1 , 2 ) ⊗ ( 2 , 2 ) (1,2) \otimes(2,2) ( 1 , 2 ) ⊗ ( 2 , 2 ) ( 2 , 2 ) ⊗ ( 1 , 2 ) (2,2) \otimes(1,2) ( 2 , 2 ) ⊗ ( 1 , 2 )
Δ k = 1 = h 1 , 2 + h 2 , 2 = ( p − 1 ) 2 + 2 4 p ( p + 1 ) ( 11 80 at p = 4 , 3 20 at p = 5 ) , (34) \Delta_{k=1}=h_{1,2}+h_{2,2}=\frac{(p-1)^2+2}{4p(p+1)}\quad\left(\frac{11}{80}\text{ at }p=4,\ \frac{3}{20}\text{ at }p=5\right), \tag*{(34)} Δ k = 1 = h 1 , 2 + h 2 , 2 = 4 p ( p + 1 ) ( p − 1 ) 2 + 2 ( 80 11 at p = 4 , 20 3 at p = 5 ) , (34) exchanged by the swap of the two sides of the defect and distinguished by their left labels, hence by their ( Y ( 2 , 1 ) , η ) (Y_{(2,1)},\eta) ( Y ( 2 , 1 ) , η ) ( 1 , 2 ) (1,2) ( 1 , 2 ) Y ( 2 , 1 ) Y_{(2,1)} Y ( 2 , 1 ) S ( 2 , 1 ) , ( 1 , 2 ) / S ( 1 , 1 ) , ( 1 , 2 ) = − 2 cos ( π / p ) S_{(2,1),(1,2)}/S_{(1,1),(1,2)}=-2\cos(\pi/p) S ( 2 , 1 ) , ( 1 , 2 ) / S ( 1 , 1 ) , ( 1 , 2 ) = − 2 cos ( π / p ) ( Y ( 2 , 1 ) , η ) = ( − 2 , + 1 ) (Y_{(2,1)},\eta)=(-\sqrt{2},+1) ( Y ( 2 , 1 ) , η ) = ( − 2 , + 1 ) p = 4 p=4 p = 4 ( − φ , − 1 ) (-\varphi,-1) ( − φ , − 1 ) d − d- d − § 5.3 , at $p=5, and call it the k = 1 k=1 k = 1 § 6 it is the ground state of the sector whose non-contractible loops carry the weight − 2 cos ( π / p ) -2\cos(\pi/p) − 2 cos ( π / p ) p p p K K K E ( k = 1 ) ( L , K ) E^{(k=1)}(L,K) E ( k = 1 ) ( L , K ) δ E ( L , K ) = E ( k = 1 ) ( L , K ) − E ( k = 1 ) ( L , 0 ) \delta E(L,K)=E^{(k=1)}(L,K)-E^{(k=1)}(L,0) δ E ( L , K ) = E ( k = 1 ) ( L , K ) − E ( k = 1 ) ( L , 0 ) ℓ \ell ℓ ℓ ∓ 1 \ell\mp1 ℓ ∓ 1 L L L ℓ = 7 , 9 , 11 , … \ell=7,9,11,\ldots ℓ = 7 , 9 , 11 , … ℓ \ell ℓ
u C ( ℓ , K ) = ℓ 2 − 1 4 π v [ δ E ( ℓ − 1 , K ) − δ E ( ℓ + 1 , K ) ] , (35) u_C(\ell,K)=\frac{\ell^2-1}{4\pi v}\left[\delta E(\ell-1,K)-\delta E(\ell+1,K)\right], \tag*{(35)} u C ( ℓ , K ) = 4 π v ℓ 2 − 1 [ δ E ( ℓ − 1 , K ) − δ E ( ℓ + 1 , K ) ] , (35) a lattice version of − ℓ 2 ∂ ℓ δ E / 2 π v -\ell^2\partial_\ell\delta E/2\pi v − ℓ 2 ∂ ℓ δ E /2 π v ℓ \ell ℓ E ( L , K ) = e b u l k L + e d e f ( K ) + 2 π v L ( x − c 12 ) + … E(L,K)=e_{\mathrm{bulk}}L+e_{\mathrm{def}}(K)+\frac{2\pi v}{L}\left(x-\frac{c}{12}\right)+\ldots E ( L , K ) = e bulk L + e def ( K ) + L 2 π v ( x − 12 c ) + … [42 , 43 , 44 ] , subtracting K = 0 K=0 K = 0 u C → x ( K ) − x ( 0 ) u_C\to x(K)-x(0) u C → x ( K ) − x ( 0 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) η \eta η p = 4 p=4 p = 4 ( 1 10 , 3 20 ) \left(\frac{1}{10},\frac{3}{20}\right) ( 10 1 , 20 3 ) Z 2 \mathbb{Z}_2 Z 2 u C ∗ = 0.7 − 11 80 = 0.5625 u_C^*=0.7-\frac{11}{80}=0.5625 u C ∗ = 0.7 − 80 11 = 0.5625 u C u_C u C K K K
B C ( K ; ℓ 1 , ℓ 2 ) = d u C d ln ℓ ≃ u C ( ℓ 2 , K ) − u C ( ℓ 1 , K ) ln ( ℓ 2 / ℓ 1 ) , (36) B_C(K;\ell_1,\ell_2)=\frac{d u_C}{d\ln\ell}\simeq\frac{u_C(\ell_2,K)-u_C(\ell_1,K)}{\ln(\ell_2/\ell_1)}, \tag*{(36)} B C ( K ; ℓ 1 , ℓ 2 ) = d ln ℓ d u C ≃ ln ( ℓ 2 / ℓ 1 ) u C ( ℓ 2 , K ) − u C ( ℓ 1 , K ) , (36) is a lattice β \beta β u C u_C u C u C u_C u C u C u_C u C u C ∝ K ℓ 1 − Δ ϕ u_C\propto K\ell^{1-\Delta_\phi} u C ∝ K ℓ 1 − Δ ϕ 1 − Δ ϕ 1-\Delta_\phi 1 − Δ ϕ B C B_C B C A C = 1 2 [ u C ( ℓ 1 ) + u C ( ℓ 2 ) ] A_C=\frac{1}{2}[u_C(\ell_1)+u_C(\ell_2)] A C = 2 1 [ u C ( ℓ 1 ) + u C ( ℓ 2 )]
Improved points. A lattice Hamiltonian of the family (29) sits at a fixed point only up to irrelevant operators, whose amplitudes depend on ( K , λ ) (K,\lambda) ( K , λ ) 1 / L 1/L 1/ L improved point a choice of ( K , λ ) (K,\lambda) ( K , λ ) [67 ] , as judged by the flatness of the finite-size spectrum in 1 / L 1/L 1/ L
Numerical methods. Exact diagonalisation (ED) uses the symmetry-reduced blocks of § 3.5 for L ≤ 14 L \le14 L ≤ 14 [49 , 50 ] treat the same ring at L = 24 , 32 , 48 L = 24,32,48 L = 24 , 32 , 48 p = 4 p = 4 p = 4 L = 24 , 32 , 48 L = 24,32,48 L = 24 , 32 , 48 p = 5 p = 5 p = 5 χ = 300 \chi= 300 χ = 300 H H H S D S_D S D S b u l k S_{\mathrm{bulk}} S bulk
Results at p = 4 p = 4 p = 4 D ( 2 , 1 ) → η D_{(2,1)} \to\eta D ( 2 , 1 ) → η
For the tricritical Ising model the category is Ising, D ( 2 , 1 ) = σ ′ D_{(2,1)} = \sigma' D ( 2 , 1 ) = σ ′ g = 2 g = \sqrt{2} g = 2 ( r , 1 ) (r,1) ( r , 1 ) g g g η \eta η g = 1 g = 1 g = 1 g = 1 g = 1 g = 1 p = 4 p = 4 p = 4 C C C M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) M ( 4 , 5 ) M(4,5) M ( 4 , 5 ) [6 ] , §6.2.4 and § 7 ). They do note that at p = 4 p = 4 p = 4 D ( 2 , 1 ) F 1 ∣ 1 D_{(2,1)}F^{1|1} D ( 2 , 1 ) F 1∣1 g g g D ( 1 , 2 ) D_{(1,2)} D ( 1 , 2 ) [6 ] , § 4 ). The invertible line η \eta η § 2 combined with the g g g g g g 2 \sqrt{2} 2 ( r , 1 ) (r,1) ( r , 1 ) g ≤ 2 g \le\sqrt{2} g ≤ 2 η \eta η η \eta η [11 ] for the conformal defects of the tricritical Ising model.) The bulk chain used here, the dilute A 3 A_3 A 3 [15 ] , belongs to the same class of Ising-fusion-category chains (labels 1 , σ , ψ 1,\sigma,\psi 1 , σ , ψ 1 , w , η 1,w,\eta 1 , w , η [24 ] . The integrable weights (13) place us directly at that point with no tuning.
Calibration of the g g g We first apply (31) to the two undeformed topological rings, with strands η \eta η σ ′ \sigma' σ ′
η = D ( 3 , 1 ) : ln g = − 0.01 ( g = 0.99 ) , σ ′ = D ( 2 , 1 ) : ln g = + 0.335 ( g = 1.40 ) , \eta= D_{(3,1)}:\ \ln g = -0.01\ (g = 0.99), \qquad\sigma' = D_{(2,1)}:\ \ln g = +0.335\ (g = 1.40), η = D ( 3 , 1 ) : ln g = − 0.01 ( g = 0.99 ) , σ ′ = D ( 2 , 1 ) : ln g = + 0.335 ( g = 1.40 ) , in agreement with g η = 1 g_{\eta}=1 g η = 1 g σ ′ = 2 = 1.414 g_{\sigma'}=\sqrt{2}=1.414 g σ ′ = 2 = 1.414 ± 0.01 \pm0.01 ± 0.01
Endpoint of the flow. Driving D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) Φ c l \Phi_{\mathrm{cl}} Φ cl ln g \ln g ln g g g g ln 2 \ln\sqrt{2} ln 2 ln g = − 0.04 \ln g=-0.04 ln g = − 0.04 g = 1.00 g=1.00 g = 1.00 K K K g = 1 g=1 g = 1 η \eta η
Improved point. The endpoint η \eta η (29) . We can tune away the leading finite-size effect by searching for a point with good conformal tower. The couplings are
( K , λ ) η = ( 4.82 , 0.17 ) . (37) (K,\lambda)_{\eta}=(4.82,0.17). \tag*{(37)} ( K , λ ) η = ( 4.82 , 0.17 ) . (37) g g g Figure 12 shows the three observables of § 3.6 at the improved point as functions of 1 / L 1/L 1/ L C 5 C_5 C 5 L = 8 L=8 L = 8 L = 24 L=24 L = 24 H ′ = H − 1.0 Y ( 2 , 1 ) − 0.8 η H'=H-1.0\,Y_{(2,1)}-0.8\,\eta H ′ = H − 1.0 Y ( 2 , 1 ) − 0.8 η H H H ⟨ Y ( 2 , 1 ) ⟩ = 1.4142 \langle Y_{(2,1)}\rangle=1.4142 ⟨ Y ( 2 , 1 ) ⟩ = 1.4142 ⟨ η ⟩ = 1.0000 \langle\eta\rangle=1.0000 ⟨ η ⟩ = 1.0000 p = 4 p=4 p = 4 χ = 160 \chi=160 χ = 160 2 × 10 − 9 2\times10^{-9} 2 × 1 0 − 9 g g g (31) is flat at g = 0.99 ( 2 ) g=0.99(2) g = 0.99 ( 2 ) L = 8 L=8 L = 8 g = 0.9965 g=0.9965 g = 0.9965 L = 24 L=24 L = 24 g η = 1 g_{\eta}=1 g η = 1 ± 0.01 \pm0.01 ± 0.01
Figure 12. p = 4 p=4 p = 4 ( K , λ ) = ( 4.82 , 0.17 ) (K,\lambda)=(4.82,0.17) ( K , λ ) = ( 4.82 , 0.17 ) g g g T T T d = L / 2 d=L/2 d = L /2 L = 8 , … , 14 L=8,\ldots,14 L = 8 , … , 14 L = 24 L=24 L = 24 g η = 1 g_{\eta}=1 g η = 1 g D ( 2 , 1 ) = 2 g_{D_{(2,1)}}=\sqrt{2} g D ( 2 , 1 ) = 2 T = 1 T=1 T = 1 Δ ε ′ = 1.2 \Delta_{\varepsilon'}=1.2 Δ ε ′ = 1.2 1 + c L − 2.7 1+cL^{-2.7} 1 + c L − 2.7 1 / L 1/L 1/ L L ≥ 12 L\geq12 L ≥ 12
and far from g D ( 2 , 1 ) = 2 g_{D_{(2,1)}}=\sqrt{2} g D ( 2 , 1 ) = 2 (33) at d = L / 2 d=L/2 d = L /2 L = 10 L=10 L = 10 L = 24 L=24 L = 24 T − 1 ∝ L − 2.7 T-1\propto L^{-2.7} T − 1 ∝ L − 2.7 1 1 1 η \eta η σ ′ \sigma' σ ′ L = 12 L = 12 L = 12 T = 1.000 \mathcal{T} = 1.000 T = 1.000 1 1 1 p = 5 p = 5 p = 5 T \mathcal{T} T 1 1 1 § 5.2 ). (iii) The leading irrelevant defect field in the neutral sector (Y ( 2 , 1 ) = + 2 Y_{(2,1)} = +\sqrt{2} Y ( 2 , 1 ) = + 2 η = + 1 \eta= +1 η = + 1 Δ = 1.238 , 1.245 , 1.247 , 1.248 \Delta= 1.238, 1.245, 1.247, 1.248 Δ = 1.238 , 1.245 , 1.247 , 1.248 L = 8 L = 8 L = 8 14 14 14 1.244 , 1.241 , 1.236 1.244, 1.241, 1.236 1.244 , 1.241 , 1.236 L = 24 , 32 , 48 L = 24, 32, 48 L = 24 , 32 , 48 1.23 1.23 1.23 ϵ ′ \epsilon' ϵ ′ 1.2 1.2 1.2 ∼ 3 % \sim3\% ∼ 3% Figure 13 .
Figure 13. p = 4 p = 4 p = 4 ( K , λ ) = ( 4.82 , 0.17 ) (K,\lambda) = (4.82, 0.17) ( K , λ ) = ( 4.82 , 0.17 ) Δ \Delta Δ 1 / L 1/L 1/ L ( Y ( 2 , 1 ) , η ) (Y_{(2,1)},\eta) ( Y ( 2 , 1 ) , η ) L = 8 , 10 , 12 , 14 L = 8, 10, 12, 14 L = 8 , 10 , 12 , 14 L = 24 L = 24 L = 24 η \eta η T T T T ‾ \overline{T} T 2 2 2 L − 1 , L ‾ − 1 L_{-1},\overline{L}_{-1} L − 1 , L − 1 1.2 1.2 1.2 ϵ \epsilon ϵ 1.075 1.075 1.075 σ \sigma σ 2 2 2 ϵ \epsilon ϵ 2.2 2.2 2.2 1 / L = 0 1/L = 0 1/ L = 0 1 / L 1/L 1/ L η \eta η D ( 2 , 1 ) → η D_{(2,1)} \to\eta D ( 2 , 1 ) → η
Sector-resolved spectrum. At the IR endpoint each strand has become η \eta η η × η = 1 \eta\times\eta= 1 η × η = 1
Y ( 2 , 1 ) = + 2 , η = + 1 : { 1 , ϵ ′ } , Y ( 2 , 1 ) = − 2 , η = + 1 : { ϵ , ϵ ′ ′ } , Y_{(2,1)} = +\sqrt{2},\ \eta= +1:\ \{\mathbf{1},\epsilon'\}, \qquad Y_{(2,1)} = -\sqrt{2},\ \eta= +1:\ \{\epsilon,\epsilon''\}, Y ( 2 , 1 ) = + 2 , η = + 1 : { 1 , ϵ ′ } , Y ( 2 , 1 ) = − 2 , η = + 1 : { ϵ , ϵ ′′ } , Y ( 2 , 1 ) = 0 , η = − 1 : { σ , σ ′ } . Y_{(2,1)} = 0,\ \eta= -1:\ \{\sigma,\sigma'\}. Y ( 2 , 1 ) = 0 , η = − 1 : { σ , σ ′ } . where ϵ , ϵ ′ , ϵ ′ ′ , σ , σ ′ \epsilon,\epsilon',\epsilon'',\sigma,\sigma' ϵ , ϵ ′ , ϵ ′′ , σ , σ ′ Δ = 0.2 , 1.2 , 3 , 0.075 , 0.875 \Delta= 0.2, 1.2, 3, 0.075, 0.875 Δ = 0.2 , 1.2 , 3 , 0.075 , 0.875 Figure 13 shows the three sectors at the improved point for L = 8 , … , 14 L = 8,\ldots,14 L = 8 , … , 14 L = 24 L = 24 L = 24 H H H § 5.3 ). Linear-in-1 / L 1/L 1/ L
Δ → 0.08 ( σ , 0.075 ) , 0.24 ( ϵ , 0.2 ) , 0.87 ( σ ′ , 0.875 ) , 1.24 ( ϵ ′ , 1.2 ) , \Delta\to0.08\ (\sigma,\,0.075), \qquad0.24\ (\epsilon,\,0.2), \qquad0.87\ (\sigma',\,0.875), \qquad1.24\ (\epsilon',\,1.2), Δ → 0.08 ( σ , 0.075 ) , 0.24 ( ϵ , 0.2 ) , 0.87 ( σ ′ , 0.875 ) , 1.24 ( ϵ ′ , 1.2 ) , and the descendants behave as they should. Together with g → 1 g \to1 g → 1 D ( 2 , 1 ) → η D_{(2,1)} \to\eta D ( 2 , 1 ) → η D ( 2 , 1 ) → 1 D_{(2,1)} \to1 D ( 2 , 1 ) → 1 p = 4 p = 4 p = 4
Lattice β \beta β We now apply the step-scaling construction of § 3.6 : the Casimir coupling u C u_C u C (35) , built from the k = 1 k=1 k = 1 p = 4 p=4 p = 4 ( Y ( 2 , 1 ) , η ) = ( − 2 , + 1 ) (Y_{(2,1)},\eta)=(-\sqrt{2},+1) ( Y ( 2 , 1 ) , η ) = ( − 2 , + 1 ) ( 0 , − 1 ) (0,-1) ( 0 , − 1 ) η \eta η ( 1 10 , 3 5 ) (\frac{1}{10},\frac{3}{5}) ( 10 1 , 5 3 ) Z 2 \mathbb{Z}_2 Z 2 u C ∗ = 0.7 − 11 80 = 0.5625 u_C^*=0.7-\frac{11}{80}=0.5625 u C ∗ = 0.7 − 80 11 = 0.5625 ( K , λ ) η = ( 4.82 , 0.17 ) (K,\lambda)_\eta=(4.82,0.17) ( K , λ ) η = ( 4.82 , 0.17 ) H = H 0 + K Φ c l + λ ( n L + n R ) H=H_0+K\Phi_{\rm cl}+\lambda(n_L+n_R) H = H 0 + K Φ cl + λ ( n L + n R ) λ = 0.035 K \lambda=0.035K λ = 0.035 K 0 ≤ K ≤ 3 0\le K\le3 0 ≤ K ≤ 3 K ≤ 0.2 K\le0.2 K ≤ 0.2 E ( k = 1 ) ( L , K ) E^{(k=1)}(L,K) E ( k = 1 ) ( L , K ) § 3.5 at L = 6 , 8 , … , 16 L=6,8,\ldots,16 L = 6 , 8 , … , 16 ℓ = 7 , 9 , … , 15 \ell=7,9,\ldots,15 ℓ = 7 , 9 , … , 15 ( 7 , 9 ) , … , ( 13 , 15 ) (7,9),\ldots,(13,15) ( 7 , 9 ) , … , ( 13 , 15 )
Figure 14. Lattice β \beta β p = 4 p=4 p = 4 H = H 0 + K Φ c l + λ ( n L + n R ) H=H_0+K\Phi_{\rm cl}+\lambda(n_L+n_R) H = H 0 + K Φ cl + λ ( n L + n R ) λ = 0.035 K \lambda=0.035K λ = 0.035 K k = 1 k=1 k = 1 L = 6 , … , 16 L=6,\ldots,16 L = 6 , … , 16 Left: B C = d u C / d ln ℓ B_C=du_C/d\ln\ell B C = d u C / d ln ℓ (36) versus the running coordinate A C = 1 2 [ u C ( ℓ 1 ) + u C ( ℓ 2 ) ] A_C=\frac{1}{2}[u_C(\ell_1)+u_C(\ell_2)] A C = 2 1 [ u C ( ℓ 1 ) + u C ( ℓ 2 )] ℓ = ( 7 , 9 ) , … , ( 13 , 15 ) \ell=(7,9),\ldots,(13,15) ℓ = ( 7 , 9 ) , … , ( 13 , 15 ) 0 ≤ K ≤ 3 0\le K\le3 0 ≤ K ≤ 3 D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) u C = 0 u_C=0 u C = 0 1 − Δ ϕ = 0.3 1-\Delta_\phi=0.3 1 − Δ ϕ = 0.3 η \eta η u C ∗ = 0.5625 u_C^*=0.5625 u C ∗ = 0.5625 Δ ϵ ′ − 1 = − 0.2 \Delta_{\epsilon'}-1=-0.2 Δ ϵ ′ − 1 = − 0.2 Right: the raw coupling u C ( ℓ , K ) u_C(\ell,K) u C ( ℓ , K ) K K K ℓ = 7 , … , 15 \ell=7,\ldots,15 ℓ = 7 , … , 15 u C ∗ u_C^* u C ∗
Figure 14 shows the result. The collapse of the successive size pairs is excellent—from ( 7 , 9 ) (7,9) ( 7 , 9 ) ( 13 , 15 ) (13,15) ( 13 , 15 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) B C ≃ 0.065 B_C\simeq0.065 B C ≃ 0.065 K ≃ 1.2 K\simeq1.2 K ≃ 1.2 A C ≃ 0.3 A_C\simeq0.3 A C ≃ 0.3
K ∗ = 1.900 , 1.892 , 1.900 , 1.912 , K^*=1.900,\ 1.892,\ 1.900,\ 1.912, K ∗ = 1.900 , 1.892 , 1.900 , 1.912 , A C ∗ = 0.516 , 0.514 , 0.515 , 0.518 for ℓ = ( 7 , 9 ) , ( 9 , 11 ) , ( 11 , 13 ) , ( 13 , 15 ) , A_C^*=0.516,\ 0.514,\ 0.515,\ 0.518 \qquad\text{for }\ell=(7,9),\ (9,11),\ (11,13),\ (13,15), A C ∗ = 0.516 , 0.514 , 0.515 , 0.518 for ℓ = ( 7 , 9 ) , ( 9 , 11 ) , ( 11 , 13 ) , ( 13 , 15 ) , within 8% of the η \eta η u C ∗ = 0.5625 u_C^*=0.5625 u C ∗ = 0.5625 u C ∗ u_C^* u C ∗ K ≃ 2.1 K\simeq2.1 K ≃ 2.1 K = 3 K=3 K = 3 ℓ \ell ℓ B C / A C B_C/A_C B C / A C 1 / ℓ 1/\ell 1/ ℓ 1 − Δ ϕ = 0.3 1-\Delta_\phi=0.3 1 − Δ ϕ = 0.3 u C u_C u C ℓ \ell ℓ − 0.59 -0.59 − 0.59 − 0.63 -0.63 − 0.63 Δ ε ′ − 1 = − 0.2 \Delta_{\varepsilon'}-1=-0.2 Δ ε ′ − 1 = − 0.2 η \eta η p = 5 p=5 p = 5 ε ′ \varepsilon' ε ′ L − 0.2 L^{-0.2} L − 0.2 β \beta β D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) η \eta η
Single-strand ring: η \eta η The two-strand spectrum above is the untwisted spectrum H D × D ‾ \mathcal{H}_{D\times\overline{D}} H D × D g = 1 g=1 g = 1 η \eta η η × η = 1 \eta\times\eta=1 η × η = 1 1 × 1 = 1 1\times1=1 1 × 1 = 1 single strand distinguishes them. In radial quantisation its spectrum is the twisted Hilbert space H D \mathcal{H}_D H D Δ = 0 \Delta=0 Δ = 0 η \eta η Z 2 \mathbb{Z}_2 Z 2 H η = ⨁ ( r , s ) V ( r , s ) ⊗ V ‾ ( 4 − r , s ) \mathcal{H}_{\eta}=\bigoplus_{(r,s)}V_{(r,s)}\otimes\overline{V}_{(4-r,s)} H η = ⨁ ( r , s ) V ( r , s ) ⊗ V ( 4 − r , s ) ( 3 80 , 3 80 ) (\frac{3}{80},\frac{3}{80}) ( 80 3 , 80 3 ) ( 1 10 , 3 5 ) (\frac{1}{10},\frac{3}{5}) ( 10 1 , 5 3 ) ( 3 5 , 1 10 ) (\frac{3}{5},\frac{1}{10}) ( 5 3 , 10 1 ) ( 7 16 , 7 16 ) (\frac{7}{16},\frac{7}{16}) ( 16 7 , 16 7 ) ( 0 , 3 2 ) (0,\frac{3}{2}) ( 0 , 2 3 ) ( 3 2 , 0 ) (\frac{3}{2},0) ( 2 3 , 0 ) 0.075 0.075 0.075 0.7 0.7 0.7 0.7 0.7 0.7 0.875 0.875 0.875 1.5 1.5 1.5 1.5 1.5 1.5 ± 1 2 \pm\frac{1}{2} ± 2 1 ± 3 2 \pm\frac{3}{2} ± 2 3 D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) L = 8 L=8 L = 8 14 14 14 L = 24 L=24 L = 24 E b u l k ( L ) + e D E_{\mathrm{bulk}}(L)+e_D E bulk ( L ) + e D e D = 1 2 [ E 0 t w o s t r a n d s ( L ) − E b u l k ( L ) ] e_D=\frac{1}{2}[E^{\mathrm{two\ strands}}_0(L)-E_{\mathrm{bulk}}(L)] e D = 2 1 [ E 0 two strands ( L ) − E bulk ( L )] L L L e D = − 3.5921 e_D=-3.5921 e D = − 3.5921 L = 14 L=14 L = 14 − 3.5915 -3.5915 − 3.5915 L = 24 L=24 L = 24 48 48 48 H H H Figure 15 ) is unambiguous: there is no state at Δ = 0 \Delta=0 Δ = 0 ( 3 80 , 3 80 ) (\frac{3}{80},\frac{3}{80}) ( 80 3 , 80 3 ) Δ = 0.081 → 0.08 \Delta=0.081\to0.08 Δ = 0.081 → 0.08 1 2 \frac{1}{2} 2 1 0.69 / 0.67 → 0.68 / 0.67 0.69/0.67\to0.68/0.67 0.69/0.67 → 0.68/0.67 σ ′ \sigma' σ ′ Y ( 2 , 1 ) = ± 2 Y_{(2,1)}=\pm\sqrt{2} Y ( 2 , 1 ) = ± 2 § 2.3 ); ( 7 16 , 7 16 ) (\frac{7}{16},\frac{7}{16}) ( 16 7 , 16 7 ) 0.877 → 0.87 0.877\to0.87 0.877 → 0.87 3 2 \frac{3}{2} 2 3 1.46 / 1.52 → 1.51 / 1.53 1.46/1.52\to1.51/1.53 1.46/1.52 → 1.51/1.53 1.69 / 1.67 1.69/1.67 1.69/1.67 1.03 1.03 1.03 p = 4 p=4 p = 4 η \eta η
Figure 15. p = 4 p = 4 p = 4 D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) ( K , λ ) = ( 4.82 , 0.17 ) (K,\lambda) = (4.82, 0.17) ( K , λ ) = ( 4.82 , 0.17 ) 1 / L 1/L 1/ L Δ = ( E − E b u l k − e D ) L / 2 π v \Delta= (E - E_{\mathrm{bulk}} - e_D)L/2\pi v Δ = ( E − E bulk − e D ) L /2 π v L = 8 , … , 14 L = 8,\ldots,14 L = 8 , … , 14 L = 24 L = 24 L = 24 1 / L = 0 1/L = 0 1/ L = 0 η \eta η ( h , h ˉ ) (h,\bar{h}) ( h , h ˉ ) Δ = 0 \Delta= 0 Δ = 0 η \eta η
Results at p = 5 p=5 p = 5 D ( 2 , 1 ) → C 5 D_{(2,1)}\to C_5 D ( 2 , 1 ) → C 5
At p = 5 p=5 p = 5 F i b ⊠ Z 2 \mathrm{Fib}\boxtimes\mathbb{Z}_2 Fib ⊠ Z 2 D ( 2 , 1 ) = τ F i b ⊗ η D_{(2,1)}=\tau_{\mathrm{Fib}}\otimes\eta D ( 2 , 1 ) = τ Fib ⊗ η g = φ = 1.618 g=\varphi=1.618 g = φ = 1.618 η \eta η g = 1 g=1 g = 1 C 5 C_5 C 5
g C 5 = 1.503 ( 2 ) ( ln g = 0.408 ) , T = 0.665 ( 10 ) , g_{C_5}=1.503(2)\quad(\ln g=0.408),\qquad\mathcal{T}=0.665(10), g C 5 = 1.503 ( 2 ) ( ln g = 0.408 ) , T = 0.665 ( 10 ) , to be compared with ln g = ln φ = 0.481 \ln g=\ln\varphi=0.481 ln g = ln φ = 0.481 T = 1 \mathcal{T}=1 T = 1 D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) C 5 C_5 C 5 1 / L 1/L 1/ L β \beta β
Location of the fixed point
We use two parametrisations of the family (29) , with Φ c l = φ ( e L + e R ) − ( n L + n R ) \Phi_{\mathrm{cl}} = \varphi(e_L + e_R) - (n_L + n_R) Φ cl = φ ( e L + e R ) − ( n L + n R ) (28) at p = 5 p = 5 p = 5 clean ray λ = 0 \lambda= 0 λ = 0
H = H 0 + K Φ c l , (38) H = H_0 + K\Phi_{\mathrm{cl}}, \tag*{(38)} H = H 0 + K Φ cl , (38) on which a single coupling K K K ( K = 0 ) (K = 0) ( K = 0 ) C 5 C_5 C 5 β \beta β § 3.6 ), which for the clean ray is computed in the loop-model formulation of § 6 (identical to the RSOS chain at p = 5 p = 5 p = 5 § 6.1 ): its zeros for the successive size pairs converge with geometrically decreasing steps (§7.1) to
K C 5 ∗ = 0.95 ( 5 ) ( λ = 0 ) (39) K_{C_5}^{*} = 0.95(5)\quad(\lambda= 0) \tag*{(39)} K C 5 ∗ = 0.95 ( 5 ) ( λ = 0 ) (39) The second is a two-parameter tower-improved point,
( K , λ ) = ( 0.84 , 0.34 ) , (40) (K,\lambda) = (0.84,\ 0.34), \tag*{(40)} ( K , λ ) = ( 0.84 , 0.34 ) , (40) the same clean field with an irrelevant counterterm in the channel-blind occupation, which gives the flattest finite-size spectrum and is used for the spectra of § 5.3 . The two are distinct lattice representatives of the same continuum interface, differing only in irrelevant content, and they give consistent values of g g g T T T Δ X \Delta_X Δ X g g g § 5.2 , the sector-resolved spectrum and its 1 / L 1/L 1/ L § 5.3 , and the Potts-interface spectrum of § 5.5 —is computed at the tower-improved point (40) , where the corrections to scaling are smallest, with ED for L ≤ 16 L \le16 L ≤ 16 L = 24 L = 24 L = 24 32 32 32 48 48 48 β \beta β § 5.4 requires a one-parameter family that starts at the ultraviolet defect D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) K = 0 K = 0 K = 0 λ = 0.405 K \lambda= 0.405K λ = 0.405 K p = 4 p = 4 p = 4 (38) , whose step-scaling curves collapse better and which fixes the location (39) of C 5 C_5 C 5 Δ X \Delta_X Δ X
g-function, transmission and the leading irrelevant dimension
Figure 17 and Table 4 collect the three defining numbers of C 5 C_5 C 5 L ≤ 16 L \le16 L ≤ 16 L = 24 , 32 L = 24, 32 L = 24 , 32 χ = 200 \chi= 200 χ = 200 L = 48 L = 48 L = 48 χ = 300 \chi= 300 χ = 300
Table 4. g g g C 5 C_5 C 5 ( K , λ ) = ( 0.84 , 0.34 ) (K,\lambda) = (0.84, 0.34) ( K , λ ) = ( 0.84 , 0.34 ) L ≤ 16 L \le16 L ≤ 16 L = 24 , 32 L = 24, 32 L = 24 , 32 χ = 200 \chi= 200 χ = 200 L = 48 L = 48 L = 48 χ = 300 \chi= 300 χ = 300 ln g \ln g ln g (31) for the interval of 2 k + 1 2k+1 2 k + 1 T \mathcal{T} T (33) at antipodal separation d = L / 2 d = L/2 d = L /2 1 / L 1/L 1/ L L ≥ 14 L \ge14 L ≥ 14 Figure 17 ).
Figure 17. Finite-size flow of the three C 5 C_5 C 5 1 / L 1/L 1/ L g g g T \mathcal{T} T d = L / 2 d=L/2 d = L /2 Δ X \Delta_{\mathrm{X}} Δ X L = 24 , 32 L=24,32 L = 24 , 32 χ = 200 \chi=200 χ = 200 L = 48 L=48 L = 48 χ = 300 \chi=300 χ = 300 g g g T \mathcal{T} T 1 / L 1/L 1/ L g = φ g=\varphi g = φ T = 1 \mathcal{T}=1 T = 1
(i) The g g g (31) is already flat by L = 12 L = 12 L = 12 ln g = 0.4085 \ln g = 0.4085 ln g = 0.4085 0.4089 0.4089 0.4089 0.4084 0.4084 0.4084 0.4076 0.4076 0.4076 L = 14 L = 14 L = 14 24 24 24 32 32 32 48 48 48 g = 1.503 ( 2 ) g = 1.503(2) g = 1.503 ( 2 ) L = 32 L = 32 L = 32 48 48 48 10 − 3 10^{-3} 1 0 − 3 ln g \ln g ln g χ \chi χ φ \varphi φ
(ii) Figure 16 shows how the transmission is measured. The estimator (33) is averaged over all admissible pairs of current blocks at the same separation d d d L = 48 L = 48 L = 48 d d d d ≲ 8 d \lesssim8 d ≲ 8 L L L d = L / 2 d = L/2 d = L /2 L = 10 L = 10 L = 10 L = 24 , 32 , 48 L = 24, 32, 48 L = 24 , 32 , 48 1 / L 1/L 1/ L T = 0.663 \mathcal{T} = 0.663 T = 0.663 L ≥ 14 L \ge14 L ≥ 14 1 / L + 1 / L 2 1/L + 1/L^2 1/ L + 1/ L 2 L ≥ 12 L \ge12 L ≥ 12 T = 0.665 ( 10 ) \mathcal{T} = 0.665(10) T = 0.665 ( 10 ) C 5 C_5 C 5
Figure 16. Measurement of the transmission coefficient of C 5 C_5 C 5 (33) , i.e. the connected current correlator across the strand divided by the bulk correlator at the same separation, versus the separation d d d L = 48 L = 48 L = 48 χ = 300 \chi= 300 χ = 300 d d d ≲ 2 × 10 − 3 \lesssim2 \times10^{-3} ≲ 2 × 1 0 − 3 L → ∞ L \to\infty L → ∞ d = L / 2 d = L/2 d = L /2 1 / L 1/L 1/ L T = 0.665 ( 10 ) T = 0.665(10) T = 0.665 ( 10 )
(iii) The leading irrelevant defect field of C 5 C_5 C 5 X X X Y τ F i b = φ Y_{\tau\mathrm{Fib}} = \varphi Y τ Fib = φ η = + 1 \eta= +1 η = + 1 L L L
Δ X = 1.12 ( 1 ) , ϵ X ≡ Δ X − 1 = 0.12 ( 1 ) . (41) \Delta_X = 1.12(1), \qquad\epsilon_X \equiv\Delta_X - 1 = 0.12(1). \tag*{(41)} Δ X = 1.12 ( 1 ) , ϵ X ≡ Δ X − 1 = 0.12 ( 1 ) . (41) The DMRG points at L = 24 , 32 , 48 L = 24, 32, 48 L = 24 , 32 , 48 L L L
Sector-resolved spectrum and its 1 / L 1/L 1/ L
The four charge sectors of the ring are labelled by the Fibonacci charge (1 1 1 τ F i b \tau_{\mathrm{Fib}} τ Fib Y τ F i b = φ Y_{\tau\mathrm{Fib}} = \varphi Y τ Fib = φ − φ − 1 -\varphi^{-1} − φ − 1 Z 2 \mathbb{Z}_2 Z 2 η = ± 1 \eta= \pm1 η = ± 1
a + = ( 1 , + ) , b − = ( τ F i b , − ) , c + = ( τ F i b , + ) , d − = ( 1 , − ) , a+ = (1,+), \qquad b- = (\tau_{\mathrm{Fib}},-), \qquad c+ = (\tau_{\mathrm{Fib}},+), \qquad d- = (1,-), a + = ( 1 , + ) , b − = ( τ Fib , − ) , c + = ( τ Fib , + ) , d − = ( 1 , − ) , which are the four combinations that occur (the wrapped Y ( 2 , 1 ) = Y τ F i b η Y_{(2,1)} = Y_{\tau\mathrm{Fib}}\,\eta Y ( 2 , 1 ) = Y τ Fib η φ \varphi φ φ − 1 \varphi^{-1} φ − 1 − φ − 1 -\varphi^{-1} − φ − 1 − φ -\varphi − φ X X X a + a+ a + b − b- b − Table 5 lists the leading levels of each sector at the tower-improved point for L = 8 , 10 , 12 L = 8, 10, 12 L = 8 , 10 , 12 L = 24 , 32 L = 24, 32 L = 24 , 32 1 / L 1/L 1/ L Figure 18 shows all levels obtained from the symmetry-reduced diagonalisation together with integer-shifted guide lines, as a check that the finite-size spectrum organises into conformal towers.
Table 5. C 5 C_5 C 5 ( K , λ ) = ( 0.84 , 0.34 ) (K,\lambda)=(0.84,0.34) ( K , λ ) = ( 0.84 , 0.34 ) v = 5 / 3 v=5/3 v = 5/3 § 5.3 . DMRG column: L = 24 L=24 L = 24 X X X L = 32 L=32 L = 32 1 / L 1/L 1/ L
Figure 18. C 5 C_{5} C 5 1 / L 1/L 1/ L ( K , λ ) = ( 0.84 , 0.34 ) (K,\lambda) = (0.84, 0.34) ( K , λ ) = ( 0.84 , 0.34 ) L = 8 , 10 , 12 L = 8, 10, 12 L = 8 , 10 , 12 L = 14 L = 14 L = 14 L = 24 L = 24 L = 24 X X X X ′ X' X ′ a + a+ a + b − b- b − c + c+ c + d − d- d − L = 32 L = 32 L = 32 X X X Table 5 ), grey dotted lines their integer-shifted descendants Δ + n \Delta+ n Δ + n 1 / L 1/L 1/ L a + a+ a + Δ = 1 \Delta= 1 Δ = 1 Δ ≃ 2 \Delta\simeq2 Δ ≃ 2 T T T T ˉ \bar{T} T ˉ X X X X ′ X' X ′ b − b- b − c + c+ c + d − d- d − Δ + 1 \Delta+ 1 Δ + 1 Δ ≃ 1.15 \Delta\simeq1.15 Δ ≃ 1.15 c + c+ c + L − 1 L_{-1} L − 1 L ˉ − 1 \bar{L}_{-1} L ˉ − 1 h , h ˉ ≠ 0 h,\bar{h} \ne0 h , h ˉ = 0 c + c+ c + d − d- d − → \to → → \to →
L → ∞ L \to\infty L → ∞ g = 1.503 ( 2 ) g = 1.503(2) g = 1.503 ( 2 ) T = 0.665 ( 10 ) \mathcal{T} = 0.665(10) T = 0.665 ( 10 )
Lattice β \beta β
Construction. We apply the step-scaling construction of § 3.6 on the single-strand ring. At p = 5 p = 5 p = 5 k = 1 k = 1 k = 1 (34) is the lowest state of the sector d − d- d − Y ( 2 , 1 ) = − φ Y_{(2,1)} = -\varphi Y ( 2 , 1 ) = − φ η = − 1 \eta= -1 η = − 1 u C u_C u C (35) is formed from its energy at consecutive even sizes, and the β \beta β B C B_C B C (36) from u C u_C u C λ = 0.405 K \lambda= 0.405K λ = 0.405 K K = 0.84 K = 0.84 K = 0.84 (40) . The energies come from the symmetry-reduced ED of § 3.5 (η = − 1 \eta=-1 η = − 1 L = 6 , … , 16 L=6,\ldots,16 L = 6 , … , 16 [49 , 50 ] for L = 18 , 20 , 22 L=18,20,22 L = 18 , 20 , 22 d − d- d − H H H H ′ = H + 0.6 Y ( 2 , 1 ) + 0.5 η H'=H+0.6Y_{(2,1)}+0.5\eta H ′ = H + 0.6 Y ( 2 , 1 ) + 0.5 η χ = 120 \chi=120 χ = 120 ≲ 4 × 10 − 8 \lesssim4\times10^{-8} ≲ 4 × 1 0 − 8 L = 12 L=12 L = 12 4 × 10 − 7 4\times10^{-7} 4 × 1 0 − 7 ℓ = ( 7 , 9 ) , … , ( 13 , 15 ) \ell=(7,9),\ldots,(13,15) ℓ = ( 7 , 9 ) , … , ( 13 , 15 ) ( 15 , 17 ) (15,17) ( 15 , 17 ) ( 17 , 19 ) (17,19) ( 17 , 19 ) ( 19 , 21 ) (19,21) ( 19 , 21 )
Result. Figure 19 shows B C B_{\mathrm{C}} B C A C A_{\mathrm{C}} A C D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) K = 1.6 K=1.6 K = 1.6
Figure 19. Lattice β \beta β p = 5 p=5 p = 5 H = H 0 + K Φ c l + λ ( n L + n R ) H=H_0+K\Phi_{\mathrm{cl}}+\lambda(n_L+n_R) H = H 0 + K Φ cl + λ ( n L + n R ) λ = 0.405 K \lambda=0.405K λ = 0.405 K K = 0.84 K=0.84 K = 0.84 (40) ), from the Casimir coupling u C u_C u C k = 1 k=1 k = 1 B C = d u C / d ln ℓ B_C=du_C/d\ln\ell B C = d u C / d ln ℓ (36) versus the running coordinate A C A_C A C 0 < K ≤ 1.6 0<K\leq1.6 0 < K ≤ 1.6 L = 6 , … , 16 L=6,\ldots,16 L = 6 , … , 16 ℓ = ( 7 , 9 ) , … , ( 13 , 15 ) \ell=(7,9),\ldots,(13,15) ℓ = ( 7 , 9 ) , … , ( 13 , 15 ) L = 18 , 20 , 22 L=18,20,22 L = 18 , 20 , 22 ( 15 , 17 ) (15,17) ( 15 , 17 ) ( 17 , 19 ) (17,19) ( 17 , 19 ) ( 19 , 21 ) (19,21) ( 19 , 21 ) K = 0.5 , 0.75 , 0.95 , 1.1 ) K=0.5,0.75,0.95,1.1) K = 0.5 , 0.75 , 0.95 , 1.1 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) u C = 0 u_C=0 u C = 0 1 − Δ ϕ = 0.267 1-\Delta_\phi=0.267 1 − Δ ϕ = 0.267 C 5 C_5 C 5 K K K
K ∗ = 0.345 , 0.441 , 0.526 , 0.600 ( E D , ℓ = ( 7 , 9 ) , … , ( 13 , 15 ) ) , K^*=0.345,\ 0.441,\ 0.526,\ 0.600 \qquad(\mathrm{ED},\ \ell=(7,9),\ldots,(13,15)), K ∗ = 0.345 , 0.441 , 0.526 , 0.600 ( ED , ℓ = ( 7 , 9 ) , … , ( 13 , 15 )) , K ∗ = 0.663 , 0.734 , 0.793 K^{*} = 0.663, 0.734, 0.793 K ∗ = 0.663 , 0.734 , 0.793 ℓ = ( 15 , 17 ) , ( 17 , 19 ) , ( 19 , 21 ) \ell= (15, 17), (17, 19), (19, 21) ℓ = ( 15 , 17 ) , ( 17 , 19 ) , ( 19 , 21 ) A C ∗ A_{C}^{*} A C ∗ − 0.14 -0.14 − 0.14 − 0.18 -0.18 − 0.18 Δ X − 1 = − 0.12 \Delta_{X} - 1 = -0.12 Δ X − 1 = − 0.12 B C = 0 B_{C} = 0 B C = 0
C5 as an interface of the three-state Potts model
The tetracritical Ising model M ( 5 , 6 ) M(5, 6) M ( 5 , 6 ) Z 2 \mathbb{Z}_{2} Z 2 [68 , 69 , 70 ] : the Potts model is the orbifold of M ( 5 , 6 ) M(5, 6) M ( 5 , 6 ) η = D ( 4 , 1 ) \eta=D_{(4,1)} η = D ( 4 , 1 ) M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) C C C S 3 = Z 3 ⋊ Z 2 S_3=\mathbb{Z}_3\rtimes\mathbb{Z}_2 S 3 = Z 3 ⋊ Z 2 Rep ( S 3 ) \operatorname{Rep}(S_3) Rep ( S 3 ) M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) η \eta η ω + ω 2 \omega+\omega^2 ω + ω 2 Z 3 \mathbb{Z}_3 Z 3 D ( 1 , 3 ) D_{(1,3)} D ( 1 , 3 ) D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) η \eta η τ F i b \tau_{\mathrm{Fib}} τ Fib φ \varphi φ h = 2 / 5 h=2/5 h = 2/5 [5 ] . Any η \eta η M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) C 5 C_5 C 5 D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) η \eta η D ( 2 , 1 ) → C 5 D_{(2,1)}\to C_5 D ( 2 , 1 ) → C 5 τ F i b → C 5 P o t t s \tau_{\mathrm{Fib}}\to C^{\mathrm{Potts}}_5 τ Fib → C 5 Potts C 5 P o t t s C^{\mathrm{Potts}}_5 C 5 Potts
Orbifold rule and lattice implementation
Let I I I [3 , 70 ] , with M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) 2 \sqrt{2} 2 I I † = 1 ⊕ η II^\dagger=1\oplus\eta I I † = 1 ⊕ η M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) I † I = 1 ⊕ C I^\dagger I=1\oplus C I † I = 1 ⊕ C C 5 C_5 C 5 X = I † C 5 I X=I^\dagger C_5I X = I † C 5 I X X X X ˉ \bar{X} X ˉ X X ˉ = I † C 5 ( I I † ) C ˉ 5 I X\bar{X}=I^\dagger C_5(II^\dagger)\bar{C}_5I X X ˉ = I † C 5 ( I I † ) C ˉ 5 I M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) η \eta η η \eta η C 5 C_5 C 5 η \eta η η \eta η C 5 C_5 C 5 η \eta η I I † I I^\dagger I I † 1 2 ( 1 + η ) \frac{1}{2}(1+\eta) 2 1 ( 1 + η )
H X X ˉ = [ H C 5 C ˉ 5 ⊕ H C 5 η C ˉ 5 ] η -even (42) \mathcal{H}_{X\bar{X}}=\left[\mathcal{H}_{C_5\bar{C}_5}\oplus\mathcal{H}_{C_5\eta\bar{C}_5}\right]^{\eta\text{-even}} \tag*{(42)} H X X ˉ = [ H C 5 C ˉ 5 ⊕ H C 5 η C ˉ 5 ] η -even (42) the usual orbifold prescription: the η \eta η η \eta η η \eta η η \eta η Z 2 \mathbb{Z}_2 Z 2 [69 , 70 ] : gauging η \eta η η \eta η 1 2 ( 1 + η ) \frac{1}{2}(1+\eta) 2 1 ( 1 + η ) Z 2 \mathbb{Z}_2 Z 2 + 1 +1 + 1 (42) and as − 1 -1 − 1 η \eta η η \eta η M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) Z 2 \mathbb{Z}_2 Z 2 C C C C C C η \eta η M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) ( 2 , 2 ) (2,2) ( 2 , 2 ) h = 1 / 40 h=1/40 h = 1/40 Z 2 \mathbb{Z}_2 Z 2 μ \mu μ C C C C = + 1 C=+1 C = + 1 η × C 5 \eta\times C_5 η × C 5 C 5 C_5 C 5 X X X η \eta η I I I M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) (42) are two-strand rings of § 3.5 : the untwisted one is the ring used throughout § 5 , the twisted one is the same ring with a third, η = ( 4 , 1 ) \eta=(4,1) η = ( 4 , 1 ) C 5 C_5 C 5 § 3.2 ; the η \eta η q q q Y ( 2 , 1 ) Y_{(2,1)} Y ( 2 , 1 ) Y ( 3 , 1 ) Y_{(3,1)} Y ( 3 , 1 ) η \eta η q = 1 q=1 q = 1 φ \varphi φ q = τ F i b q=\tau_{\mathrm{Fib}} q = τ Fib − φ − 1 -\varphi^{-1} − φ − 1 Z 2 \mathbb{Z}_2 Z 2 C C C + 1 +1 + 1 − 1 -1 − 1 Z 3 \mathbb{Z}_3 Z 3 Z 3 \mathbb{Z}_3 Z 3 Z 3 \mathbb{Z}_3 Z 3 C = + C=+ C = + C = − C=- C = − ω \omega ω ω 2 \omega^2 ω 2 Z 3 \mathbb{Z}_3 Z 3 C C C C C C
Check at the topological point. At K = λ = 0 K=\lambda=0 K = λ = 0 τ F i b \tau_{\mathrm{Fib}} τ Fib η \eta η L = 10 L=10 L = 10 0 0 0 2 / 15 2/15 2/15 q = τ F i b q=\tau_{\mathrm{Fib}} q = τ Fib C = + C=+ C = + C = − C=- C = − 2 / 5 2/5 2/5 q = 1 q=1 q = 1 q = τ F i b q=\tau_{\mathrm{Fib}} q = τ Fib C = − C=- C = − 11 / 15 11/15 11/15 q = 1 , τ F i b q=1,\tau_{\mathrm{Fib}} q = 1 , τ Fib C = ± C=\pm C = ± 0.80 0.80 0.80 q = τ F i b q=\tau_{\mathrm{Fib}} q = τ Fib C = + C=+ C = + Z 3 \mathbb{Z}_3 Z 3 σ \sigma σ h = 1 / 15 h=1/15 h = 1/15 h = 2 / 3 h=2/3 h = 2/3 C = ± C=\pm C = ± h = 2 / 5 h=2/5 h = 2/5 Z 3 \mathbb{Z}_3 Z 3 C C C
Spectrum of C 5 P o t t s C_5^{\mathrm{Potts}} C 5 Potts Table 6 and Figure 20 give the interface-operator spectrum at the improved point ( K , λ ) = ( 0.84 , 0.34 ) (K,\lambda)=(0.84,0.34) ( K , λ ) = ( 0.84 , 0.34 ) L = 8 , 10 , 12 , 14 , 16 L=8,10,12,14,16 L = 8 , 10 , 12 , 14 , 16 L = 24 L=24 L = 24 L = 24 L=24 L = 24 § 5.3 , with the η \eta η ( 2 , 1 ) (2,1) ( 2 , 1 ) L = 8 , 12 L=8,12 L = 8 , 12 L = 14 , 16 , 24 L=14,16,24 L = 14 , 16 , 24 [ τ F i b , − ] [\tau_{\mathrm{Fib}},-] [ τ Fib , − ] [ τ F i b , + ] [\tau_{\mathrm{Fib}},+] [ τ Fib , + ] [ 1 , − ] [1,-] [ 1 , − ] X = 1.116 X=1.116 X = 1.116 [ 1 , + ] [1,+] [ 1 , + ] τ F i b \tau_{\mathrm{Fib}} τ Fib L = 12 , 14 , 16 L=12,14,16 L = 12 , 14 , 16 Z 3 \mathbb{Z}_3 Z 3 2 / 15 2/15 2/15 C = − C=- C = − C = + C=+ C = + C = + C=+ C = + C = − C=- C = − C 5 P o t t s C_5^{\mathrm{Potts}} C 5 Potts Z 3 \mathbb{Z}_3 Z 3 φ ( 1 , 3 ) ⊗ φ ˉ ( 3 , 3 ) \varphi_{(1,3)}\otimes\bar{\varphi}_{(3,3)} φ ( 1 , 3 ) ⊗ φ ˉ ( 3 , 3 ) D ( 1 , 3 ) D_{(1,3)} D ( 1 , 3 ) S ( 1 , 3 ) ( 1 , 3 ) / S ( 1 , 1 ) ( 1 , 3 ) = − 1 = ω + ω 2 S_{(1,3)(1,3)}/S_{(1,1)(1,3)}=-1=\omega+\omega^2 S ( 1 , 3 ) ( 1 , 3 ) / S ( 1 , 1 ) ( 1 , 3 ) = − 1 = ω + ω 2 Z 3 \mathbb{Z}_3 Z 3 τ F i b \tau_{\mathrm{Fib}} τ Fib Z 3 \mathbb{Z}_3 Z 3 C 5 P o t t s C_5^{\mathrm{Potts}} C 5 Potts Z 2 = ⟨ C ⟩ \mathbb{Z}_2=\langle C\rangle Z 2 = ⟨ C ⟩ S 3 S_3 S 3 η \eta η τ F i b \tau_{\mathrm{Fib}} τ Fib
Table 6. Interface operators of C 5 P o t t s C_5^{\mathrm{Potts}} C 5 Potts ( K , λ ) = ( 0.84 , 0.34 ) (K,\lambda)=(0.84,0.34) ( K , λ ) = ( 0.84 , 0.34 ) [ q , C ] [q,C] [ q , C ] L = 12 L=12 L = 12 L = 24 L=24 L = 24 χ = 120 \chi=120 χ = 120 C = + C=+ C = + η \eta η C = − C=- C = − C C C X X X L = 24 L=24 L = 24 M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) § 5.3 ).
Figure 20. Sector-resolved interface spectrum of C 5 P o t t s C_5^{\mathrm{Potts}} C 5 Potts ( K , λ ) = ( 0.84 , 0.34 ) (K,\lambda)=(0.84,0.34) ( K , λ ) = ( 0.84 , 0.34 ) 1 / L 1/L 1/ L [ q , C ] [q,C] [ q , C ] Figure 18 : circles are the ED levels at L = 8 , 10 , 12 , 14 , 16 L=8,10,12,14,16 L = 8 , 10 , 12 , 14 , 16 C = + C=+ C = + η \eta η C = − C=- C = − L = 24 L=24 L = 24 1 / L 1/L 1/ L L = 14 , 16 , 24 L=14,16,24 L = 14 , 16 , 24 L = 12 , 14 , 16 L=12,14,16 L = 12 , 14 , 16 τ F i b \tau_{\mathrm{Fib}} τ Fib τ F i b \tau_{\mathrm{Fib}} τ Fib Δ = 1.2 \Delta=1.2 Δ = 1.2 [ 1 , + ] [1,+] [ 1 , + ] X ≃ 1.11 X\simeq1.11 X ≃ 1.11 M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) [ τ F i b , − ] [\tau_{\mathrm{Fib}},-] [ τ Fib , − ] Δ + 1 \Delta+1 Δ + 1 Δ = 2 / 15 \Delta=2/15 Δ = 2/15 17 / 15 17/15 17/15 L − 1 L_{-1} L − 1 L = 8 L=8 L = 8 → \to → → \to → X X X Δ X ≃ 1.11 \Delta_X\simeq1.11 Δ X ≃ 1.11 ∝ L − 0.11 \propto L^{-0.11} ∝ L − 0.11
Stability. An interface is stable under a symmetry if none of its relevant operators is neutral under that symmetry. Reading Table 6 : the relevant operators of C 5 P o t t s C_5^{\mathrm{Potts}} C 5 Potts [ τ F i b , − ] [\tau_{\mathrm{Fib}},-] [ τ Fib , − ] [ τ F i b , + ] [\tau_{\mathrm{Fib}},+] [ τ Fib , + ] [ 1 , − ] [1,-] [ 1 , − ] [ 1 , + ] [1,+] [ 1 , + ] X ∼ 1.11 X\sim1.11 X ∼ 1.11 C 5 P o t t s C_5^{\mathrm{Potts}} C 5 Potts τ F i b \tau_{\mathrm{Fib}} τ Fib C C C C C C C C C τ F i b \tau_{\mathrm{Fib}} τ Fib C C C Z 3 \mathbb{Z}_3 Z 3 S 3 S_3 S 3 Z 3 \mathbb{Z}_3 Z 3 τ F i b \tau_{\mathrm{Fib}} τ Fib C C C C 5 C_5 C 5 Y ( 2 , 1 ) Y_{(2,1)} Y ( 2 , 1 ) η \eta η M ( 5 , 6 ) M(5,6) M ( 5 , 6 ) g g g ⟨ 0 ∣ D ∣ 0 ⟩ \langle0|D|0\rangle ⟨ 0∣ D ∣0 ⟩ g C 5 P o t t s = 1.503 ( 2 ) g_{C_5}^{\mathrm{Potts}} = 1.503(2) g C 5 Potts = 1.503 ( 2 ) T = 0.665 ( 10 ) T = 0.665(10) T = 0.665 ( 10 )
The loop-model formulation at continuous p p p
The RSOS construction fixes p p p [71 ] with loop weight n = 2 cos ( π / p ) = d w n = 2\cos(\pi/p) = d_w n = 2 cos ( π / p ) = d w O ( n ) O(n) O ( n ) [35 , 72 , 73 , 74 ] , where p p p C p C_p C p p = 5 p = 5 p = 5 p = 4 p = 4 p = 4
Loop model with seams
Hilbert space. We use the oriented-strand (three-state vertex) representation of the dilute O ( n ) O(n) O ( n ) [35 , 73 , 74 ] , Figure 21 (a). Each slot i i i s i ∈ { 0 , + , − } s_i \in\{0, +, -\} s i ∈ { 0 , + , − } + + + − - − L L L M = L + 1 M = L + 1 M = L + 1 c c c a L − 1 a_{L-1} a L − 1 a 0 a_0 a 0 Figure 22 (a)) which is never empty : it is the world-line of the ( 2 , 1 ) (2,1) ( 2 , 1 ) S z = ∑ i s i S^z = \sum_i s_i S z = ∑ i s i S z = 0 S^z = 0 S z = 0 L = 12 L = 12 L = 12 3 13 3^{13} 3 13 dim = 139 152 \dim= 139\,152 dim = 139 152 s c = ± s_c = \pm s c = ±
Figure 21. The loop model in the oriented-strand representation. (a) Slot states. (b) The six face moves of (45) , drawn with time upwards; the coefficients are the RSOS weights (13) . (c) Orientation phases: a cup or cap carries q ± 1 / 2 q^{\pm1/2} q ± 1/2 ∣ c ⟩ = q 1 / 2 ∣ + − ⟩ + q − 1 / 2 ∣ − + ⟩ \lvert c\rangle= q^{1/2}\lvert+-\rangle+ q^{-1/2}\lvert-+\rangle ∣ c ⟩ = q 1/2 ∣ + − ⟩ + q − 1/2 ∣ − + ⟩ ⟨ c ~ ∣ \langle\tilde{c}\rvert ⟨ c ~ ∣ ⟨ c ~ ∣ c ⟩ = q + q − 1 = n \langle\tilde{c}\vert c\rangle= q + q^{-1} = n ⟨ c ~ ∣ c ⟩ = q + q − 1 = n e 2 = n e e^2 = ne e 2 = n e
Figure 22. Seam and twist. (a) The single-seam ring: L L L c c c L − 1 L-1 L − 1 (46) : the strand is braided in front of the content of a 0 a_0 a 0 h h h a 0 a_0 a 0 (47) . (d) A twist ϕ \phi ϕ 2 cos ϕ 2\cos\phi 2 cos ϕ n n n Y ( 2 , 1 ) Y_{(2,1)} Y ( 2 , 1 ) (49) .
Loop weight from orientations. Loops acquire their weight from the phases attached to their turns (Figure 21 (c)). A strand that turns by ± π \pm\pi ± π q ± 1 / 2 q^{\pm1/2} q ± 1/2 q = e i π / p q = e^{i\pi/p} q = e iπ / p ± 2 π \pm2\pi ± 2 π q ± 1 q^{\pm1} q ± 1 q + q − 1 = 2 cos ( π / p ) = n q + q^{-1} = 2\cos(\pi/p) = n q + q − 1 = 2 cos ( π / p ) = n ∣ c ⟩ = q 1 / 2 ∣ + − ⟩ + q − 1 / 2 ∣ − + ⟩ \lvert c\rangle= q^{1/2}\lvert+-\rangle+ q^{-1/2}\lvert-+\rangle ∣ c ⟩ = q 1/2 ∣ + − ⟩ + q − 1/2 ∣ − + ⟩ ⟨ c ~ ∣ = q 1 / 2 ⟨ + − ∣ + q − 1 / 2 ⟨ − + ∣ \langle\widetilde{c}\rvert= q^{1/2}\langle+-\rvert+ q^{-1/2}\langle-+\rvert ⟨ c ∣ = q 1/2 ⟨ + − ∣ + q − 1/2 ⟨ − + ∣
e = ∣ c ⟩ ⟨ c ~ ∣ , c u p = ∣ c ⟩ ⟨ 00 ∣ , c a p = ∣ 00 ⟩ ⟨ c ~ ∣ , e 2 = ⟨ c ~ ∣ c ⟩ e = n e , (43) e = \lvert c\rangle\langle\widetilde{c}\rvert,\qquad\mathrm{cup} = \lvert c\rangle\langle00\rvert,\qquad\mathrm{cap} = \lvert00\rangle\langle\widetilde{c}\rvert,\qquad e^2 = \langle\widetilde{c}\vert c\rangle e = n e, \tag*{(43)} e = ∣ c ⟩ ⟨ c ∣ , cup = ∣ c ⟩ ⟨ 00 ∣ , cap = ∣ 00 ⟩ ⟨ c ∣ , e 2 = ⟨ c ∣ c ⟩ e = n e , (43) so that e e e ⟨ c ~ ∣ \langle\widetilde{c}\rvert ⟨ c ∣ not the Hermitian conjugate of ∣ c ⟩ \lvert c\rangle ∣ c ⟩ q ∓ 1 / 2 q^{\mp1/2} q ∓ 1/2 q q q p p p [75 ] ), and we diagonalise with left and right eigenvectors throughout. The eigenvalues are nevertheless real on all the Kac levels , by which we mean the states that belong to the minimal model proper: those whose scaling limit lies in a representation V ( r , s ) ⊗ V ( r ′ , s ′ ) V_{(r,s)} \otimes V_{(r',s')} V ( r , s ) ⊗ V ( r ′ , s ′ ) M ( p , p + 1 ) M(p,p+1) M ( p , p + 1 ) 1 ≤ r ≤ p − 1 1 \le r \le p - 1 1 ≤ r ≤ p − 1 1 ≤ s ≤ p 1 \le s \le p 1 ≤ s ≤ p h r , s + h r ′ , s h_{r,s} + h_{r',s} h r , s + h r ′ , s p p p U q ( s l 2 ) U_q(sl_2) U q ( s l 2 ) § 6.2 .
Heights and orientations. The RSOS chain of § 3 is related to this vertex chain by a map of configurations. A domain wall of the height model is a point where a loop crosses the chain, and the sign of the height step across it, a i − a i − 1 = ± 1 a_i-a_{i-1}=\pm1 a i − a i − 1 = ± 1 ± 1 \pm1 ± 1 s i = a i − a i − 1 ∈ { 0 , + , − } s_i=a_i-a_{i-1}\in\{0,+,-\} s i = a i − a i − 1 ∈ { 0 , + , − } oriented loop configuration, injectively (given a 0 a_0 a 0 1 ≤ a i ≤ p − 1 1\leq a_i\leq p-1 1 ≤ a i ≤ p − 1 ∑ i s i = 0 \sum_i s_i=0 ∑ i s i = 0 S z = 0 S^z=0 S z = 0 c j = a j ± 1 c_j=a_j\pm1 c j = a j ± 1 s c = ± s_c=\pm s c = ± (43) act, on a triple of heights with equal ends l l l
e ∣ l a l ⟩ = S a S a ′ S l ∣ l a ′ l ⟩ , cup ∣ l l l ⟩ = S a S l ∣ l a l ⟩ , cap ∣ l a l ⟩ = S a S l ∣ l l l ⟩ , (44) e\lvert l\,a\,l\rangle=\sqrt{\frac{S_aS_{a'}}{S_l}}\lvert l\,a'\,l\rangle,\qquad\operatorname{cup}\lvert l\,l\,l\rangle=\sqrt{\frac{S_a}{S_l}}\lvert l\,a\,l\rangle,\qquad\operatorname{cap}\lvert l\,a\,l\rangle=\sqrt{\frac{S_a}{S_l}}\lvert l\,l\,l\rangle, \tag*{(44)} e ∣ l a l ⟩ = S l S a S a ′ ∣ l a ′ l ⟩ , cup ∣ l l l ⟩ = S l S a ∣ l a l ⟩ , cap ∣ l a l ⟩ = S l S a ∣ l l l ⟩ , (44) with a , a ′ a,a' a , a ′ l l l w e w_e w e w ∪ ∩ w_{\cup\cap} w ∪∩ (13) and the projector P 1 = e / d w P^1=e/d_w P 1 = e / d w Figure 10 ). A contractible loop with outside height a a a a ± 1 a\pm1 a ± 1 S a ± 1 / S a S_{a\pm1}/S_a S a ± 1 / S a q ± 1 q^{\pm1} q ± 1 ∑ a ′ a d j a S a ′ / S a = n = q + q − 1 \sum_{a'\,\mathrm{adj}\,a}S_{a'}/S_a=n=q+q^{-1} ∑ a ′ adj a S a ′ / S a = n = q + q − 1 ⟨ c ~ ∣ c ⟩ = n \langle\widetilde c|c\rangle=n ⟨ c ∣ c ⟩ = n [33 , 76 ] : the RSOS sectors are reproduced by twists of the S z = 0 S^z=0 S z = 0 p p p
Hamiltonian. The pair term is the element of the dilute Temperley–Lieb algebra that defines the RSOS face operator of § 3.1 , Figure 21 (b), now in the vertex representation:
h = w ∅ 1 ∅ + w ∥ 1 ∥ + w = 1 = + w ⇄ ( hop ) + w ∪ ∩ ( cup + cap ) + w e e , H 0 = − ∑ j h j , j + 1 , (45) h=w_{\varnothing}1_{\varnothing}+w_{\parallel}1_{\parallel}+w_{=}1_{=}+w_{\rightleftarrows}(\operatorname{hop})+w_{\cup\cap}(\operatorname{cup}+\operatorname{cap})+w_e e,\qquad H_0=-\sum_j h_{j,j+1}, \tag*{(45)} h = w ∅ 1 ∅ + w ∥ 1 ∥ + w = 1 = + w ⇄ ( hop ) + w ∪∩ ( cup + cap ) + w e e , H 0 = − j ∑ h j , j + 1 , (45) where 1 ∅ 1_{\varnothing} 1 ∅ 1 ∥ 1_{\parallel} 1 ∥ 1 = 1_{=} 1 = e e e (43) . The six scalar coefficients are the same functions of λ = π ( p − 1 ) / 4 p \lambda=\pi(p-1)/4p λ = π ( p − 1 ) /4 p (13) , now with p p p w ∪ ∩ = sin 2 λ / N 0 w_{\cup\cap}=\sin2\lambda/\sqrt{\mathcal N_0} w ∪∩ = sin 2 λ / N 0 w e = − sin λ / N 0 w_e=-\sin\lambda/\sqrt{\mathcal N_0} w e = − sin λ / N 0 (13) being the matrix elements (44) of the generators rather than part of the coefficients. In the vertex language h h h 9 × 9 9\times9 9 × 9
The seam. The pair term that straddles the seam is built exactly as in § 3.2 : the strand is braided out of the way, the ordinary h h h Figure 22 (b)),
h x c y s e a m = B c y − 1 h x c B c y , x = a L − 1 , y = a 0 , (46) h^{\mathrm{seam}}_{xcy}=B^{-1}_{cy}h_{xc}B_{cy}, \qquad x=a_{L-1},\ y=a_0, \tag*{(46)} h x cy seam = B cy − 1 h x c B cy , x = a L − 1 , y = a 0 , (46) a 27 × 27 27 \times27 27 × 27 ( x , c , y ) (x,c,y) ( x , c , y ) B B B [77 ] , Figure 22 (c): it is 1 1 1 P P P
B = A 1 + A − 1 e , A = i q 1 / 2 , n = − A 2 − A − 2 = q + q − 1 , (47) B=A\mathbf{1}+A^{-1}e,\qquad A=iq^{1/2},\qquad n=-A^2-A^{-2}=q+q^{-1}, \tag*{(47)} B = A 1 + A − 1 e , A = i q 1/2 , n = − A 2 − A − 2 = q + q − 1 , (47) when both slots carry a strand. Equation (47) is the F R F − 1 FRF^{-1} F R F − 1 (15) of the RSOS seam written in the skein language, and the two blocks of B B B P P P 1 1 1 A A A D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) L L L h s e a m h^{\mathrm{seam}} h seam h h h
One difference of bookkeeping between the two representations should be kept in mind. In the RSOS chain the seam term changes a j a_j a j c j c_j c j § 3.2 ); those heights are fusion labels, not the contents of slots. In the vertex chain there is no such spectator: (46) acts on all three slots ( x , c , y ) (x,c,y) ( x , c , y ) y y y y y y y y y h h h ( x , c ) (x,c) ( x , c ) ( c , y ) (c,y) ( c , y ) S z S^z S z c c c
The clean field. The deformation (28) lives on the same three slots,
Φ c l = n ( e x c + e c y ) − ( n x + n y ) , n i = 1 − ∣ 0 ⟩ ⟨ 0 ∣ i , (48) \Phi_{\mathrm{cl}}=n\left(e_{xc}+e_{cy}\right)-\left(n_x+n_y\right),\qquad n_i=1-\lvert0\rangle\langle0\rvert_i, \tag*{(48)} Φ cl = n ( e x c + e cy ) − ( n x + n y ) , n i = 1 − ∣ 0 ⟩ ⟨ 0 ∣ i , (48) where e x c e_{xc} e x c e c y e_{cy} e cy (43) on the strand and its left (right) neighbour and n i n_i n i i i i d w ( e L + e R ) − ( n L + n R ) d_w(e_L+e_R)-(n_L+n_R) d w ( e L + e R ) − ( n L + n R ) d w = n d_w=n d w = n H = H 0 − h s e a m + K Φ c l H=H_0-h^{\mathrm{seam}}+K\Phi_{\mathrm{cl}} H = H 0 − h seam + K Φ cl
H 0 = − ∑ o r d i n a r y p a i r s h H_0=-\sum_{\mathrm{ordinary\ pairs}}h H 0 = − ordinary pairs ∑ h containing the L − 1 L-1 L − 1 Figure 22 (a)).
Twist sectors. In the RSOS calculation the symmetry sectors were obtained by diagonalising the wrapped ( 2 , 1 ) (2,1) ( 2 , 1 ) p p p twist , in three steps.
(i) Conserved orientation and a boundary-condition parameter. On the ring the total orientation S z = ∑ k s k S^z=\sum_k s_k S z = ∑ k s k S z S^z S z S z = 0 S^z=0 S z = 0 S z S^z S z c c c a 0 a_0 a 0 Figure 22 (a)), and every matrix element in which a strand of orientation s = ± 1 s = \pm1 s = ± 1 e i s ϕ e^{is\phi} e i s ϕ ϕ \phi ϕ (46) ) and Φ c l \Phi_{\mathrm{cl}} Φ cl H ( ϕ ) H(\phi) H ( ϕ ) c c c a 0 a_0 a 0 e i ϕ e^{i\phi} e i ϕ e − i ϕ e^{-i\phi} e − i ϕ H H H H ( ϕ ) H(\phi) H ( ϕ ) ϕ \phi ϕ S z = 0 S^z = 0 S z = 0
(ii) What the twist weighs. Consider the loops traced out in space-time by the transfer matrix. A contractible loop crosses the cut an even number of times with opposite orientations, so the twist phases cancel and its weight stays n = 2 cos ( π / p ) n = 2\cos(\pi/p) n = 2 cos ( π / p ) 1 + 1 = 2 1 + 1 = 2 1 + 1 = 2 e i ϕ + e − i ϕ = 2 cos ϕ e^{i\phi} + e^{-i\phi} = 2\cos\phi e i ϕ + e − i ϕ = 2 cos ϕ Figure 22 (d)). The twist is therefore a knob that sets the weight of non-contractible loops independently of n n n ϕ = π / p \phi= \pi/p ϕ = π / p n n n [76 , 78 ] .
(iii) Which ϕ \phi ϕ The decomposition of the RSOS torus partition function into loop (Coulomb-gas) partition functions with twisted boundary conditions is standard: summing over the heights of the plateaus between loops that wind around the ring turns the sum over configurations into a sum over the eigenvalues λ r = 2 cos ( π r / p ) \lambda_r = 2\cos(\pi r/p) λ r = 2 cos ( π r / p ) r = 1 , … , p − 1 r = 1,\ldots,p - 1 r = 1 , … , p − 1 A p − 1 A_{p-1} A p − 1 λ r \lambda_r λ r [33 , 76 , 79 ] ; in transfer-matrix language the RSOS levels are recovered from the corresponding twisted loop representations with 2 cos ϕ = λ r 2\cos\phi= \lambda_r 2 cos ϕ = λ r p p p r r r ( 2 , 1 ) (2,1) ( 2 , 1 ) ( − 1 ) r + s (-1)^{r+s} ( − 1 ) r + s η \eta η (4) ), the Kac state with holomorphic label ( r , s ) (r,s) ( r , s ) Y ( 2 , 1 ) = ( − 1 ) r + s 2 cos ( π r / p ) Y_{(2,1)} = (-1)^{r+s} 2\cos(\pi r/p) Y ( 2 , 1 ) = ( − 1 ) r + s 2 cos ( π r / p )
ϕ = π r p ( r + s even ) , ϕ = π − π r p ( r + s odd ) , (49) \phi= \frac{\pi r}{p}\quad(r+s\ \text{even}),\qquad\phi= \pi- \frac{\pi r}{p}\quad(r+s\ \text{odd}), \tag*{(49)} ϕ = p π r ( r + s even ) , ϕ = π − p π r ( r + s odd ) , (49) where the second case is the sign flip r → p − r r \to p-r r → p − r η \eta η ϕ \phi ϕ π − ϕ \pi-\phi π − ϕ p p p a , b , c , d a,b,c,d a , b , c , d § 5.3 , ( Y ( 2 , 1 ) , η ) = ( φ , + ) , ( φ − 1 , − ) , ( φ − 1 , + ) , ( − φ , − ) (Y_{(2,1)},\eta) = (\varphi,+),(\varphi^{-1},-),(\varphi^{-1},+),(-\varphi,-) ( Y ( 2 , 1 ) , η ) = ( φ , + ) , ( φ − 1 , − ) , ( φ − 1 , + ) , ( − φ , − ) p = 5 p=5 p = 5 ϕ = π / 5 , 2 π / 5 , 3 π / 5 , 4 π / 5 \phi= \pi/5, 2\pi/5, 3\pi/5, 4\pi/5 ϕ = π /5 , 2 π /5 , 3 π /5 , 4 π /5 k = 1 k = 1 k = 1 − 2 cos ( π / p ) -2\cos(\pi/p) − 2 cos ( π / p ) ϕ = π − π / p \phi= \pi- \pi/p ϕ = π − π / p ϕ \phi ϕ p p p p p p
Validation. The construction was validated at p = 5 p = 5 p = 5 10 − 13 10^{-13} 1 0 − 13 D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) u C u_C u C (35) along the clean ray, recomputed on the RSOS chain for L ≤ 14 L \le14 L ≤ 14 [73 , 76 , 80 ] , so besides the Kac levels they contain states that are null at integer p p p Δ = 0.20 \Delta= 0.20 Δ = 0.20 ( 1 , 1 ) ⊗ ( 2 , 1 ) (1, 1) \otimes(2, 1) ( 1 , 1 ) ⊗ ( 2 , 1 ) a a a p = 5 p = 5 p = 5 Δ = 1 \Delta= 1 Δ = 1 p p p
The running coupling at continuous p p p
For continuous p p p p p p u C u_C u C (35) built from the k = 1 k = 1 k = 1 v ( p ) = 4 p / ( 3 ( p − 1 ) ) v(p) = 4p/(3(p - 1)) v ( p ) = 4 p / ( 3 ( p − 1 )) v ( 4 ) = 16 / 9 v(4) = 16/9 v ( 4 ) = 16/9 v ( 5 ) = 5 / 3 v(5) = 5/3 v ( 5 ) = 5/3 ℓ = 7 , 9 , 11 , 13 \ell= 7, 9, 11, 13 ℓ = 7 , 9 , 11 , 13 L = 6 , … , 14 L = 6, \ldots, 14 L = 6 , … , 14 p p p K K K 0.05 0.05 0.05 K = 0 K=0 K = 0 − 2 cos ( π / p ) -2\cos(\pi/p) − 2 cos ( π / p ) ( 1 , 2 ) ⊗ ( 2 , 2 ) (1,2)\otimes(2,2) ( 1 , 2 ) ⊗ ( 2 , 2 ) Δ k = 1 \Delta_{k=1} Δ k = 1 (34) . Two properties make it suitable at continuous p p p p = 4 p=4 p = 4 C p C_p C p K ≃ 0.3 K\simeq0.3 K ≃ 0.3 p p p K ≳ 3.5 K\gtrsim3.5 K ≳ 3.5 β \beta β B C B_C B C (36) is then formed from w C w_C w C A C A_C A C § 4 and § 5 .
Results at non-integer p: the family C p C_p C p
The motivation for following the flow at non-integer p p p p = 5 p=5 p = 5 C 5 C_5 C 5 p = 4 p=4 p = 4 η \eta η C 5 C_5 C 5 C p C_p C p p p p η \eta η p p p C p C_p C p p p p [81 , 82 , 83 ] —and the flow out of D ( 2 , 1 ) D(2,1) D ( 2 , 1 ) C p C_p C p p = 4 p = 4 p = 4 η \eta η C 5 C_5 C 5 η \eta η p = 4 p = 4 p = 4 p = 4 p = 4 p = 4 p p p β \beta β p p p [ 4 , 5 ] [4, 5] [ 4 , 5 ] β \beta β § 6.2 at each of them. The result, presented in this section, is the first scenario: the zero of the β \beta β C 5 C_5 C 5 η \eta η η \eta η p = 4 p = 4 p = 4 § 7.2 .
C p C_p C p 4 ≤ p ≤ 5 4 \leq p \leq5 4 ≤ p ≤ 5 Figures 23 and 24 show B C B_C B C p = 5 , 4.75 , 4.5 , 4.25 , 4 p = 5, 4.75, 4.5, 4.25, 4 p = 5 , 4.75 , 4.5 , 4.25 , 4 L = 6 L = 6 L = 6 14 14 14 ℓ = 7 , 9 , 11 , 13 \ell= 7, 9, 11, 13 ℓ = 7 , 9 , 11 , 13 L = 14 L = 14 L = 14 1.17 × 10 6 1.17 \times10^6 1.17 × 1 0 6 p p p D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 )
Table 1 (PDF p. 53).
The stable fixed point C p C_p C p p p p K ∗ K^* K ∗ A C ∗ A_C^* A C ∗ p = 5 p = 5 p = 5 p = 4 p = 4 p = 4 β \beta β p = 4 p = 4 p = 4 K ∗ = 1.87 K^* = 1.87 K ∗ = 1.87 A C ∗ = 0.517 A_C^* = 0.517 A C ∗ = 0.517 p = 5 p = 5 p = 5 K ∗ K^* K ∗ K C 5 ∗ = 0.95 ( 5 ) K_{C_5}^* = 0.95(5) K C 5 ∗ = 0.95 ( 5 ) (39) , and the slope at the zero, −0.21, is close to the improved-ray value −0.16(2); the improved ray, by contrast, drifts much more because of the K K K § 5.4 ). The crossing value A C ∗ = 0.27 A_C^* = 0.27 A C ∗ = 0.27 0.30 0.30 0.30 L − 0.12 L^{-0.12} L − 0.12 Δ X − 1 \Delta_X - 1 Δ X − 1 p = 5 p = 5 p = 5 − 0.64 -0.64 − 0.64 p = 4 p = 4 p = 4 L Δ X − 1 L^{\Delta_X-1} L Δ X − 1 p = 5 p = 5 p = 5 p = 4 p = 4 p = 4
The endpoint at p = 4 p = 4 p = 4 D ( 1 , 4 ) D_{(1,4)} D ( 1 , 4 )
The family C p C_p C p p = 4 p = 4 p = 4 η \eta η M ( p , p + 1 ) M(p,p+1) M ( p , p + 1 ) η = D ( p − 1 , 1 ) ≅ D ( 1 , p ) \eta= D_{(p-1,1)} \cong D_{(1,p)} η = D ( p − 1 , 1 ) ≅ D ( 1 , p ) p = 4 p = 4 p = 4 D ( 1 , 4 ) D_{(1,4)} D ( 1 , 4 ) ( p − 1 , 1 ) (p-1,1) ( p − 1 , 1 ) ( 1 , 4 ) (1,4) ( 1 , 4 ) p p p c = 1 − 6 / p ( p + 1 ) c = 1 - 6/p(p+1) c = 1 − 6/ p ( p + 1 ) [84 , 85 ] has topological lines for all degenerate labels ( r , s ) (r,s) ( r , s ) r r r s s s d ( r , s ) = [ r ] q [ s ] q ′ d_{(r,s)} = [r]_q[s]_{q^\prime} d ( r , s ) = [ r ] q [ s ] q ′ q = e i π / p q = e^{i\pi/p} q = e iπ / p q ′ = e i π / ( p + 1 ) q^\prime= e^{i\pi/(p+1)} q ′ = e iπ / ( p + 1 ) D ( 1 , 4 ) D_{(1,4)} D ( 1 , 4 ) ( r , 1 ) (r,1) ( r , 1 )
d ( 1 , 4 ) = sin ( 4 π / ( p + 1 ) ) sin ( π / ( p + 1 ) ) = 1.73 , 1.57 , 1.40 , 1.20 , 1 at p = 5 , 4.75 , 4.5 , 4.25 , 4 , (50) d_{(1,4)} = \frac{\sin\left(4\pi/(p+1)\right)}{\sin\left(\pi/(p+1)\right)} = 1.73,\ 1.57,\ 1.40,\ 1.20,\ 1 \quad\text{at } p = 5,\ 4.75,\ 4.5,\ 4.25,\ 4, \tag*{(50)} d ( 1 , 4 ) = sin ( π / ( p + 1 ) ) sin ( 4 π / ( p + 1 ) ) = 1.73 , 1.57 , 1.40 , 1.20 , 1 at p = 5 , 4.75 , 4.5 , 4.25 , 4 , (50) so it is invertible only at p = 4 p = 4 p = 4 p = 5 p = 5 p = 5 D ( 1 , 4 ) ≅ D ( 4 , 2 ) D_{(1,4)} \cong D_{(4,2)} D ( 1 , 4 ) ≅ D ( 4 , 2 ) 3 \sqrt{3} 3 p > 4 p > 4 p > 4 D ( 1 , 4 ) D_{(1,4)} D ( 1 , 4 ) C p C_p C p g C 5 = 1.503 g_{C_5} = 1.503 g C 5 = 1.503 T = 0.665 \mathcal{T} = 0.665 T = 0.665 p = 5 p = 5 p = 5 3 \sqrt{3} 3 1 1 1 D ( 1 , 4 ) = η D_{(1,4)} = \eta D ( 1 , 4 ) = η p = 4 p = 4 p = 4
Defect operators on D ( 1 , 4 ) D_{(1,4)} D ( 1 , 4 ) The operators on D ( 1 , 4 ) D_{(1,4)} D ( 1 , 4 ) D ( 1 , 4 ) ⊗ D ‾ ( 1 , 4 ) D_{(1,4)} \otimes\overline{D}_{(1,4)} D ( 1 , 4 ) ⊗ D ( 1 , 4 ) c ∈ ( 1 , 4 ) × ( 1 , 4 ) = ( 1 , 1 ) + ( 1 , 3 ) + ( 1 , 5 ) + ( 1 , 7 ) c \in(1,4) \times(1,4) = (1,1) + (1,3) + (1,5) + (1,7) c ∈ ( 1 , 4 ) × ( 1 , 4 ) = ( 1 , 1 ) + ( 1 , 3 ) + ( 1 , 5 ) + ( 1 , 7 ) c c c ( r , 1 ) (r,1) ( r , 1 ) ( 1 , s ) (1,s) ( 1 , s ) s s s § 4 ) are the ( 1 , s ) ⊗ ( 1 , s ′ ) (1,s) \otimes(1,s^\prime) ( 1 , s ) ⊗ ( 1 , s ′ ) s , s ′ s,s^\prime s , s ′
Table 7.
For generic p p p relevant symmetric operators, the chiral defect fields ϕ L = ϕ ( 1 , 3 ) ⊗ 1 ˉ \phi^L = \phi_{(1,3)} \otimes\bar{1} ϕ L = ϕ ( 1 , 3 ) ⊗ 1 ˉ ϕ R = 1 ⊗ ϕ ˉ ( 1 , 3 ) \phi^R = 1 \otimes\bar{\phi}_{(1,3)} ϕ R = 1 ⊗ ϕ ˉ ( 1 , 3 ) ( h 1 , 3 , 0 ) (h_{1,3},0) ( h 1 , 3 , 0 ) ( 0 , h 1 , 3 ) (0,h_{1,3}) ( 0 , h 1 , 3 ) Δ = ( p − 1 ) / ( p + 1 ) \Delta= (p-1)/(p+1) Δ = ( p − 1 ) / ( p + 1 ) c = ( 1 , 3 ) c = (1,3) c = ( 1 , 3 ) p = 4 p = 4 p = 4 ( 1 , 4 ) ≅ ( 3 , 1 ) (1,4) \cong(3,1) ( 1 , 4 ) ≅ ( 3 , 1 ) ( 3 , 1 ) × ( 3 , 1 ) = ( 1 , 1 ) (3,1) \times(3,1) = (1,1) ( 3 , 1 ) × ( 3 , 1 ) = ( 1 , 1 ) ( 1 , s ) ⊗ ( 1 , s ) (1,s) \otimes(1,s) ( 1 , s ) ⊗ ( 1 , s ) ( 1 , 3 ) ⊗ ( 1 , 3 ) (1,3) \otimes(1,3) ( 1 , 3 ) ⊗ ( 1 , 3 ) Δ = 6 / 5 \Delta= 6/5 Δ = 6/5 η \eta η § 4 . For p > 4 p > 4 p > 4 ( 1 , 3 ) (1,3) ( 1 , 3 ) D ( 1 , 4 ) D_{(1,4)} D ( 1 , 4 ) p = 5 p = 5 p = 5 D ( 1 , 4 ) ≅ D ( 4 , 2 ) D_{(1,4)} \cong D_{(4,2)} D ( 1 , 4 ) ≅ D ( 4 , 2 ) p > 4 p > 4 p > 4
The relevant fields ϕ L , R \phi^{L,R} ϕ L , R D ( 1 , 4 ) D_{(1,4)} D ( 1 , 4 ) p = 4 p = 4 p = 4 D ( 1 , 4 ) = η D_{(1,4)} = \eta D ( 1 , 4 ) = η p = 4 p = 4 p = 4 O ( n ) O(n) O ( n ) n n n Q \mathbb{Q} Q [80 , 86 ] .
Summary and discussion
The dilute A p − 1 A_{p-1} A p − 1 C p C_p C p S U ( 2 ) p − 2 SU(2)_{p-2} S U ( 2 ) p − 2 ( r , 1 ) (r,1) ( r , 1 ) M ( p , p + 1 ) M(p,p+1) M ( p , p + 1 ) F F F R R R (28) between the two fusion channels of the defect anyon and its neighbouring domain wall is the clean lattice representative of the KRW field ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) \phi_{(1,3)} \otimes\bar{\phi}_{(3,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 3 , 3 ) ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 1 , 3 ) \phi_{(1,3)} \otimes\bar{\phi}_{(1,3)} ϕ ( 1 , 3 ) ⊗ ϕ ˉ ( 1 , 3 )
With the clean operator the KRW flow is reproduced from first principles. At p = 4 p = 4 p = 4 η \eta η g g g 2 \sqrt{2} 2 Z 2 \mathbb{Z}_2 Z 2 p = 5 p = 5 p = 5 C 5 C_5 C 5 g = 1.503 ( 2 ) g = 1.503(2) g = 1.503 ( 2 ) T = 0.665 ( 10 ) \mathcal{T} = 0.665(10) T = 0.665 ( 10 ) Δ X = 1.12 ( 1 ) \Delta_X = 1.12(1) Δ X = 1.12 ( 1 ) η \eta η C 5 C_5 C 5 τ F i b → C 5 P o t t s \tau_{\mathrm{Fib}} \to C^{\mathrm{Potts}}_5 τ Fib → C 5 Potts Z 3 \mathbb{Z}_3 Z 3 τ F i b \tau_{\mathrm{Fib}} τ Fib C C C § 5.5 ). Followed at continuous p p p § 6 ), the family C p C_p C p C 5 C_5 C 5 η \eta η p = 4 p = 4 p = 4 C p C_p C p ϕ ( 1 , 3 ) \phi_{(1,3)} ϕ ( 1 , 3 ) D ( 1 , 4 ) D_{(1,4)} D ( 1 , 4 ) ( 1 , 3 ) (1,3) ( 1 , 3 ) p = 4 p = 4 p = 4 η = D ( 1 , 4 ) \eta= D_{(1,4)} η = D ( 1 , 4 ) § 7.2 ).
Figure 24. The ℓ = ( 11 , 13 ) \ell=(11,13) ℓ = ( 11 , 13 ) Figure 23 (L = 10 , 12 , 14 L=10,12,14 L = 10 , 12 , 14 C p C_p C p p = 5 p=5 p = 5 p = 4 p=4 p = 4
Figure 23. Loop model on the clean ray H 0 + K Φ c l H_0 + K\Phi_{\mathrm{cl}} H 0 + K Φ cl β \beta β B C B_C B C (35) versus A C A_C A C p = 5 , 4.75 , 4.5 , 4.25 , 4 p = 5, 4.75, 4.5, 4.25, 4 p = 5 , 4.75 , 4.5 , 4.25 , 4 L = 6 L = 6 L = 6 ℓ = ( 7 , 9 ) \ell= (7, 9) ℓ = ( 7 , 9 ) ( 9 , 11 ) (9, 11) ( 9 , 11 ) ( 11 , 13 ) (11, 13) ( 11 , 13 ) 1.35 K ∗ 1.35 K^* 1.35 K ∗ D ( 2 , 1 ) D_{(2,1)} D ( 2 , 1 ) K ∗ K^* K ∗ A C ∗ A_C^* A C ∗ p p p
Comparison with conformal perturbation theory and with integrable defect flows. The transmission of C C C [12 ] : for the diagonal KRW fixed point they find the reflection coefficient R C = 1 − T = 8 9 π 2 y 4 + O ( y 5 ) R_C=1-T=\frac{8}{9}\pi^2y^4+O(y^5) R C = 1 − T = 9 8 π 2 y 4 + O ( y 5 ) y = 1 − h 1 , 3 = 2 / ( p + 1 ) y=1-h_{1,3}=2/(p+1) y = 1 − h 1 , 3 = 2/ ( p + 1 ) R = 0.284 R=0.284 R = 0.284 p = 4 p=4 p = 4 0.366 0.366 0.366 [11 ] ) and R = π 2 / 72 = 0.137 R=\pi^2/72=0.137 R = π 2 /72 = 0.137 p = 5 p=5 p = 5 R = 1 − T = 0.335 ( 10 ) R=1-T=0.335(10) R = 1 − T = 0.335 ( 10 ) g g g y = 1 / 3 y=1/3 y = 1/3 g g g [13 ] . On the lattice side, the chiral KRW flows—the axes of the ( λ l , λ r ) (\lambda_l,\lambda_r) ( λ l , λ r ) g g g [23 , 87 ] ; those flows end on topological lines, whose identity is now fixed in general by the functional-relation analysis of [27 ] , and the present construction is the complementary one, in which the non-chiral deformation of an exactly topological seam reaches the non-topological interface C C C
Comparison with open string field theory. The g g g C C C [14 ] as a level-truncated solution of open string field theory in the doubled theory M ( p , p + 1 ) ⊗ M ( p , p + 1 ) M(p,p+1)\otimes M(p,p+1) M ( p , p + 1 ) ⊗ M ( p , p + 1 )
g C O S F T ≃ 1.08 – 1.10 ( p = 4 ) , g C O S F T ≃ 1.489 ( p = 5 ) , g_C^{\mathrm{OSFT}} \simeq1.08\text{--}1.10 \quad(p=4), \qquad g_C^{\mathrm{OSFT}} \simeq1.489 \quad(p=5), g C OSFT ≃ 1.08 – 1.10 ( p = 4 ) , g C OSFT ≃ 1.489 ( p = 5 ) , with the p = 4 p=4 p = 4 ln g C − ln g D ( 2 , 1 ) = − π 2 4 y 3 + O ( y 4 ) \ln g_C-\ln g_{D(2,1)}=-\frac{\pi^2}{4}y^3+O(y^4) ln g C − ln g D ( 2 , 1 ) = − 4 π 2 y 3 + O ( y 4 ) y = 2 p + 1 y=\frac{2}{p+1} y = p + 1 2 p = 5 p=5 p = 5 g = 1.503 ( 2 ) g=1.503(2) g = 1.503 ( 2 ) C 5 C_5 C 5 g = 1.00 g=1.00 g = 1.00 p = 4 p=4 p = 4
Acknowledgments
This work was heavily assisted by Claude Fable 5 (Anthropic): the lattice constructions, the numerical codes, the computations reported here and the drafting of the text were developed in collaboration with it, with the author directing the questions and checking the results. The work of the author is supported by the European Union (ERC, QFTinAdS, project number 101087025, PI Balt van Rees). Views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency.
