Introduction
Defects play crucial roles in quantum and classical many-body systems and field theories: they elucidate the structure of the underlying theories and determine physical properties. Any quantum system with global (zero-form) internal symmetry comes with a distinguished type of codimension-two defects, known as monodromy (or twist) defects. Upon encircling the defects, the fundamental fields are transformed by the corresponding symmetry action. In other words, the boundary conditions of the fields on the two-dimensional transverse plane are twisted by a symmetry of the system.
Physically, a monodromy defect can be created by threading a localized “magnetic” flux of the symmetry, analogous to an Aharonov-Bohm flux, producing a defect extending in the time direction. Alternatively, such defects can be generated by acting on the system with a disorder operator, i.e., a partial symmetry transformation [1 , 2 , 3 , 4 , 5 ] . In a system with an exact or emergent space-time symmetry, these two constructions are equivalent and are related by a space-time rotation. Such defects, or extended operators, often provide probes of the underlying phase of matter beyond conventional local measurements. It is also worth noting that certain entanglement measures, such as Rényi entropies, can be formulated as the free energy of the system with replica symmetry defects [6 , 7 , 8 ] .
The presence of a defect modifies the local correlations in its vicinity. This effect is particularly pronounced in a gapless system, where the correlation functions on the defect exhibit asymptotic long-distance behaviors distinct from those in the bulk. In particular, when the bulk is described by a conformal field theory (CFT), e.g. when the system is tuned to a critical point, the defect often preserves the residual conformal symmetry and is therefore called a conformal defect. In this case, the properties of local observables near the defect can be organized as a defect CFT (DCFT) [9 , 10 , 11 ] , which essentially describes the universality class, or the “phase,” of the defect theory.
In this work, we study monodromy defects in the O(N) Wilson-Fisher theories in space-time dimension d ≥ 3 d \ge3 d ≥ 3 [12 , 13 , 14 , 15 , 16 , 17 , 18 ] , which we briefly review below.
We will denote the fundamental O(N) vector field by Φ = ( Φ 1 , Φ 2 , … , Φ N ) T \boldsymbol{\Phi} = (\Phi_1,\Phi_2,\ldots,\Phi_N)^\mathsf{T} Φ = ( Φ 1 , Φ 2 , … , Φ N ) T 2 × 2 2 \times2 2 × 2 ± 1 \pm1 ± 1
We mostly consider the following type of monodromy. Group the first M M M M / 2 M/2 M /2 ψ \psi ψ ψ j = Φ 2 j − 1 + i Φ 2 j \psi_j = \Phi_{2j-1} + i\Phi_{2j} ψ j = Φ 2 j − 1 + i Φ 2 j j = 1 , … , M / 2 j = 1,\ldots,M/2 j = 1 , … , M /2 M ≤ N M \le N M ≤ N N − M N-M N − M ϕ \phi ϕ
ψ ( x ⃗ , r , θ + 2 π ) = e 2 π i ν ψ ( x ⃗ , r , θ ) . (1) \psi(\vec{x},r,\theta+2\pi) = e^{2\pi i\nu}\psi(\vec{x},r,\theta). \tag*{(1)} ψ ( x , r , θ + 2 π ) = e 2 π i ν ψ ( x , r , θ ) . (1) Here r r r θ \theta θ x ⃗ \vec{x} x ( d − 2 ) (d-2) ( d − 2 ) 0 ≤ ν ≤ 1 2 0 \le\nu\le\frac{1}{2} 0 ≤ ν ≤ 2 1 ν \nu ν 1 − ν 1-\nu 1 − ν
When M = N M=N M = N [12 ] , and analytical approaches for general N N N [13 , 14 , 15 , 19 ] . An infrared (IR) stable DCFT fixed point (FP) is found in all these studies, which can be accessed perturbatively in the ϵ \epsilon ϵ [13 , 14 , 20 , 21 ] , or through the large N N N [15 , 19 ] .
In this work, we study monodromy defects with M < N M < N M < N ϵ \epsilon ϵ N N N
We then study a large N N N M , N → ∞ M,N \to\infty M , N → ∞ γ = M / N \gamma= M/N γ = M / N N N N ϵ \epsilon ϵ d d d γ \gamma γ d = 3 d = 3 d = 3 γ \gamma γ v v v d = 3 d = 3 d = 3 d = 3 d = 3 d = 3 v < v c s t d , a l t ( γ ) v < v_c^{\mathrm{std,alt}}(\gamma) v < v c std , alt ( γ ) N N N v v v γ \gamma γ N N N Table I .
Table I. Evolution of defect FPs in the large N N N d : 4 → 3 d:4\to3 d : 4 → 3 ( v , γ ) (v,\gamma) ( v , γ ) ( ϕ , ψ ) (\phi,\psi) ( ϕ , ψ ) W W W S S S d = 4 d=4 d = 4
Lastly, in d = 3 d = 3 d = 3 N = 3 N = 3 N = 3 Z 2 \mathbb{Z}_2 Z 2 Φ 3 → − Φ 3 \Phi_3 \to-\Phi_3 Φ 3 → − Φ 3 Φ 1 \Phi_1 Φ 1 Φ 2 \Phi_2 Φ 2 v = 1 / 2 v = 1/2 v = 1/2 v = 1 / 4 v = 1/4 v = 1/4 ∼ c ~ Φ ~ 2 \sim\tilde{c}\tilde{\Phi}^2 ∼ c ~ Φ ~ 2 c ~ < 0 \tilde{c} < 0 c ~ < 0 ∣ c ~ ∣ |\tilde{c}| ∣ c ~ ∣ Z 2 \mathbb{Z}_2 Z 2 ∣ c ~ ∣ |\tilde{c}| ∣ c ~ ∣ v = 1 / 2 v = 1/2 v = 1/2 Δ ϕ \Delta_\phi Δ ϕ 0.409 ± 0.011 0.409\pm0.011 0.409 ± 0.011 c ~ = 0 \tilde{c}=0 c ~ = 0 0.194 ± 0.006 0.194\pm0.006 0.194 ± 0.006 c ~ ∗ = − 1.8 \tilde{c}_*=-1.8 c ~ ∗ = − 1.8 v = 1 / 4 v=1/4 v = 1/4 Z 2 \mathbb{Z}_2 Z 2
Renormalization group analysis
In this section we study the defect dynamics using the renormalization group analysis.
It is helpful to review the case of the free theory for our choice of monodromy. The fields can be decomposed according to the angular momentum around the defect in Eq. (1) , with the spin ℓ \ell ℓ v v v ψ ( x ⃗ , r , θ ) = ∑ ℓ ∈ Z + v ψ ℓ ( x ⃗ , r ) e i ℓ θ \psi(\vec{x},r,\theta)=\sum_{\ell\in\mathbb{Z}+v}\psi_\ell(\vec{x},r)e^{i\ell\theta} ψ ( x , r , θ ) = ∑ ℓ ∈ Z + v ψ ℓ ( x , r ) e i ℓ θ ϕ ( x ⃗ , r , θ ) = ∑ ℓ ∈ Z ϕ ℓ ( x ⃗ , r ) e i ℓ θ \phi(\vec{x},r,\theta)=\sum_{\ell\in\mathbb{Z}}\phi_\ell(\vec{x},r)e^{i\ell\theta} ϕ ( x , r , θ ) = ∑ ℓ ∈ Z ϕ ℓ ( x , r ) e i ℓ θ
Due to conformal invariance, each mode obeys a bulk-to-defect operator product expansion (OPE) in terms of defect primaries (denoted with a hat), as r → 0 r\to0 r → 0
ψ ℓ ( x ⃗ , r ) ∼ r Δ ℓ − Δ ψ ^ ℓ ψ ^ ℓ ( x ⃗ ) , ϕ ℓ ( x ⃗ , r ) ∼ r Δ ℓ − Δ ϕ ^ ℓ ϕ ^ ℓ ( x ⃗ ) . (2) \psi_\ell(\vec{x},r)\sim r^{\Delta_\ell-\Delta_{\hat{\psi}_\ell}}\hat{\psi}_\ell(\vec{x}),\quad\phi_\ell(\vec{x},r)\sim r^{\Delta_\ell-\Delta_{\hat{\phi}_\ell}}\hat{\phi}_\ell(\vec{x}). \tag*{(2)} ψ ℓ ( x , r ) ∼ r Δ ℓ − Δ ψ ^ ℓ ψ ^ ℓ ( x ) , ϕ ℓ ( x , r ) ∼ r Δ ℓ − Δ ϕ ^ ℓ ϕ ^ ℓ ( x ) . (2) The bulk scaling dimension for these fields is the usual value, Δ = d 2 − 1 \Delta=\frac{d}{2}-1 Δ = 2 d − 1 ψ ^ ℓ \hat{\psi}_\ell ψ ^ ℓ ϕ ^ ℓ \hat{\phi}_\ell ϕ ^ ℓ Δ ℓ = d 2 − 1 + ∣ ℓ ∣ \Delta_\ell=\frac{d}{2}-1+|\ell| Δ ℓ = 2 d − 1 + ∣ ℓ ∣ ℓ ∈ Z + v \ell\in\mathbb{Z}+v ℓ ∈ Z + v ψ ^ ℓ = v \hat{\psi}_{\ell=v} ψ ^ ℓ = v Δ ℓ = v = d 2 − 1 + v \Delta_{\ell=v}=\frac{d}{2}-1+v Δ ℓ = v = 2 d − 1 + v ϕ ^ ℓ = 0 \hat{\phi}_{\ell=0} ϕ ^ ℓ = 0 Δ ℓ = 0 = d 2 − 1 \Delta_{\ell=0}=\frac{d}{2}-1 Δ ℓ = 0 = 2 d − 1 ϕ ^ ℓ = 0 2 \hat{\phi}_{\ell=0}^2 ϕ ^ ℓ = 0 2 2 Δ ℓ = 0 = d − 2 2\Delta_{\ell=0}=d-2 2 Δ ℓ = 0 = d − 2 ψ \psi ψ ϕ \phi ϕ
In d ≥ 3 d\ge3 d ≥ 3 Δ ℓ = v = d 2 − 1 − v \Delta_{\ell=v}=\frac{d}{2}-1-v Δ ℓ = v = 2 d − 1 − v Δ ℓ > Δ \Delta_\ell>\Delta Δ ℓ > Δ Δ ℓ < Δ \Delta_\ell<\Delta Δ ℓ < Δ ψ ℓ = v \psi_{\ell=v} ψ ℓ = v
We now turn to the interacting theory. The action of the Wilson-Fisher theory in d d d
S = S Φ + S d e f e c t , S Φ = ∫ d d x [ 1 2 ( ∂ Φ ) 2 + λ 4 ! ( Φ 2 ) 2 ] , S d e f e c t = ∫ d e f e c t d d − 2 x 1 2 ( c ϕ ^ ℓ = 0 2 + h ∣ ψ ^ ℓ = v ∣ 2 ) . (3) \begin{aligned}
S &= S_{\Phi}+S_{\mathrm{defect}},\\
S_{\Phi} &= \int\mathrm{d}^{d}x\left[\frac{1}{2}(\partial\Phi)^2+\frac{\lambda}{4!}(\Phi^2)^2\right],\\
S_{\mathrm{defect}} &= \int_{\mathrm{defect}}\mathrm{d}^{d-2}x\ \frac{1}{2}\left(c\hat{\phi}_{\ell=0}^{2}+h\left|\hat{\psi}_{\ell=v}\right|^2\right). \tag*{(3)}
\end{aligned} S S Φ S defect = S Φ + S defect , = ∫ d d x [ 2 1 ( ∂ Φ ) 2 + 4 ! λ ( Φ 2 ) 2 ] , = ∫ defect d d − 2 x 2 1 ( c ϕ ^ ℓ = 0 2 + h ψ ^ ℓ = v 2 ) . (3) Here we include the classically marginal mass term for the untwisted fields on the defect, as well as the mass for the twisted field with the lowest spin ℓ = v \ell=v ℓ = v v ≲ ϵ v\lesssim\epsilon v ≲ ϵ [7 ] . For d = 4 − ϵ d=4-\epsilon d = 4 − ϵ ϵ ≪ 1 \epsilon\ll1 ϵ ≪ 1 λ ∗ = 48 π 2 ϵ N + 8 \lambda_*=\frac{48\pi^2\epsilon}{N+8} λ ∗ = N + 8 48 π 2 ϵ
We find that the β \beta β c c c h h h
β c = c 2 2 π + M κ λ 24 π + ( N − M + 2 ) λ c 48 π 2 + M λ h 48 π 2 , (4) \beta_c=\frac{c^2}{2\pi}+\frac{M\kappa\lambda}{24\pi}+\frac{(N-M+2)\lambda c}{48\pi^2}+\frac{M\lambda h}{48\pi^2}, \tag*{(4)} β c = 2 π c 2 + 24 π M κλ + 48 π 2 ( N − M + 2 ) λ c + 48 π 2 M λh , (4) β h = h 2 2 π + ( M + 2 ) κ λ 24 π + ( N − M ) λ c 48 π 2 + ( M + 2 ) λ h 48 π 2 + 2 v h . \beta_h=\frac{h^2}{2\pi}+\frac{(M+2)\kappa\lambda}{24\pi}+\frac{(N-M)\lambda c}{48\pi^2}+\frac{(M+2)\lambda h}{48\pi^2}+2vh. β h = 2 π h 2 + 24 π ( M + 2 ) κλ + 48 π 2 ( N − M ) λ c + 48 π 2 ( M + 2 ) λh + 2 v h . Here we use the convention β c = d c d log μ \beta_c=\frac{\mathrm{d}c}{\mathrm{d}\log\mu} β c = d l o g μ d c β h = d h d log μ \beta_h=\frac{\mathrm{d}h}{\mathrm{d}\log\mu} β h = d l o g μ d h κ = v ( 1 − v ) \kappa=v(1-v) κ = v ( 1 − v ) β c \beta_c β c β h \beta_h β h
To better understand the landscape of FPs of the defect RG flow, we analyze the 1 ≫ v ≫ ϵ 1\gg v\gg\epsilon 1 ≫ v ≫ ϵ v ≲ ϵ v\lesssim\epsilon v ≲ ϵ ϕ ^ ℓ = 0 \hat{\phi}_{\ell=0} ϕ ^ ℓ = 0 ψ ^ ℓ = v \hat{\psi}_{\ell=v} ψ ^ ℓ = v
ϕ ^ ℓ = 0 : Δ ℓ = 0 = d 2 − 1 + α ℓ = 0 , α ℓ = 0 = c ∗ ± 2 π , ψ ^ ℓ = v : Δ ℓ = v = d 2 − 1 + v + α ℓ = v , α ℓ = v = h ∗ ± 2 π . (5) \begin{aligned}
\hat{\phi}_{\ell=0}: \quad\Delta_{\ell=0}&=\frac{d}{2}-1+\alpha_{\ell=0}, \qquad\alpha_{\ell=0}=\frac{c_*^{\pm}}{2\pi},\\
\hat{\psi}_{\ell=v}: \quad\Delta_{\ell=v}&=\frac{d}{2}-1+v+\alpha_{\ell=v}, \qquad\alpha_{\ell=v}=\frac{h_*^{\pm}}{2\pi}. \tag*{(5)}
\end{aligned} ϕ ^ ℓ = 0 : Δ ℓ = 0 ψ ^ ℓ = v : Δ ℓ = v = 2 d − 1 + α ℓ = 0 , α ℓ = 0 = 2 π c ∗ ± , = 2 d − 1 + v + α ℓ = v , α ℓ = v = 2 π h ∗ ± . (5) Accordingly, ϕ ℓ = 0 \phi_{\ell=0} ϕ ℓ = 0 c ∗ > 0 c_*>0 c ∗ > 0 c ∗ < 0 c_*<0 c ∗ < 0 ψ ℓ = v \psi_{\ell=v} ψ ℓ = v h ∗ + 2 π v > 0 h_*+2\pi v>0 h ∗ + 2 π v > 0 h ∗ + 2 π v < 0 h_*+2\pi v<0 h ∗ + 2 π v < 0
Defect FPs in the 1 ≫ v ≫ ϵ 1\gg v\gg\epsilon 1 ≫ v ≫ ϵ
In this regime, the RG analysis can be simplified. To leading order β h = h 2 2 π + 2 v h \beta_h=\frac{h^2}{2\pi}+2vh β h = 2 π h 2 + 2 v h h ∗ = 0 h_*=0 h ∗ = 0 h ∗ = − 4 π v h_*=-4\pi v h ∗ = − 4 π v
field in the bare theory. This analysis is equivalent to dropping the defect mass perturbation for the twisted fields in Eq. (3) and choosing their boundary condition by hand. For standard boundary conditions,
β c 1 ≫ v ≫ ϵ = c 2 2 π + M κ λ 24 π + ( N − M + 2 ) λ c 48 π 2 . (6) \beta_c^{1\gg v\gg\epsilon}=\frac{c^2}{2\pi}+\frac{M\kappa\lambda}{24\pi}+\frac{(N-M+2)\lambda c}{48\pi^2}. \tag*{(6)} β c 1 ≫ v ≫ ϵ = 2 π c 2 + 24 π M κλ + 48 π 2 ( N − M + 2 ) λ c . (6) We define the critical value κ c \kappa_c κ c
κ c = ( N − M + 2 ) 2 λ 192 π 2 M = ( N − M + 2 ) 2 4 M ( N + 8 ) ϵ . (7) \kappa_c=\frac{(N-M+2)^2\lambda}{192\pi^2M}=\frac{(N-M+2)^2}{4M(N+8)}\epsilon. \tag*{(7)} κ c = 192 π 2 M ( N − M + 2 ) 2 λ = 4 M ( N + 8 ) ( N − M + 2 ) 2 ϵ . (7) Because v ( 1 − v ) ≤ 1 / 4 v(1-v)\leq1/4 v ( 1 − v ) ≤ 1/4 κ ~ c = min ( κ c , 1 4 ) \tilde{\kappa}_c=\min(\kappa_c,\frac{1}{4}) κ ~ c = min ( κ c , 4 1 ) κ < κ ~ c \kappa<\tilde{\kappa}_c κ < κ ~ c
c ∗ ± = − π N − M + 2 N + 8 ϵ ( 1 ± 1 − κ κ c ) . (8) c_*^{\pm}=-\pi\frac{N-M+2}{N+8}\epsilon\left(1\pm\sqrt{1-\frac{\kappa}{\kappa_c}}\right). \tag*{(8)} c ∗ ± = − π N + 8 N − M + 2 ϵ ( 1 ± 1 − κ c κ ) . (8) As a result, when κ < κ c \kappa<\kappa_c κ < κ c 1 ≫ v ≫ ϵ 1\gg v\gg\epsilon 1 ≫ v ≫ ϵ γ = M N ≲ ϵ 4 κ ( N + 2 ) 2 N ( N + 8 ) ≪ 1 \gamma=\frac{M}{N}\lesssim\frac{\epsilon}{4\kappa}\frac{(N+2)^2}{N(N+8)}\ll1 γ = N M ≲ 4 κ ϵ N ( N + 8 ) ( N + 2 ) 2 ≪ 1 c ∗ ± < 0 c_*^{\pm}<0 c ∗ ± < 0 ϕ ℓ = 0 \phi_{\ell=0} ϕ ℓ = 0 α ℓ = 0 \alpha_{\ell=0} α ℓ = 0 c ∗ − c_*^- c ∗ − c ∗ + c_*^+ c ∗ + ϕ ^ ℓ = 0 2 \hat{\phi}_{\ell=0}^{2} ϕ ^ ℓ = 0 2 c ∗ − c_*^- c ∗ − c ∗ + c_*^+ c ∗ + c ∗ − c_*^- c ∗ − c ∗ + c_*^+ c ∗ +
When κ > κ c \kappa>\kappa_c κ > κ c γ > ϵ 4 κ ( N + 2 ) 2 N ( N + 8 ) \gamma>\frac{\epsilon}{4\kappa}\frac{(N+2)^2}{N(N+8)} γ > 4 κ ϵ N ( N + 8 ) ( N + 2 ) 2 c → − ∞ c\to-\infty c → − ∞ N N N N = 3 N = 3 N = 3 1 ≫ v ≫ ϵ 1 \gg v \gg\epsilon 1 ≫ v ≫ ϵ κ ~ c = 1 / 4 \tilde{\kappa}_c = 1/4 κ ~ c = 1/4 γ \gamma γ v v v [22 ] .
We have thus seen that for κ c < 1 / 4 \kappa_c < 1/4 κ c < 1/4 v c a l t , s t d ( γ ) v_c^{\mathrm{alt,std}}(\gamma) v c alt , std ( γ ) v < v c a l t , s t d v < v_c^{\mathrm{alt,std}} v < v c alt , std v > v c a l t , s t d v > v_c^{\mathrm{alt,std}} v > v c alt , std v v v
As already mentioned, the runaway flow should lead to a non-perturbative FP where c c c O ( 1 ) \mathcal{O}(1) O ( 1 ) ϕ \phi ϕ O ( N − M ) O(N-M) O ( N − M ) N − M > 1 N-M > 1 N − M > 1 ( 2 − ϵ ) (2-\epsilon) ( 2 − ϵ ) [23 ] . A possible endpoint of the runaway RG flow is a new DCFT when c ∗ c_* c ∗ O ( 1 ) \mathcal{O}(1) O ( 1 ) N N N N − M = 1 N-M=1 N − M = 1 O ( 1 ) = Z 2 O(1)=\mathbb{Z}_2 O ( 1 ) = Z 2
The leading correction to the defect scaling dimension for the twisted field can be extracted as well. From expansion of the solution to β h \beta_h β h (4) in ϵ \epsilon ϵ ψ ^ ℓ = v \hat{\psi}_{\ell=v} ψ ^ ℓ = v
α ℓ = v = h ∗ 2 π ≃ − ( M + 2 ) ϵ 2 ( N + 8 ) v ( κ + N − M M + 2 c ∗ ± 2 π ) . (9) \alpha_{\ell=v} = \frac{h_*}{2\pi} \simeq-\frac{(M+2)\epsilon}{2(N+8)v}\left(\kappa+ \frac{N-M}{M+2}\frac{c_*^{\pm}}{2\pi}\right). \tag*{(9)} α ℓ = v = 2 π h ∗ ≃ − 2 ( N + 8 ) v ( M + 2 ) ϵ ( κ + M + 2 N − M 2 π c ∗ ± ) . (9) As h ∗ + 2 π v ≥ 0 h_*+2\pi v \ge0 h ∗ + 2 π v ≥ 0 ψ ^ ℓ = v \hat{\psi}_{\ell=v} ψ ^ ℓ = v M = N M=N M = N [14 , 15 ] . It also agrees with results in the large N N N ℓ ≠ 0 \ell\ne0 ℓ = 0
Defect FPs in the v v v
To accurately capture the physics in this regime with RG, it is imperative to keep the defect mass perturbation for the twisted fields in the action. The FPs are complicated, but we may comment on some new qualitative features of the solutions:
The same two perturbative FPs with alternate boundary conditions for ϕ ℓ = 0 \phi_{\ell=0} ϕ ℓ = 0 ψ ℓ = v \psi_{\ell=v} ψ ℓ = v v < v c a l t , s t d v < v_c^{\mathrm{alt,std}} v < v c alt , std γ ∼ O ( 1 ) \gamma\sim\mathcal{O}(1) γ ∼ O ( 1 ) N N N γ \gamma γ v c a l t , s t d v_c^{\mathrm{alt,std}} v c alt , std v c a l t , s t d ≫ ϵ v_c^{\mathrm{alt,std}} \gg\epsilon v c alt , std ≫ ϵ v c a l t , s t d ≲ ϵ v_c^{\mathrm{alt,std}} \lesssim\epsilon v c alt , std ≲ ϵ Figure 1 .
Figure 1. Prototypical RG flow (shown for v ≲ ϵ v\lesssim\epsilon v ≲ ϵ v < v c a l t , s t d v<v_c^{\mathrm{alt,std}} v < v c alt , std ϕ ℓ = 0 \phi_{\ell=0} ϕ ℓ = 0 ψ ℓ = v \psi_{\ell=v} ψ ℓ = v
For finite N N N v c a l t , s t d v_c^{\mathrm{alt,std}} v c alt , std γ \gamma γ γ → 1 \gamma\to1 γ → 1 v v v γ \gamma γ γ \gamma γ v v v
For N → ∞ N \to\infty N → ∞ γ ≥ 1 / 2 \gamma\ge1/2 γ ≥ 1/2 v c a l t , s t d → 0 v_c^{\mathrm{alt,std}} \to0 v c alt , std → 0 v < v c a l t , s t d v < v_c^{\mathrm{alt,std}} v < v c alt , std γ → 1 / 2 \gamma\to1/2 γ → 1/2
The regions of parameters in which these FPs exist for finite N N N N → ∞ N \to\infty N → ∞ Figure 2 . We leave discussion of when the twisted field has alternate boundary conditions to the SM S1.
Figure 2. Landscape of FPs for ϵ ≪ 1 \epsilon\ll1 ϵ ≪ 1 ψ ℓ = v \psi_{\ell=v} ψ ℓ = v h ∗ + 2 π v > 0 h_*+2\pi v>0 h ∗ + 2 π v > 0 N N N c ∗ < 0 c_*<0 c ∗ < 0 ϕ ℓ = 0 \phi_{\ell=0} ϕ ℓ = 0 N N N N N N v c a l t , s t d ( γ ) v_c^{\mathrm{alt,std}}(\gamma) v c alt , std ( γ ) γ = 1 / 2 \gamma=1/2 γ = 1/2 ( + 1 ) (+1) ( + 1 )
Large N N N
Next we analyze the problem in the limit of large N N N N N N M , N → ∞ M,N \to\infty M , N → ∞ γ = M / N \gamma=M/N γ = M / N ( Φ 2 ) 2 (\Phi^2)^2 ( Φ 2 ) 2 σ ( x ) = σ g \sigma(x)=\frac{\sigma}{g} σ ( x ) = g σ σ ∗ \sigma_* σ ∗ [7 , 15 ] . Details can be found in the SM S3.
At this point, one has to specify the boundary conditions of the fields near the defect. As reviewed above, for each mode there are potentially two options, standard or alternate. The defect scaling dimension is given by
Δ ℓ = d 2 − 1 ± σ ∗ + ℓ 2 + O ( N − 1 ) , (10) \Delta_\ell= \frac{d}{2}-1 \pm\sqrt{\sigma_*+\ell^2}+\mathcal{O}(N^{-1}), \tag*{(10)} Δ ℓ = 2 d − 1 ± σ ∗ + ℓ 2 + O ( N − 1 ) , (10) where + / − +/- + / − ℓ \ell ℓ σ ∗ ≥ 0 \sigma_* \ge0 σ ∗ ≥ 0 ℓ = 0 \ell= 0 ℓ = 0 ℓ \ell ℓ d 2 − 1 − σ ∗ + ℓ 2 > 0 \frac{d}{2} - 1 - \sqrt{\sigma_* + \ell^2} > 0 2 d − 1 − σ ∗ + ℓ 2 > 0
One advantage of the large N N N d d d d d d d = 3 d = 3 d = 3 ℓ = 0 \ell= 0 ℓ = 0 ℓ ≠ 0 \ell\ne0 ℓ = 0 σ ∗ \sigma_* σ ∗
0 ≤ σ ∗ < ( d 2 − 1 ) 2 . (11) 0 \le\sigma_* < \left(\frac{d}{2} - 1\right)^2 . \tag*{(11)} 0 ≤ σ ∗ < ( 2 d − 1 ) 2 . (11) Details of the derivation and more discussions on other choices of boundary conditions can be found in the SM S3.
When ϵ = 4 − d \epsilon= 4-d ϵ = 4 − d σ ∗ \sigma_* σ ∗ c ∗ c_* c ∗ σ ∗ = ∣ c ∗ ∣ / 2 π \sqrt{\sigma_*} = |c_*|/2\pi σ ∗ = ∣ c ∗ ∣/2 π ϵ \epsilon ϵ N N N
In addition, one can also find a “strong-pinning” solution with
σ ∗ = 1 − ϵ 2 − ( 1 − γ ) ϵ 2 + O ( ϵ 3 ) . (12) \sqrt{\sigma_*} = 1 - \frac{\epsilon}{2} - (1-\gamma)\epsilon^2 + O(\epsilon^3). \tag*{(12)} σ ∗ = 1 − 2 ϵ − ( 1 − γ ) ϵ 2 + O ( ϵ 3 ) . (12) Notably, σ ∗ \sigma_* σ ∗ Δ ℓ = 0 = ( 1 − γ ) ϵ 2 \Delta_{\ell=0} = (1-\gamma)\epsilon^2 Δ ℓ = 0 = ( 1 − γ ) ϵ 2 N N N ∣ c ∣ ∝ σ ∗ |c| \propto\sqrt{\sigma_*} ∣ c ∣ ∝ σ ∗ O ( 1 ) O(1) O ( 1 ) σ ∗ \sigma_* σ ∗ ϵ \epsilon ϵ
as “strong-pinning” FPs, as opposed to the perturbative FPs with “weak-pinning”. The parametrically small Δ ℓ = 0 \Delta_{\ell=0} Δ ℓ = 0 [7 ] .
Having understood the small ϵ \epsilon ϵ ϵ \epsilon ϵ O ( 1 ) O(1) O ( 1 ) v v v O ( 1 ) O(1) O ( 1 ) γ \gamma γ v ≫ ϵ v \gg\epsilon v ≫ ϵ ϵ < ϵ c = 1 − γ 2 \epsilon< \epsilon_c = \frac{1-\gamma}{2} ϵ < ϵ c = 2 1 − γ ϵ \epsilon ϵ ϵ > ϵ c \epsilon> \epsilon_c ϵ > ϵ c ϵ \epsilon ϵ ϵ ∼ O ( 1 ) \epsilon\sim O(1) ϵ ∼ O ( 1 ) v = 0.05 , γ = 1 / 3 v = 0.05, \gamma= 1/3 v = 0.05 , γ = 1/3 Figure 3 . We see that the (continuation of the) strong-pinning FP annihilates with the unstable perturbative FP, while the stable perturbative FP continues to exist all the way to d = 3 d = 3 d = 3 σ ∗ \sigma_* σ ∗ ϵ \epsilon ϵ v v v γ < 1 / 2 \gamma< 1/2 γ < 1/2
Figure 3. Solutions to the large N N N d d d v = 0.05 , γ = 1 / 3 v = 0.05, \gamma= 1/3 v = 0.05 , γ = 1/3 v = 0.5 , γ = 2 / 3 v = 0.5, \gamma= 2/3 v = 0.5 , γ = 2/3 γ = 1 / 3 , v = 0.05 \gamma= 1/3, v = 0.05 γ = 1/3 , v = 0.05 d d d
To summarize, the large N N N γ \gamma γ ϵ \epsilon ϵ d = 3 d = 3 d = 3 γ ≪ 1 \gamma\ll1 γ ≪ 1 v v v d = 3 d = 3 d = 3
σ ∗ = γ π ( 1 − 2 v ) tan π v + O ( γ 2 ) . (13) \sqrt{\sigma_*} = \frac{\gamma}{\pi}(1-2v)\tan\pi v + O(\gamma^2). \tag*{(13)} σ ∗ = π γ ( 1 − 2 v ) tan π v + O ( γ 2 ) . (13) The expression for other d d d
Moreover, when γ < 1 / 2 \gamma< 1/2 γ < 1/2 v ≪ 1 , σ ∗ = O ( v 2 ) v \ll1,\sigma_* = O(v^2) v ≪ 1 , σ ∗ = O ( v 2 ) ϵ = 4 − d ≫ v \epsilon= 4-d \gg v ϵ = 4 − d ≫ v
σ ∗ = γ 1 − 2 γ v + O ( v 2 ) . (14) \sqrt{\sigma_*} = \frac{\gamma}{\sqrt{1-2\gamma}}v + O(v^2). \tag*{(14)} σ ∗ = 1 − 2 γ γ v + O ( v 2 ) . (14) Now we turn to γ > 1 / 2 \gamma> 1/2 γ > 1/2 d ∈ [ 3 , 4 ) d \in[3,4) d ∈ [ 3 , 4 ) Figure 3 for an example. The one near d = 4 d = 4 d = 4 d = 3 d = 3 d = 3 Figure 4 shows the dependence of Δ ℓ = 0 , Δ ℓ = v \Delta_{\ell=0}, \Delta_{\ell=v} Δ ℓ = 0 , Δ ℓ = v v v v γ = 1 / 3 \gamma= 1/3 γ = 1/3 2 / 3 2/3 2/3 γ = 2 / 3 \gamma= 2/3 γ = 2/3 σ ∗ \sigma_* σ ∗ v → 0 v \to0 v → 0 v → 0 v \to0 v → 0 γ > 1 / 2 \gamma> 1/2 γ > 1/2 γ < 1 / 2 \gamma< 1/2 γ < 1/2 (14) the v → 0 v \to0 v → 0
Figure 4. Δ ℓ = 0 \Delta_{\ell=0} Δ ℓ = 0 Δ ℓ = v \Delta_{\ell=v} Δ ℓ = v v v v γ = 1 / 3 \gamma= 1/3 γ = 1/3 σ ∗ → 0 \sigma_* \to0 σ ∗ → 0 v → 0 v \to0 v → 0 γ = 2 / 3 \gamma= 2/3 γ = 2/3 v → 0 v \to0 v → 0 σ ∗ \sigma_* σ ∗ d = 3.0001 d = 3.0001 d = 3.0001
So far we have focused on the (alt for ℓ = 0 \ell= 0 ℓ = 0 ℓ = v \ell= v ℓ = v d = 3 d = 3 d = 3 d ∈ [ 3 , 4 ) d \in[3,4) d ∈ [ 3 , 4 ) v c s t d , a l t ( d , γ ) ≥ 0 v_c^{\mathrm{std,alt}}(d,\gamma) \ge0 v c std , alt ( d , γ ) ≥ 0 v < v c s t d , a l t ( d , γ ) v < v_c^{\mathrm{std,alt}}(d,\gamma) v < v c std , alt ( d , γ ) v > v c s t d , a l t ( d , γ ) v > v_c^{\mathrm{std,alt}}(d,\gamma) v > v c std , alt ( d , γ ) d = 3 d = 3 d = 3 Figure 5 . Note that in this case v c s t d , a l t ( d = 3 , γ ) = 0 v_c^{\mathrm{std,alt}}(d = 3,\gamma) = 0 v c std , alt ( d = 3 , γ ) = 0 γ ≥ 1 / 2 \gamma\ge1/2 γ ≥ 1/2 d = 3 d = 3 d = 3 v ≪ 1 v \ll1 v ≪ 1 γ ≪ 1 \gamma\ll1 γ ≪ 1 d → 4 d \to4 d → 4 d = 3 d = 3 d = 3 γ → 0 \gamma\to0 γ → 0 v → 1 / 2 v \to1/2 v → 1/2 d = 3 d = 3 d = 3
Figure 5. Landscape of solutions at large N N N d = 3.0001 d = 3.0001 d = 3.0001 v < v c s t d , a l t ( d = 3 , γ ) v < v_c^{\mathrm{std,alt}}(d = 3,\gamma) v < v c std , alt ( d = 3 , γ )
Therefore, together with the earlier analysis of (alt, std) solutions, we find that there are three (one) physical solutions to the gap equation at d = 3 d = 3 d = 3 v < v c s t d , a l t ( γ ) v < v_c^{\mathrm{std,alt}}(\gamma) v < v c std , alt ( γ ) v > v c s t d , a l t ( γ ) v > v_c^{\mathrm{std,alt}}(\gamma) v > v c std , alt ( γ ) d = 4 − ϵ d = 4-\epsilon d = 4 − ϵ v < v c s t d , a l t ( γ ) v < v_c^{\mathrm{std,alt}}(\gamma) v < v c std , alt ( γ ) N N N Table I .
Monte Carlo simulations
In this section, we study monodromy line defects in the three-dimensional O ( N ) O(N) O ( N ) N = 3 N = 3 N = 3 ϕ 4 \phi^4 ϕ 4 L × L × L L \times L \times L L × L × L
The dynamical variable at each site x \mathbf{x} x N N N Φ x \boldsymbol{\Phi}_{\mathbf{x}} Φ x O ( N ) O(N) O ( N ) U x , μ U_{x,\mu} U x , μ
H [ Φ ; U , c ~ ] = − β ∑ x ∑ μ = 1 3 Φ x ⋅ U x , μ Φ x + μ ^ + ∑ x [ Φ x 2 + λ ( Φ x 2 − 1 ) 2 ] + H b i n [ Φ ; c ~ ] , (15) \mathcal{H}[\boldsymbol{\Phi};U,\widetilde{c}] = -\beta\sum_{\mathbf{x}}\sum_{\mu=1}^{3}\boldsymbol{\Phi}_{\mathbf{x}}\cdot U_{\mathbf{x},\mu}\boldsymbol{\Phi}_{\mathbf{x}+\hat{\mu}} + \sum_{\mathbf{x}}\left[\boldsymbol{\Phi}_{\mathbf{x}}^2+\lambda\left(\boldsymbol{\Phi}_{\mathbf{x}}^2-1\right)^2\right] + H_{\mathrm{bin}}[\boldsymbol{\Phi};\widetilde{c}], \tag*{(15)} H [ Φ ; U , c ] = − β x ∑ μ = 1 ∑ 3 Φ x ⋅ U x , μ Φ x + μ ^ + x ∑ [ Φ x 2 + λ ( Φ x 2 − 1 ) 2 ] + H bin [ Φ ; c ] , (15) and the partition function is Z [ U , c ~ ] = ∫ ∏ x d N Φ x exp ( − H [ Φ ; U , c ~ ] ) Z[U,\widetilde{c}] = \int\prod_x d^N\Phi_x\exp\left(-\mathcal{H}[\boldsymbol{\Phi};U,\widetilde{c}]\right) Z [ U , c ] = ∫ ∏ x d N Φ x exp ( − H [ Φ ; U , c ] ) λ → ∞ \lambda\to\infty λ → ∞ (15) reduces to the hard O ( N ) O(N) O ( N ) ( β , λ ) (\beta,\lambda) ( β , λ ) λ \lambda λ ( λ , β ) = ( 4.5 , 0.6862385 ) (\lambda,\beta) = (4.5,0.6862385) ( λ , β ) = ( 4.5 , 0.6862385 ) N = 3 N = 3 N = 3 [24 ] .
The monodromy defect is introduced by modifying the nearest-neighbor couplings on an open, codimension-one sheet of bonds. Specifically, the y y y ( x , y c , z ) (x,y_c,z) ( x , y c , z ) ( x , y c + 1 , z ) (x,y_c+1,z) ( x , y c + 1 , z ) x 1 ≤ x < x 2 x_1 \le x < x_2 x 1 ≤ x < x 2 z z z g ∈ O ( N ) g \in O(N) g ∈ O ( N ) O ( N ) O(N) O ( N )
U ( x , y , z ) , y = g , y = y c , x 1 ≤ x < x 2 , (16) U_{(x,y,z),y} = g,\quad y = y_c,\quad x_1 \le x < x_2, \tag*{(16)} U ( x , y , z ) , y = g , y = y c , x 1 ≤ x < x 2 , (16) while all other link fields are unity. The open sheet with twisted link fields has two boundaries, located on dual lattice lines ( x 1 − 1 / 2 , y c + 1 / 2 ) (x_1-1/2,y_c+1/2) ( x 1 − 1/2 , y c + 1/2 ) ( x 2 − 1 / 2 , y c + 1 / 2 ) (x_2-1/2,y_c+1/2) ( x 2 − 1/2 , y c + 1/2 ) g g g g − 1 g^{-1} g − 1
We will study two types of monodromy defects:
For a rotational monodromy, the twist acts in the ( Φ 1 , Φ 2 ) (\Phi_1,\Phi_2) ( Φ 1 , Φ 2 )
g ( v ) = ( cos ( 2 π v ) − sin ( 2 π v ) sin ( 2 π v ) cos ( 2 π v ) ) ⊕ 1 N − 2 . (17) g(v) =
\begin{pmatrix}
\cos(2\pi v) & -\sin(2\pi v) \\
\sin(2\pi v) & \cos(2\pi v)
\end{pmatrix}
\oplus\mathbf{1}_{N-2}. \tag*{(17)} g ( v ) = ( cos ( 2 π v ) sin ( 2 π v ) − sin ( 2 π v ) cos ( 2 π v ) ) ⊕ 1 N − 2 . (17) Equivalently, for the complex field ψ = Φ 1 + i Φ 2 \psi= \Phi_1 + i\Phi_2 ψ = Φ 1 + i Φ 2 (1) .
For a reflection monodromy, we have g = 1 N − 1 ⊕ ( − 1 ) g = \mathbf{1}_{N-1} \oplus(-1) g = 1 N − 1 ⊕ ( − 1 )
Only the location of the two boundary lines is physical, because the sheet connecting them can be deformed by a local O ( N ) O(N) O ( N )
To represent the defect perturbation S d e f e c t S_{\mathrm{defect}} S defect (3) , we introduce the projector P u P_u P u a a a a = 1 , 2 a = 1, 2 a = 1 , 2
S a = { ( x a − 1 , y c ) , ( x a , y c ) , ( x a − 1 , y c + 1 ) , ( x a , y c + 1 ) } . (18) S_a = \{(x_a - 1,y_c),(x_a,y_c),(x_a - 1,y_c + 1),(x_a,y_c + 1)\}. \tag*{(18)} S a = {( x a − 1 , y c ) , ( x a , y c ) , ( x a − 1 , y c + 1 ) , ( x a , y c + 1 )} . (18) The lattice pinning term is chosen to be
H p i n [ Φ ; c ~ ] = c ~ 4 ∑ z = 0 L − 1 ∑ p ∈ S 1 , 2 ∣ P u Φ ( p , z ) ∣ 2 . (19) H_{\mathrm{pin}}[\boldsymbol{\Phi};\tilde{c}] = \frac{\tilde{c}}{4}\sum_{z=0}^{L-1}\sum_{p\in S_{1,2}} \lvert P_u\boldsymbol{\Phi}_{(p,z)}\rvert^2. \tag*{(19)} H pin [ Φ ; c ~ ] = 4 c ~ z = 0 ∑ L − 1 p ∈ S 1 , 2 ∑ ∣ P u Φ ( p , z ) ∣ 2 . (19) Thus Eq. (19) is a lattice representation of the continuum coupling c c c S d e f e c t S_{\mathrm{defect}} S defect (3) . For the rotation monodromy Eq. (17) in N = 3 N = 3 N = 3 P u Φ = Φ 3 P_u\boldsymbol{\Phi} = \Phi_3 P u Φ = Φ 3 P u Φ = Φ 1 + i Φ 2 P_u\boldsymbol{\Phi} = \Phi_1 + i\Phi_2 P u Φ = Φ 1 + i Φ 2 c ~ = 0 \tilde{c} = 0 c ~ = 0
Because the defect lies on the dual lattice, defect operators are naturally represented by superposing those on the four sites on the elementary transverse plaquette (see Eq. (18) ).
(see Eq. (18) ) We always measure observables around the first line S 1 S_1 S 1
For a rotational monodromy in N = 3 N = 3 N = 3
ψ l a t ( z ) = 1 4 ∑ p ∈ S e − i ν θ p [ Φ 1 ( p , z ) + i Φ 2 ( p , z ) ] , (20) \psi^{\mathrm{lat}}(z) = \frac{1}{4}\sum_{p\in S}e^{-i\nu\theta_p}\left[\Phi_1(p,z)+i\Phi_2(p,z)\right], \tag*{(20)} ψ lat ( z ) = 4 1 p ∈ S ∑ e − i ν θ p [ Φ 1 ( p , z ) + i Φ 2 ( p , z ) ] , (20) ϕ l a t ( z ) = 1 4 ∑ p ∈ S Φ 3 ( p , z ) . (21) \phi^{\mathrm{lat}}(z) = \frac{1}{4}\sum_{p\in S}\Phi_3(p,z). \tag*{(21)} ϕ lat ( z ) = 4 1 p ∈ S ∑ Φ 3 ( p , z ) . (21) Here θ p = arg [ ( x p − x d ) + i ( y p − y d ) ] \theta_p = \arg[(x_p-x_d)+i(y_p-y_d)] θ p = arg [( x p − x d ) + i ( y p − y d )] ( x d , y d ) = ( x 1 − 1 / 2 , y c + 1 / 2 ) (x_d,y_d) = (x_1 - 1/2,y_c + 1/2) ( x d , y d ) = ( x 1 − 1/2 , y c + 1/2 ) (20) and (21) denote the twisted and untwisted field respectively, where the phase factors are introduced to enforce the expected behavior under S O ( 2 ) SO(2) S O ( 2 )
For the reflection monodromy defect, we measure the defect operators
ϕ l a t ( z ) = 1 4 ∑ p ∈ S [ Φ 1 ( p , z ) + i Φ 2 ( p , z ) ] , (22) \phi^{\mathrm{lat}}(z) = \frac{1}{4}\sum_{p\in S}\left[\Phi_1(p,z)+i\Phi_2(p,z)\right], \tag*{(22)} ϕ lat ( z ) = 4 1 p ∈ S ∑ [ Φ 1 ( p , z ) + i Φ 2 ( p , z ) ] , (22) ψ l a t ( z ) = 1 4 ∑ p ∈ S e − i θ p / 2 Φ 3 ( p , z ) . (23) \psi^{\mathrm{lat}}(z) = \frac{1}{4}\sum_{p\in S}e^{-i\theta_p/2}\Phi_3(p,z). \tag*{(23)} ψ lat ( z ) = 4 1 p ∈ S ∑ e − i θ p /2 Φ 3 ( p , z ) . (23) Eqs. (22) and (23) denote the untwisted and twisted field respectively.
In these expressions, the superscript “lat” indicates that these are interpolating lattice operators. Their long-distance correlation functions are dominated by the corresponding lowest defect primaries, although the microscopic operators may also overlap with descendants and higher-dimensional primaries. For each run, we measure the two-point correlator for defect operator O ^ ( z ) \hat{O}(z) O ^ ( z )
G ( z , L ) = ⟨ O ^ † ( 0 ) O ^ ( z ) ⟩ , (24) G(z,L) = \langle\hat{O}^{\dagger}(0)\hat{O}(z)\rangle, \tag*{(24)} G ( z , L ) = ⟨ O ^ † ( 0 ) O ^ ( z )⟩ , (24) where O ^ ( z ) \hat{O}(z) O ^ ( z ) (20) , (21) , (22) and (23) , and ⟨ ⋯ ⟩ \langle\cdots\rangle ⟨ ⋯ ⟩
O ^ † ( 0 ) O ^ ( z ) ‾ = 1 L ∑ z 0 = 0 L − 1 O ( z 0 ) ∗ O ( z 0 + z ) . (25) \overline{\hat{O}^{\dagger}(0)\hat{O}(z)} = \frac{1}{L}\sum_{z_0=0}^{L-1}O(z_0)^{*}O(z_0+z). \tag*{(25)} O ^ † ( 0 ) O ^ ( z ) = L 1 z 0 = 0 ∑ L − 1 O ( z 0 ) ∗ O ( z 0 + z ) . (25) where 0 ≤ z ≤ L / 2 0 \le z \le L/2 0 ≤ z ≤ L /2 z z z r z = ( L / π ) sin ( π z / L ) r_z = (L/\pi)\sin(\pi z/L) r z = ( L / π ) sin ( π z / L )
G ( r z , L ) = L − 2 Δ F ( r z L ) , (26) G(r_z,L) = L^{-2\Delta}\mathcal{F}\left(\frac{r_z}{L}\right), \tag*{(26)} G ( r z , L ) = L − 2Δ F ( L r z ) , (26) where Δ \Delta Δ F \mathcal{F} F (26) , and statistical errors are reported using jackknife method. In the figures we also include the cost functions S S S (129) for the definition). Moreover, the nonzero pinning strengths c ~ ∗ \tilde{c}_* c ~ ∗ L ≤ 64 L \le64 L ≤ 64 c ~ ∗ = − 1.8 \tilde{c}_* = -1.8 c ~ ∗ = − 1.8 − 1.5 -1.5 − 1.5 − 1.75 -1.75 − 1.75 v = 1 / 2 v = 1/2 v = 1/2 v = 1 / 4 v = 1/4 v = 1/4
Figures 6 and 7 show the results for the v = 1 / 2 v = 1/2 v = 1/2 c ~ = 0 \tilde{c} = 0 c ~ = 0 (19) ) we obtain Δ ψ = 0.920 ± 0.036 \Delta_{\psi} = 0.920 \pm0.036 Δ ψ = 0.920 ± 0.036 Figure 6 ), and Δ ϕ = 0.409 ± 0.011 \Delta_{\phi} = 0.409 \pm0.011 Δ ϕ = 0.409 ± 0.011 Figure 7 ).
Figure 6. Determination of the scaling dimension of the twisted defect primary ψ ^ \hat{\psi} ψ ^ ν = 1 / 2 \nu= 1/2 ν = 1/2 N = 3 N = 3 N = 3 c ~ = 0 \tilde{c} = 0 c ~ = 0 c ~ ∗ = − 1.8 \tilde{c}_* = -1.8 c ~ ∗ = − 1.8 ⟨ ψ ^ † ( 0 ) ψ ^ ( r z ) ⟩ \langle\hat{\psi}^{\dagger}(0)\hat{\psi}(r_z)\rangle ⟨ ψ ^ † ( 0 ) ψ ^ ( r z )⟩ r z = ( L / π ) sin ( π z / L ) r_z = (L/\pi)\sin(\pi z/L) r z = ( L / π ) sin ( π z / L ) L L L (26) , from which we get Δ ψ = 0.920 ± 0.036 \Delta_{\psi} = 0.920 \pm0.036 Δ ψ = 0.920 ± 0.036 c ~ = 0 \tilde{c} = 0 c ~ = 0 Δ ψ = 0.998 ± 0.061 \Delta_{\psi} = 0.998 \pm0.061 Δ ψ = 0.998 ± 0.061 c ~ ∗ = − 1.8 \tilde{c}_* = -1.8 c ~ ∗ = − 1.8 S = 1.21 S = 1.21 S = 1.21 1.03 1.03 1.03
Figure 7. Determination of the scaling dimensions of the untwisted defect primary ϕ ^ \hat{\phi} ϕ ^ v = 1 / 2 v = 1/2 v = 1/2 N = 3 N = 3 N = 3 c ~ = 0 \tilde{c} = 0 c ~ = 0 c ~ ∗ = − 1.8 \tilde{c}_* = -1.8 c ~ ∗ = − 1.8 Δ ϕ = 0.409 ± 0.011 \Delta_{\phi} = 0.409 \pm0.011 Δ ϕ = 0.409 ± 0.011 c ~ = 0 \tilde{c} = 0 c ~ = 0 Δ ϕ = 0.194 ± 0.006 \Delta_{\phi} = 0.194 \pm0.006 Δ ϕ = 0.194 ± 0.006 c ~ ∗ = − 1.8 \tilde{c}_* = -1.8 c ~ ∗ = − 1.8 S = 2.52 S = 2.52 S = 2.52
Next we turn on the lattice pinning. In general, we find that when ∣ c ~ ∣ |\tilde{c}| ∣ c ~ ∣ (26) to extract scaling dimensions. We find that as ∣ c ~ ∣ |\tilde{c}| ∣ c ~ ∣ Δ ϕ \Delta_{\phi} Δ ϕ Δ ψ \Delta_{\psi} Δ ψ c ~ ∗ = − 1.8 \tilde{c}_* = -1.8 c ~ ∗ = − 1.8 Δ ϕ = 0.194 ± 0.006 \Delta_{\phi} = 0.194 \pm0.006 Δ ϕ = 0.194 ± 0.006 c ~ = 0 \tilde{c} = 0 c ~ = 0 Δ ψ = 0.998 ± 0.061 \Delta_{\psi} = 0.998 \pm0.061 Δ ψ = 0.998 ± 0.061 N N N γ = 2 / 3 \gamma= 2/3 γ = 2/3 v = 1 / 2 v = 1/2 v = 1/2 Δ ℓ = 0 ≈ 0.228 \Delta_{\ell=0} \approx0.228 Δ ℓ = 0 ≈ 0.228 Δ ℓ = 1 / 2 ≈ 1.069 \Delta_{\ell=1/2} \approx1.069 Δ ℓ = 1/2 ≈ 1.069
Importantly, the pinning field can not be made too strong: when ∣ c ~ ∣ > 2.0 |\tilde{c}| > 2.0 ∣ c ~ ∣ > 2.0 Z 2 \mathbb{Z}_2 Z 2 ϕ \phi ϕ Figure S6 . Across the range of c ~ \tilde{c} c ~ c ~ = 0 \tilde{c} = 0 c ~ = 0 Δ ϕ \Delta_{\phi} Δ ϕ Δ ψ \Delta_{\psi} Δ ψ Δ Φ ≈ 0.519 \Delta_{\Phi} \approx0.519 Δ Φ ≈ 0.519 [24 , 25 ] .
Figure S6. Evidence of spontaneously symmetry breaking on the remaining Z 2 \mathbb{Z}_2 Z 2 v = 1 / 2 v=1/2 v = 1/2 N = 3 N=3 N = 3 m 0 2 m_0^2 m 0 2 c ~ \tilde{c} c ~ m 0 2 m_0^2 m 0 2 Z 2 \mathbb{Z}_2 Z 2 ∣ c ~ ∣ ≲ 2 |\tilde{c}|\lesssim2 ∣ c ~ ∣ ≲ 2 Z 2 \mathbb{Z}_2 Z 2 ∣ c ~ ∣ ≳ 2 |\tilde{c}|\gtrsim2 ∣ c ~ ∣ ≳ 2 Z 2 \mathbb{Z}_2 Z 2
We have also studied the v = 1 / 4 v = 1/4 v = 1/4 v = 1 / 2 v = 1/2 v = 1/2
the measured scaling dimensions can be found in Table II.
While we do not fully understand the reasons for the drifting in the fitted scaling dimensions, there are two natural mechanisms: First, it is possible that the drifting is a crossover behavior caused by slow RG flows. Second, the close proximity to the SSB phase can also affect the behavior of the correlation function of the untwisted field.
Finally, we consider the reflection defect corresponding to g = 1 2 ⊕ ( − 1 ) g = 1_2 \oplus(-1) g = 1 2 ⊕ ( − 1 ) O ( 2 ) O(2) O ( 2 ) [23 ] . Hence this setup should be less prone from the influence of the proximate defect SSB phase.
Indeed, in this case the drifting of the scaling dimensions with the pinning field appears to be much weaker: For the twisted field ψ \psi ψ Δ ψ = 0.934 ± 0.036 \Delta_{\psi} = 0.934 \pm0.036 Δ ψ = 0.934 ± 0.036 c ~ = 0 \tilde{c} = 0 c ~ = 0 Δ ψ = 0.976 ± 0.066 \Delta_{\psi} = 0.976 \pm0.066 Δ ψ = 0.976 ± 0.066 c ~ ∗ = − 1.75 \tilde{c}_* = -1.75 c ~ ∗ = − 1.75 Figure 8 ). For the untwisted field ϕ \phi ϕ Δ ϕ = 0.472 ± 0.006 \Delta_{\phi} = 0.472 \pm0.006 Δ ϕ = 0.472 ± 0.006 c ~ = 0 \tilde{c} = 0 c ~ = 0 Δ ϕ = 0.411 ± 0.006 \Delta_{\phi} = 0.411 \pm0.006 Δ ϕ = 0.411 ± 0.006 c ~ ∗ = − 1.75 \tilde{c}_* = -1.75 c ~ ∗ = − 1.75 Figure 9 ). The discrepancy is larger for the untwisted field, which we believe is likely due to finite-size effects. For comparison, the large N N N γ = 1 / 3 \gamma= 1/3 γ = 1/3 v = 1 / 2 v = 1/2 v = 1/2 Δ ℓ = 0 ≈ 0.410 \Delta_{\ell=0} \approx0.410 Δ ℓ = 0 ≈ 0.410 Δ ℓ = 1 / 2 ≈ 1.008 \Delta_{\ell=1/2} \approx1.008 Δ ℓ = 1/2 ≈ 1.008
Figure 8. Determination of the scaling dimension of the twisted defect primary ψ ^ \hat{\psi} ψ ^ N = 3 N = 3 N = 3 c ~ = 0 \tilde{c} = 0 c ~ = 0 c ~ ∗ = − 1.75 \tilde{c}_* = -1.75 c ~ ∗ = − 1.75 Δ ψ = 0.934 ± 0.036 \Delta_{\psi} = 0.934 \pm0.036 Δ ψ = 0.934 ± 0.036 c ~ = 0 \tilde{c} = 0 c ~ = 0 Δ ψ = 0.976 ± 0.066 \Delta_{\psi} = 0.976 \pm0.066 Δ ψ = 0.976 ± 0.066 c ~ ∗ = − 1.75 \tilde{c}_* = -1.75 c ~ ∗ = − 1.75 S = 1.09 S = 1.09 S = 1.09 S = 1.71 S = 1.71 S = 1.71
Figure 9. Determination of the scaling dimension of the untwisted defect primary ϕ ^ \hat{\phi} ϕ ^ N = 3 N=3 N = 3 c ~ = 0 \tilde{c}=0 c ~ = 0 c ~ ∗ = − 1.75 \tilde{c}_{*}=-1.75 c ~ ∗ = − 1.75 Δ ϕ = 0.472 ± 0.006 \Delta_{\phi}=0.472\pm0.006 Δ ϕ = 0.472 ± 0.006 c ~ = 0 \tilde{c}=0 c ~ = 0 Δ ϕ = 0.411 ± 0.006 \Delta_{\phi}=0.411\pm0.006 Δ ϕ = 0.411 ± 0.006 c ~ ∗ = − 1.75 \tilde{c}_{*}=-1.75 c ~ ∗ = − 1.75 S = 2.47 S=2.47 S = 2.47 S = 3.20 S=3.20 S = 3.20
We summarize the defect scaling dimensions from the Monte Carlo simulations of the three types of monodromy defects in Table. II.
Table 2. Scaling dimensions of untwisted (ϕ \phi ϕ ψ \psi ψ N = 3 N=3 N = 3 N N N d = 3 d=3 d = 3 c ~ ∗ = − 1.8 , − 1.5 , − 1.75 \tilde{c}_{*}=-1.8,-1.5,-1.75 c ~ ∗ = − 1.8 , − 1.5 , − 1.75 N N N γ = 2 / 3 , 2 / 3 , 1 / 3 \gamma=2/3,2/3,1/3 γ = 2/3 , 2/3 , 1/3 Z 2 \mathbb{Z}_{2} Z 2 v = 1 / 2 v=1/2 v = 1/2 γ = 1 / 3 \gamma=1/3 γ = 1/3 N N N
Conclusions and Discussion
In this work, we have explored the possible DCFT FPs of partial-monodromy defects in O(N N N d d d ϵ \epsilon ϵ N N N ψ \psi ψ ϕ \phi ϕ d = 3 d=3 d = 3 N N N N = 3 N=3 N = 3 v v v γ \gamma γ
The drifting behavior of scaling dimensions seen in the Monte Carlo simulation for the rotational monodromy is peculiar. Considering this drift is caused by pinning of the untwisted field, a completely symmetric defect perturbation, one possible explanation is that this operator is generated in RG but the flow is extremely slow. If so, we would expect that increasing the pinning field strength on the lattice might drive us closer towards reaching the expected stable DCFT, but for sufficiently large ∣ c ~ ∣ |\tilde{c}| ∣ c ~ ∣ Z 2 \mathbb{Z}_{2} Z 2 N > 3 N>3 N > 3
The large N N N d = 3 d=3 d = 3 v < v c s t d , a l t ( γ ) v<v_{c}^{\mathrm{std,alt}}(\gamma) v < v c std , alt ( γ ) c ~ \tilde{c} c ~
We have focused on the existence of different possible partial-monodromy DCFT FPs and, in particular, their local properties such as defect scaling dimensions of the leading O ( N ) O(N) O ( N ) O ( N ) O(N) O ( N ) ∼ Φ 2 \sim\Phi^{2} ∼ Φ 2 d = 3 d=3 d = 3 N N N σ \sigma σ [26 ] . In the future, it would also be interesting to compute universal contributions to the free energy of these defects, such as the defect entropy and cusp anomalous dimension of monodromy line defects in d = 3 d=3 d = 3 [1 , 2 , 5 ] . We leave these investigations to future work. A complementary study of monodromy defects in three-dimensional O(N N N [27 ] .
Acknowledgments
MC acknowledges enlightening conversations with Yichul Choi, Max Metlitski, Zimo Sun, Yifan Wang and Xiao-Chuan Wu, and is grateful for Chunhao Guo and Zheng Yan for sharing unpublished results. MC would like to thank the hospitality from the Institute for Advanced Study where part of the work was done. JF acknowledges helpful discussion with Michael P. Zaletel. We thank Wei Zhu for informing us about upcoming related work. MC is supported by NSF grant DMR-2424315. GS and NMJ acknowledge support through the Yale Prize Postdoctoral Fellowship in Condensed Matter Theory. TW acknowledges support from the Harvard Quantum Initiative Fellowship.
JF acknowledges support by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-AC02-05-CH11231 (Theory of Materials program KC2301). This research uses the Lawrencium computational cluster provided by the Lawrence Berkeley National Laboratory (supported by the U.S. Department of Energy, Office of Basic Energy Sciences under Contract No. DE-AC02-05-CH11231). Generative AI tools have been used to assist coding, checking derivations and creating figures.
Note added: During the late stages of finalizing the manuscript, we became aware of related works [28 , 29 ] that study monodromy defects in O(N N N
* jiechao_feng@berkeley.edu
† nayan.myerson-jain@yale.edu
‡ shankar.ganesh@yale.edu
§ taige_wang@fas.harvard.edu
¶ m.cheng@yale.edu
Supplemental Material
Renormalization group calculation and additional details
We return to the Wilson-Fisher theory described in the main text. The action can be rewritten as
S = S Φ + S d e f e c t S = S_{\Phi} + S_{\mathrm{defect}} S = S Φ + S defect S Φ = ∫ d d x [ 1 2 ( ∂ Φ ) 2 + λ 4 ! ( Φ 2 ) 2 ] = ∫ d d x [ 1 2 ∣ ∂ ψ ∣ 2 + 1 2 ( ∂ ϕ ) 2 + λ 4 ! ( ϕ 2 ) 2 + λ 4 ! ( ∣ ψ ∣ 2 ) 2 + 2 λ 4 ! ∣ ψ ∣ 2 ϕ 2 ] S_{\Phi} = \int\mathrm{d}^{d}x \left[\frac{1}{2}(\partial\Phi)^2 + \frac{\lambda}{4!}(\Phi^2)^2\right] = \int\mathrm{d}^{d}x \left[\frac{1}{2}|\partial\psi|^2 + \frac{1}{2}(\partial\phi)^2 + \frac{\lambda}{4!}(\phi^2)^2 + \frac{\lambda}{4!}(|\psi|^2)^2 + \frac{2\lambda}{4!}|\psi|^2\phi^2\right] S Φ = ∫ d d x [ 2 1 ( ∂ Φ ) 2 + 4 ! λ ( Φ 2 ) 2 ] = ∫ d d x [ 2 1 ∣ ∂ ψ ∣ 2 + 2 1 ( ∂ ϕ ) 2 + 4 ! λ ( ϕ 2 ) 2 + 4 ! λ ( ∣ ψ ∣ 2 ) 2 + 4 ! 2 λ ∣ ψ ∣ 2 ϕ 2 ] S d e f e c t = ∫ d d − 2 x [ c 2 ϕ ^ ℓ = 0 2 + h 2 ∣ ψ ^ ℓ = v ∣ 2 ] . (27) S_{\mathrm{defect}} = \int\mathrm{d}^{d-2}x \left[\frac{c}{2}\hat{\phi}_{\ell=0}^{2} + \frac{h}{2}|\hat{\psi}_{\ell=v}|^2\right]. \tag*{(27)} S defect = ∫ d d − 2 x [ 2 c ϕ ^ ℓ = 0 2 + 2 h ∣ ψ ^ ℓ = v ∣ 2 ] . (27) The free bulk-to-bulk propagator in d = 4 − ϵ d=4-\epsilon d = 4 − ϵ [14 , 15 ]
G ψ ( x ⃗ − y ⃗ , r x , r y , θ ) = ∑ ℓ ∈ Z + v G ℓ ψ ( x ⃗ − y ⃗ , r x , r y ) e i ℓ θ . (28) G^{\psi}(\vec{x}-\vec{y},r_x,r_y,\theta)=\sum_{\ell\in\mathbb{Z}+v}G_{\ell}^{\psi}(\vec{x}-\vec{y},r_x,r_y)e^{i\ell\theta}. \tag*{(28)} G ψ ( x − y , r x , r y , θ ) = ℓ ∈ Z + v ∑ G ℓ ψ ( x − y , r x , r y ) e i ℓ θ . (28) G ℓ ∈ Z + v ψ ( x ⃗ − y ⃗ , r x , r y ) = ⟨ ψ i , ℓ † ( x ⃗ , r x ) ψ i , ℓ ( y ⃗ , r y ) ⟩ = 1 2 π d / 2 ( r x r y ) Δ Γ ( Δ ℓ ) Γ ( Δ ℓ + 2 − d 2 ) 2 F 1 ( Δ ℓ , Δ ℓ + 3 − d 2 , 2 Δ ℓ + 3 − d , − ζ − 1 ) ( 4 ζ ) Δ ℓ . G_{\ell\in\mathbb{Z}+v}^{\psi}(\vec{x}-\vec{y},r_x,r_y)=\langle\psi_{i,\ell}^{\dagger}(\vec{x},r_x)\psi_{i,\ell}(\vec{y},r_y)\rangle=\frac{1}{2\pi^{d/2}(r_xr_y)^{\Delta}}\frac{\Gamma(\Delta_{\ell})}{\Gamma(\Delta_{\ell}+2-\frac{d}{2})}\frac{{}_2F_1(\Delta_{\ell},\Delta_{\ell}+\frac{3-d}{2},2\Delta_{\ell}+3-d,-\zeta^{-1})}{(4\zeta)^{\Delta_{\ell}}}. G ℓ ∈ Z + v ψ ( x − y , r x , r y ) = ⟨ ψ i , ℓ † ( x , r x ) ψ i , ℓ ( y , r y )⟩ = 2 π d /2 ( r x r y ) Δ 1 Γ ( Δ ℓ + 2 − 2 d ) Γ ( Δ ℓ ) ( 4 ζ ) Δ ℓ 2 F 1 ( Δ ℓ , Δ ℓ + 2 3 − d , 2 Δ ℓ + 3 − d , − ζ − 1 ) . The propagator for the untwisted fields we denote as G ℓ ϕ ( x ⃗ − y ⃗ , r x , r y ) G_{\ell}^{\phi}(\vec{x}-\vec{y},r_x,r_y) G ℓ ϕ ( x − y , r x , r y ) ℓ ∈ Z \ell\in\mathbb{Z} ℓ ∈ Z ζ = ( ( ∣ x ⃗ − y ⃗ ∣ 2 + ( r x − r y ) 2 ) / 4 r x r y ) \zeta=((|\vec{x}-\vec{y}|^2+(r_x-r_y)^2)/4r_xr_y) ζ = (( ∣ x − y ∣ 2 + ( r x − r y ) 2 ) /4 r x r y ) ζ → 0 \zeta\to0 ζ → 0
G ψ ( x ⃗ − x ⃗ , r x , r x , 0 ) = ⟨ ∣ ψ i ( x ⃗ , r x , θ ) ∣ 2 ⟩ = C ψ ± ( ϵ ) r x 2 Δ G^{\psi}(\vec{x}-\vec{x},r_x,r_x,0)=\langle|\psi_i(\vec{x},r_x,\theta)|^2\rangle=\frac{C_{\psi}^{\pm}(\epsilon)}{r_x^{2\Delta}} G ψ ( x − x , r x , r x , 0 ) = ⟨ ∣ ψ i ( x , r x , θ ) ∣ 2 ⟩ = r x 2Δ C ψ ± ( ϵ ) and
C ψ ± ( 0 ) = v ( v ∓ 1 ) 4 π 2 . (29) C_{\psi}^{\pm}(0)=\frac{v(v\mp1)}{4\pi^2}. \tag*{(29)} C ψ ± ( 0 ) = 4 π 2 v ( v ∓ 1 ) . (29) with C ψ + C_{\psi}^{+} C ψ + C ψ − C_{\psi}^{-} C ψ − ψ ℓ = v \psi_{\ell=v} ψ ℓ = v
⟨ ϕ ^ i , ℓ † ( x ⃗ ) ϕ i ( y ⃗ , r y , θ ) ⟩ = e i ℓ θ K ℓ ϕ ( x ⃗ − y ⃗ , r y ) \langle\hat{\phi}_{i,\ell}^{\dagger}(\vec{x})\phi_i(\vec{y},r_y,\theta)\rangle=e^{i\ell\theta}K_{\ell}^{\phi}(\vec{x}-\vec{y},r_y) ⟨ ϕ ^ i , ℓ † ( x ) ϕ i ( y , r y , θ )⟩ = e i ℓ θ K ℓ ϕ ( x − y , r y ) with
K ℓ ϕ ( x ⃗ − y ⃗ , r y ) = lim r x → 0 r x Δ − Δ ℓ G ℓ ϕ ( x ⃗ − y ⃗ , r x , r y ) = 1 4 π d / 2 r y Δ Γ ( Δ ℓ ) Γ ( Δ ℓ + 2 − d 2 ) ( r y ∣ x ⃗ − y ⃗ ∣ 2 + r y 2 ) Δ ℓ . (30) K_{\ell}^{\phi}(\vec{x}-\vec{y},r_y)=\lim_{r_x\to0}r_x^{\Delta-\Delta_{\ell}}G_{\ell}^{\phi}(\vec{x}-\vec{y},r_x,r_y)=\frac{1}{4\pi^{d/2}r_y^{\Delta}}\frac{\Gamma(\Delta_{\ell})}{\Gamma(\Delta_{\ell}+2-\frac{d}{2})}\left(\frac{r_y}{|\vec{x}-\vec{y}|^2+r_y^2}\right)^{\Delta_{\ell}}. \tag*{(30)} K ℓ ϕ ( x − y , r y ) = r x → 0 lim r x Δ − Δ ℓ G ℓ ϕ ( x − y , r x , r y ) = 4 π d /2 r y Δ 1 Γ ( Δ ℓ + 2 − 2 d ) Γ ( Δ ℓ ) ( ∣ x − y ∣ 2 + r y 2 r y ) Δ ℓ . (30) Likewise the defect-to-defect propagator for the untwisted fields is given by G ^ ℓ ϕ ( x ⃗ − y ⃗ ) = lim r y → 0 r y Δ − Δ ℓ K ℓ ϕ ( x ⃗ − y ⃗ , r y ) \hat{G}_{\ell}^{\phi}(\vec{x}-\vec{y})=\lim_{r_y\to0}r_y^{\Delta-\Delta_{\ell}}K_{\ell}^{\phi}(\vec{x}-\vec{y},r_y) G ^ ℓ ϕ ( x − y ) = lim r y → 0 r y Δ − Δ ℓ K ℓ ϕ ( x − y , r y )
G ^ ℓ ϕ ( x ⃗ − y ⃗ ) = 1 4 π d / 2 Γ ( Δ ℓ ) Γ ( Δ ℓ + 2 − d 2 ) 1 ∣ x ⃗ − y ⃗ ∣ 2 Δ ℓ . (31) \hat{G}_{\ell}^{\phi}(\vec{x}-\vec{y})=\frac{1}{4\pi^{d/2}}\frac{\Gamma(\Delta_{\ell})}{\Gamma(\Delta_{\ell}+2-\frac{d}{2})}\frac{1}{|\vec{x}-\vec{y}|^{2\Delta_{\ell}}}. \tag*{(31)} G ^ ℓ ϕ ( x − y ) = 4 π d /2 1 Γ ( Δ ℓ + 2 − 2 d ) Γ ( Δ ℓ ) ∣ x − y ∣ 2 Δ ℓ 1 . (31) The bulk-to-defect and defect-to-defect propagators for the twisted fields take the same functional form but with twice the normalization and the appropriately shifted spin.
We renormalize the Wilson-Fisher theory by rewriting bare quantities in terms of renormalized quantities and counterterms as ϕ 0 = Z ϕ ϕ \phi_0=\sqrt{Z_{\phi}}\phi ϕ 0 = Z ϕ ϕ λ 0 = μ ϵ ( λ + δ λ ) / Z ϕ 2 \lambda_0=\mu^{\epsilon}(\lambda+\delta\lambda)/Z_{\phi}^2 λ 0 = μ ϵ ( λ + δ λ ) / Z ϕ 2 c 0 = μ ξ ( c + δ c ) / Z ϕ c_0=\mu^{\xi}(c+\delta c)/Z_{\phi} c 0 = μ ξ ( c + δ c ) / Z ϕ h 0 = μ ξ − 2 v ( h + δ h ) / Z ϕ h_0=\mu^{\xi-2v}(h+\delta h)/Z_{\phi} h 0 = μ ξ − 2 v ( h + δ h ) / Z ϕ ξ \xi ξ Δ ℓ = 0 = d 2 − 1 − ξ 2 \Delta_{\ell=0}=\frac{d}{2}-1-\frac{\xi}{2} Δ ℓ = 0 = 2 d − 1 − 2 ξ Δ ℓ = v = d 2 − 1 + v − ξ 2 \Delta_{\ell=v}=\frac{d}{2}-1+v-\frac{\xi}{2} Δ ℓ = v = 2 d − 1 + v − 2 ξ ϕ \phi ϕ ψ \psi ψ Z ϕ = 1 Z_{\phi}=1 Z ϕ = 1 v ≪ 1 v\ll1 v ≪ 1 ϵ ≪ 1 \epsilon\ll1 ϵ ≪ 1 ψ ℓ = v \psi_{\ell=v} ψ ℓ = v C ψ + ( 0 ) = − v 4 π 2 C_{\psi}^{+}(0)=-\frac{v}{4\pi^2} C ψ + ( 0 ) = − 4 π 2 v
The β \beta β c c c h h h δ c \delta c δ c Figure S1 . The form of the correction to the bulk-to-bulk propagators from the counterterms are
Figure S1. UV divergent diagrams for the bulk-to-bulk propagators G ℓ = 0 ϕ ( x ⃗ − y ⃗ , r x , r y ) G^{\phi}_{\ell=0}(\vec{x}-\vec{y},r_x,r_y) G ℓ = 0 ϕ ( x − y , r x , r y ) G ℓ = ν ψ ( x ⃗ − y ⃗ , r x , r y ) G^{\psi}_{\ell=\nu}(\vec{x}-\vec{y},r_x,r_y) G ℓ = ν ψ ( x − y , r x , r y ) c , h c,h c , h λ \lambda λ c c c h h h
δ c . t . G ℓ = 0 ϕ ( x ⃗ − y ⃗ , r x , r y ) = − δ c ∫ d 2 z ⃗ K ℓ = 0 ϕ ( x ⃗ − z ⃗ , r x ) K ℓ = 0 ϕ ( z ⃗ − y ⃗ , r y ) , (32) \delta^{\mathrm{c.t.}}G^{\phi}_{\ell=0}(\vec{x}-\vec{y},r_x,r_y)=-\delta c\int\mathrm{d}^{2}\vec{z}\,K^{\phi}_{\ell=0}(\vec{x}-\vec{z},r_x)K^{\phi}_{\ell=0}(\vec{z}-\vec{y},r_y), \tag*{(32)} δ c.t. G ℓ = 0 ϕ ( x − y , r x , r y ) = − δ c ∫ d 2 z K ℓ = 0 ϕ ( x − z , r x ) K ℓ = 0 ϕ ( z − y , r y ) , (32) δ c . t . G ℓ = ν ψ ( x ⃗ − y ⃗ , r x , r y ) = − δ h 2 ∫ d 2 z ⃗ K ℓ = ν ψ ( x ⃗ − z ⃗ , r x ) K ℓ = ν ψ ( z ⃗ − y ⃗ , r y ) . (33) \delta^{\mathrm{c.t.}}G^{\psi}_{\ell=\nu}(\vec{x}-\vec{y},r_x,r_y)=-\frac{\delta h}{2}\int\mathrm{d}^{2}\vec{z}\,K^{\psi}_{\ell=\nu}(\vec{x}-\vec{z},r_x)K^{\psi}_{\ell=\nu}(\vec{z}-\vec{y},r_y). \tag*{(33)} δ c.t. G ℓ = ν ψ ( x − y , r x , r y ) = − 2 δ h ∫ d 2 z K ℓ = ν ψ ( x − z , r x ) K ℓ = ν ψ ( z − y , r y ) . (33) In computing the UV divergent part of diagrams we evaluate any prefactors of integrals at ϵ = 0 \epsilon=0 ϵ = 0 ξ = 0 \xi=0 ξ = 0 ν = 0 \nu=0 ν = 0
The first diagram a ) a) a )
δ 1 G ℓ = 0 ϕ ( x ⃗ − y ⃗ , r x , r y ) = − M λ μ ϵ 2 × 3 ! ∫ d d z ⃗ G ℓ = 0 ϕ ( x ⃗ − z ⃗ , r x , r z ) G ψ ( 0 , r z , r z , 0 ) G ℓ = 0 ϕ ( z ⃗ − y ⃗ , r z , r y ) ∼ − π M λ C ϕ + ( 0 ) 6 ∫ d d − 2 z ⃗ d r z r z 2 Δ − 1 G ℓ = 0 ϕ ( x ⃗ − z ⃗ , r x , r z ) G ℓ = 0 ϕ ( z ⃗ − y ⃗ , r z , r y ) . \begin{aligned}
\delta^{1}G^{\phi}_{\ell=0}(\vec{x}-\vec{y},r_x,r_y)=-\frac{M\lambda\mu^\epsilon}{2\times3!}\int\mathrm{d}^{d}\vec{z}\,G^{\phi}_{\ell=0}(\vec{x}-\vec{z},r_x,r_z)G^{\psi}(0,r_z,r_z,0)G^{\phi}_{\ell=0}(\vec{z}-\vec{y},r_z,r_y) \\
&\sim-\frac{\pi M\lambda C^{+}_{\phi}(0)}{6}\int\frac{\mathrm{d}^{d-2}\vec{z}\,\mathrm{d}r_z}{r_z^{2\Delta-1}}G^{\phi}_{\ell=0}(\vec{x}-\vec{z},r_x,r_z)G^{\phi}_{\ell=0}(\vec{z}-\vec{y},r_z,r_y).
\end{aligned} δ 1 G ℓ = 0 ϕ ( x − y , r x , r y ) = − 2 × 3 ! M λ μ ϵ ∫ d d z G ℓ = 0 ϕ ( x − z , r x , r z ) G ψ ( 0 , r z , r z , 0 ) G ℓ = 0 ϕ ( z − y , r z , r y ) ∼ − 6 π M λ C ϕ + ( 0 ) ∫ r z 2Δ − 1 d d − 2 z d r z G ℓ = 0 ϕ ( x − z , r x , r z ) G ℓ = 0 ϕ ( z − y , r z , r y ) . As r z → 0 r_z\to0 r z → 0 G ℓ = 0 ϕ ( x ⃗ − z ⃗ , r x , r z ) → r z − ξ / 2 K ℓ = 0 ϕ ( x ⃗ − z ⃗ , r x ) G^{\phi}_{\ell=0}(\vec{x}-\vec{z},r_x,r_z)\to r_z^{-\xi/2}K^{\phi}_{\ell=0}(\vec{x}-\vec{z},r_x) G ℓ = 0 ϕ ( x − z , r x , r z ) → r z − ξ /2 K ℓ = 0 ϕ ( x − z , r x ) Δ = d 2 − 1 = 1 − ϵ 2 \Delta=\frac{d}{2}-1=1-\frac{\epsilon}{2} Δ = 2 d − 1 = 1 − 2 ϵ ϵ \epsilon ϵ ξ \xi ξ
δ 1 G ℓ = 0 ϕ ( x ⃗ − y ⃗ , r x , r y ) = − π M λ C ϕ + ( 0 ) 6 ( ϵ − ξ ) ∫ d 2 z ⃗ K ℓ = 0 ϕ ( x ⃗ − z ⃗ , r x ) K ℓ = 0 ϕ ( z ⃗ − y ⃗ , r y ) (34) \delta^{1}G^{\phi}_{\ell=0}(\vec{x}-\vec{y},r_x,r_y)=-\frac{\pi M\lambda C^{+}_{\phi}(0)}{6(\epsilon-\xi)}\int\mathrm{d}^{2}\vec{z}\,K^{\phi}_{\ell=0}(\vec{x}-\vec{z},r_x)K^{\phi}_{\ell=0}(\vec{z}-\vec{y},r_y) \tag*{(34)} δ 1 G ℓ = 0 ϕ ( x − y , r x , r y ) = − 6 ( ϵ − ξ ) π M λ C ϕ + ( 0 ) ∫ d 2 z K ℓ = 0 ϕ ( x − z , r x ) K ℓ = 0 ϕ ( z − y , r y ) (34) which is canceled by choosing counterterm δ 1 c = − π M λ C ϕ + ( 0 ) 6 ( ϵ − ξ ) = M λ 24 π ( ϵ − ξ ) \delta^{1}c=-\frac{\pi M\lambda C^{+}_{\phi}(0)}{6(\epsilon-\xi)}=\frac{M\lambda}{24\pi(\epsilon-\xi)} δ 1 c = − 6 ( ϵ − ξ ) π M λ C ϕ + ( 0 ) = 24 π ( ϵ − ξ ) M λ
The second diagram b ) b) b )
δ 2 G ℓ = 0 ϕ ( x ⃗ − y ⃗ , r x , r y ) = ( N − M + 2 ) λ c μ ϵ + ξ 6 ∫ d d − 2 z ⃗ 1 d d − 2 z ⃗ 2 G ℓ = 0 ϕ ( x ⃗ − z ⃗ 1 , r x , r z 1 ) G ℓ = 0 ϕ ( z ⃗ 1 − z ⃗ 2 , r z 1 ) 2 G ℓ = 0 ϕ ( z ⃗ 1 − y ⃗ , r z 1 , r y ) ∼ π ( N − M + 2 ) λ c 48 π 4 ∫ d 2 z ⃗ 1 d r z 1 r z 1 1 − ξ [ ∫ d d − 2 z ⃗ 2 ( ∣ z ⃗ 1 − z ⃗ 2 ∣ 2 + r z 1 2 ) d − 2 − ξ ] G ℓ = 0 ϕ ( x ⃗ − z ⃗ 1 , r x , r z 1 ) G ℓ = 0 ϕ ( z ⃗ 1 − y ⃗ , r z 1 , r y ) . \begin{aligned}
\delta^{2}G^{\phi}_{\ell=0}(\vec{x}-\vec{y},r_x,r_y)=\frac{(N-M+2)\lambda c\mu^{\epsilon+\xi}}{6}\int\mathrm{d}^{d-2}\vec{z}_1\mathrm{d}^{d-2}\vec{z}_2\,G^{\phi}_{\ell=0}(\vec{x}-\vec{z}_1,r_x,r_{z_1})G^{\phi}_{\ell=0}(\vec{z}_1-\vec{z}_2,r_{z_1})^2G^{\phi}_{\ell=0}(\vec{z}_1-\vec{y},r_{z_1},r_y) \\
&\sim\frac{\pi(N-M+2)\lambda c}{48\pi^4}\int\mathrm{d}^{2}\vec{z}_1\mathrm{d}r_{z_1}\,r_{z_1}^{1-\xi}\left[\int\frac{\mathrm{d}^{d-2}\vec{z}_2}{\left(|\vec{z}_1-\vec{z}_2|^2+r_{z_1}^2\right)^{d-2-\xi}}\right]G^{\phi}_{\ell=0}(\vec{x}-\vec{z}_1,r_x,r_{z_1})G^{\phi}_{\ell=0}(\vec{z}_1-\vec{y},r_{z_1},r_y).
\end{aligned} δ 2 G ℓ = 0 ϕ ( x − y , r x , r y ) = 6 ( N − M + 2 ) λ c μ ϵ + ξ ∫ d d − 2 z 1 d d − 2 z 2 G ℓ = 0 ϕ ( x − z 1 , r x , r z 1 ) G ℓ = 0 ϕ ( z 1 − z 2 , r z 1 ) 2 G ℓ = 0 ϕ ( z 1 − y , r z 1 , r y ) ∼ 48 π 4 π ( N − M + 2 ) λ c ∫ d 2 z 1 d r z 1 r z 1 1 − ξ [ ∫ ( ∣ z 1 − z 2 ∣ 2 + r z 1 2 ) d − 2 − ξ d d − 2 z 2 ] G ℓ = 0 ϕ ( x − z 1 , r x , r z 1 ) G ℓ = 0 ϕ ( z 1 − y , r z 1 , r y ) . The integral over z ⃗ 2 \vec{z}_2 z 2 ∫ d d − 2 z ⃗ 2 ( ∣ z ⃗ 1 − z ⃗ 2 ∣ 2 + r z 1 2 ) d − 2 − ξ = π d / 2 − 1 Γ ( d − 2 2 − ξ ) Γ ( d − 2 − ξ ) r z 1 d − 2 − 2 ξ \int\frac{\mathrm{d}^{d-2}\vec{z}_2}{\left(|\vec{z}_1-\vec{z}_2|^2+r_{z_1}^2\right)^{d-2-\xi}}=\frac{\pi^{d/2-1}\Gamma\left(\frac{d-2}{2}-\xi\right)}{\Gamma(d-2-\xi)r_{z_1}^{d-2-2\xi}} ∫ ( ∣ z 1 − z 2 ∣ 2 + r z 1 2 ) d − 2 − ξ d d − 2 z 2 = Γ ( d − 2 − ξ ) r z 1 d − 2 − 2 ξ π d /2 − 1 Γ ( 2 d − 2 − ξ )
δ 2 G ℓ = 0 ϕ ( x ⃗ − y ⃗ , r x , r y ) = ( N − M + 2 ) λ c 48 π 2 ∫ d d − 2 z ⃗ 1 d r z 1 r z 1 d − 3 − ξ G ℓ = 0 ϕ ( x ⃗ − z ⃗ 1 , r x , r z 1 ) G ℓ = 0 ϕ ( z ⃗ 1 − y ⃗ , r z 1 , r y ) . \delta^{2}G^{\phi}_{\ell=0}(\vec{x}-\vec{y},r_x,r_y)=\frac{(N-M+2)\lambda c}{48\pi^2}\int\frac{\mathrm{d}^{d-2}\vec{z}_1\mathrm{d}r_{z_1}}{r_{z_1}^{d-3-\xi}}G^{\phi}_{\ell=0}(\vec{x}-\vec{z}_1,r_x,r_{z_1})G^{\phi}_{\ell=0}(\vec{z}_1-\vec{y},r_{z_1},r_y). δ 2 G ℓ = 0 ϕ ( x − y , r x , r y ) = 48 π 2 ( N − M + 2 ) λ c ∫ r z 1 d − 3 − ξ d d − 2 z 1 d r z 1 G ℓ = 0 ϕ ( x − z 1 , r x , r z 1 ) G ℓ = 0 ϕ ( z 1 − y , r z 1 , r y ) . Again because as we approach the defect, G ℓ = 0 ϕ ( x ⃗ − z ⃗ , r x , r z ) → r z − ξ / 2 K ℓ = 0 ϕ ( x ⃗ − z ⃗ , r x ) G_{\ell=0}^{\phi}(\vec{x}-\vec{z},r_x,r_z)\to r_z^{-\xi/2}K_{\ell=0}^{\phi}(\vec{x}-\vec{z},r_x) G ℓ = 0 ϕ ( x − z , r x , r z ) → r z − ξ /2 K ℓ = 0 ϕ ( x − z , r x )
δ 2 G ℓ = 0 ϕ ( x ⃗ − y ⃗ , r x , r y ) = ( N − M + 2 ) λ c 48 π 2 ϵ ∫ d 2 z ⃗ 1 K ℓ = 0 ϕ ( x ⃗ − z ⃗ 1 , r x ) K ℓ = 0 ϕ ( z ⃗ 1 − y ⃗ , r y ) (35) \delta^{2}G_{\ell=0}^{\phi}(\vec{x}-\vec{y},r_x,r_y)=\frac{(N-M+2)\lambda c}{48\pi^{2}\epsilon}\int\mathrm{d}^{2}\vec{z}_1 K_{\ell=0}^{\phi}(\vec{x}-\vec{z}_1,r_x)K_{\ell=0}^{\phi}(\vec{z}_1-\vec{y},r_y) \tag*{(35)} δ 2 G ℓ = 0 ϕ ( x − y , r x , r y ) = 48 π 2 ϵ ( N − M + 2 ) λ c ∫ d 2 z 1 K ℓ = 0 ϕ ( x − z 1 , r x ) K ℓ = 0 ϕ ( z 1 − y , r y ) (35) which is canceled by choosing δ 2 c = λ c ( N − M + 2 ) 48 π 2 ϵ \delta^{2}c=\frac{\lambda c(N-M+2)}{48\pi^{2}\epsilon} δ 2 c = 48 π 2 ϵ λ c ( N − M + 2 )
We now compute the third diagram c). This correction is at leading order
δ 3 G ℓ = 0 ϕ ( x ⃗ − y ⃗ , r x , r y ) = c 2 μ 2 ξ ∫ d d − 2 z ⃗ 1 d d − 2 z ⃗ 2 K ℓ = 0 ϕ ( x ⃗ − z ⃗ 1 , r x ) G ^ ℓ = 0 ϕ ( z ⃗ 1 − z ⃗ 2 ) K ℓ = 0 ϕ ( z ⃗ 2 − y ⃗ , r y ) . \delta^{3}G_{\ell=0}^{\phi}(\vec{x}-\vec{y},r_x,r_y)=c^{2}\mu^{2\xi}\int\mathrm{d}^{d-2}\vec{z}_1\mathrm{d}^{d-2}\vec{z}_2 K_{\ell=0}^{\phi}(\vec{x}-\vec{z}_1,r_x)\widehat{G}_{\ell=0}^{\phi}(\vec{z}_1-\vec{z}_2)K_{\ell=0}^{\phi}(\vec{z}_2-\vec{y},r_y). δ 3 G ℓ = 0 ϕ ( x − y , r x , r y ) = c 2 μ 2 ξ ∫ d d − 2 z 1 d d − 2 z 2 K ℓ = 0 ϕ ( x − z 1 , r x ) G ℓ = 0 ϕ ( z 1 − z 2 ) K ℓ = 0 ϕ ( z 2 − y , r y ) . The integral over z ⃗ 2 \vec{z}_2 z 2 z ⃗ 2 → z ⃗ 2 + z ⃗ 1 \vec{z}_2\to\vec{z}_2+\vec{z}_1 z 2 → z 2 + z 1 z ⃗ 2 = 0 \vec{z}_2=0 z 2 = 0 ∫ d 2 z ⃗ 2 G ^ ℓ = 0 ϕ ( z ⃗ 2 ) = 1 4 π 2 ∫ d 2 z ⃗ 2 ∣ z ⃗ 2 ∣ − ( 2 − ξ ) = 1 2 π ξ \int\mathrm{d}^{2}\vec{z}_2\widehat{G}_{\ell=0}^{\phi}(\vec{z}_2)=\frac{1}{4\pi^{2}}\int\mathrm{d}^{2}\vec{z}_2|\vec{z}_2|^{-(2-\xi)}=\frac{1}{2\pi\xi} ∫ d 2 z 2 G ℓ = 0 ϕ ( z 2 ) = 4 π 2 1 ∫ d 2 z 2 ∣ z 2 ∣ − ( 2 − ξ ) = 2 π ξ 1
The final result is that the third diagram is set to be
δ 3 G ℓ = 0 ϕ ( x ⃗ − y ⃗ , r x , r y ) = c 2 2 π ξ ∫ d 2 z ⃗ 1 K ℓ = 0 ϕ ( x ⃗ − z ⃗ 1 , r x ) K ℓ = 0 ϕ ( z ⃗ 1 − y ⃗ , r y ) (36) \delta^{3}G_{\ell=0}^{\phi}(\vec{x}-\vec{y},r_x,r_y)=\frac{c^{2}}{2\pi\xi}\int\mathrm{d}^{2}\vec{z}_1 K_{\ell=0}^{\phi}(\vec{x}-\vec{z}_1,r_x)K_{\ell=0}^{\phi}(\vec{z}_1-\vec{y},r_y) \tag*{(36)} δ 3 G ℓ = 0 ϕ ( x − y , r x , r y ) = 2 π ξ c 2 ∫ d 2 z 1 K ℓ = 0 ϕ ( x − z 1 , r x ) K ℓ = 0 ϕ ( z 1 − y , r y ) (36) which is canceled by choosing counterterm δ 3 c = c 2 2 π ξ \delta^{3}c=\frac{c^{2}}{2\pi\xi} δ 3 c = 2 π ξ c 2
The fourth diagram d) contributes a correction after expanding finite prefactors accordingly
δ 4 G ℓ = 0 ϕ ( x ⃗ − y ⃗ , r x , r y ) = M λ h μ ϵ − 2 v + ξ 4 ! ∫ d d z ⃗ 1 d d − 2 z ⃗ 2 G ℓ = 0 ϕ ( x ⃗ − z ⃗ 1 , r x , r z 1 ) K ℓ = 0 ψ ( z ⃗ 1 − z ⃗ 2 , r z 1 ) 2 G ℓ = 0 ϕ ( z ⃗ 1 − y ⃗ , r z 1 , r y ) . ∼ π M λ h 48 π 4 ∫ d d − 2 z ⃗ 1 d r z 1 r z 1 2 v + 1 − ξ [ ∫ d d − 2 z ⃗ 2 ( ∣ z ⃗ 1 − z ⃗ 2 ∣ 2 + r z 1 2 ) d − 2 + 2 v − ξ ] G ℓ = 0 ϕ ( x ⃗ − z ⃗ 1 , r x , r z 1 ) G ℓ = 0 ϕ ( z ⃗ 1 − y ⃗ , r z 1 , r y ) . \begin{aligned}
\delta^{4}G_{\ell=0}^{\phi}(\vec{x}-\vec{y},r_x,r_y)=\frac{M\lambda h\mu^{\epsilon-2v+\xi}}{4!}\int\mathrm{d}^{d}\vec{z}_1\mathrm{d}^{d-2}\vec{z}_2 G_{\ell=0}^{\phi}(\vec{x}-\vec{z}_1,r_x,r_{z_1})K_{\ell=0}^{\psi}(\vec{z}_1-\vec{z}_2,r_{z_1})^{2}G_{\ell=0}^{\phi}(\vec{z}_1-\vec{y},r_{z_1},r_y). \\
&\sim\frac{\pi M\lambda h}{48\pi^{4}}\int\mathrm{d}^{d-2}\vec{z}_1\mathrm{d}r_{z_1}\,r_{z_1}^{2v+1-\xi}\left[\int\frac{\mathrm{d}^{d-2}\vec{z}_2}{(|\vec{z}_1-\vec{z}_2|^{2}+r_{z_1}^{2})^{d-2+2v-\xi}}\right]G_{\ell=0}^{\phi}(\vec{x}-\vec{z}_1,r_x,r_{z_1})G_{\ell=0}^{\phi}(\vec{z}_1-\vec{y},r_{z_1},r_y).
\end{aligned} δ 4 G ℓ = 0 ϕ ( x − y , r x , r y ) = 4 ! M λh μ ϵ − 2 v + ξ ∫ d d z 1 d d − 2 z 2 G ℓ = 0 ϕ ( x − z 1 , r x , r z 1 ) K ℓ = 0 ψ ( z 1 − z 2 , r z 1 ) 2 G ℓ = 0 ϕ ( z 1 − y , r z 1 , r y ) . ∼ 48 π 4 π M λh ∫ d d − 2 z 1 d r z 1 r z 1 2 v + 1 − ξ [ ∫ ( ∣ z 1 − z 2 ∣ 2 + r z 1 2 ) d − 2 + 2 v − ξ d d − 2 z 2 ] G ℓ = 0 ϕ ( x − z 1 , r x , r z 1 ) G ℓ = 0 ϕ ( z 1 − y , r z 1 , r y ) . This becomes the following
δ 4 G ℓ = 0 ϕ ( x ⃗ − y ⃗ , r x , r y ) = M λ h 48 π 2 ∫ d d − 2 z ⃗ 1 d r z 1 r z 1 d − 3 + 2 v − ξ G ℓ = 0 ϕ ( x ⃗ − z ⃗ 1 , r x , r z 1 ) G ℓ = 0 ϕ ( z ⃗ 1 − y ⃗ , r z 1 , r y ) . \delta^{4}G_{\ell=0}^{\phi}(\vec{x}-\vec{y},r_x,r_y)=\frac{M\lambda h}{48\pi^{2}}\int\frac{\mathrm{d}^{d-2}\vec{z}_1\,\mathrm{d}r_{z_1}}{r_{z_1}^{d-3+2v-\xi}}G_{\ell=0}^{\phi}(\vec{x}-\vec{z}_1,r_x,r_{z_1})G_{\ell=0}^{\phi}(\vec{z}_1-\vec{y},r_{z_1},r_y). δ 4 G ℓ = 0 ϕ ( x − y , r x , r y ) = 48 π 2 M λh ∫ r z 1 d − 3 + 2 v − ξ d d − 2 z 1 d r z 1 G ℓ = 0 ϕ ( x − z 1 , r x , r z 1 ) G ℓ = 0 ϕ ( z 1 − y , r z 1 , r y ) . The UV divergent part of this integral is therefore merely
δ 4 G ℓ = 0 ϕ ( x ⃗ − y ⃗ , r x , r y ) = M λ h 48 π 2 ( ϵ − 2 v ) ∫ d 2 z ⃗ 1 K ℓ = 0 ϕ ( x ⃗ − z ⃗ 1 , r x ) K ℓ = 0 ϕ ( z ⃗ 1 − y ⃗ , r y ) (37) \delta^{4}G_{\ell=0}^{\phi}(\vec{x}-\vec{y},r_x,r_y)=\frac{M\lambda h}{48\pi^{2}(\epsilon-2v)}\int\mathrm{d}^{2}\vec{z}_1 K_{\ell=0}^{\phi}(\vec{x}-\vec{z}_1,r_x)K_{\ell=0}^{\phi}(\vec{z}_1-\vec{y},r_y) \tag*{(37)} δ 4 G ℓ = 0 ϕ ( x − y , r x , r y ) = 48 π 2 ( ϵ − 2 v ) M λh ∫ d 2 z 1 K ℓ = 0 ϕ ( x − z 1 , r x ) K ℓ = 0 ϕ ( z 1 − y , r y ) (37) which is canceled by choosing counterterm δ 4 c = − M λ h 48 π 2 ( ϵ − 2 v ) \delta^{4}c=-\frac{M\lambda h}{48\pi^{2}(\epsilon-2v)} δ 4 c = − 48 π 2 ( ϵ − 2 v ) M λh
δ c = δ 1 c + δ 2 c + δ 3 c + δ 4 c = M κ λ 24 π ( ϵ − ξ ) + ( N − M + 2 ) λ c 48 π 2 ϵ + c 2 2 π ξ + M λ h 48 π 2 ( ϵ − 2 v ) . (38) \delta c=\delta^{1}c+\delta^{2}c+\delta^{3}c+\delta^{4}c=\frac{M\kappa\lambda}{24\pi(\epsilon-\xi)}+\frac{(N-M+2)\lambda c}{48\pi^{2}\epsilon}+\frac{c^{2}}{2\pi\xi}+\frac{M\lambda h}{48\pi^{2}(\epsilon-2v)}. \tag*{(38)} δ c = δ 1 c + δ 2 c + δ 3 c + δ 4 c = 24 π ( ϵ − ξ ) M κλ + 48 π 2 ϵ ( N − M + 2 ) λ c + 2 π ξ c 2 + 48 π 2 ( ϵ − 2 v ) M λh . (38) The diagrams that correct the bulk-to-bulk propagator G ℓ = v ψ G_{\ell=v}^{\psi} G ℓ = v ψ h h h Figure S1 . The evaluation of these diagrams, including the method of regularization, mirrors the calculation above but with different symmetry factors since the twisted field ψ \psi ψ
δ h = δ 1 h + δ 2 h + δ 3 h + δ 4 h = ( M + 2 ) κ λ 24 π ( ϵ + 2 v − ξ ) + ( M + 2 ) λ h 48 π 2 ϵ + h 2 2 π ( ξ − 2 v ) + ( N − M ) λ c 48 π 2 ( ϵ + 2 v ) . (39) \delta h=\delta^{1}h+\delta^{2}h+\delta^{3}h+\delta^{4}h=\frac{(M+2)\kappa\lambda}{24\pi(\epsilon+2v-\xi)}+\frac{(M+2)\lambda h}{48\pi^{2}\epsilon}+\frac{h^{2}}{2\pi(\xi-2v)}+\frac{(N-M)\lambda c}{48\pi^{2}(\epsilon+2v)}. \tag*{(39)} δ h = δ 1 h + δ 2 h + δ 3 h + δ 4 h = 24 π ( ϵ + 2 v − ξ ) ( M + 2 ) κλ + 48 π 2 ϵ ( M + 2 ) λh + 2 π ( ξ − 2 v ) h 2 + 48 π 2 ( ϵ + 2 v ) ( N − M ) λ c . (39) The counterterms Eq. (38) and Eq. (39) determine the β \beta β
d c d log μ = β c = c 2 2 π + M κ λ 24 π + ( N − M + 2 ) λ c 48 π 2 + M λ h 48 π 2 , \frac{\mathrm{d}c}{\mathrm{d}\log\mu}=\beta_c=\frac{c^{2}}{2\pi}+\frac{M\kappa\lambda}{24\pi}+\frac{(N-M+2)\lambda c}{48\pi^{2}}+\frac{M\lambda h}{48\pi^{2}}, d log μ d c = β c = 2 π c 2 + 24 π M κλ + 48 π 2 ( N − M + 2 ) λ c + 48 π 2 M λh , d h d log μ = β h = h 2 2 π + ( M + 2 ) κ λ 24 π + ( N − M ) λ c 48 π 2 + ( M + 2 ) λ h 48 π 2 + 2 v h . (40) \frac{\mathrm{d}h}{\mathrm{d}\log\mu}=\beta_h=\frac{h^{2}}{2\pi}+\frac{(M+2)\kappa\lambda}{24\pi}+\frac{(N-M)\lambda c}{48\pi^{2}}+\frac{(M+2)\lambda h}{48\pi^{2}}+2vh. \tag*{(40)} d log μ d h = β h = 2 π h 2 + 24 π ( M + 2 ) κλ + 48 π 2 ( N − M ) λ c + 48 π 2 ( M + 2 ) λh + 2 v h . (40) Figure S2. In our calculation, we have neglected some additional higher-order diagrams at 1-loop and at 2-loop. Examples of such diagrams that may lend to the renormalization of c c c v ≪ 1 v \ll1 v ≪ 1
Additionally, in the N → ∞ N \to\infty N → ∞ h ∗ ± = − 2 π v ∓ c ∗ 2 + ( 2 π v ) 2 h_\ast^\pm=-2\pi v\mp\sqrt{c_\ast^2+(2\pi v)^2} h ∗ ± = − 2 π v ∓ c ∗ 2 + ( 2 π v ) 2
c ∗ 2 2 π ϵ + ( 1 − γ ) sgn ( c ∗ ) ∣ c ∗ ∣ + γ [ sgn ( h ∗ + 2 π v ) c ∗ 2 + ( 2 π v ) 2 − 2 π v 2 ] = 0. (41) \frac{c_\ast^2}{2\pi\epsilon}+(1-\gamma)\operatorname{sgn}(c_\ast)|c_\ast|+\gamma\left[\operatorname{sgn}(h_\ast+2\pi v)\sqrt{c_\ast^2+(2\pi v)^2}-2\pi v^2\right]=0. \tag*{(41)} 2 π ϵ c ∗ 2 + ( 1 − γ ) sgn ( c ∗ ) ∣ c ∗ ∣ + γ [ sgn ( h ∗ + 2 π v ) c ∗ 2 + ( 2 π v ) 2 − 2 π v 2 ] = 0. (41) The fact that the defect anomalous dimensions for ϕ ^ ℓ = 0 \hat{\phi}_{\ell=0} ϕ ^ ℓ = 0 ψ ^ ℓ = v \hat{\psi}_{\ell=v} ψ ^ ℓ = v α ℓ = 0 = c ∗ 2 π \alpha_{\ell=0}=\frac{c_\ast}{2\pi} α ℓ = 0 = 2 π c ∗ α ℓ = v = h ∗ 2 π \alpha_{\ell=v}=\frac{h_\ast}{2\pi} α ℓ = v = 2 π h ∗ c ∗ c_\ast c ∗ d = 4 d=4 d = 4
δ G ^ ℓ = 0 ϕ ( x ⃗ − y ⃗ ) = − c ∗ lim r → 0 ∫ d 2 z ⃗ K ℓ = 0 ϕ ( x ⃗ − z ⃗ , r ) K ℓ = 0 ϕ ( z ⃗ − y ⃗ , r ) = − c ∗ ( 4 π 2 ) 2 lim r → 0 ∫ d 2 z ⃗ [ ∣ x ⃗ − z ⃗ ∣ 2 + r 2 ] [ ∣ z ⃗ − y ⃗ ∣ 2 + r 2 ] . \delta\hat{G}^{\phi}_{\ell=0}(\vec{x}-\vec{y})=-c_\ast\lim_{r\to0}\int d^2\vec{z}\,K^{\phi}_{\ell=0}(\vec{x}-\vec{z},r)K^{\phi}_{\ell=0}(\vec{z}-\vec{y},r)=-\frac{c_\ast}{(4\pi^2)^2}\lim_{r\to0}\int\frac{d^2\vec{z}}{\left[|\vec{x}-\vec{z}|^2+r^2\right]\left[|\vec{z}-\vec{y}|^2+r^2\right]}. δ G ^ ℓ = 0 ϕ ( x − y ) = − c ∗ r → 0 lim ∫ d 2 z K ℓ = 0 ϕ ( x − z , r ) K ℓ = 0 ϕ ( z − y , r ) = − ( 4 π 2 ) 2 c ∗ r → 0 lim ∫ [ ∣ x − z ∣ 2 + r 2 ] [ ∣ z − y ∣ 2 + r 2 ] d 2 z . This can be rewritten directly using Feynman parameters as follows and by shifting z ⃗ → z ⃗ + y ⃗ \vec{z}\to\vec{z}+\vec{y} z → z + y
∫ d 2 z ⃗ [ ∣ x ⃗ − z ⃗ ∣ 2 + r 2 ] [ ∣ z ⃗ − y ⃗ ∣ 2 + r 2 ] = π ∫ 0 1 d u u ( 1 − u ) ∣ x ⃗ − y ⃗ ∣ 2 + r 2 . \int\frac{d^2\vec{z}}{\left[|\vec{x}-\vec{z}|^2+r^2\right]\left[|\vec{z}-\vec{y}|^2+r^2\right]}=\pi\int_0^1\frac{du}{u(1-u)|\vec{x}-\vec{y}|^2+r^2}. ∫ [ ∣ x − z ∣ 2 + r 2 ] [ ∣ z − y ∣ 2 + r 2 ] d 2 z = π ∫ 0 1 u ( 1 − u ) ∣ x − y ∣ 2 + r 2 d u . Evaluating the integral over the Feynman parameter and taking the small r r r
δ G ^ ℓ = 0 ϕ ( x ⃗ − y ⃗ ) ∼ − c ∗ 2 π 1 4 π 2 log ( ∣ x ⃗ − y ⃗ ∣ 2 / r 2 ) ∣ x ⃗ − y ⃗ ∣ 2 ⟹ α ℓ = 0 = c ∗ 2 π . (42) \delta\hat{G}^{\phi}_{\ell=0}(\vec{x}-\vec{y})\sim-\frac{c_\ast}{2\pi}\frac{1}{4\pi^2}\frac{\log\left(|\vec{x}-\vec{y}|^2/r^2\right)}{|\vec{x}-\vec{y}|^2}\Longrightarrow\alpha_{\ell=0}=\frac{c_\ast}{2\pi}. \tag*{(42)} δ G ^ ℓ = 0 ϕ ( x − y ) ∼ − 2 π c ∗ 4 π 2 1 ∣ x − y ∣ 2 log ( ∣ x − y ∣ 2 / r 2 ) ⟹ α ℓ = 0 = 2 π c ∗ . (42) as in the free theory, G ^ ℓ = 0 ϕ ( x ⃗ − y ⃗ ) = 1 4 π 2 1 ∣ x ⃗ − y ⃗ ∣ 2 \hat{G}^{\phi}_{\ell=0}(\vec{x}-\vec{y})=\frac{1}{4\pi^2}\frac{1}{|\vec{x}-\vec{y}|^2} G ^ ℓ = 0 ϕ ( x − y ) = 4 π 2 1 ∣ x − y ∣ 2 1 α ℓ = v = h ∗ 2 π \alpha_{\ell=v}=\frac{h_\ast}{2\pi} α ℓ = v = 2 π h ∗ ϵ \epsilon ϵ v v v
Further discussion of monodromy defect fixed points
Defect fixed points in the 1 ≫ v ≫ ϵ 1\gg v\gg\epsilon 1 ≫ v ≫ ϵ Here, as was stated in the main text, β h = h 2 2 π + 2 v h \beta_h=\frac{h^2}{2\pi}+2vh β h = 2 π h 2 + 2 v h h ∗ − = 0 h_\ast^-=0 h ∗ − = 0 h ∗ + = − 4 π v h_\ast^+=-4\pi v h ∗ + = − 4 π v c ∗ ± < 0 c_\ast^\pm<0 c ∗ ± < 0 (8) ):
c ∗ ± , s t d = − π ( N − M + 2 N + 8 ϵ ± ( N − M + 2 N + 8 ϵ ) 2 − 4 M v ( 1 − v ) ϵ N + 8 ) = − π N − M + 2 N + 8 ϵ ( 1 ± 1 − κ κ c ) . (43) c_\ast^{\pm,\mathrm{std}}=-\pi\left(\frac{N-M+2}{N+8}\epsilon\pm\sqrt{\left(\frac{N-M+2}{N+8}\epsilon\right)^2-\frac{4Mv(1-v)\epsilon}{N+8}}\right)=-\pi\frac{N-M+2}{N+8}\epsilon\left(1\pm\sqrt{1-\frac{\kappa}{\kappa_c}}\right). \tag*{(43)} c ∗ ± , std = − π N + 8 N − M + 2 ϵ ± ( N + 8 N − M + 2 ϵ ) 2 − N + 8 4 M v ( 1 − v ) ϵ = − π N + 8 N − M + 2 ϵ ( 1 ± 1 − κ c κ ) . (43) with κ = v ( 1 − v ) \kappa=v(1-v) κ = v ( 1 − v ) κ c = ( N − M + 2 ) 2 ϵ / 4 M ( N + 8 ) \kappa_c=(N-M+2)^2\epsilon/4M(N+8) κ c = ( N − M + 2 ) 2 ϵ /4 M ( N + 8 ) ϕ ^ ℓ = 0 2 \hat{\phi}^2_{\ell=0} ϕ ^ ℓ = 0 2 β c \beta_c β c β c ′ ( c ∗ ± , s t d ) = c ∗ ± , s t d π + ( N − M + 2 ) λ 48 π 2 = ∓ N − M + 2 N + 8 ϵ 1 − κ / κ c \beta'_c(c_\ast^{\pm,\mathrm{std}})=\frac{c_\ast^{\pm,\mathrm{std}}}{\pi}+\frac{(N-M+2)\lambda}{48\pi^2}=\mp\frac{N-M+2}{N+8}\epsilon\sqrt{1-\kappa/\kappa_c} β c ′ ( c ∗ ± , std ) = π c ∗ ± , std + 48 π 2 ( N − M + 2 ) λ = ∓ N + 8 N − M + 2 ϵ 1 − κ / κ c c ∗ − , s t d c_\ast^{-,\mathrm{std}} c ∗ − , std c ∗ + , s t d c_\ast^{+,\mathrm{std}} c ∗ + , std
As discussed, the requirement that there are perturbatively accessible two fixed-points is κ < κ c \kappa<\kappa_c κ < κ c 1 ≫ v ≫ ϵ 1\gg v\gg\epsilon 1 ≫ v ≫ ϵ γ ≲ ϵ 4 κ ( N + 2 ) 2 N ( N + 8 ) ≪ 1 \gamma\lesssim\frac{\epsilon}{4\kappa}\frac{(N+2)^2}{N(N+8)}\ll1 γ ≲ 4 κ ϵ N ( N + 8 ) ( N + 2 ) 2 ≪ 1 γ c = ϵ ( N + 2 ) 2 N ( N + 8 ) \gamma_c=\epsilon\frac{(N+2)^2}{N(N+8)} γ c = ϵ N ( N + 8 ) ( N + 2 ) 2 γ < γ c \gamma<\gamma_c γ < γ c v v v γ > γ c \gamma>\gamma_c γ > γ c v v v N N N 1 ≫ v ≫ ϵ 1 \gg v \gg\epsilon 1 ≫ v ≫ ϵ N N N N > N c ≃ ϵ 2 + O ( 1 ) N > N_c \simeq\frac{\epsilon}{2} + \mathcal{O}(1) N > N c ≃ 2 ϵ + O ( 1 ) γ < γ c \gamma< \gamma_c γ < γ c
Instead, for h ∗ + = − 4 π v h_*^+ = -4\pi v h ∗ + = − 4 π v ψ ℓ = v \psi_{\ell=v} ψ ℓ = v
c ∗ ± , a l t . = − π ( N − M + 2 N + 8 ϵ ± ( N − M + 2 N + 8 ϵ ) 2 + 4 M v ( v + 1 ) ϵ N + 8 ) = − π N − M + 2 N + 8 ϵ ( 1 ± 1 + κ ′ ′ κ c ) (44) c_*^{\pm,\mathrm{alt.}}=-\pi\left(\frac{N-M+2}{N+8}\epsilon\pm\sqrt{\left(\frac{N-M+2}{N+8}\epsilon\right)^2+\frac{4Mv(v+1)\epsilon}{N+8}}\right)=-\pi\frac{N-M+2}{N+8}\epsilon\left(1\pm\sqrt{1+\frac{\kappa''}{\kappa_c}}\right) \tag*{(44)} c ∗ ± , alt. = − π N + 8 N − M + 2 ϵ ± ( N + 8 N − M + 2 ϵ ) 2 + N + 8 4 M v ( v + 1 ) ϵ = − π N + 8 N − M + 2 ϵ ( 1 ± 1 + κ c κ ′′ ) (44) with κ ′ ′ = v ( 1 + v ) \kappa''=v(1+v) κ ′′ = v ( 1 + v ) ϕ ℓ = 0 \phi_{\ell=0} ϕ ℓ = 0 c ∗ + , a l t . < 0 c_*^{+,\mathrm{alt.}}<0 c ∗ + , alt. < 0 standard boundary conditions as (c ∗ − , a l t . > 0 c_*^{-,\mathrm{alt.}}>0 c ∗ − , alt. > 0 0 < v < 1 / 2 0<v<1/2 0 < v < 1/2 0 < γ < 1 0<\gamma<1 0 < γ < 1 β \beta β β ′ ( c ∗ ± , a l t . ) = c ∗ ± , a l t . π + ( N − M + 2 ) λ 48 π 2 = ∓ N − M + 2 N + 8 ϵ 1 + κ ′ ′ / κ c \beta'(c_*^{\pm,\mathrm{alt.}})=\frac{c_*^{\pm,\mathrm{alt.}}}{\pi}+\frac{(N-M+2)\lambda}{48\pi^2}=\mp\frac{N-M+2}{N+8}\epsilon\sqrt{1+\kappa''/\kappa_c} β ′ ( c ∗ ± , alt. ) = π c ∗ ± , alt. + 48 π 2 ( N − M + 2 ) λ = ∓ N + 8 N − M + 2 ϵ 1 + κ ′′ / κ c c ∗ + , a l t . c_*^{+,\mathrm{alt.}} c ∗ + , alt. c ∗ − , a l t . c_*^{-,\mathrm{alt.}} c ∗ − , alt. h h h h ∗ + 2 π \frac{h_*^+}{2\pi} 2 π h ∗ + (9) with substitution κ → − κ ′ ′ \kappa\to-\kappa'' κ → − κ ′′
Defect fixed points in the v ≲ ϵ v\lesssim\epsilon v ≲ ϵ In this regime, we have commented on the behavior of the two fixed points when the twisted field has standard boundary conditions in the main text. For our two new fixed points when the twisted field instead has alternate boundary conditions, they of course also appear in this regime, and never annihilate for any choice of parameters v v v γ \gamma γ Figure S3 , we show a schematic picture of the general classification of all four fixed points in RG that we have discussed.
Figure S3. Landscape of monodromy defect fixed points. Figure a) is a schematic picture of all four possible perturbative fixed points classified by boundary condition for ( ϕ ℓ = 0 , ψ ℓ = v ) (\phi_{\ell=0},\psi_{\ell=v}) ( ϕ ℓ = 0 , ψ ℓ = v ) v ≲ ϵ v \lesssim\epsilon v ≲ ϵ v < v c a l t , s t d v < v_{c}^{\mathrm{alt,std}} v < v c alt , std ϕ ℓ = 0 \phi_{\ell=0} ϕ ℓ = 0 ψ ℓ = v \psi_{\ell=v} ψ ℓ = v
Calculation of defect scaling dimensions of higher-spin fields
In this section, we compute the scaling dimension of ϕ ^ ℓ ≠ 0 \hat{\phi}_{\ell\ne0} ϕ ^ ℓ = 0 ψ ^ ℓ ≠ v \hat{\psi}_{\ell\ne v} ψ ^ ℓ = v Figure S1 ): diagrams a , b , d a,b,d a , b , d e , f , h e,f,h e , f , h
We begin with the untwisted field. The first diagram that corrects the defect-to-defect propagator a ) a) a )
δ 1 G ^ ℓ ϕ ^ ( x ⃗ − y ⃗ ) = − 2 λ 4 ! ∫ d d z ⟨ ϕ ^ i , ℓ † ( x ⃗ ) ϕ ( z ) 2 ∣ ψ ( z ) ∣ 2 ϕ ^ i , ℓ ( y ⃗ ) ⟩ = − π M λ C ψ + ( ϵ ) 6 ∫ d d − 2 z ⃗ d r z r z 2 Δ − 1 K ℓ ϕ ( x ⃗ − z ⃗ , r z ) K ℓ ϕ ( z ⃗ − y ⃗ , r z ) = − π M λ C ψ + ( ϵ ) 6 ( Γ ( Δ ℓ ) ( 4 π d / 2 Γ ( Δ ℓ + 2 − d 2 ) ) ) 2 ∫ d d − 2 z ⃗ d r z r z 4 Δ − 1 r z 2 Δ ℓ [ ∣ x ⃗ − z ⃗ ∣ 2 + r z 2 ] Δ ℓ [ ∣ z ⃗ − y ⃗ ∣ 2 + r z 2 ] Δ ℓ . (45) \begin{aligned}
\delta^1 \hat{G}^{\hat{\phi}}_{\ell}(\vec{x}-\vec{y})=-\frac{2\lambda}{4!}\int d^d z\left\langle\hat{\phi}^{\dagger}_{i,\ell}(\vec{x})\phi(z)^2|\psi(z)|^2\hat{\phi}_{i,\ell}(\vec{y})\right\rangle\\
&=-\frac{\pi M\lambda C_{\psi}^{+}(\epsilon)}{6}\int\frac{d^{d-2}\vec{z}d r_z}{r_z^{2\Delta-1}}K_{\ell}^{\phi}(\vec{x}-\vec{z},r_z)K_{\ell}^{\phi}(\vec{z}-\vec{y},r_z) \\
&=-\frac{\pi M\lambda C_{\psi}^{+}(\epsilon)}{6}\left(\frac{\Gamma(\Delta_{\ell})}{(4\pi^{d/2}\Gamma(\Delta_{\ell}+2-\frac{d}{2}))}\right)^2\int\frac{d^{d-2}\vec{z}d r_z}{r_z^{4\Delta-1}}\frac{r_z^{2\Delta_{\ell}}}{[|\vec{x}-\vec{z}|^2+r_z^2]^{\Delta_{\ell}}[|\vec{z}-\vec{y}|^2+r_z^2]^{\Delta_{\ell}}}. \tag*{(45)}
\end{aligned} δ 1 G ^ ℓ ϕ ^ ( x − y ) = − 4 ! 2 λ ∫ d d z ⟨ ϕ ^ i , ℓ † ( x ) ϕ ( z ) 2 ∣ ψ ( z ) ∣ 2 ϕ ^ i , ℓ ( y ) ⟩ = − 6 π M λ C ψ + ( ϵ ) ∫ r z 2Δ − 1 d d − 2 z d r z K ℓ ϕ ( x − z , r z ) K ℓ ϕ ( z − y , r z ) = − 6 π M λ C ψ + ( ϵ ) ( ( 4 π d /2 Γ ( Δ ℓ + 2 − 2 d )) Γ ( Δ ℓ ) ) 2 ∫ r z 4Δ − 1 d d − 2 z d r z [ ∣ x − z ∣ 2 + r z 2 ] Δ ℓ [ ∣ z − y ∣ 2 + r z 2 ] Δ ℓ r z 2 Δ ℓ . (45) Computing this integral can be done by again using Feynman parameters. We rewrite
1 [ ∣ x ⃗ − z ⃗ ∣ 2 + r z 2 ] Δ ℓ [ ∣ z ⃗ − y ⃗ ∣ 2 + r z 2 ] Δ ℓ = Γ ( 2 Δ ℓ ) Γ ( Δ ℓ ) 2 ∫ 0 1 d u u Δ ℓ − 1 ( 1 − u ) Δ ℓ − 1 ( u [ ∣ x ⃗ − z ⃗ ∣ 2 + r z 2 ] + ( 1 − u ) [ ∣ z ⃗ − y ⃗ ∣ 2 + r z 2 ] ) 2 Δ ℓ . (46) \frac{1}{[|\vec{x}-\vec{z}|^2+r_z^2]^{\Delta_{\ell}}[|\vec{z}-\vec{y}|^2+r_z^2]^{\Delta_{\ell}}}
=\frac{\Gamma(2\Delta_{\ell})}{\Gamma(\Delta_{\ell})^2}\int_0^1du\frac{u^{\Delta_{\ell}-1}(1-u)^{\Delta_{\ell}-1}}{(u[|\vec{x}-\vec{z}|^2+r_z^2]+(1-u)[|\vec{z}-\vec{y}|^2+r_z^2])^{2\Delta_{\ell}}}. \tag*{(46)} [ ∣ x − z ∣ 2 + r z 2 ] Δ ℓ [ ∣ z − y ∣ 2 + r z 2 ] Δ ℓ 1 = Γ ( Δ ℓ ) 2 Γ ( 2 Δ ℓ ) ∫ 0 1 d u ( u [ ∣ x − z ∣ 2 + r z 2 ] + ( 1 − u ) [ ∣ z − y ∣ 2 + r z 2 ] ) 2 Δ ℓ u Δ ℓ − 1 ( 1 − u ) Δ ℓ − 1 . (46) We perform the integral over the coordinates along the defect first after an appropriate shift of variables. The integrals needed are
∫ d d − 2 z ⃗ ( ∣ z ⃗ ∣ 2 + u ( 1 − u ) ∣ s ⃗ ∣ 2 + r z 2 ) 2 Δ ℓ = π d / 2 − 1 Γ ( 2 Δ ℓ − Δ ) Γ ( 2 Δ ℓ ) 1 ( u ( 1 − u ) ∣ s ⃗ ∣ 2 + r z 2 ) 2 Δ ℓ − Δ . \int\frac{d^{d-2}\vec{z}}{(|\vec{z}|^2+u(1-u)|\vec{s}|^2+r_z^2)^{2\Delta_{\ell}}
=\frac{\pi^{d/2-1}\Gamma(2\Delta_{\ell}-\Delta)}{\Gamma(2\Delta_{\ell})}\frac{1}{(u(1-u)|\vec{s}|^2+r_z^2)^{2\Delta_{\ell}-\Delta}}.} ∫ ( ∣ z ∣ 2 + u ( 1 − u ) ∣ s ∣ 2 + r z 2 ) 2 Δ ℓ = Γ ( 2 Δ ℓ ) π d /2 − 1 Γ ( 2 Δ ℓ − Δ ) ( u ( 1 − u ) ∣ s ∣ 2 + r z 2 ) 2 Δ ℓ − Δ 1 . d d − 2 z ∫ d r z r z 2 Δ ℓ + 1 − 4 Δ ( u ( 1 − u ) ∣ s ⃗ ∣ 2 + r z 2 ) 2 Δ ℓ − Δ = Γ ( Δ ℓ − ϵ 2 ) Γ ( Δ ℓ − Δ + ϵ 2 ) 2 Γ ( 2 Δ ℓ − Δ ) 1 ( u ( 1 − u ) ∣ s ⃗ ∣ 2 ) Δ ℓ − ϵ / 2 . \int\frac{d r_z\,r_z^{2\Delta_{\ell}+1-4\Delta}}{(u(1-u)|\vec{s}|^2+r_z^2)^{2\Delta_{\ell}-\Delta}}
=\frac{\Gamma(\Delta_{\ell}-\frac{\epsilon}{2})\Gamma(\Delta_{\ell}-\Delta+\frac{\epsilon}{2})}{2\Gamma(2\Delta_{\ell}-\Delta)}\frac{1}{(u(1-u)|\vec{s}|^2)^{\Delta_{\ell}-\epsilon/2}}. ∫ ( u ( 1 − u ) ∣ s ∣ 2 + r z 2 ) 2 Δ ℓ − Δ d r z r z 2 Δ ℓ + 1 − 4Δ = 2Γ ( 2 Δ ℓ − Δ ) Γ ( Δ ℓ − 2 ϵ ) Γ ( Δ ℓ − Δ + 2 ϵ ) ( u ( 1 − u ) ∣ s ∣ 2 ) Δ ℓ − ϵ /2 1 . ∫ 0 1 d u u Δ ℓ − 1 ( 1 − u ) Δ ℓ − 1 [ u ( 1 − u ) ] Δ ℓ − ϵ / 2 = ∫ 0 1 d u [ u ( 1 − u ) ] 1 − ϵ / 2 = Γ ( ϵ 2 ) 2 Γ ( ϵ ) . \int_0^1du\frac{u^{\Delta_{\ell}-1}(1-u)^{\Delta_{\ell}-1}}{[u(1-u)]^{\Delta_{\ell}-\epsilon/2}}
=\int_0^1\frac{du}{[u(1-u)]^{1-\epsilon/2}}
=\frac{\Gamma(\frac{\epsilon}{2})^2}{\Gamma(\epsilon)}. ∫ 0 1 d u [ u ( 1 − u ) ] Δ ℓ − ϵ /2 u Δ ℓ − 1 ( 1 − u ) Δ ℓ − 1 = ∫ 0 1 [ u ( 1 − u ) ] 1 − ϵ /2 d u = Γ ( ϵ ) Γ ( 2 ϵ ) 2 . Putting everything together,
δ G ^ ℓ ϕ ^ ( x ⃗ − y ⃗ ) = − π Δ + 1 M λ C ψ + ( ϵ ) 12 ( 4 π d / 2 ) 2 Γ ( Δ ℓ − ϵ 2 ) Γ ( Δ ℓ − Δ + ϵ 2 ) Γ ( Δ ℓ + 2 − d 2 ) 2 Γ ( ϵ 2 ) 2 Γ ( ϵ ) e ϵ log ∣ s ⃗ ∣ ∣ s ⃗ ∣ 2 Δ ℓ . (47) \delta\hat{G}^{\hat{\phi}}_{\ell}(\vec{x}-\vec{y})
=-\frac{\pi^{\Delta+1}M\lambda C_{\psi}^{+}(\epsilon)}{12(4\pi^{d/2})^2}
\frac{\Gamma(\Delta_{\ell}-\frac{\epsilon}{2})\Gamma(\Delta_{\ell}-\Delta+\frac{\epsilon}{2})}{\Gamma(\Delta_{\ell}+2-\frac{d}{2})^2}
\frac{\Gamma(\frac{\epsilon}{2})^2}{\Gamma(\epsilon)}
\frac{e^{\epsilon\log|\vec{s}|}}{|\vec{s}|^{2\Delta_{\ell}}}. \tag*{(47)} δ G ^ ℓ ϕ ^ ( x − y ) = − 12 ( 4 π d /2 ) 2 π Δ + 1 M λ C ψ + ( ϵ ) Γ ( Δ ℓ + 2 − 2 d ) 2 Γ ( Δ ℓ − 2 ϵ ) Γ ( Δ ℓ − Δ + 2 ϵ ) Γ ( ϵ ) Γ ( 2 ϵ ) 2 ∣ s ∣ 2 Δ ℓ e ϵ l o g ∣ s ∣ . (47) Now we expand this in ϵ → 0 \epsilon\to0 ϵ → 0
δ 1 G ^ ℓ ϕ ^ ( x ⃗ − y ⃗ ) = − π 2 M λ C ψ + ( 0 ) 12 ( 4 π 2 ) 2 ∣ ℓ ∣ ( 4 ϵ − π 2 ϵ 6 + ⋯ ) ( 1 + ϵ log ∣ s ⃗ ∣ + ⋯ ) 1 ∣ s ⃗ ∣ 2 Δ ℓ = − M λ C ψ + ( 0 ) 96 π 2 ∣ ℓ ∣ log ∣ s ⃗ ∣ 2 ∣ s ⃗ ∣ 2 Δ ℓ + ⋯ (48) \begin{aligned}
\delta^1\hat{G}^{\hat{\phi}}_{\ell}(\vec{x}-\vec{y})
=-\frac{\pi^2M\lambda C_{\psi}^{+}(0)}{12(4\pi^2)^2|\ell|}
\left(\frac{4}{\epsilon}-\frac{\pi^2\epsilon}{6}+\cdots\right)(1+\epsilon\log|\vec{s}|+\cdots)\frac{1}{|\vec{s}|^{2\Delta_{\ell}}} \\
&=-\frac{M\lambda C_{\psi}^{+}(0)}{96\pi^2|\ell|}\frac{\log|\vec{s}|^2}{|\vec{s}|^{2\Delta_{\ell}}}+\cdots \tag*{(48)}
\end{aligned} δ 1 G ^ ℓ ϕ ^ ( x − y ) = − 12 ( 4 π 2 ) 2 ∣ ℓ ∣ π 2 M λ C ψ + ( 0 ) ( ϵ 4 − 6 π 2 ϵ + ⋯ ) ( 1 + ϵ log ∣ s ∣ + ⋯ ) ∣ s ∣ 2 Δ ℓ 1 = − 96 π 2 ∣ ℓ ∣ M λ C ψ + ( 0 ) ∣ s ∣ 2 Δ ℓ log ∣ s ∣ 2 + ⋯ (48) The next correction to the defect-to-defect propagator from b ) b) b )
δ 2 G ^ ℓ ϕ ^ ( x ⃗ − y ⃗ ) = λ c 2 × 4 ! ∫ d d z 1 d d − 2 z ⃗ 2 ⟨ ϕ ^ i , ℓ † ( x ⃗ ) ( ϕ ( z 1 ) 2 ) 2 ϕ ^ ℓ = 0 ( z ⃗ 2 ) 2 ϕ ^ i , ℓ ( y ⃗ ) ⟩ = ( N − M + 2 ) λ c 6 ∫ d d z 1 d d − 2 z ⃗ 2 K ℓ ϕ ( x ⃗ − z ⃗ 1 , r z 1 ) K ℓ = 0 ϕ ( z ⃗ 1 − z ⃗ 2 , r z 1 ) 2 K ℓ ϕ ( z ⃗ 1 − y ⃗ , r z 1 ) . (49) \begin{aligned}
\delta^2\hat{G}^{\hat{\phi}}_{\ell}(\vec{x}-\vec{y})
=\frac{\lambda c}{2\times4!}\int d^d z_1d^{d-2}\vec{z}_2\left\langle\hat{\phi}^{\dagger}_{i,\ell}(\vec{x})(\phi(z_1)^2)^2\hat{\phi}_{\ell=0}(\vec{z}_2)^2\hat{\phi}_{i,\ell}(\vec{y})\right\rangle\\
&=\frac{(N-M+2)\lambda c}{6}\int d^d z_1d^{d-2}\vec{z}_2 K_{\ell}^{\phi}(\vec{x}-\vec{z}_1,r_{z_1})K_{\ell=0}^{\phi}(\vec{z}_1-\vec{z}_2,r_{z_1})^2K_{\ell}^{\phi}(\vec{z}_1-\vec{y},r_{z_1}). \tag*{(49)}
\end{aligned} δ 2 G ^ ℓ ϕ ^ ( x − y ) = 2 × 4 ! λ c ∫ d d z 1 d d − 2 z 2 ⟨ ϕ ^ i , ℓ † ( x ) ( ϕ ( z 1 ) 2 ) 2 ϕ ^ ℓ = 0 ( z 2 ) 2 ϕ ^ i , ℓ ( y ) ⟩ = 6 ( N − M + 2 ) λ c ∫ d d z 1 d d − 2 z 2 K ℓ ϕ ( x − z 1 , r z 1 ) K ℓ = 0 ϕ ( z 1 − z 2 , r z 1 ) 2 K ℓ ϕ ( z 1 − y , r z 1 ) . (49) We perform the integral over z ⃗ 2 \vec{z}_2 z 2
δ 2 G ^ ℓ ϕ ^ ( x ⃗ − y ⃗ ) = ( N − M + 2 ) λ c log ∣ s ⃗ ∣ 2 768 π 5 ∣ ℓ ∣ 1 ∣ s ⃗ ∣ 2 Δ ℓ + ⋯ (50) \delta^2\hat{G}^{\hat{\phi}}_{\ell}(\vec{x}-\vec{y})
=\frac{(N-M+2)\lambda c\log|\vec{s}|^2}{768\pi^5|\ell|}\frac{1}{|\vec{s}|^{2\Delta_{\ell}}}+\cdots \tag*{(50)} δ 2 G ^ ℓ ϕ ^ ( x − y ) = 768 π 5 ∣ ℓ ∣ ( N − M + 2 ) λ c log ∣ s ∣ 2 ∣ s ∣ 2 Δ ℓ 1 + ⋯ (50) The third and final correction to the defect-to-defect propagator from d ) d) d )
δ 3 G ^ ℓ ϕ ^ ( x ⃗ − y ⃗ ) = λ h 4 ! ∫ d d z 1 d d − 2 z ⃗ 2 ⟨ ϕ ^ i , ℓ † ( x ⃗ ) ϕ ( z 1 ) 2 ∣ ψ ( z 1 ) ∣ 2 ψ ^ ℓ = v ( z ⃗ 2 ) 2 ϕ ^ i , ℓ ( y ⃗ ) ⟩ = M λ h 24 ∫ d d z 1 d d − 2 z ⃗ 2 K ℓ ϕ ( x ⃗ − z ⃗ 1 , r z 1 ) K ℓ = v ψ ( z ⃗ 1 − z ⃗ 2 , r z 1 ) 2 K ℓ ϕ ( z ⃗ 1 − y ⃗ , r z 1 ) . (51) \begin{aligned}
\delta^3\hat{G}^{\hat{\phi}}_{\ell}(\vec{x}-\vec{y})
=\frac{\lambda h}{4!}\int d^d z_1d^{d-2}\vec{z}_2\left\langle\hat{\phi}^{\dagger}_{i,\ell}(\vec{x})\phi(z_1)^2|\psi(z_1)|^2\hat{\psi}_{\ell=v}(\vec{z}_2)^2\hat{\phi}_{i,\ell}(\vec{y})\right\rangle\\
&=\frac{M\lambda h}{24}\int d^d z_1d^{d-2}\vec{z}_2 K_{\ell}^{\phi}(\vec{x}-\vec{z}_1,r_{z_1})K_{\ell=v}^{\psi}(\vec{z}_1-\vec{z}_2,r_{z_1})^2K_{\ell}^{\phi}(\vec{z}_1-\vec{y},r_{z_1}). \tag*{(51)}
\end{aligned} δ 3 G ^ ℓ ϕ ^ ( x − y ) = 4 ! λh ∫ d d z 1 d d − 2 z 2 ⟨ ϕ ^ i , ℓ † ( x ) ϕ ( z 1 ) 2 ∣ ψ ( z 1 ) ∣ 2 ψ ^ ℓ = v ( z 2 ) 2 ϕ ^ i , ℓ ( y ) ⟩ = 24 M λh ∫ d d z 1 d d − 2 z 2 K ℓ ϕ ( x − z 1 , r z 1 ) K ℓ = v ψ ( z 1 − z 2 , r z 1 ) 2 K ℓ ϕ ( z 1 − y , r z 1 ) . (51) We again perform the integral over z ⃗ 2 \vec{z}_2 z 2 v v v
δ 3 G ^ ϕ ( x ⃗ − y ⃗ ) = M λ h 768 π 5 ∣ ℓ ∣ log ∣ s ⃗ ∣ 2 ∣ s ⃗ ∣ 2 Δ ℓ + … (52) \delta^3 \hat{G}^{\phi}(\vec{x}-\vec{y})=\frac{M\lambda h}{768\pi^5|\ell|}\frac{\log|\vec{s}|^2}{|\vec{s}|^{2\Delta_\ell}}+\ldots \tag*{(52)} δ 3 G ^ ϕ ( x − y ) = 768 π 5 ∣ ℓ ∣ M λh ∣ s ∣ 2 Δ ℓ log ∣ s ∣ 2 + … (52) The free defect-to-defect propagator in d = 4 d=4 d = 4 G ^ ℓ ϕ ( x ⃗ − y ⃗ ) = 1 4 π 2 ∣ x ⃗ − y ⃗ ∣ 2 Δ ℓ \hat{G}^{\phi}_{\ell}(\vec{x}-\vec{y})=\frac{1}{4\pi^2|\vec{x}-\vec{y}|^{2\Delta_\ell}} G ^ ℓ ϕ ( x − y ) = 4 π 2 ∣ x − y ∣ 2 Δ ℓ 1
α ℓ = ( M λ C ϕ + ( 0 ) 24 ∣ ℓ ∣ − ( N − M + 2 ) λ c 192 π 3 ∣ ℓ ∣ − M λ h 192 π 3 ∣ ℓ ∣ ) = − M ϵ 2 ( N + 8 ) ∣ ℓ ∣ ( κ + ( N − M + 2 ) c ∗ M 1 2 π + h ∗ 2 π ) . (53) \alpha_\ell=\left(\frac{M\lambda\mathcal{C}^{+}_{\phi}(0)}{24|\ell|}-\frac{(N-M+2)\lambda_c}{192\pi^3|\ell|}-\frac{M\lambda h}{192\pi^3|\ell|}\right)=-\frac{M\epsilon}{2(N+8)|\ell|}\left(\kappa+\frac{(N-M+2)c_*}{M}\frac{1}{2\pi}+\frac{h_*}{2\pi}\right). \tag*{(53)} α ℓ = ( 24∣ ℓ ∣ M λ C ϕ + ( 0 ) − 192 π 3 ∣ ℓ ∣ ( N − M + 2 ) λ c − 192 π 3 ∣ ℓ ∣ M λh ) = − 2 ( N + 8 ) ∣ ℓ ∣ M ϵ ( κ + M ( N − M + 2 ) c ∗ 2 π 1 + 2 π h ∗ ) . (53) A similar calculation shows that for the anomalous dimension of the twisted fields, one instead finds from computing diagrams e , f , h ) e,f,h) e , f , h )
α ℓ = − ( M + 2 ) ϵ 2 ( N + 8 ) ∣ ℓ ∣ ( κ + N − M M + 2 c ∗ 2 π + h ∗ 2 π ) . (54) \alpha_\ell=-\frac{(M+2)\epsilon}{2(N+8)|\ell|}\left(\kappa+\frac{N-M}{M+2}\frac{c_*}{2\pi}+\frac{h_*}{2\pi}\right). \tag*{(54)} α ℓ = − 2 ( N + 8 ) ∣ ℓ ∣ ( M + 2 ) ϵ ( κ + M + 2 N − M 2 π c ∗ + 2 π h ∗ ) . (54) Note that in the N → ∞ N\to\infty N → ∞ h ∗ 2 π = − v ∓ ( c ∗ / 2 π ) 2 + v 2 \frac{h_*}{2\pi}=-v\mp\sqrt{(c_*/2\pi)^2+v^2} 2 π h ∗ = − v ∓ ( c ∗ /2 π ) 2 + v 2
α ℓ = − ϵ 2 ∣ ℓ ∣ [ ( 1 − γ ) c ∗ 2 π ± γ ( c ∗ / 2 π ) 2 + v 2 − γ v 2 ] = 1 2 ∣ ℓ ∣ ( c ∗ 2 π ) 2 \alpha_\ell=-\frac{\epsilon}{2|\ell|}\left[(1-\gamma)\frac{c_*}{2\pi}\pm\gamma\sqrt{(c_*/2\pi)^2+v^2}-\gamma v^2\right]=\frac{1}{2|\ell|}\left(\frac{c_*}{2\pi}\right)^2 α ℓ = − 2∣ ℓ ∣ ϵ [ ( 1 − γ ) 2 π c ∗ ± γ ( c ∗ /2 π ) 2 + v 2 − γ v 2 ] = 2∣ ℓ ∣ 1 ( 2 π c ∗ ) 2 by Eq. (41) . This is consistent with results found in large N N N σ ∗ = ( c ∗ / 2 π ) 2 \sigma_*=(c_*/2\pi)^2 σ ∗ = ( c ∗ /2 π ) 2 1 ≫ v ≫ ϵ 1\gg v\gg\epsilon 1 ≫ v ≫ ϵ N = M N=M N = M α ℓ = − N + 2 N + 8 κ ϵ 2 ∣ ℓ ∣ \alpha_\ell=-\frac{N+2}{N+8}\frac{\kappa\epsilon}{2|\ell|} α ℓ = − N + 8 N + 2 2∣ ℓ ∣ κ ϵ h ∗ = 0 h_*=0 h ∗ = 0 α ℓ = N + 2 N + 8 κ ′ ′ ϵ 2 ∣ ℓ ∣ \alpha_\ell=\frac{N+2}{N+8}\frac{\kappa''\epsilon}{2|\ell|} α ℓ = N + 8 N + 2 2∣ ℓ ∣ κ ′′ ϵ h ∗ = − 4 π v h_*=-4\pi v h ∗ = − 4 π v [14 , 15 ] .
AdS/CFT and boundary conditions
Here we will review the AdS/CFT techniques pertinent to the discussion of the large N N N
Any DCFT with dimension p < d p<d p < d [30 , 31 ] . This can be seen directly by choosing spherical coordinates for the ( d − p ) (d-p) ( d − p )
d s 2 = d x ⃗ p 2 + d r 2 + r 2 d Ω d − p − 1 2 = r 2 ( d x ⃗ p 2 + d r 2 r 2 + d Ω d − p − 1 2 ) . (55) ds^2=d\vec{x}_p^{\,2}+dr^2+r^2d\Omega_{d-p-1}^2=r^2\left(\frac{d\vec{x}_p^{\,2}+dr^2}{r^2}+d\Omega_{d-p-1}^2\right). \tag*{(55)} d s 2 = d x p 2 + d r 2 + r 2 d Ω d − p − 1 2 = r 2 ( r 2 d x p 2 + d r 2 + d Ω d − p − 1 2 ) . (55) Hence, up to a Weyl transformation that peels off the Weyl factor Ω w e y l = r − 1 \Omega_{\mathrm{weyl}}=r^{-1} Ω weyl = r − 1 S d − p − 1 × H p + 1 S^{d-p-1}\times H^{p+1} S d − p − 1 × H p + 1 d = D + 1 d=D+1 d = D + 1 S 1 × H D S^1\times H^D S 1 × H D r = 0 r=0 r = 0 O ^ \hat{O} O ^ Δ \Delta Δ O ^ → Ω − Δ O ^ = r Δ O ^ \hat{O}\to\Omega^{-\Delta}\hat{O}=r^\Delta\hat{O} O ^ → Ω − Δ O ^ = r Δ O ^
Consider the following example of a free real scalar on H D H^D H D d s 2 = d x ⃗ D − 1 2 + d r 2 r 2 ds^2=\frac{d\vec{x}_{D-1}^{\,2}+dr^2}{r^2} d s 2 = r 2 d x D − 1 2 + d r 2 ℓ 2 \ell^2 ℓ 2
S = 1 2 ∫ d D x g [ ( ∂ ϕ ) 2 − ( D − 1 ) 2 4 ϕ 2 + ℓ 2 ϕ 2 ] . (56) S=\frac{1}{2}\int d^D x\sqrt{g}\left[(\partial\phi)^2-\frac{(D-1)^2}{4}\phi^2+\ell^2\phi^2\right]. \tag*{(56)} S = 2 1 ∫ d D x g [ ( ∂ ϕ ) 2 − 4 ( D − 1 ) 2 ϕ 2 + ℓ 2 ϕ 2 ] . (56) When ℓ 2 = 0 \ell^2=0 ℓ 2 = 0
ϕ ( x ⃗ , r → 0 ) ∼ r D − 1 2 − ∣ ℓ ∣ ϕ ^ ( 0 ) ( x ⃗ ) + r D − 1 2 + ∣ ℓ ∣ ϕ ^ ( 1 ) ( x ⃗ ) + … (57) \phi(\vec{x},r\to0)\sim r^{\frac{D-1}{2}-|\ell|}\hat{\phi}^{(0)}(\vec{x})+r^{\frac{D-1}{2}+|\ell|}\hat{\phi}^{(1)}(\vec{x})+\ldots \tag*{(57)} ϕ ( x , r → 0 ) ∼ r 2 D − 1 − ∣ ℓ ∣ ϕ ^ ( 0 ) ( x ) + r 2 D − 1 + ∣ ℓ ∣ ϕ ^ ( 1 ) ( x ) + … (57) There are two possible choices of consistent boundary conditions corresponding to fixing one of the two leading asymptotic solutions: ϕ ^ ( 0 ) ( x ⃗ ) \hat{\phi}^{(0)}(\vec{x}) ϕ ^ ( 0 ) ( x ) ϕ ^ ( 1 ) ( x ⃗ ) \hat{\phi}^{(1)}(\vec{x}) ϕ ^ ( 1 ) ( x ) [32 ] . In the quantum theory,
Standard boundary condition: ϕ ( x ⃗ , r → 0 ) ∼ r D − 1 2 + ∣ ℓ ∣ ϕ ^ ( 1 ) ( x ⃗ ) \phi(\vec{x},r\to0)\sim r^{\frac{D-1}{2}+|\ell|}\hat{\phi}^{(1)}(\vec{x}) ϕ ( x , r → 0 ) ∼ r 2 D − 1 + ∣ ℓ ∣ ϕ ^ ( 1 ) ( x ) Δ ^ = D − 1 2 + ∣ ℓ ∣ \hat{\Delta}=\frac{D-1}{2}+|\ell| Δ ^ = 2 D − 1 + ∣ ℓ ∣
Alternate boundary condition: ϕ ( x ⃗ , r → 0 ) ∼ r D − 1 2 − ∣ ℓ ∣ ϕ ^ ( 0 ) ( x ⃗ ) \phi(\vec{x},r\to0)\sim r^{\frac{D-1}{2}-|\ell|}\hat{\phi}^{(0)}(\vec{x}) ϕ ( x , r → 0 ) ∼ r 2 D − 1 − ∣ ℓ ∣ ϕ ^ ( 0 ) ( x ) Δ ^ = D − 1 2 − ∣ ℓ ∣ \hat{\Delta}=\frac{D-1}{2}-|\ell| Δ ^ = 2 D − 1 − ∣ ℓ ∣
With standard boundary conditions, the leading boundary operator has dimension greater than the bulk scaling dimension while for alternate boundary conditions the leading boundary operator has dimension less than the bulk scaling dimension. We use this terminology throughout the paper to distinguish different monodromy defect fixed points. Note that these boundary conditions are degenerate when ℓ = 0 \ell=0 ℓ = 0 r − Δ r^{-\Delta} r − Δ
More details on the large N N N
Here we closely follow Ref. [15 ] . In the following we write d = D + 1 d=D+1 d = D + 1 S 1 × H D S^1\times\mathbb{H}^D S 1 × H D
S = 1 2 ∫ d D + 1 x g [ g μ ν ∂ μ Φ ∂ ν Φ − ( D − 1 ) 2 4 Φ 2 + λ N ( Φ 2 ) 2 ] (58) S=\frac{1}{2}\int d^{D+1}x\sqrt{g}\left[g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi-\frac{(D-1)^2}{4}\Phi^2+\frac{\lambda}{N}(\Phi^2)^2\right] \tag*{(58)} S = 2 1 ∫ d D + 1 x g [ g μν ∂ μ Φ ∂ ν Φ − 4 ( D − 1 ) 2 Φ 2 + N λ ( Φ 2 ) 2 ] (58) Introducing a Hubbard-Stratonovich field σ \sigma σ N N N
S = 1 2 ∫ d D + 1 x g [ g μ ν ∂ μ Φ ∂ ν Φ − ( D − 1 ) 2 4 Φ 2 + σ Φ 2 ] . (59) S=\frac{1}{2}\int d^{D+1}x\sqrt{g}\left[g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi-\frac{(D-1)^2}{4}\Phi^2+\sigma\Phi^2\right]. \tag*{(59)} S = 2 1 ∫ d D + 1 x g [ g μν ∂ μ Φ ∂ ν Φ − 4 ( D − 1 ) 2 Φ 2 + σ Φ 2 ] . (59) where an IR-irrelevant σ 2 \sigma^2 σ 2 [33 ] . Integrating out Φ \Phi Φ
Z = ∫ D σ exp [ − N 2 Tr ln ( − ∇ 2 + σ − ( D − 1 ) 2 4 ) ] . (60) Z=\int\mathcal{D}\sigma\exp\left[-\frac{N}{2}\operatorname{Tr}\ln\left(-\nabla^2+\sigma-\frac{(D-1)^2}{4}\right)\right]. \tag*{(60)} Z = ∫ D σ exp [ − 2 N Tr ln ( − ∇ 2 + σ − 4 ( D − 1 ) 2 ) ] . (60) At large N N N σ \sigma σ σ ∗ \sigma_\ast σ ∗ ⟨ σ ( x ) ⟩ = σ ∗ \langle\sigma(x)\rangle=\sigma_\ast ⟨ σ ( x )⟩ = σ ∗ σ ∗ / r Δ σ \sigma_\ast/r^{\Delta_\sigma} σ ∗ / r Δ σ Δ σ = 2 \Delta_\sigma=2 Δ σ = 2 N N N N N N
F N ∝ ( 1 − γ ) ∑ ℓ ∈ Z ∫ 0 ∞ d ν μ D ( ν ) log ( ν 2 + ℓ 2 + σ ∗ ) + γ ∑ ℓ ∈ Z + v ∫ 0 ∞ d ν μ D ( ν ) log ( ν 2 + ℓ 2 + σ ∗ ) , (61) \frac{F}{N}\propto(1-\gamma)\sum_{\ell\in\mathbb{Z}}\int_0^\infty d\nu\,\mu_D(\nu)\log\left(\nu^2+\ell^2+\sigma_\ast\right)+\gamma\sum_{\ell\in\mathbb{Z}+v}\int_0^\infty d\nu\,\mu_D(\nu)\log\left(\nu^2+\ell^2+\sigma_\ast\right), \tag*{(61)} N F ∝ ( 1 − γ ) ℓ ∈ Z ∑ ∫ 0 ∞ d ν μ D ( ν ) log ( ν 2 + ℓ 2 + σ ∗ ) + γ ℓ ∈ Z + v ∑ ∫ 0 ∞ d ν μ D ( ν ) log ( ν 2 + ℓ 2 + σ ∗ ) , (61) where μ D ( ν ) \mu_D(\nu) μ D ( ν ) H D \mathbb{H}^D H D ν \nu ν [15 ] to find
( 1 − γ ) T 0 ( σ ∗ ) + γ T v ( σ ∗ ) = 0 , (62) (1-\gamma)T_0(\sigma_\ast)+\gamma T_v(\sigma_\ast)=0, \tag*{(62)} ( 1 − γ ) T 0 ( σ ∗ ) + γ T v ( σ ∗ ) = 0 , (62) where we have defined
T η ( σ ) = ∑ ℓ ∈ Z + η Γ ( Δ ℓ ) Γ ( 2 − D + Δ ℓ ) , Δ ℓ = D − 1 2 + τ ℓ σ ∗ + ℓ 2 (63) T_\eta(\sigma)=\sum_{\ell\in\mathbb{Z}+\eta}\frac{\Gamma(\Delta_\ell)}{\Gamma(2-D+\Delta_\ell)},\qquad\Delta_\ell=\frac{D-1}{2}+\tau_\ell\sqrt{\sigma_\ast+\ell^2} \tag*{(63)} T η ( σ ) = ℓ ∈ Z + η ∑ Γ ( 2 − D + Δ ℓ ) Γ ( Δ ℓ ) , Δ ℓ = 2 D − 1 + τ ℓ σ ∗ + ℓ 2 (63) are the dimensions of defect operators of transverse spin ℓ \ell ℓ τ ℓ = ± 1 \tau_\ell=\pm1 τ ℓ = ± 1 ℓ \ell ℓ σ ∗ \sigma_\ast σ ∗ ℓ 2 \ell^2 ℓ 2 (56) .
Unitarity bounds restrict the set of spin ℓ \ell ℓ q q q SO ( d + 1 , 1 ) \operatorname{SO}(d+1,1) SO ( d + 1 , 1 ) SO ( d − q + 1 , 1 ) × SO ( q ) \operatorname{SO}(d-q+1,1)\times\operatorname{SO}(q) SO ( d − q + 1 , 1 ) × SO ( q ) SO ( d − q ) \operatorname{SO}(d-q) SO ( d − q ) Δ ℓ ≥ ( d − q − 2 ) / 2 \Delta_\ell\geq(d-q-2)/2 Δ ℓ ≥ ( d − q − 2 ) /2 ( d − q ) (d-q) ( d − q ) Δ ℓ = 0 \Delta_{\ell=0} Δ ℓ = 0 σ ∗ > 0 \sigma_\ast>0 σ ∗ > 0 γ = 1 \gamma=1 γ = 1 [15 ] . Moreover, positivity of σ ∗ \sigma_\ast σ ∗ d = 3 d=3 d = 3 ℓ = 0 \ell=0 ℓ = 0 ℓ = v \ell=v ℓ = v v ≤ 1 / 2 v\leq1/2 v ≤ 1/2
The sum in Eq. (63) is divergent and can be regularized by defining
R η ( σ ) = ∑ k = 0 ∞ [ f ( σ + ( k + η ) 2 ) − ( k + η ) D − 2 ] + ζ ( 2 − D , η ) , (64) R_\eta(\sigma)=\sum_{k=0}^{\infty}\left[f\left(\sqrt{\sigma+(k+\eta)^2}\right)-(k+\eta)^{D-2}\right]+\zeta(2-D,\eta), \tag*{(64)} R η ( σ ) = k = 0 ∑ ∞ [ f ( σ + ( k + η ) 2 ) − ( k + η ) D − 2 ] + ζ ( 2 − D , η ) , (64) where ζ ( s , η ) \zeta(s,\eta) ζ ( s , η )
f ( x ) = Γ ( D − 1 2 + x ) Γ ( 3 − D 2 + x ) = Γ ( 1 − ϵ / 2 + x ) Γ ( ϵ / 2 + x ) . (65) f(x)=\frac{\Gamma\left(\frac{D-1}{2}+x\right)}{\Gamma\left(\frac{3-D}{2}+x\right)}=\frac{\Gamma(1-\epsilon/2+x)}{\Gamma(\epsilon/2+x)}. \tag*{(65)} f ( x ) = Γ ( 2 3 − D + x ) Γ ( 2 D − 1 + x ) = Γ ( ϵ /2 + x ) Γ ( 1 − ϵ /2 + x ) . (65) Then we write the series in the gap equation as
T 0 ( σ ) = f ( τ 0 σ ) + 2 R 1 ( σ ) , T v ( σ ) = f ( τ v v 2 + σ ) + R 1 + v ( σ ) + R 1 − v ( σ ) , (66) \begin{aligned}
T_0(\sigma)&=f(\tau_0\sqrt{\sigma})+2R_1(\sigma),\\
T_v(\sigma)&=f(\tau_v\sqrt{v^2+\sigma})+R_{1+v}(\sigma)+R_{1-v}(\sigma), \tag*{(66)}
\end{aligned} T 0 ( σ ) T v ( σ ) = f ( τ 0 σ ) + 2 R 1 ( σ ) , = f ( τ v v 2 + σ ) + R 1 + v ( σ ) + R 1 − v ( σ ) , (66) where the smallest ℓ \ell ℓ τ v = 1 \tau_v=1 τ v = 1 T v ( σ ) = R v ( σ ) + R 1 − v ( σ ) T_v(\sigma)=R_v(\sigma)+R_{1-v}(\sigma) T v ( σ ) = R v ( σ ) + R 1 − v ( σ )
This dimensional regularization scheme guarantees that σ ∗ = 0 \sigma_\ast=0 σ ∗ = 0 v = 0 v=0 v = 0 d d d T 0 ( 0 ) = 0 T_0(0)=0 T 0 ( 0 ) = 0
f ( 0 ) = Γ ( D − 1 2 ) Γ ( 3 − D 2 ) . (67) f(0)=\frac{\Gamma\left(\frac{D-1}{2}\right)}{\Gamma\left(\frac{3-D}{2}\right)}. \tag*{(67)} f ( 0 ) = Γ ( 2 3 − D ) Γ ( 2 D − 1 ) . (67) Recall the regularized expression for R η ( 0 ) R_\eta(0) R η ( 0 )
R η ( 0 ) = ∑ n = 0 ∞ [ Γ ( n + η + a ) Γ ( n + η + b ) − ( n + η ) D − 2 ] + ζ ( 2 − D , η ) . (68) R_\eta(0)=\sum_{n=0}^{\infty}\left[\frac{\Gamma(n+\eta+a)}{\Gamma(n+\eta+b)}-(n+\eta)^{D-2}\right]+\zeta(2-D,\eta). \tag*{(68)} R η ( 0 ) = n = 0 ∑ ∞ [ Γ ( n + η + b ) Γ ( n + η + a ) − ( n + η ) D − 2 ] + ζ ( 2 − D , η ) . (68) Here a = D − 1 2 a=\frac{D-1}{2} a = 2 D − 1 b = 1 − a b=1-a b = 1 − a
R η ( 0 ) = Γ ( η + a ) ( b − a − 1 ) Γ ( η + b − 1 ) = Γ ( η + D − 1 2 ) ( 1 − D ) Γ ( η + 1 − D 2 ) . (69) R_\eta(0)=\frac{\Gamma(\eta+a)}{(b-a-1)\Gamma(\eta+b-1)}=\frac{\Gamma\left(\eta+\frac{D-1}{2}\right)}{(1-D)\Gamma\left(\eta+\frac{1-D}{2}\right)}. \tag*{(69)} R η ( 0 ) = ( b − a − 1 ) Γ ( η + b − 1 ) Γ ( η + a ) = ( 1 − D ) Γ ( η + 2 1 − D ) Γ ( η + 2 D − 1 ) . (69) Therefore
T 0 ( 0 ) = f ( 0 ) + 2 R 1 ( 0 ) = 2 ( D − 1 ) Γ ( 3 − D 2 ) [ D − 1 2 Γ ( D − 1 2 ) − Γ ( D + 1 2 ) ] = 0. (70) T_0(0)=f(0)+2R_1(0)=\frac{2}{(D-1)\Gamma\left(\frac{3-D}{2}\right)}\left[\frac{D-1}{2}\Gamma\left(\frac{D-1}{2}\right)-\Gamma\left(\frac{D+1}{2}\right)\right]=0. \tag*{(70)} T 0 ( 0 ) = f ( 0 ) + 2 R 1 ( 0 ) = ( D − 1 ) Γ ( 2 3 − D ) 2 [ 2 D − 1 Γ ( 2 D − 1 ) − Γ ( 2 D + 1 ) ] = 0. (70) In general, the regularized gap equation has to be solved numerically. One subtlety is that the gap equation can not be directly applied to d = 3 d=3 d = 3 d → 3 + d\to3^+ d → 3 + d − 3 d-3 d − 3 d = 3 d=3 d = 3 d = 3 + d=3^+ d = 3 +
Below we present several analytically solvable limits of the gap equation:
The ϵ = 4 − d ≪ 1 \epsilon=4-d\ll1 ϵ = 4 − d ≪ 1 σ ∗ ≪ 1 \sigma_\ast\ll1 σ ∗ ≪ 1
The γ ≪ 1 \gamma\ll1 γ ≪ 1 d d d d = 3 d=3 d = 3 v → 1 / 2 v\to1/2 v → 1/2 v c s t d , a l t ( γ ) v_c^{\mathrm{std,alt}}(\gamma) v c std , alt ( γ )
The v ≪ 1 v\ll1 v ≪ 1 d d d
Perturbative solutions in the ϵ = 4 − d ≪ 1 \epsilon= 4-d \ll1 ϵ = 4 − d ≪ 1
In this section, we show that perturbative solutions (σ ∗ ≪ 1 \sigma_* \ll1 σ ∗ ≪ 1 N N N ϵ \epsilon ϵ (66) and evaluate R v R_v R v R 1 − v R_{1-v} R 1 − v R 1 R_1 R 1 σ \sigma σ ℓ ≫ 1 \ell\gg1 ℓ ≫ 1 σ ≪ v \sigma\ll v σ ≪ v f ( v + t ) f(\sqrt{v+t}) f ( v + t ) t t t
R 1 ± v = ∑ ℓ = 0 ∞ f ( ( ℓ + 1 ± v ) 2 + σ ) = ζ ( ϵ − 1 , 1 ± v ) + ( σ 2 + ϵ 12 − ϵ σ 2 ) ζ ( ϵ + 1 , 1 ± v ) + … (71) \begin{aligned}
R_{1\pm v}=\sum_{\ell=0}^{\infty}f\left(\sqrt{(\ell+1\pm v)^2+\sigma}\right) \\
&=\zeta(\epsilon-1,1\pm v)+\left(\frac{\sigma}{2}+\frac{\epsilon}{12}-\frac{\epsilon\sigma}{2}\right)\zeta(\epsilon+1,1\pm v)+\ldots \tag*{(71)}
\end{aligned} R 1 ± v = ℓ = 0 ∑ ∞ f ( ( ℓ + 1 ± v ) 2 + σ ) = ζ ( ϵ − 1 , 1 ± v ) + ( 2 σ + 12 ϵ − 2 ϵ σ ) ζ ( ϵ + 1 , 1 ± v ) + … (71) where the ellipses indicate higher order terms in σ \sigma σ ϵ \epsilon ϵ R 1 R_1 R 1 v = 0 v=0 v = 0 (66) ,
T v ( σ ) = f ( τ v v 2 + σ ) + [ ζ ( ϵ − 1 , 1 + v ) + ζ ( ϵ − 1 , 1 − v ) ] + ( σ 2 + ϵ 12 ) [ ζ ( ϵ + 1 , 1 + v ) + ζ ( ϵ + 1 , 1 − v ) ] + O ( σ ϵ ) , ≃ τ v v 2 + σ − v 2 + σ ϵ + O ( σ , ϵ ) , (72) \begin{aligned}
T_v(\sigma)=f\left(\tau_v\sqrt{v^2+\sigma}\right)+\left[\zeta(\epsilon-1,1+v)+\zeta(\epsilon-1,1-v)\right]+\left(\frac{\sigma}{2}+\frac{\epsilon}{12}\right)\left[\zeta(\epsilon+1,1+v)+\zeta(\epsilon+1,1-v)\right]+O(\sigma\epsilon), \\
&\simeq\tau_v\sqrt{v^2+\sigma}-v^2+\frac{\sigma}{\epsilon}+O(\sigma,\epsilon), \tag*{(72)}
\end{aligned} T v ( σ ) = f ( τ v v 2 + σ ) + [ ζ ( ϵ − 1 , 1 + v ) + ζ ( ϵ − 1 , 1 − v ) ] + ( 2 σ + 12 ϵ ) [ ζ ( ϵ + 1 , 1 + v ) + ζ ( ϵ + 1 , 1 − v ) ] + O ( σ ϵ ) , ≃ τ v v 2 + σ − v 2 + ϵ σ + O ( σ , ϵ ) , (72) where the O ( ϵ , σ ) O(\epsilon,\sigma) O ( ϵ , σ ) O ( ϵ 2 , ϵ σ ) O(\epsilon^2,\epsilon\sigma) O ( ϵ 2 , ϵ σ ) ϵ \epsilon ϵ O ( ϵ ) O(\epsilon) O ( ϵ )
s 2 ϵ + ( 1 − γ ) τ 0 s + γ ( τ v v 2 + s 2 − v 2 ) = 0 , s ≡ σ . (73) \frac{s^2}{\epsilon}+(1-\gamma)\tau_0s+\gamma\left(\tau_v\sqrt{v^2+s^2}-v^2\right)=0,\qquad s\equiv\sqrt{\sigma}. \tag*{(73)} ϵ s 2 + ( 1 − γ ) τ 0 s + γ ( τ v v 2 + s 2 − v 2 ) = 0 , s ≡ σ . (73) As discussed earlier, unitarity requires σ ∗ > 0 \sigma_*>0 σ ∗ > 0 γ < 1 \gamma<1 γ < 1 s ≥ 0 s\geq0 s ≥ 0 ϵ \epsilon ϵ
s = ∣ c ∣ 2 π , τ 0 = sgn ( c ) , τ v = sgn ( h + 2 π v ) , s=\frac{|c|}{2\pi},\qquad\tau_0=\operatorname{sgn}(c),\qquad\tau_v=\operatorname{sgn}(h+2\pi v), s = 2 π ∣ c ∣ , τ 0 = sgn ( c ) , τ v = sgn ( h + 2 π v ) , in the large N N N (41) .
The ``strong-pinning'' (runaway) saddle points
Here we look for a solution of σ ∗ \sigma_* σ ∗ ℓ = 0 \ell=0 ℓ = 0 ℓ = v \ell=v ℓ = v ℓ \ell ℓ R η R_\eta R η
R η = ∑ ℓ = 0 ∞ [ f ( ( ℓ + η ) 2 + σ ) − ( ℓ + η ) 1 − ϵ − K ( ℓ + η ) − 1 − ϵ ] + ζ ( ϵ − 1 , η ) + K ζ ( ϵ + 1 , η ) . (74) R_\eta=\sum_{\ell=0}^{\infty}\left[f\left(\sqrt{(\ell+\eta)^2+\sigma}\right)-(\ell+\eta)^{1-\epsilon}-K(\ell+\eta)^{-1-\epsilon}\right]+\zeta(\epsilon-1,\eta)+K\zeta(\epsilon+1,\eta). \tag*{(74)} R η = ℓ = 0 ∑ ∞ [ f ( ( ℓ + η ) 2 + σ ) − ( ℓ + η ) 1 − ϵ − K ( ℓ + η ) − 1 − ϵ ] + ζ ( ϵ − 1 , η ) + K ζ ( ϵ + 1 , η ) . (74) Here
K = σ 2 + ϵ 12 − ϵ σ 2 + O ( ϵ 2 ) . (75) K=\frac{\sigma}{2}+\frac{\epsilon}{12}-\frac{\epsilon\sigma}{2}+O(\epsilon^2). \tag*{(75)} K = 2 σ + 12 ϵ − 2 ϵ σ + O ( ϵ 2 ) . (75) The sum over ℓ \ell ℓ ζ ( ϵ − 1 , η ) \zeta(\epsilon-1,\eta) ζ ( ϵ − 1 , η ) ϵ → 0 \epsilon\to0 ϵ → 0
ζ ( 1 + ϵ , η ) = 1 ϵ − ψ ( η ) + O ( ϵ ) , (76) \zeta(1+\epsilon,\eta)=\frac{1}{\epsilon}-\psi(\eta)+O(\epsilon), \tag*{(76)} ζ ( 1 + ϵ , η ) = ϵ 1 − ψ ( η ) + O ( ϵ ) , (76) where ψ ( η ) \psi(\eta) ψ ( η )
R η = ∑ ℓ = 0 ∞ [ ( ℓ + η ) 2 + σ − ( ℓ + η ) − σ 2 ( ℓ + η ) ] + ζ ( − 1 , η ) + ( σ 2 + ϵ 12 − ϵ σ 2 ) ( 1 ϵ − ψ ( η ) ) + O ( ϵ ) (77) R_\eta=\sum_{\ell=0}^{\infty}\left[\sqrt{(\ell+\eta)^2+\sigma}-(\ell+\eta)-\frac{\sigma}{2(\ell+\eta)}\right]+\zeta(-1,\eta)+\left(\frac{\sigma}{2}+\frac{\epsilon}{12}-\frac{\epsilon\sigma}{2}\right)\left(\frac{1}{\epsilon}-\psi(\eta)\right)+O(\epsilon) \tag*{(77)} R η = ℓ = 0 ∑ ∞ [ ( ℓ + η ) 2 + σ − ( ℓ + η ) − 2 ( ℓ + η ) σ ] + ζ ( − 1 , η ) + ( 2 σ + 12 ϵ − 2 ϵ σ ) ( ϵ 1 − ψ ( η ) ) + O ( ϵ ) (77) = σ 2 ϵ + r η ( σ ) + O ( ϵ ) . (78) =\frac{\sigma}{2\epsilon}+r_\eta(\sigma)+O(\epsilon). \tag*{(78)} = 2 ϵ σ + r η ( σ ) + O ( ϵ ) . (78) Solutions in the 1 limit
Here
r η ( σ ) = ∑ ℓ = 0 ∞ [ ( ℓ + η ) 2 + σ − ( ℓ + η ) − σ 2 ( ℓ + η ) ] + ζ ( − 1 , η ) + 1 12 − σ 2 − σ ψ ( η ) 2 . (79) r_\eta(\sigma)=\sum_{\ell=0}^{\infty}\left[\sqrt{(\ell+\eta)^2+\sigma}-(\ell+\eta)-\frac{\sigma}{2(\ell+\eta)}\right]+\zeta(-1,\eta)+\frac{1}{12}-\frac{\sigma}{2}-\frac{\sigma\psi(\eta)}{2}. \tag*{(79)} r η ( σ ) = ℓ = 0 ∑ ∞ [ ( ℓ + η ) 2 + σ − ( ℓ + η ) − 2 ( ℓ + η ) σ ] + ζ ( − 1 , η ) + 12 1 − 2 σ − 2 σ ψ ( η ) . (79) Therefore we have
T 0 ( σ ) = f ( τ 0 σ ) + σ ϵ + 2 r 1 ( σ ) + O ( ϵ ) = f ( τ 0 σ ) + σ ϵ + 2 r 1 ( σ ) + O ( ϵ ) , (80) T_0(\sigma)=f(\tau_0\sqrt{\sigma})+\frac{\sigma}{\epsilon}+2r_1(\sigma)+O(\epsilon)=f(\tau_0\sqrt{\sigma})+\frac{\sigma}{\epsilon}+2r_1(\sigma)+O(\epsilon), \tag*{(80)} T 0 ( σ ) = f ( τ 0 σ ) + ϵ σ + 2 r 1 ( σ ) + O ( ϵ ) = f ( τ 0 σ ) + ϵ σ + 2 r 1 ( σ ) + O ( ϵ ) , (80) T v ( σ ) = σ ϵ + f ( τ v σ + v 2 ) + r 1 + v ( σ ) + r 1 − v ( σ ) + O ( ϵ ) (81) T_v(\sigma)=\frac{\sigma}{\epsilon}+f(\tau_v\sqrt{\sigma+v^2})+r_{1+v}(\sigma)+r_{1-v}(\sigma)+O(\epsilon) \tag*{(81)} T v ( σ ) = ϵ σ + f ( τ v σ + v 2 ) + r 1 + v ( σ ) + r 1 − v ( σ ) + O ( ϵ ) (81) So the gap equation becomes
( 1 − γ ) f ( τ 0 σ ) + γ f ( τ v σ + v 2 ) + σ ϵ + 2 ( 1 − γ ) r 1 ( σ ) + γ [ r 1 + v ( σ ) + r 1 − v ( σ ) ] = 0. (82) (1-\gamma)f(\tau_0\sqrt{\sigma})+\gamma f(\tau_v\sqrt{\sigma+v^2})+\frac{\sigma}{\epsilon}+2(1-\gamma)r_1(\sigma)+\gamma[r_{1+v}(\sigma)+r_{1-v}(\sigma)]=0. \tag*{(82)} ( 1 − γ ) f ( τ 0 σ ) + γ f ( τ v σ + v 2 ) + ϵ σ + 2 ( 1 − γ ) r 1 ( σ ) + γ [ r 1 + v ( σ ) + r 1 − v ( σ )] = 0. (82) The last three terms are finite, so they can be neglected to the leading order in ϵ \epsilon ϵ ℓ = 0 \ell=0 ℓ = 0 ℓ = v \ell=v ℓ = v v = O ( ϵ ) v=O(\epsilon) v = O ( ϵ ) v v v Δ ℓ = 1 − ϵ 2 + τ ℓ σ + ℓ 2 \Delta_\ell=1-\frac{\epsilon}{2}+\tau_\ell\sqrt{\sigma+\ell^2} Δ ℓ = 1 − 2 ϵ + τ ℓ σ + ℓ 2 f ( τ ℓ σ + ℓ 2 ) f(\tau_\ell\sqrt{\sigma+\ell^2}) f ( τ ℓ σ + ℓ 2 ) Δ ℓ \Delta_\ell Δ ℓ O ( ϵ ) O(\epsilon) O ( ϵ )
f ( τ ℓ σ + ℓ 2 ) = Γ ( Δ ℓ ) Γ ( − 1 + ϵ + Δ ℓ ) = ( − 1 + ϵ + Δ ℓ ) Γ ( Δ ℓ ) Γ ( Δ ℓ + ϵ ) ≈ − Δ ℓ + ϵ Δ ℓ . (83) f(\tau_\ell\sqrt{\sigma+\ell^2})=\frac{\Gamma(\Delta_\ell)}{\Gamma(-1+\epsilon+\Delta_\ell)}=(-1+\epsilon+\Delta_\ell)\frac{\Gamma(\Delta_\ell)}{\Gamma(\Delta_\ell+\epsilon)}\approx-\frac{\Delta_\ell+\epsilon}{\Delta_\ell}. \tag*{(83)} f ( τ ℓ σ + ℓ 2 ) = Γ ( − 1 + ϵ + Δ ℓ ) Γ ( Δ ℓ ) = ( − 1 + ϵ + Δ ℓ ) Γ ( Δ ℓ + ϵ ) Γ ( Δ ℓ ) ≈ − Δ ℓ Δ ℓ + ϵ . (83) Now suppose the ℓ = 0 \ell=0 ℓ = 0 τ 0 = − 1 \tau_0=-1 τ 0 = − 1 Δ ℓ = 0 = 1 − ϵ 2 − σ \Delta_{\ell=0}=1-\frac{\epsilon}{2}-\sqrt{\sigma} Δ ℓ = 0 = 1 − 2 ϵ − σ σ = 1 + O ( ϵ ) \sigma=1+O(\epsilon) σ = 1 + O ( ϵ ) τ v = 1 \tau_v=1 τ v = 1 ℓ = v \ell=v ℓ = v
− ( 1 − γ ) ( ϵ Δ 0 ) + 1 ϵ = 0. (84) -(1-\gamma)\left(\frac{\epsilon}{\Delta_0}\right)+\frac{1}{\epsilon}=0. \tag*{(84)} − ( 1 − γ ) ( Δ 0 ϵ ) + ϵ 1 = 0. (84) Therefore Δ ℓ = 0 = ( 1 − γ ) ϵ 2 + O ( ϵ 3 ) \Delta_{\ell=0}=(1-\gamma)\epsilon^2+O(\epsilon^3) Δ ℓ = 0 = ( 1 − γ ) ϵ 2 + O ( ϵ 3 )
σ ∗ = 1 − ϵ 2 − ( 1 − γ ) ϵ 2 , (85) \sqrt{\sigma_*}=1-\frac{\epsilon}{2}-(1-\gamma)\epsilon^2, \tag*{(85)} σ ∗ = 1 − 2 ϵ − ( 1 − γ ) ϵ 2 , (85) which gives σ ∗ = 1 − ϵ + ( 2 γ − 7 4 ) ϵ 2 + O ( ϵ 3 ) \sigma_*=1-\epsilon+\left(2\gamma-\frac{7}{4}\right)\epsilon^2+O(\epsilon^3) σ ∗ = 1 − ϵ + ( 2 γ − 4 7 ) ϵ 2 + O ( ϵ 3 ) ϵ 2 \epsilon^2 ϵ 2 v v v ℓ = v \ell=v ℓ = v σ = 1 − v 2 + O ( ϵ ) \sigma=1-v^2+O(\epsilon) σ = 1 − v 2 + O ( ϵ )
− γ ϵ Δ v + 1 − v 2 ϵ = 0. (86) -\gamma\frac{\epsilon}{\Delta_v}+\frac{1-v^2}{\epsilon}=0. \tag*{(86)} − γ Δ v ϵ + ϵ 1 − v 2 = 0. (86) Therefore Δ v = γ 1 − v 2 ϵ 2 + O ( ϵ 3 ) \Delta_v=\frac{\gamma}{1-v^2}\epsilon^2+O(\epsilon^3) Δ v = 1 − v 2 γ ϵ 2 + O ( ϵ 3 ) τ v = − 1 \tau_v=-1 τ v = − 1 τ 0 = ± 1 \tau_0=\pm1 τ 0 = ± 1 σ ∗ \sigma_* σ ∗
σ ∗ = 1 − v 2 − ϵ + ( 1 4 − 2 γ 1 − v 2 ) ϵ 2 + O ( ϵ 3 ) , (87) \sigma_*=1-v^2-\epsilon+\left(\frac{1}{4}-\frac{2\gamma}{1-v^2}\right)\epsilon^2+O(\epsilon^3), \tag*{(87)} σ ∗ = 1 − v 2 − ϵ + ( 4 1 − 1 − v 2 2 γ ) ϵ 2 + O ( ϵ 3 ) , (87) To summarize, a strong-pinning solution exists for i) τ 0 = − 1 , τ v = 1 \tau_0=-1,\tau_v=1 τ 0 = − 1 , τ v = 1 ℓ = 0 \ell=0 ℓ = 0 τ 0 = ± 1 , τ v = − 1 \tau_0=\pm1,\tau_v=-1 τ 0 = ± 1 , τ v = − 1 ℓ = v \ell=v ℓ = v τ 0 = τ v = 1 \tau_0=\tau_v=1 τ 0 = τ v = 1
Solutions in the γ ≪ 1 \gamma\ll1 γ ≪ 1
We now consider yet another analytically solvable limit with γ ≪ 1 \gamma\ll1 γ ≪ 1 σ ∗ = O ( γ 2 ) \sigma_*=O(\gamma^2) σ ∗ = O ( γ 2 ) s = σ ∗ s=\sqrt{\sigma_*} s = σ ∗ T 0 T_0 T 0 f ( τ 0 s ) f(\tau_0s) f ( τ 0 s )
f ( τ 0 s ) = f ( 0 ) + τ 0 f ′ ( 0 ) s + O ( s 2 ) . (88) f(\tau_0s)=f(0)+\tau_0f'(0)s+O(s^2). \tag*{(88)} f ( τ 0 s ) = f ( 0 ) + τ 0 f ′ ( 0 ) s + O ( s 2 ) . (88) Now for R 1 R_1 R 1 σ = s 2 \sigma=s^2 σ = s 2
R 1 ( s 2 ) = R 1 ( 0 ) + O ( s 2 ) . (89) R_1(s^2)=R_1(0)+O(s^2). \tag*{(89)} R 1 ( s 2 ) = R 1 ( 0 ) + O ( s 2 ) . (89) So we have
T 0 ( σ ) = f ( τ 0 s ) + 2 R 1 ( s 2 ) = f ( 0 ) + 2 R 1 ( 0 ) + τ 0 f ′ ( 0 ) s + O ( s 2 ) = τ 0 f ′ ( 0 ) s + O ( s 2 ) . (90) T_0(\sigma)=f(\tau_0s)+2R_1(s^2)=f(0)+2R_1(0)+\tau_0f'(0)s+O(s^2)=\tau_0f'(0)s+O(s^2). \tag*{(90)} T 0 ( σ ) = f ( τ 0 s ) + 2 R 1 ( s 2 ) = f ( 0 ) + 2 R 1 ( 0 ) + τ 0 f ′ ( 0 ) s + O ( s 2 ) = τ 0 f ′ ( 0 ) s + O ( s 2 ) . (90) Here
f ′ ( 0 ) = π Γ ( D − 1 2 ) Γ ( 3 − D 2 ) tan π D 2 , (91) f'(0)=\pi\frac{\Gamma\left(\frac{D-1}{2}\right)}{\Gamma\left(\frac{3-D}{2}\right)}\tan\frac{\pi D}{2}, \tag*{(91)} f ′ ( 0 ) = π Γ ( 2 3 − D ) Γ ( 2 D − 1 ) tan 2 π D , (91) which vanishes at D = 2 D=2 D = 2 D ∈ ( 2 , 3 ) D\in(2,3) D ∈ ( 2 , 3 ) D = 2 D=2 D = 2
f ′ ( 0 ) = π 2 2 ( D − 2 ) + O ( ( D − 2 ) 2 ) . (92) f'(0)=\frac{\pi^2}{2}(D-2)+O\left((D-2)^2\right). \tag*{(92)} f ′ ( 0 ) = 2 π 2 ( D − 2 ) + O ( ( D − 2 ) 2 ) . (92) Now we evaluate T v ( s 2 ) T_v(s^2) T v ( s 2 ) R R R s 2 s^2 s 2 v v v s = 0 s=0 s = 0 T 0 ( σ ) T_0(\sigma) T 0 ( σ ) O ( s ) O(s) O ( s ) (69) :
R η ( 0 ) = Γ ( η + D − 1 2 ) ( 1 − D ) Γ ( η + 1 − D 2 ) . (93) R_\eta(0)=\frac{\Gamma\left(\eta+\frac{D-1}{2}\right)}{(1-D)\Gamma\left(\eta+\frac{1-D}{2}\right)}. \tag*{(93)} R η ( 0 ) = ( 1 − D ) Γ ( η + 2 1 − D ) Γ ( η + 2 D − 1 ) . (93) We can write
T v ( s 2 ) = f ( τ v v 2 + s 2 ) + R 1 + v ( 0 ) + R 1 − v ( 0 ) + O ( s 2 ) (94) T_v(s^2)=f\left(\tau_v\sqrt{v^2+s^2}\right)+R_{1+v}(0)+R_{1-v}(0)+O(s^2) \tag*{(94)} T v ( s 2 ) = f ( τ v v 2 + s 2 ) + R 1 + v ( 0 ) + R 1 − v ( 0 ) + O ( s 2 ) (94) = Γ ( τ v v 2 + s 2 + D − 1 2 ) Γ ( τ v v 2 + s 2 + 3 − D 2 ) + 1 1 − D ( Γ ( 1 + v + D − 1 2 ) Γ ( 1 + v + 1 − D 2 ) + Γ ( 1 − v + D − 1 2 ) Γ ( 1 − v + 1 − D 2 ) ) + O ( s 2 ) . (95) =\frac{\Gamma\left(\tau_v\sqrt{v^2+s^2}+\frac{D-1}{2}\right)}{\Gamma\left(\tau_v\sqrt{v^2+s^2}+\frac{3-D}{2}\right)}
+\frac{1}{1-D}\left(\frac{\Gamma\left(1+v+\frac{D-1}{2}\right)}{\Gamma\left(1+v+\frac{1-D}{2}\right)}+\frac{\Gamma\left(1-v+\frac{D-1}{2}\right)}{\Gamma\left(1-v+\frac{1-D}{2}\right)}\right)+O(s^2). \tag*{(95)} = Γ ( τ v v 2 + s 2 + 2 3 − D ) Γ ( τ v v 2 + s 2 + 2 D − 1 ) + 1 − D 1 ( Γ ( 1 + v + 2 1 − D ) Γ ( 1 + v + 2 D − 1 ) + Γ ( 1 − v + 2 1 − D ) Γ ( 1 − v + 2 D − 1 ) ) + O ( s 2 ) . (95) The gap equation is now given by
τ 0 ( 1 − γ ) f ′ ( 0 ) s + γ T v ( s 2 ) = 0. (96) \tau_0(1-\gamma)f'(0)s+\gamma T_v(s^2)=0. \tag*{(96)} τ 0 ( 1 − γ ) f ′ ( 0 ) s + γ T v ( s 2 ) = 0. (96) For D > 2 D>2 D > 2 f ( τ v v 2 + s 2 ) f\left(\tau_v\sqrt{v^2+s^2}\right) f ( τ v v 2 + s 2 ) s 2 s^2 s 2 s = 0 s=0 s = 0 T v T_v T v
s = − τ 0 γ T v ( 0 ) ( 1 − γ ) f ′ ( 0 ) . (97) s=-\frac{\tau_0\gamma T_v(0)}{(1-\gamma)f'(0)}. \tag*{(97)} s = − ( 1 − γ ) f ′ ( 0 ) τ 0 γ T v ( 0 ) . (97) This result holds if v v v ϵ = 4 − d \epsilon=4-d ϵ = 4 − d γ \gamma γ v ≪ 1 v\ll1 v ≪ 1
Now we evaluate the expression for D = 2 D=2 D = 2 D = 2 + δ D=2+\delta D = 2 + δ O ( δ ) O(\delta) O ( δ )
T v ( 0 ) = δ [ π ( τ v 2 − v ) tan π v + ψ ( 1 2 + τ v v 2 + s 2 ) − ψ ( 1 2 + τ v v ) ] + O ( δ 2 ) . (98) T_v(0)=\delta\left[\pi\left(\frac{\tau_v}{2}-v\right)\tan\pi v+\psi\left(\frac{1}{2}+\tau_v\sqrt{v^2+s^2}\right)-\psi\left(\frac{1}{2}+\tau_v v\right)\right]+O(\delta^2). \tag*{(98)} T v ( 0 ) = δ [ π ( 2 τ v − v ) tan π v + ψ ( 2 1 + τ v v 2 + s 2 ) − ψ ( 2 1 + τ v v ) ] + O ( δ 2 ) . (98) Recall f ′ ( 0 ) = π 2 2 δ f'(0)=\frac{\pi^2}{2}\delta f ′ ( 0 ) = 2 π 2 δ
For τ v = 1 \tau_v=1 τ v = 1 ψ ( 1 / 2 + v 2 + s 2 ) \psi(1/2+\sqrt{v^2+s^2}) ψ ( 1/2 + v 2 + s 2 ) s s s s = 0 s=0 s = 0 d = 3 d=3 d = 3
s = − 2 γ τ 0 π ( τ v 2 − v ) tan π v + O ( γ 2 ) . (99) s=-\frac{2\gamma\tau_0}{\pi}\left(\frac{\tau_v}{2}-v\right)\tan\pi v+O(\gamma^2). \tag*{(99)} s = − π 2 γ τ 0 ( 2 τ v − v ) tan π v + O ( γ 2 ) . (99) For τ 0 = τ v = ± 1 \tau_0=\tau_v=\pm1 τ 0 = τ v = ± 1 s s s d = 3 d=3 d = 3
For τ 0 = − 1 , τ v = 1 \tau_0=-1,\tau_v=1 τ 0 = − 1 , τ v = 1
s = 2 γ π ( 1 2 − v ) tan π v + O ( γ 2 ) . (100) s=\frac{2\gamma}{\pi}\left(\frac{1}{2}-v\right)\tan\pi v+O(\gamma^2). \tag*{(100)} s = π 2 γ ( 2 1 − v ) tan π v + O ( γ 2 ) . (100) s s s 0 0 0 v = 0 v=0 v = 0 2 γ π 2 \frac{2\gamma}{\pi^2} π 2 2 γ v = 1 2 v=\frac{1}{2} v = 2 1
However, when τ v = − 1 \tau_v=-1 τ v = − 1 Δ v = 1 / 2 − v 2 + s 2 \Delta_v=1/2-\sqrt{v^2+s^2} Δ v = 1/2 − v 2 + s 2 v v v 1 / 2 1/2 1/2 ∣ v − 1 / 2 ∣ \lvert v-1/2\rvert ∣ v − 1/2 ∣ γ → 0 \gamma\to0 γ → 0 s = 0 s=0 s = 0 τ 0 = 1 , τ v = − 1 \tau_0=1,\tau_v=-1 τ 0 = 1 , τ v = − 1
s = 2 γ π ( 1 2 + v ) tan π v + O ( γ 2 ) . (101) s=\frac{2\gamma}{\pi}\left(\frac{1}{2}+v\right)\tan\pi v+O(\gamma^2). \tag*{(101)} s = π 2 γ ( 2 1 + v ) tan π v + O ( γ 2 ) . (101) Indeed, setting v → 1 / 2 v \to1/2 v → 1/2 ℓ = v \ell= v ℓ = v 1 / 2 − v ≫ O ( γ 2 / 3 ) 1/2 - v \gg O(\gamma^{2/3}) 1/2 − v ≫ O ( γ 2/3 )
Now let us consider v v v 1 / 2 1/2 1/2 v = 1 / 2 − t v = 1/2 - t v = 1/2 − t Δ v − ≈ t − s 2 \Delta_v^- \approx t - s^2 Δ v − ≈ t − s 2 ψ ( x ) = − 1 / x − γ E + O ( x ) \psi(x) = -1/x - \gamma_E + O(x) ψ ( x ) = − 1/ x − γ E + O ( x ) x → 0 x \to0 x → 0
ψ ( Δ v − ) − ψ ( 1 2 − v ) ≈ − 1 Δ v − + 1 t (102) \psi(\Delta_v^-) - \psi\left(\frac{1}{2} - v\right) \approx-\frac{1}{\Delta_v^-} + \frac{1}{t} \tag*{(102)} ψ ( Δ v − ) − ψ ( 2 1 − v ) ≈ − Δ v − 1 + t 1 (102) On the other hand, the first term in T v ( 0 ) T_v(0) T v ( 0 ) − π ( 1 − t ) cot π t ≈ − 1 / t + 1 -\pi(1-t)\cot\pi t \approx-1/t + 1 − π ( 1 − t ) cot π t ≈ − 1/ t + 1
T v ′ ( 0 ) ≈ − 1 Δ v − + 1 + O ( t , s 2 ) . (103) T_v'(0) \approx-\frac{1}{\Delta_v^-} + 1 + O(t,s^2). \tag*{(103)} T v ′ ( 0 ) ≈ − Δ v − 1 + 1 + O ( t , s 2 ) . (103) The gap equation becomes
s ( t − s 2 ) = 2 τ 0 γ π 2 . (104) s(t-s^2)=\frac{2\tau_0\gamma}{\pi^2}. \tag*{(104)} s ( t − s 2 ) = π 2 2 τ 0 γ . (104) The unitarity bound requires Δ v − ≈ t − s 2 > 0 \Delta_v^- \approx t-s^2 > 0 Δ v − ≈ t − s 2 > 0 τ 0 = 1 \tau_0=1 τ 0 = 1 0 < s < t 0 < s < \sqrt{t} 0 < s < t s ( t − s 2 ) s(t-s^2) s ( t − s 2 ) s = t / 3 s=\sqrt{t/3} s = t /3 2 π 2 γ < 2 ( t 3 ) 3 / 2 \frac{2}{\pi^2}\gamma< 2\left(\frac{t}{3}\right)^{3/2} π 2 2 γ < 2 ( 3 t ) 3/2 t > 3 ( γ / π 2 ) 2 / 3 t > 3(\gamma/\pi^2)^{2/3} t > 3 ( γ / π 2 ) 2/3 t = 3 ( γ / π 2 ) 2 / 3 t = 3(\gamma/\pi^2)^{2/3} t = 3 ( γ / π 2 ) 2/3 v s t d , a l t ( γ ) = 1 / 2 − 3 ( γ / π 2 ) 2 / 3 v^{\mathrm{std,alt}}(\gamma)=1/2-3(\gamma/\pi^2)^{2/3} v std , alt ( γ ) = 1/2 − 3 ( γ / π 2 ) 2/3 γ → 0 \gamma\to0 γ → 0
When γ 2 / 3 ≪ t \gamma^{2/3} \ll t γ 2/3 ≪ t (104) is s 1 ≈ 2 γ π 2 t s_1 \approx\frac{2\gamma}{\pi^2t} s 1 ≈ π 2 t 2 γ (101) in the v → 1 / 2 v \to1/2 v → 1/2 s 2 ≈ t + O ( γ / t ) s_2 \approx\sqrt{t}+O(\gamma/t) s 2 ≈ t + O ( γ / t ) Figure S4 (where s 1 ( s 2 ) s_1(s_2) s 1 ( s 2 )
Figure S4. Solutions σ ∗ \sigma_* σ ∗ d d d ℓ = v \ell= v ℓ = v ℓ = 0 \ell= 0 ℓ = 0
The v1 limit
Here we show that in the v ≪ 1 v \ll1 v ≪ 1 σ ∗ ∼ v 2 \sigma_* \sim v^2 σ ∗ ∼ v 2 γ < 1 2 \gamma< \frac{1}{2} γ < 2 1 γ ≪ 1 \gamma\ll1 γ ≪ 1 f ( v 2 + σ ) f\left(\sqrt{v^2+\sigma}\right) f ( v 2 + σ ) T v ( σ ) T_v(\sigma) T v ( σ )
In the following we write σ ∗ = x 2 v 2 \sigma_* = x^2v^2 σ ∗ = x 2 v 2 x ≥ 0 x \ge0 x ≥ 0 O ( 1 ) O(1) O ( 1 ) v → 0 v \to0 v → 0 T 0 T_0 T 0 γ ≪ 1 \gamma\ll1 γ ≪ 1
T 0 ( σ ) = τ 0 f ′ ( 0 ) x v + O ( v 2 ) . (105) T_0(\sigma)=\tau_0f'(0)xv+O(v^2). \tag*{(105)} T 0 ( σ ) = τ 0 f ′ ( 0 ) xv + O ( v 2 ) . (105) A caveat here is that if one also considers small ϵ \epsilon ϵ O ( v 2 ) O(v^2) O ( v 2 ) R 1 R_1 R 1 O ( v 2 / ϵ ) O(v^2/\epsilon) O ( v 2 / ϵ ) v ≪ ϵ v \ll\epsilon v ≪ ϵ v , ϵ v,\epsilon v , ϵ ϵ \epsilon ϵ v v v ϵ \epsilon ϵ
Next is T v ( σ ) = f ( τ v v 1 + x 2 ) + R 1 + v ( σ ) + R 1 − v ( σ ) T_v(\sigma)=f(\tau_vv\sqrt{1+x^2})+R_{1+v}(\sigma)+R_{1-v}(\sigma) T v ( σ ) = f ( τ v v 1 + x 2 ) + R 1 + v ( σ ) + R 1 − v ( σ ) R η = 1 ± v ( σ ) R_{\eta=1\pm v}(\sigma) R η = 1 ± v ( σ ) v v v v v v η \eta η σ ∼ v 2 \sigma\sim v^2 σ ∼ v 2
R 1 + v ( σ ) + R 1 − v ( σ ) = 2 R 1 ( 0 ) + O ( v 2 ) . (106) R_{1+v}(\sigma)+R_{1-v}(\sigma)=2R_1(0)+O(v^2). \tag*{(106)} R 1 + v ( σ ) + R 1 − v ( σ ) = 2 R 1 ( 0 ) + O ( v 2 ) . (106) Therefore
T v ( σ ) = f ( τ v v 1 + x 2 ) + R 1 + v ( 0 ) + R 1 − v ( 0 ) = f ( 0 ) + 2 R 1 ( 0 ) + τ v f ′ ( 0 ) v 1 + x 2 + O ( v 2 ) = τ v f ′ ( 0 ) v 1 + x 2 + O ( v 2 ) . (107) \begin{aligned}
T_v(\sigma) &= f(\tau_vv\sqrt{1+x^2})+R_{1+v}(0)+R_{1-v}(0) \\
&= f(0)+2R_1(0)+\tau_vf'(0)v\sqrt{1+x^2}+O(v^2) \\
&= \tau_vf'(0)v\sqrt{1+x^2}+O(v^2). \tag*{(107)}
\end{aligned} T v ( σ ) = f ( τ v v 1 + x 2 ) + R 1 + v ( 0 ) + R 1 − v ( 0 ) = f ( 0 ) + 2 R 1 ( 0 ) + τ v f ′ ( 0 ) v 1 + x 2 + O ( v 2 ) = τ v f ′ ( 0 ) v 1 + x 2 + O ( v 2 ) . (107) Thus altogether to O ( v ) O(v) O ( v )
( 1 − γ ) τ 0 f ′ ( 0 ) x v + γ τ v f ′ ( 0 ) v 1 + x 2 = 0. (108) (1-\gamma)\tau_0f'(0)xv+\gamma\tau_vf'(0)v\sqrt{1+x^2}=0. \tag*{(108)} ( 1 − γ ) τ 0 f ′ ( 0 ) xv + γ τ v f ′ ( 0 ) v 1 + x 2 = 0. (108) Since f ′ ( 0 ) ≠ 0 f'(0)\ne0 f ′ ( 0 ) = 0 d ∈ ( 3 , 4 ) d\in(3,4) d ∈ ( 3 , 4 )
( 1 − γ ) τ 0 x + γ τ v 1 + x 2 = 0 , (109) (1-\gamma)\tau_0x+\gamma\tau_v\sqrt{1+x^2}=0, \tag*{(109)} ( 1 − γ ) τ 0 x + γ τ v 1 + x 2 = 0 , (109) and the solution is
x = − τ 0 τ v γ 1 − 2 γ . (110) x=-\frac{\tau_0\tau_v\gamma}{\sqrt{1-2\gamma}}. \tag*{(110)} x = − 1 − 2 γ τ 0 τ v γ . (110) When τ 0 = τ v \tau_0=\tau_v τ 0 = τ v x x x τ 0 = − τ v \tau_0=-\tau_v τ 0 = − τ v
σ ∗ = γ 2 1 − 2 γ v 2 + O ( v 3 ) . (111) \sigma_*=\frac{\gamma^2}{1-2\gamma}v^2+\mathcal{O}(v^3). \tag*{(111)} σ ∗ = 1 − 2 γ γ 2 v 2 + O ( v 3 ) . (111) Note that the result is independent of d d d
Evaluation of R η ( 0 ) R_\eta(0) R η ( 0 )
Recall the definition of R η ( 0 ) R_\eta(0) R η ( 0 )
R η ( 0 ) = ∑ n = 0 ∞ [ Γ ( n + η + a ) Γ ( n + η + b ) − ( n + η ) D − 2 ] + ζ ( 2 − D , η ) . (112) R_\eta(0)=\sum_{n=0}^{\infty}\left[\frac{\Gamma(n+\eta+a)}{\Gamma(n+\eta+b)}-(n+\eta)^{D-2}\right]+\zeta(2-D,\eta). \tag*{(112)} R η ( 0 ) = n = 0 ∑ ∞ [ Γ ( n + η + b ) Γ ( n + η + a ) − ( n + η ) D − 2 ] + ζ ( 2 − D , η ) . (112) Define S n ( a , b ) = Γ ( n + η + a ) / Γ ( n + η + b ) S_n(a,b)=\Gamma(n+\eta+a)/\Gamma(n+\eta+b) S n ( a , b ) = Γ ( n + η + a ) /Γ ( n + η + b ) Γ ( z + 1 ) = z Γ ( z ) \Gamma(z+1)=z\Gamma(z) Γ ( z + 1 ) = z Γ ( z )
S n + 1 ( a , b ) − S n ( a , b ) = ( n + η + a n + η + b − 1 ) S n ( a , b ) = a − b n + η + b S n ( a , b ) = ( a − b ) S n ( a , b + 1 ) . (113) S_{n+1}(a,b)-S_n(a,b)=\left(\frac{n+\eta+a}{n+\eta+b}-1\right)S_n(a,b)=\frac{a-b}{n+\eta+b}S_n(a,b)=(a-b)S_n(a,b+1). \tag*{(113)} S n + 1 ( a , b ) − S n ( a , b ) = ( n + η + b n + η + a − 1 ) S n ( a , b ) = n + η + b a − b S n ( a , b ) = ( a − b ) S n ( a , b + 1 ) . (113) Hence we have the recursion relation
S n ( a , b ) = S n + 1 ( a , b − 1 ) − S n ( a , b − 1 ) a − b + 1 . (114) S_n(a,b)=\frac{S_{n+1}(a,b-1)-S_n(a,b-1)}{a-b+1}. \tag*{(114)} S n ( a , b ) = a − b + 1 S n + 1 ( a , b − 1 ) − S n ( a , b − 1 ) . (114) And the partial sum
∑ n = 0 N S n ( a , b ) = 1 a − b + 1 [ S N + 1 ( a , b − 1 ) − S 0 ( a , b − 1 ) ] . (115) \sum_{n=0}^{N}S_n(a,b)=\frac{1}{a-b+1}\left[S_{N+1}(a,b-1)-S_0(a,b-1)\right]. \tag*{(115)} n = 0 ∑ N S n ( a , b ) = a − b + 1 1 [ S N + 1 ( a , b − 1 ) − S 0 ( a , b − 1 ) ] . (115) S N + 1 S_{N+1} S N + 1 N N N q = N + 1 + η q=N+1+\eta q = N + 1 + η
S N + 1 ( a , b − 1 ) = Γ ( q + a ) Γ ( q + b − 1 ) (116) S_{N+1}(a,b-1)=\frac{\Gamma(q+a)}{\Gamma(q+b-1)} \tag*{(116)} S N + 1 ( a , b − 1 ) = Γ ( q + b − 1 ) Γ ( q + a ) (116) = q a − b + 1 [ 1 + ( a − b + 1 ) ( a + b − 2 ) 2 q + O ( q − 2 ) ] (117) =q^{a-b+1}\left[1+\frac{(a-b+1)(a+b-2)}{2q}+\mathcal{O}(q^{-2})\right] \tag*{(117)} = q a − b + 1 [ 1 + 2 q ( a − b + 1 ) ( a + b − 2 ) + O ( q − 2 ) ] (117) = q D − 1 [ 1 − D − 1 2 q + O ( q − 2 ) ] . (118) =q^{D-1}\left[1-\frac{D-1}{2q}+\mathcal{O}(q^{-2})\right]. \tag*{(118)} = q D − 1 [ 1 − 2 q D − 1 + O ( q − 2 ) ] . (118) Next we can evaluate the partial sum of ( n + η ) D − 2 (n+\eta)^{D-2} ( n + η ) D − 2
∑ n = 0 N ( n + η ) D − 2 = q D − 1 D − 1 − 1 2 q D − 2 + ζ ( 2 − D , η ) + O ( q D − 3 ) . (119) \sum_{n=0}^{N}(n+\eta)^{D-2}=\frac{q^{D-1}}{D-1}-\frac{1}{2}q^{D-2}+\zeta(2-D,\eta)+\mathcal{O}(q^{D-3}). \tag*{(119)} n = 0 ∑ N ( n + η ) D − 2 = D − 1 q D − 1 − 2 1 q D − 2 + ζ ( 2 − D , η ) + O ( q D − 3 ) . (119) Since we are interested in D < 3 D<3 D < 3 q D − 3 q^{D-3} q D − 3 N → ∞ N\to\infty N → ∞ R η ( 0 ) R_\eta(0) R η ( 0 ) N → ∞ N\to\infty N → ∞
R η ( 0 ) = lim N → ∞ ( ∑ n = 0 N S n ( a , b ) − ∑ n = 0 N ( n + η ) D − 2 ) + ζ ( 2 − D , η ) (120) R_\eta(0)=\lim_{N\to\infty}\left(\sum_{n=0}^{N}S_n(a,b)-\sum_{n=0}^{N}(n+\eta)^{D-2}\right)+\zeta(2-D,\eta) \tag*{(120)} R η ( 0 ) = N → ∞ lim ( n = 0 ∑ N S n ( a , b ) − n = 0 ∑ N ( n + η ) D − 2 ) + ζ ( 2 − D , η ) (120) = − S 0 ( a , b − 1 ) a − b + 1 (121) =-\frac{S_0(a,b-1)}{a-b+1} \tag*{(121)} = − a − b + 1 S 0 ( a , b − 1 ) (121) = Γ ( η + a ) ( b − a − 1 ) Γ ( η + b − 1 ) . (122) =\frac{\Gamma(\eta+a)}{(b-a-1)\Gamma(\eta+b-1)}. \tag*{(122)} = ( b − a − 1 ) Γ ( η + b − 1 ) Γ ( η + a ) . (122) Additional numerical results
Figure S4 illustrates the evolutions of the saddle points with τ 0 = 1 , τ v = − 1 \tau_0 = 1,\tau_v = -1 τ 0 = 1 , τ v = − 1 d d d v < v c s t d , a l t ( γ ) v < v_c^{\mathrm{std,alt}}(\gamma) v < v c std , alt ( γ ) d = 4 d = 4 d = 4 d = 3 d = 3 d = 3 v > v c s t d , a l t ( γ ) v > v_c^{\mathrm{std,alt}}(\gamma) v > v c std , alt ( γ ) d c > 3 d_c > 3 d c > 3 d d d 3 3 3
In Figure S5 we show a similar plot for v = 0.4 v = 0.4 v = 0.4 γ = 0.3 \gamma= 0.3 γ = 0.3 τ 0 = τ v = − 1 \tau_0 = \tau_v = -1 τ 0 = τ v = − 1 d = 3 d = 3 d = 3 d d d d c ≈ 3.76 d_c \approx3.76 d c ≈ 3.76 v v v γ \gamma γ γ \gamma γ v v v
Figure S5. σ ∗ \sigma_* σ ∗ d d d v = 0.4 v = 0.4 v = 0.4 γ = 0.3 \gamma= 0.3 γ = 0.3 ℓ = 0 , v \ell= 0,v ℓ = 0 , v
More data on Monte Carlo simulations
Details on the update algorithm and the pinning strength selection
Here we describe the Monte Carlo update algorithm in our simulation. Configurations are generated using a local hybrid update consisting of one Metropolis sweep followed by deterministic microcanonical overrelaxation sweeps. For a site x x x
B x = ∑ μ = 1 3 ( U x , μ Φ x + μ ^ + U x − μ ^ , μ T Φ x − μ ^ ) . (123) B_x = \sum_{\mu=1}^{3} \left(U_{x,\mu}\Phi_{x+\hat{\mu}} + U^{T}_{x-\hat{\mu},\mu}\Phi_{x-\hat{\mu}}\right). \tag*{(123)} B x = μ = 1 ∑ 3 ( U x , μ Φ x + μ ^ + U x − μ ^ , μ T Φ x − μ ^ ) . (123) We also define
w x = 1 4 ∑ a = 1 2 1 x ∈ S a , V ( Φ ) = Φ 2 + λ ( Φ 2 − 1 ) 2 . (124) w_x = \frac{1}{4}\sum_{a=1}^{2}\mathbf{1}_{x\in\mathcal{S}_a}, \qquad V(\Phi) = \Phi^2 + \lambda\left(\Phi^2-1\right)^2. \tag*{(124)} w x = 4 1 a = 1 ∑ 2 1 x ∈ S a , V ( Φ ) = Φ 2 + λ ( Φ 2 − 1 ) 2 . (124) So that H p i n = c ~ ∑ x w x ∣ P u Φ x ∣ 2 H_{\rm pin}=\tilde{c}\sum_x w_x|P_u\Phi_x|^2 H pin = c ~ ∑ x w x ∣ P u Φ x ∣ 2
Φ x ′ = Φ x + ϵ η , η ∼ N ( 0 , 1 ) , (125) \Phi'_x = \Phi_x + \epsilon\eta,\qquad\eta\sim\mathcal{N}(0,1), \tag*{(125)} Φ x ′ = Φ x + ϵη , η ∼ N ( 0 , 1 ) , (125) with ϵ = 0.55 \epsilon= 0.55 ϵ = 0.55
Δ H = V ( Φ x ′ ) − V ( Φ x ) − β ( Φ x ′ − Φ x ) ⋅ B x + c ~ w x [ ∣ P u Φ x ′ ∣ 2 − ∣ P u Φ x ∣ 2 ] . (126) \Delta H = V(\Phi'_x)-V(\Phi_x)-\beta\left(\Phi'_x-\Phi_x\right)\cdot B_x+\tilde{c}w_x\left[\left|P_u\Phi'_x\right|^2-\left|P_u\Phi_x\right|^2\right]. \tag*{(126)} Δ H = V ( Φ x ′ ) − V ( Φ x ) − β ( Φ x ′ − Φ x ) ⋅ B x + c ~ w x [ ∣ P u Φ x ′ ∣ 2 − ∣ P u Φ x ∣ 2 ] . (126) The proposal is accepted with probability min ( 1 , e − Δ H ) \min(1,e^{-\Delta H}) min ( 1 , e − Δ H ) (15) . Each Metropolis sweep is followed by reflections of the field about its local molecular field,
Φ x ′ = 2 Φ x ⋅ B x B x 2 B x − Φ x , (127) \Phi'_x = 2\frac{\Phi_x\cdot B_x}{B_x^2}B_x-\Phi_x, \tag*{(127)} Φ x ′ = 2 B x 2 Φ x ⋅ B x B x − Φ x , (127) provided B x 2 B_x^2 B x 2 c ~ = 0 \tilde{c}=0 c ~ = 0 Φ x 2 \Phi_x^2 Φ x 2 Φ x ⋅ B x \Phi_x\cdot B_x Φ x ⋅ B x c ~ ≠ 0 \tilde{c}\ne0 c ~ = 0 (127) is however accepted with probability min ( 1 , exp [ − c ~ w x ( ∣ P u Φ x ′ ∣ 2 − ∣ P u Φ x ∣ 2 ) ] ) \min\left(1,\exp\left[-\tilde{c}w_x\left(\left|P_u\Phi'_x\right|^2-\left|P_u\Phi_x\right|^2\right)\right]\right) min ( 1 , exp [ − c ~ w x ( ∣ P u Φ x ′ ∣ 2 − ∣ P u Φ x ∣ 2 ) ] )
Finally we elaborate how we choose the nonzero pinning strengths c ~ ∗ \tilde{c}_* c ~ ∗ c ~ \tilde{c} c ~ L ≤ 64 L\le64 L ≤ 64 c ~ ∗ = − 1.8 , − 1.5 , − 1.75 \tilde{c}_*=-1.8,-1.5,-1.75 c ~ ∗ = − 1.8 , − 1.5 , − 1.75 v = 1 / 2 v=1/2 v = 1/2 v = 1 / 4 v=1/4 v = 1/4
Additional data for v = 1 / 2 v=1/2 v = 1/2 N = 3 N=3 N = 3
We explain the finite-size collapse method that was used to estimate the scaling dimensions of defect primaries in Figure 6 and Figure 7 . Denoting u ≡ r z / L u \equiv r_z/L u ≡ r z / L x i = ln u i x_i=\ln u_i x i = ln u i y i = ln G i + 2 Δ ^ ln L i y_i=\ln G_i+2\widehat{\Delta}\ln L_i y i = ln G i + 2 Δ ln L i Δ ^ \widehat{\Delta} Δ (26) , when correct scaling dimension is used, all transformed data should lie on a single master curve
y i ( Δ ^ ) = ln F ( u i ) ≡ Y ( x i ) . (128) y_i(\widehat{\Delta})=\ln\mathcal{F}(u_i)\equiv Y(x_i). \tag*{(128)} y i ( Δ ) = ln F ( u i ) ≡ Y ( x i ) . (128) For each x i x_i x i Y ^ ( x i ) \widehat{Y}(x_i) Y ( x i )
S ( Δ ^ ) = 1 N ∑ i = 1 N [ y i ( Δ ^ ) − Y ^ ( x i ) ] 2 σ y i 2 + σ Y ^ i 2 . (129) S(\widehat{\Delta})=\frac{1}{N}\sum_{i=1}^{N}\frac{\left[y_i(\widehat{\Delta})-\widehat{Y}(x_i)\right]^2}{\sigma_{y_i}^2+\sigma_{\widehat{Y}_i}^2}. \tag*{(129)} S ( Δ ) = N 1 i = 1 ∑ N σ y i 2 + σ Y i 2 [ y i ( Δ ) − Y ( x i ) ] 2 . (129) S ≃ 1 S\simeq1 S ≃ 1 Δ \Delta Δ S S S Δ \Delta Δ S ( Δ ^ ) ≤ S min + 1 S(\widehat{\Delta})\leq S_{\min}+1 S ( Δ ) ≤ S m i n + 1 S S S
Next we briefly discuss the spontaneous breaking of the remaining Z 2 \mathbb{Z}_2 Z 2 M ϕ = 1 L ∑ z = 0 L − 1 ϕ l a t ( z ) \mathcal{M}_{\phi}=\frac{1}{L}\sum_{z=0}^{L-1}\phi^{\mathrm{lat}}(z) M ϕ = L 1 ∑ z = 0 L − 1 ϕ lat ( z ) ϕ l a t ( z ) \phi^{\mathrm{lat}}(z) ϕ lat ( z ) (21) . Our ansatz for squared line magnetization reads
⟨ M ϕ 2 ⟩ L = m 0 2 + A ϕ L − p ϕ + ⋯ , (130) \left\langle\mathcal{M}_{\phi}^{2}\right\rangle_L=m_0^2+A_{\phi}L^{-p_{\phi}}+\cdots, \tag*{(130)} ⟨ M ϕ 2 ⟩ L = m 0 2 + A ϕ L − p ϕ + ⋯ , (130) where m 0 2 = lim L → ∞ ⟨ M ϕ 2 ⟩ L m_0^2=\lim_{L\to\infty}\left\langle\mathcal{M}_{\phi}^{2}\right\rangle_L m 0 2 = lim L → ∞ ⟨ M ϕ 2 ⟩ L Z 2 \mathbb{Z}_2 Z 2 ϕ ^ → − ϕ ^ \hat{\phi}\to-\hat{\phi} ϕ ^ → − ϕ ^ Figure S6 shows m 0 2 m_0^2 m 0 2 c ~ \tilde{c} c ~ − 3 -3 − 3 ∣ c ~ ∣ ≲ 2 |\tilde{c}|\lesssim2 ∣ c ~ ∣ ≲ 2 Z 2 \mathbb{Z}_2 Z 2 ∣ c ~ ∣ ≳ 2 |\tilde{c}|\gtrsim2 ∣ c ~ ∣ ≳ 2 Z 2 \mathbb{Z}_2 Z 2 O ( N ) O(N) O ( N )
v = 1 / 4 v=1/4 v = 1/4 N = 3 N=3 N = 3 Figure S7 and Figure S8 show the results for the v = 1 / 4 v=1/4 v = 1/4 (20) and (21) .
Figure S7. Determination of the scaling dimensions of the twisted defect primary ψ ^ \hat{\psi} ψ ^ v = 1 / 4 v=1/4 v = 1/4 N = 3 N=3 N = 3 c ~ = 0 \tilde{c}=0 c ~ = 0 c ~ ∗ = − 1.5 \tilde{c}_{*}=-1.5 c ~ ∗ = − 1.5 Δ ψ ^ = 0.632 ± 0.011 \Delta_{\hat{\psi}}=0.632\pm0.011 Δ ψ ^ = 0.632 ± 0.011 c ~ = 0 \tilde{c}=0 c ~ = 0 Δ ψ ^ = 0.725 ± 0.016 \Delta_{\hat{\psi}}=0.725\pm0.016 Δ ψ ^ = 0.725 ± 0.016 c ~ ∗ = − 1.5 \tilde{c}_{*}=-1.5 c ~ ∗ = − 1.5
Figure S8. Determination of the scaling dimensions of the untwisted defect primary ϕ ^ \hat{\phi} ϕ ^ v = 1 / 4 v=1/4 v = 1/4 N = 3 N=3 N = 3 c ~ = 0 \tilde{c}=0 c ~ = 0 c ~ ∗ = − 1.5 \tilde{c}_{*}=-1.5 c ~ ∗ = − 1.5 Δ ϕ ^ = 0.461 ± 0.011 \Delta_{\hat{\phi}}=0.461\pm0.011 Δ ϕ ^ = 0.461 ± 0.011 c ~ = 0 \tilde{c}=0 c ~ = 0 Δ ϕ ^ = 0.270 ± 0.006 \Delta_{\hat{\phi}}=0.270\pm0.006 Δ ϕ ^ = 0.270 ± 0.006 c ~ ∗ = − 1.5 \tilde{c}_{*}=-1.5 c ~ ∗ = − 1.5
Without explicit pinning term (c ~ = 0 \tilde{c}=0 c ~ = 0 Δ ψ ^ = 0.632 ± 0.011 \Delta_{\hat{\psi}}=0.632\pm0.011 Δ ψ ^ = 0.632 ± 0.011 Δ ϕ ^ = 0.461 ± 0.011 \Delta_{\hat{\phi}}=0.461\pm0.011 Δ ϕ ^ = 0.461 ± 0.011 c ~ ∗ = − 1.5 \tilde{c}_{*}=-1.5 c ~ ∗ = − 1.5 Δ ψ ^ = 0.725 ± 0.016 \Delta_{\hat{\psi}}=0.725\pm0.016 Δ ψ ^ = 0.725 ± 0.016 Δ ϕ ^ = 0.270 ± 0.006 \Delta_{\hat{\phi}}=0.270\pm0.006 Δ ϕ ^ = 0.270 ± 0.006 S = 1.55 S=1.55 S = 1.55 Figure S7 , S = 1.20 S=1.20 S = 1.20 Figure S7 , S = 1.40 S=1.40 S = 1.40 Figure S8 , and S = 1.72 S=1.72 S = 1.72 Figure S8 , indicating that the collapse is generally within statistical errors. We confirm that with c ~ ∗ = − 1.5 \tilde{c}_{*}=-1.5 c ~ ∗ = − 1.5 Z 2 \mathbb{Z}_{2} Z 2 N N N γ = 2 3 , v = 1 4 \gamma=\frac{2}{3},v=\frac{1}{4} γ = 3 2 , v = 4 1 Δ ℓ = 0 ≈ 0.243 , Δ ℓ = 1 4 ≈ 0.860 \Delta_{\ell=0}\approx0.243,\Delta_{\ell=\frac{1}{4}}\approx0.860 Δ ℓ = 0 ≈ 0.243 , Δ ℓ = 4 1 ≈ 0.860
