Introduction
In the Kibble-Zurek (KZ) mechanism, equilibrium properties determine a nonequilibrium phenomenon. The critical exponents of a continuous phase transition fix how the density of topological defects scales with the rate at which the transition is crossed [1 , 2 , 3 , 4 , 5 ] . The validity of the KZ mechanism in the quantum domain was theoretically established in a series of seminal works, from minimal two-level settings to genuinely many-body ones [6 , 7 , 8 , 9 ] . Subsequent efforts have involved experimental tests in quantum simulators [10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 ] , which have also revealed its breakdown in frustrated lattices [19 ] , and ultracold atomic systems [20 , 21 , 22 , 23 , 24 , 25 ] , as well as numerical simulations [26 , 27 ] , including classical benchmarks of claimed quantum advantage [28 ] . The universality unveiled by the KZ scaling extends to the full counting statistics of topological defects, whereby every defect number cumulant follows the same power law as the average [29 , 30 , 31 , 32 , 33 , 34 ] . In quantum Ising chains, the defects are identical to kinks arising from spontaneous Z 2 \mathbb{Z}_2 Z 2
assume that kinks form independently of each other. In a spin chain with fractional long-range interactions, we find that the critical exponents still fix the defect density at intermediate range, while at long range, where interactions bind defects into pairs, the variance of the kink number exceeds twice its mean, the upper limit for any free-fermion state.
Long-range interactions arise naturally on a variety of quantum simulation platforms [35 ] , and power-law interacting systems have been experimentally investigated with trapped-ion crystals [36 , 37 , 38 ] , in Rydberg quantum simulators [11 ] , and in condensed matter [39 ] . In this setting, nonequilibrium scaling laws were found in a pioneering study in the Lipkin-Meshkov-Glick (LMG) model [40 ] . The KZ scaling holds in the exactly solvable long-range Kitaev chain, with anomalous deviations in parts of parameter space [41 , 42 ] , while in the strong long-range limit the kink density becomes independent of the driving rate [43 , 44 ] . Matrix-product-state simulations of the long-range Ising chain found power-law scaling with exponents that change by about 25% with the interaction range [45 ] . The KZ scaling also persists in classical regular networks of low connectivity [46 ] and under truncated counterdiabatic driving [47 , 48 , 49 ] . Experimentally, trapped-ion quantum simulators have observed the KZ mechanism in the long-range transverse-field Ising model [50 ] . Beyond the mean number of defects, their universal statistics has been characterized in free-fermion models, including long-range Kitaev chains [32 , 49 , 51 ] in agreement with previous beyond-KZM studies [29 , 30 , 31 ] , also in the presence of noise [12 , 13 , 51 , 52 ] . In Rydberg chains, domain-wall statistics were measured after a single sweep [53 ] , and the defect number distribution after slow ramps departs from that of independently formed defects [54 ] . In the LMG model the defect number follows a negative binomial distribution, which is super-Poissonian [44 ] . Despite these advances, even the average number of kinks remains poorly understood in nonintegrable long-range spin systems, while their full counting statistics has so far been obtained only for free-fermion chains and in the fully connected limit, and has not been explored in nonintegrable spin chains, especially in the most exciting long-range interacting regime. Most pressingly, it remains open whether kink-kink interactions induce a domain-wall confinement strong enough to break the independent kink formation assumed in the free-fermion description, leading to completely different counting statistics.
A prominent route to long-range quantum critical dynamics is provided by fractional calculus, which extends differentiation and integration to noninteger order and is widely used to model anomalous transport [55 , 56 , 57 ] and wave propagation in optics [58 , 59 , 60 ] . In quantum mechanics, it leads to the fractional Schrödinger equation, originally derived from Lévy-flight path integrals in which the Brownian paths of the Feynman construction are replaced by paths with heavy-tailed step-length statistics [61 , 62 ] . The fractional order acts as a Lévy index and controls the diffusive character of the quantum states. A standard realization is the Riesz fractional derivative, defined in momentum space by ∂ q f ( x ) → − ∣ k ∣ q f ~ ( k ) \partial_q f(x) \to-\lvert k\rvert^q \tilde{f}(k) ∂ q f ( x ) → − ∣ k ∣ q f ~ ( k ) f ~ ( k ) \tilde{f}(k) f ~ ( k ) f ( x ) f(x) f ( x ) q q q [63 , 64 ] . Yet, Lévy flights provide a natural origin for it. For 0 < q < 2 0 < q < 2 0 < q < 2 ∣ x ∣ − ( 1 + q ) \lvert x\rvert^{-(1+q)} ∣ x ∣ − ( 1 + q ) e − ∣ k ∣ q e^{-\lvert k\rvert^q} e − ∣ k ∣ q − ∣ k ∣ q -\lvert k\rvert^q − ∣ k ∣ q [62 , 65 ] . The nonlocality is therefore inherited from the stability of the underlying jump statistics rather than imposed by hand as a prescribed algebraic decay. By the generalized central limit theorem, sums of many heavy-tailed steps approach Lévy-stable laws, just as sums of steps with finite variance approach the Gaussian, their q = 2 q = 2 q = 2 [55 , 56 ] , so heavy-tailed transport is generic in disordered and multiscale media.
On a lattice, the coupling between two sites at distance r r r J r , q J_{r,q} J r , q r − ( 1 + q ) r^{-(1+q)} r − ( 1 + q ) q < 2 q < 2 q < 2 q = 2 q = 2 q = 2 q > 2 q > 2 q > 2 [66 , 67 ] . Quantum Ising chains with pure power-law couplings r − ( 1 + q ) r^{-(1+q)} r − ( 1 + q ) q < 2 / 3 q < 2/3 q < 2/3 [68 , 69 , 70 ] and, according to functional renormalization-group and quantum Monte Carlo studies, in the short-range class for q > 7 / 4 q > 7/4 q > 7/4 [69 , 71 ] . The fractional couplings share this tail, but whether the fractional chain follows the same thresholds has not been established; classical simulations of the fractional model are consistent with the corresponding classical thresholds [66 ] . For q ≤ 1 q \le1 q ≤ 1 [72 , 73 ] . The critical exponents vary continuously with the fractional order [66 , 67 ] , and the spreading of the entanglement follows the Lévy light cones [67 ] . These features make fractional models a natural setting in which to confront the KZM with a continuous family of exponents, which is also relevant for quantum simulators of realistic materials beyond idealized lattices.
In this work, we study the quantum critical dynamics of the fractional Ising chain with its interaction generated by a fractional derivative of continuously tunable order, by means of time-dependent variational principle (TDVP) simulations [74 ] with a matrix-product-operator (MPO) and matrix-product-state (MPS) representation of the long-range interacting Hamiltonian and its eigenstates, respectively. Driving the transverse field across the critical point, we obtain the residual kink density as a function of the driving time and find a power-law exponent that continuously varies with the fractional order. We furthermore extract the critical exponents from the equilibrium quantities alone, from the finite-size scaling of the gap, and from the entanglement light cone arising from a local perturbation at criticality. With these exponents, the Kibble-Zurek prediction accounts for the measured scaling at intermediate- and short-range order, while the measured decay is slower at small fractional order. Going beyond the mean kink number, we resolve the higher cumulants and reconstruct the full kink number distribution of a nonintegrable long-range spin chain. These results are benchmarked against the exactly solvable free-fermion approximation of the model, a fractional p p p
Within the free-fermion description, we derive the full statistics of both the excitations and the kinks. We show that the kink number relative to the free-fermion ground state coincides with the number of excitations, up to a fractional-order dependent prefactor. We further prove that in any free-fermion state the kink number variance cannot exceed twice its mean and the total correlation of a bond cannot exceed its local variance, implying a Gaussian limiting distribution for the full kink statistics.
In the TDVP simulations, both bounds are violated deep in the long-range regime, also at the center of the chain, where kinks are created close to each other, consistent with the domain-wall confinement induced by the kink-kink interactions. Nevertheless, the kink distribution tion still approaches a Gaussian for large kink numbers. The paper is organized as follows. Section II introduces the fractional Ising chain, the numerical scheme used throughout, and the numerically obtained kink-number cumulants and full counting statistics. Section III then measures z z z ν \nu ν p p p
Fractional Ising model
Model and numerical method
We consider the fractional transverse-field Ising chain
H ^ I s i n g ( t ) = − g ( t ) ∑ i = 1 L σ ^ i z − ( 1 − g ( t ) ) ∑ i = 1 L ∑ r = 1 L J r , q σ ^ i x σ ^ i + r x (1) \hat{H}_{\mathrm{Ising}}(t) = -g(t)\sum_{i=1}^{L}\hat{\sigma}_{i}^{z}
-(1-g(t))\sum_{i=1}^{L}\sum_{r=1}^{L}J_{r,q}\hat{\sigma}_{i}^{x}\hat{\sigma}_{i+r}^{x} \tag*{(1)} H ^ Ising ( t ) = − g ( t ) i = 1 ∑ L σ ^ i z − ( 1 − g ( t )) i = 1 ∑ L r = 1 ∑ L J r , q σ ^ i x σ ^ i + r x (1) driven by the linear protocol g ( t ) = 1 − t / τ Q g(t)=1-t/\tau_{Q} g ( t ) = 1 − t / τ Q t ∈ [ 0 , τ Q ] t\in[0,\tau_{Q}] t ∈ [ 0 , τ Q ] τ Q \tau_{Q} τ Q i + r ≤ L i+r\leq L i + r ≤ L ℏ = 1 \hbar=1 ℏ = 1 (1) . The couplings are given by the coefficients of the centered discretized Riesz fractional derivative of order q q q [63 ] , expressed as binomial coefficients of the noninteger argument,
J r , q = ( − 1 ) r + 1 ( q q / 2 + r ) . (2) J_{r,q}=(-1)^{r+1}\binom{q}{q/2+r}. \tag*{(2)} J r , q = ( − 1 ) r + 1 ( q /2 + r q ) . (2) For large separations the couplings decay as
J r , q ≃ Γ ( q + 1 ) π sin ( π q 2 ) r − ( 1 + q ) , r → ∞ . (3) J_{r,q}\simeq\frac{\Gamma(q+1)}{\pi}\sin\left(\frac{\pi q}{2}\right)r^{-(1+q)},\qquad r\to\infty. \tag*{(3)} J r , q ≃ π Γ ( q + 1 ) sin ( 2 π q ) r − ( 1 + q ) , r → ∞. (3) The next subleading correction scales as r − ( 3 + q ) r^{-(3+q)} r − ( 3 + q ) [67 ] . For 0 < q < 2 0<q<2 0 < q < 2 q = 2 q=2 q = 2 J 1 , 2 = 1 J_{1,2}=1 J 1 , 2 = 1 q > 2 q>2 q > 2 r − 3 r^{-3} r − 3 [66 ] .
At g = 1 g=1 g = 1 Z 2 \mathbb{Z}_{2} Z 2 σ ^ i x → − σ ^ i x \hat{\sigma}_{i}^{x}\to-\hat{\sigma}_{i}^{x} σ ^ i x → − σ ^ i x x x x q q q g = 0 g=0 g = 0 m m m
V ( m ) = 4 ∑ r ≥ 1 J r , q min ( r , m ) = V 0 + 4 A q q ( 1 − q ) m 1 − q + O ( m − 1 − q ) , (4) \begin{aligned}
V(m)=4\sum_{r\geq1}J_{r,q}\min(r,m) \\
&=V_{0}+\frac{4A_{q}}{q(1-q)}m^{1-q}+O(m^{-1-q}), \tag*{(4)}
\end{aligned} V ( m ) = 4 r ≥ 1 ∑ J r , q min ( r , m ) = V 0 + q ( 1 − q ) 4 A q m 1 − q + O ( m − 1 − q ) , (4) for q ≠ 1 q\ne1 q = 1 A q = Γ ( q + 1 ) sin ( π q / 2 ) / π A_{q}=\Gamma(q+1)\sin(\pi q/2)/\pi A q = Γ ( q + 1 ) sin ( π q /2 ) / π (3) } and $V_{0}=\frac{q}{q-1}\left(q/2\right)$, which equals $4\sum_{r}rJ_{r,q}$ for $q>1$. For $q<1$ the potential grows as $m^{1-q}$, and logarithmically at $q=1$, confining kinks into pairs [72 , 73 ] . For $1<q<2$ it saturates at $V_{0}$, i.e., the energy cost of two free walls, and for $q>2$ it decreases with $m$, as the antiferromagnetic tail repels the walls at short distance. The free-fermion approximation of {xref}Sec. IV A does not contain this kink-kink interaction {citep}[76 ] `.
Away from q = 2 q=2 q = 2 [75 ] using the open-source package ITensors.jl [77 ] , for L = 150 L=150 L = 150 J r , q σ ^ i x σ ^ i + r x J_{r,q}\hat{\sigma}_{i}^{x}\hat{\sigma}_{i+r}^{x} J r , q σ ^ i x σ ^ i + r x 1 ≤ r ≤ L − 1 1\leq r\leq L-1 1 ≤ r ≤ L − 1 (2) rather than from a fit by a sum of exponentials [78 ] . The conversion to an MPO compresses, at each bond l l l J i j J_{ij} J ij i ≤ l < j i\leq l<j i ≤ l < j 10 − 15 10^{-15} 1 0 − 15 [79 ] , so that the MPO bond dimension at bond l l l 10 − 15 10^{-15} 1 0 − 15 q q q L / 2 + 2 = 77 L/2+2=77 L /2 + 2 = 77 10 − 14 10^{-14} 1 0 − 14 [67 ] , and the couplings are reproduced to a relative accuracy of at least 10 − 8 10^{-8} 1 0 − 8 g ( t ) g(t) g ( t ) [74 ] implemented in TimeEvoMPS.jl [80 ] , starting from the paramagnetic product state, with a maximal bond dimension χ max = 200 \chi_{\max}=200 χ m a x = 200 0.01 ≤ δ t ≤ 0.4 0.01\leq\delta t\leq0.4 0.01 ≤ δ t ≤ 0.4 δ t ≤ τ Q / 10 \delta t\leq\tau_{Q}/10 δ t ≤ τ Q /10 10 − 7 10^{-7} 1 0 − 7 10 − 10 10^{-10} 1 0 − 10 χ max \chi_{\max} χ m a x
To control the truncation, each of the 356 driving times was simulated with typically four to six combinations of δ t \delta t δ t 10 − 8 10^{-8} 1 0 − 8 q q q τ Q \tau_Q τ Q κ 1 \kappa_1 κ 1 κ 2 \kappa_2 κ 2 κ 3 \kappa_3 κ 3 κ 2 \kappa_2 κ 2 10 − 7 10^{-7} 1 0 − 7 κ 2 \kappa_2 κ 2 χ max \chi_{\max} χ m a x 10 − 8 10^{-8} 1 0 − 8 q ≤ 0.9 q \leq0.9 q ≤ 0.9 q = 1.25 q = 1.25 q = 1.25 τ Q ≳ 20 \tau_Q \gtrsim20 τ Q ≳ 20 τ Q \tau_Q τ Q q = 0.2 q = 0.2 q = 0.2 τ Q = 30 \tau_Q = 30 τ Q = 30 q ≲ 0.3 q \lesssim0.3 q ≲ 0.3 κ 2 / κ 1 \kappa_2/\kappa_1 κ 2 / κ 1 κ 2 / κ 1 ≃ 3.1 \kappa_2/\kappa_1 \simeq3.1 κ 2 / κ 1 ≃ 3.1 q = 0.1 q = 0.1 q = 0.1 q = 0.2 q = 0.2 q = 0.2 q = 0.3 q = 0.3 q = 0.3 τ Q ≥ 50 \tau_Q \geq50 τ Q ≥ 50
On the open chain each spin feels, on average over the sites, 59% of the total coupling ∑ r ≠ 0 J r , q = J q \sum_{r\ne0} J_{r,q} = J_q ∑ r = 0 J r , q = J q q = 0.2 q = 0.2 q = 0.2 q = 0.5 q = 0.5 q = 0.5 q = 1.5 q = 1.5 q = 1.5 L − q L^{-q} L − q L = 200 L = 200 L = 200 L = 150 L = 150 L = 150 (9) , the gaps of all system sizes collapse onto a single curve near the minimum for every q q q q ≥ 1.5 q \geq1.5 q ≥ 1.5 g g g q q q L ≳ 140 L \gtrsim140 L ≳ 140 Figure 5 ). In addition, as the bound of App. F holds for every L L L L = 12 L = 12 L = 12
Figure 5. Scaling collapse of the low-energy gap for (a) q = 0.1 q = 0.1 q = 0.1 q = 0.2 q = 0.2 q = 0.2 q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25 q q q Δ L z / ( 1 + b L − ω ) \Delta L^z/(1+bL^{-\omega}) Δ L z / ( 1 + b L − ω ) [ g − g c ( L ) ] L 1 / ν [g-g_c(L)]L^{1/\nu} [ g − g c ( L )] L 1/ ν z z z b b b (9) and ν \nu ν g c g_c g c b ′ b' b ′ (10) . Away from the minimum the curves of small sizes separate, most visibly at small q q q L L L
Kink number cumulants and full counting statistics
To access universal signatures beyond the KZM, we now turn to the full counting statistics of kinks [12 , 13 , 29 ] . Specifically, we focus on the probability distribution of finding n n n ∣ Ψ ( τ Q ) ⟩ \lvert\Psi(\tau_Q)\rangle ∣ Ψ ( τ Q )⟩
P ( n ) = ⟨ Ψ ( τ Q ) ∣ δ N ^ , n ∣ Ψ ( τ Q ) ⟩ , (5) P(n) = \langle\Psi(\tau_Q)|\delta_{\hat{N},n}|\Psi(\tau_Q)\rangle, \tag*{(5)} P ( n ) = ⟨ Ψ ( τ Q ) ∣ δ N ^ , n ∣Ψ ( τ Q )⟩ , (5) with the kink-number operator
N ^ = 1 2 ∑ i = 1 L − 1 ( 1 − σ ^ i x σ ^ i + 1 x ) . (6) \hat{N} = \frac{1}{2}\sum_{i=1}^{L-1}\left(1-\hat{\sigma}^{x}_{i}\hat{\sigma}^{x}_{i+1}\right). \tag*{(6)} N ^ = 2 1 i = 1 ∑ L − 1 ( 1 − σ ^ i x σ ^ i + 1 x ) . (6) A complementary characterization is provided by the Fourier transform of P ( n ) P(n) P ( n ) P ~ ( θ ) = ∑ n e i θ n P ( n ) \widetilde{P}(\theta) = \sum_n e^{i\theta n}P(n) P ( θ ) = ∑ n e i θ n P ( n )
ln P ~ ( θ ) = ∑ ℓ = 1 ∞ κ ℓ ( i θ ) ℓ ℓ ! , κ ℓ = ∂ θ ℓ ln P ~ ( θ ) ∣ θ = 0 , (7) \ln\widetilde{P}(\theta) = \sum_{\ell=1}^{\infty}\kappa_\ell\frac{(i\theta)^\ell}{\ell!}, \qquad\kappa_\ell= \left.\partial_\theta^\ell\ln\widetilde{P}(\theta)\right|_{\theta=0}, \tag*{(7)} ln P ( θ ) = ℓ = 1 ∑ ∞ κ ℓ ℓ ! ( i θ ) ℓ , κ ℓ = ∂ θ ℓ ln P ( θ ) θ = 0 , (7) evaluated with the final time-evolved MPS for ∣ Ψ ( τ Q ) ⟩ \lvert\Psi(\tau_Q)\rangle ∣ Ψ ( τ Q )⟩ N ^ \hat{N} N ^ σ ^ x \hat{\sigma}^{x} σ ^ x P ~ ( θ ) \widetilde{P}(\theta) P ( θ ) θ \theta θ N ^ \hat{N} N ^ P ~ ( θ ) \widetilde{P}(\theta) P ( θ ) P ( n ) P(n) P ( n )
Besides the kink number distribution, we investigate its first three cumulants, κ 1 = ⟨ N ^ ⟩ \kappa_1 = \langle\hat{N}\rangle κ 1 = ⟨ N ^ ⟩ κ 2 = ⟨ ( N ^ − κ 1 ) 2 ⟩ \kappa_2 = \langle(\hat{N}-\kappa_1)^2\rangle κ 2 = ⟨( N ^ − κ 1 ) 2 ⟩ κ 3 = ⟨ ( N ^ − κ 1 ) 3 ⟩ \kappa_3 = \langle(\hat{N}-\kappa_1)^3\rangle κ 3 = ⟨( N ^ − κ 1 ) 3 ⟩ ⟨ ⋅ ⟩ = ⟨ Ψ ( τ Q ) ∣ ⋅ ∣ Ψ ( τ Q ) ⟩ \langle\cdot\rangle= \langle\Psi(\tau_Q)|\cdot|\Psi(\tau_Q)\rangle ⟨ ⋅ ⟩ = ⟨ Ψ ( τ Q ) ∣ ⋅ ∣Ψ ( τ Q )⟩ ⟨ N ^ m ⟩ \langle\hat{N}^m\rangle ⟨ N ^ m ⟩ m = 1 , 2 , 3 m = 1,2,3 m = 1 , 2 , 3 N ^ \hat{N} N ^
Figure 1 shows the mean defect number in the long-range regime, for q = 0.1 q = 0.1 q = 0.1 q = 0.2 q = 0.2 q = 0.2 q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25 q = 0.2 q = 0.2 q = 0.2 q = 0.1 q = 0.1 q = 0.1 q q q q > 2 q > 2 q > 2
Figure 2 shows the exponents α \alpha α τ Q − α \tau_Q^{-\alpha} τ Q − α q ≲ 1 q \lesssim1 q ≲ 1 τ Q \tau_Q τ Q q ≲ 0.3 q \lesssim0.3 q ≲ 0.3 τ Q − 1 / ( 2 q ) \tau_Q^{-1/(2q)} τ Q − 1/ ( 2 q ) q > 1 / 4 q > 1/4 q > 1/4 τ Q − 2 \tau_Q^{-2} τ Q − 2 q < 1 / 4 q < 1/4 q < 1/4 L = 150 L = 150 L = 150 r ≤ L / 2 r \leq L/2 r ≤ L /2 q ≳ 1.5 q \gtrsim1.5 q ≳ 1.5 § 2.13 ), which steepens the decay of the mean at L = 150 L = 150 L = 150 α ≃ 0.59 \alpha\simeq0.59 α ≃ 0.59 q = 1.5 q = 1.5 q = 1.5 q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25 q > 2 q > 2 q > 2 1 / 2 1/2 1/2 Figure 3 , in the same three regimes. When the mean is large, the kink number distribution approaches a Gaussian with mean κ 1 \kappa_1 κ 1 κ 2 \kappa_2 κ 2 Figure 4 ) and the third cumulant (Figure 10 in App. § 1 ) approach the free-fermion results at slightly smaller q q q q ≳ 1.25 q\gtrsim1.25 q ≳ 1.25 Figure 11 in App. § 1 ). Deep in the long-range regime, however, the range of τ Q \tau_Q τ Q (4) . At slow ramps, where the mean is small, the histograms depart from the Gaussian form.
Figure 3. Kink number histograms from the characteristic function for (a) q = 0.2 q = 0.2 q = 0.2 q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25 q = 0.2 q = 0.2 q = 0.2
Figure 4. Variance κ 2 \kappa_2 κ 2 q = 0.1 q = 0.1 q = 0.1 q = 0.2 q = 0.2 q = 0.2 q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25 Figure 1 . For q = 0.1 q = 0.1 q = 0.1 q = 0.2 q = 0.2 q = 0.2 q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25
Figure 10. Third cumulant κ 3 \kappa_3 κ 3 q = 0.1 , 0.2 q=0.1,0.2 q = 0.1 , 0.2 q = 1.5 q=1.5 q = 1.5 q = 2.25 q=2.25 q = 2.25 Figure 1 .
Figure 11. Cumulant ratios as functions of the driving time. Panels (a) and (b) show κ 2 / κ 1 \kappa_2/\kappa_1 κ 2 / κ 1 κ 3 / κ 1 \kappa_3/\kappa_1 κ 3 / κ 1 q = 0.1 q=0.1 q = 0.1 q = 0.2 q=0.2 q = 0.2 q = 1.5 q=1.5 q = 1.5 q = 2.25 q=2.25 q = 2.25 g = 0 g=0 g = 0 ( κ ℓ − κ ℓ G S ) / ( κ 1 − κ 1 G S ) (\kappa_\ell-\kappa_\ell^{\mathrm{GS}})/(\kappa_1-\kappa_1^{\mathrm{GS}}) ( κ ℓ − κ ℓ GS ) / ( κ 1 − κ 1 GS ) κ ℓ e x / κ 1 e x \kappa_\ell^{\mathrm{ex}}/\kappa_1^{\mathrm{ex}} κ ℓ ex / κ 1 ex κ ℓ / κ 1 \kappa_\ell/\kappa_1 κ ℓ / κ 1 R 2 = 2 [ 1 − 2 − 1 / ( 2 q ∗ ) ] R_2=2[1-2^{-1/(2q_*)}] R 2 = 2 [ 1 − 2 − 1/ ( 2 q ∗ ) ] R 3 = 4 [ 1 − 3 ⋅ 2 − 1 / ( 2 q ∗ ) + 2 ⋅ 3 − 1 / ( 2 q ∗ ) ] R_3=4[1-3\cdot2^{-1/(2q_*)}+2\cdot3^{-1/(2q_*)}] R 3 = 4 [ 1 − 3 ⋅ 2 − 1/ ( 2 q ∗ ) + 2 ⋅ 3 − 1/ ( 2 q ∗ ) ] (35) , with q ∗ = min ( q , 1 ) q_*=\min(q,1) q ∗ = min ( q , 1 ) q < 1 / 4 q<1/4 q < 1/4 κ 2 e x / κ 1 e x = 2 [ 1 − J 0 / ( 4 I 0 τ Q 2 q ) ] \kappa_2^{\mathrm{ex}}/\kappa_1^{\mathrm{ex}}=2[1-J_0/(4I_0\tau_Q^{2q})] κ 2 ex / κ 1 ex = 2 [ 1 − J 0 / ( 4 I 0 τ Q 2 q )] κ 3 e x / κ 1 e x = 4 [ 1 − 3 J 0 / ( 4 I 0 τ Q 2 q ) ] \kappa_3^{\mathrm{ex}}/\kappa_1^{\mathrm{ex}}=4[1-3J_0/(4I_0\tau_Q^{2q})] κ 3 ex / κ 1 ex = 4 [ 1 − 3 J 0 / ( 4 I 0 τ Q 2 q )] I 0 I_0 I 0 (76) and J 0 J_0 J 0 R ~ 2 ≃ 1.91 \widetilde R_2\simeq1.91 R 2 ≃ 1.91 R ~ 3 ≃ 3.45 \widetilde R_3\simeq3.45 R 3 ≃ 3.45 q = 0.1 q=0.1 q = 0.1 R ~ 2 ≃ 1.92 \widetilde R_2\simeq1.92 R 2 ≃ 1.92 R ~ 3 ≃ 3.53 \widetilde R_3\simeq3.53 R 3 ≃ 3.53 q = 0.2 q=0.2 q = 0.2 q ≤ 0.2 q\leq0.2 q ≤ 0.2 κ 2 / κ 1 \kappa_2/\kappa_1 κ 2 / κ 1
Gap scaling and entanglement light cone
For a linear ramp through the critical point g c g_c g c ξ ∼ ∣ g − g c ∣ − ν \xi\sim\lvert g-g_c\rvert^{-\nu} ξ ∼ ∣ g − g c ∣ − ν τ e q ∼ ξ z ∼ ∣ g − g c ∣ − z ν \tau_{\mathrm{eq}}\sim\xi^z\sim\lvert g-g_c\rvert^{-z\nu} τ eq ∼ ξ z ∼ ∣ g − g c ∣ − z ν τ e q \tau_{\mathrm{eq}} τ eq τ Q ∣ g − g c ∣ \tau_Q\lvert g-g_c\rvert τ Q ∣ g − g c ∣ ∣ g − g c ∣ ∼ τ Q − 1 / ( 1 + z ν ) \lvert g-g_c\rvert\sim\tau_Q^{-1/(1+z\nu)} ∣ g − g c ∣ ∼ τ Q − 1/ ( 1 + z ν ) ξ K Z ∼ τ Q ν / ( 1 + z ν ) \xi_{\mathrm{KZ}}\sim\tau_Q^{\nu/(1+z\nu)} ξ KZ ∼ τ Q ν / ( 1 + z ν ) κ 1 / L ∼ ξ K Z − 1 ∼ τ Q − 1 / ( 1 + z ν ) \kappa_1/L\sim\xi_{\mathrm{KZ}}^{-1}\sim\tau_Q^{-1/(1+z\nu)} κ 1 / L ∼ ξ KZ − 1 ∼ τ Q − 1/ ( 1 + z ν ) [5 ] . In this section, we test whether this KZ scaling, with ν \nu ν z z z z z z
Finite-size scaling of the critical energy gap
We extract both exponents from the universal scaling form of the critical gap,
Δ ( g , L ) = L − z F [ ( g − g c ) L 1 / ν ] , (8) \Delta(g,L)=L^{-z}\mathcal{F}\left[(g-g_c)L^{1/\nu}\right], \tag*{(8)} Δ ( g , L ) = L − z F [ ( g − g c ) L 1/ ν ] , (8) with F ( x ) ∼ x z ν \mathcal{F}(x)\sim x^{z\nu} F ( x ) ∼ x z ν x ≫ 1 x\gg1 x ≫ 1 F ( x ) ∼ 1 \mathcal{F}(x)\sim1 F ( x ) ∼ 1 x ≪ 1 x\ll1 x ≪ 1 Δ ( g c , L ) ∼ L − z \Delta(g_c,L)\sim L^{-z} Δ ( g c , L ) ∼ L − z
Δ ( g c , L ) = a L − z ( 1 + b L − ω ) , (9) \Delta(g_c,L)=aL^{-z}\left(1+bL^{-\omega}\right), \tag*{(9)} Δ ( g c , L ) = a L − z ( 1 + b L − ω ) , (9) where the subleading term with ω > 0 \omega>0 ω > 0 z z z L L L Δ ( g , L → ∞ ) ∼ ∣ g − g c ∣ z ν \Delta(g,L\to\infty)\sim\lvert g-g_c\rvert^{z\nu} Δ ( g , L → ∞ ) ∼ ∣ g − g c ∣ z ν ν \nu ν g c ( L ) g_c(L) g c ( L ) d Δ / d g = L 1 / ν − z F ′ ( x ) = 0 d\Delta/dg=L^{1/\nu-z}\mathcal{F}'(x)=0 d Δ/ d g = L 1/ ν − z F ′ ( x ) = 0 F ′ ( x ) = 0 \mathcal{F}'(x)=0 F ′ ( x ) = 0
x = ( g c ( L ) − g c ) L 1 / ν x = (g_c(L) - g_c)L^{1/\nu} x = ( g c ( L ) − g c ) L 1/ ν L L L a ′ a' a ′
g c ( L ) = g c + a ′ L − 1 / ν ( 1 + b ′ L − ω ′ ) . (10) g_c(L) = g_c + a' L^{-1/\nu}\left(1 + b' L^{-\omega'}\right). \tag*{(10)} g c ( L ) = g c + a ′ L − 1/ ν ( 1 + b ′ L − ω ′ ) . (10) This fixes ν \nu ν z z z g c ( L ) g_c(L) g c ( L ) Figure 9 and Tables II and III of App. § 2.11 .
Figure 9. (a) Minimum gap Δ ( g c , L ) \Delta(g_c,L) Δ ( g c , L ) g c ( L ) g_c(L) g c ( L ) (9) and (10) with the parameters of Tables II and III.
In the numerical realization, we used DMRG simulations [75 , 77 ] to search for the minimum gap in a sufficiently large window around the free-fermion critical point g c F F = J q / ( 2 + J q ) g_c^{\mathrm{FF}} = J_q/(2 + J_q) g c FF = J q / ( 2 + J q ) J q = ( q / 2 ) J_q = \left(q/2\right) J q = ( q /2 ) (1) divided by 1 − g 1-g 1 − g
H ^ D M R G = − h ∑ i = 1 L σ ^ i z − ∑ i = 1 L − 1 ∑ r = 1 L − i J r , q σ ^ i x σ ^ i + r x , h = g 1 − g , (11) \hat{H}_{\mathrm{DMRG}} = -h \sum_{i=1}^{L}\hat{\sigma}_i^z - \sum_{i=1}^{L-1}\sum_{r=1}^{L-i}J_{r,q}\hat{\sigma}_i^x\hat{\sigma}_{i+r}^x,\qquad h = \frac{g}{1-g}, \tag*{(11)} H ^ DMRG = − h i = 1 ∑ L σ ^ i z − i = 1 ∑ L − 1 r = 1 ∑ L − i J r , q σ ^ i x σ ^ i + r x , h = 1 − g g , (11) with a single field parameter h h h Δ = E 2 − E 0 \Delta= E_2 - E_0 Δ = E 2 − E 0 g g g L = 200 L = 200 L = 200 χ max = 200 \chi_{\max} = 200 χ m a x = 200 10 − 10 10^{-10} 1 0 − 10 max ( 20 , 0.1 ∣ E 0 ∣ ) \max(20, 0.1|E_0|) max ( 20 , 0.1∣ E 0 ∣ ) 10 − 6 10^{-6} 1 0 − 6 10 − 3 10^{-3} 1 0 − 3 χ max = 400 \chi_{\max} = 400 χ m a x = 400
less than 1%, even at q = 0.1 q = 0.1 q = 0.1 q = 1.5 q = 1.5 q = 1.5 L = 20 L = 20 L = 20 Δ = 0.282 \Delta= 0.282 Δ = 0.282 g = 0.450 g = 0.450 g = 0.450 (1) , in agreement with our DMRG value 0.283. For each L L L [81 , 82 ] to a relative accuracy of 2 × 10 − 4 2 \times10^{-4} 2 × 1 0 − 4 h h h (9) and (10) by least squares [82 ] , with next-to-leading order correction terms kept only where the data resolve them. The exponents are listed in Table I. As the free-fermion model in Sec. IV predicts z ν = 1 z\nu= 1 z ν = 1 q q q 0.4 ≤ q ≤ 2.25 0.4 \leq q \leq2.25 0.4 ≤ q ≤ 2.25 z ν ≃ 0.6 z\nu\simeq0.6 z ν ≃ 0.6 z ν ≃ 0.7 z\nu\simeq0.7 z ν ≃ 0.7 q ≥ 2.5 q \geq2.5 q ≥ 2.5 q < 0.3 q < 0.3 q < 0.3 z ν z\nu z ν z = q / 2 z = q/2 z = q /2 q = 2 / 3 q = 2/3 q = 2/3 [68 , 69 , 70 ] , and z ν ≃ 0.50 z\nu\simeq0.50 z ν ≃ 0.50 z ν = 1 / 2 z\nu= 1/2 z ν = 1/2 ν \nu ν 1 / q 1/q 1/ q 2 / ( 3 q ) 2/(3q) 2/ ( 3 q ) z z z (9) . At q = 1.5 q = 1.5 q = 1.5 z ≃ 0.82 z \simeq0.82 z ≃ 0.82 z ≃ 0.83 z \simeq0.83 z ≃ 0.83 [35 ] , and lies between the earlier estimate z ≃ 0.75 z \simeq0.75 z ≃ 0.75 [67 ] and the free-fermion value z = 1 z = 1 z = 1 L = 12 L = 12 L = 12 L L L ν \nu ν z z z Figure 5 . There, the gap is rescaled at next-to-leading order, dividing out the subleading factor of Eq. (9) and subtracting the critical point g c ( L ) g_c(L) g c ( L ) (10) at the given finite system size L L L g c g_c g c ν / ( z ν + 1 ) \nu/(z\nu+1) ν / ( z ν + 1 ) Figure 2 . Tested on the exactly solvable free-fermion chain of Sec. IV A, these fits have a systematic uncertainty of 10–20%. These KZ exponents deviate appreciably from the TDVP results for q ≲ 1 q \lesssim1 q ≲ 1 q > 1 q > 1 q > 1
Figure 2. Power-law exponents α \alpha α q > 1 q>1 q > 1 q < 1 q<1 q < 1 q > 1 q>1 q > 1 q > 2 q>2 q > 2 q q q
Table 1. Critical exponents z z z ν \nu ν (9) and (10) , the extrapolated critical point g c g_c g c (1) , and the resulting KZ exponent ν / ( z ν + 1 ) \nu/(z\nu+1) ν / ( z ν + 1 )
Entanglement light cone
As a complementary route to the dynamical exponent, z z z [83 ] , whereas algebraically decaying couplings admit nonlinear and, for sufficiently slow decay, unbounded light cones [84 , 85 , 86 ] , as observed directly in trapped-ion crystals [87 , 88 ] . We prepare the ground state of H ^ I s i n g \hat{H}_{\mathrm{Ising}} H ^ Ising g c ( L ) g_c(L) g c ( L ) L = 150 L = 150 L = 150 [74 ] under the locally perturbed Hamiltonian
H ^ ( t ) = H ^ I s i n g ( g c ( L ) ) + h 0 W B H ( t ) σ ^ j ∗ z , (12) \hat{H}(t) = \hat{H}_{\mathrm{Ising}}(g_c(L)) + h_0 W_{\mathrm{BH}}(t) \hat{\sigma}^{z}_{j^*}, \tag*{(12)} H ^ ( t ) = H ^ Ising ( g c ( L )) + h 0 W BH ( t ) σ ^ j ∗ z , (12) with j ∗ ≡ ⌊ L / 2 ⌋ j^* \equiv\lfloor L/2 \rfloor j ∗ ≡ ⌊ L /2 ⌋ z z z h 0 = 2 h_0 = 2 h 0 = 2 T w = 1.2 T_w = 1.2 T w = 1.2
W B H ( t ) = a 0 − a 1 cos 2 π t T w + a 2 cos 4 π t T w − a 3 cos 6 π t T w , (13) W_{\mathrm{BH}}(t) = a_0-a_1\cos\frac{2\pi t}{T_w}+a_2\cos\frac{4\pi t}{T_w}-a_3\cos\frac{6\pi t}{T_w}, \tag*{(13)} W BH ( t ) = a 0 − a 1 cos T w 2 π t + a 2 cos T w 4 π t − a 3 cos T w 6 π t , (13) for 0 ≤ t ≤ T w 0 \le t \le T_w 0 ≤ t ≤ T w W B H ( t ) = 0 W_{\mathrm{BH}}(t) = 0 W BH ( t ) = 0 ( a 0 , a 1 , a 2 , a 3 ) = ( 0.35875 , 0.48829 , 0.14128 , 0.01168 ) (a_0,a_1,a_2,a_3) = (0.35875, 0.48829, 0.14128, 0.01168) ( a 0 , a 1 , a 2 , a 3 ) = ( 0.35875 , 0.48829 , 0.14128 , 0.01168 ) [89 ] . The window suppresses the high-frequency content of the pulse, so that mainly low-energy quasiparticles are excited. The time evolution is carried out with two-site TDVP up to t max = 30 t_{\max} = 30 t m a x = 30 δ t = 0.02 \delta t = 0.02 δ t = 0.02 χ max = 200 \chi_{\max} = 200 χ m a x = 200 10 − 10 10^{-10} 1 0 − 10
The propagation is tracked through the bond entanglement entropy across the cut between sites j j j j + 1 j+1 j + 1
S b o n d ( j , t ) = − Tr [ ρ ^ ≤ j ( t ) ln ρ ^ ≤ j ( t ) ] , (14) S_{\mathrm{bond}}(j,t) = -\operatorname{Tr}\left[\hat{\rho}_{\leq j}(t)\ln\hat{\rho}_{\leq j}(t)\right], \tag*{(14)} S bond ( j , t ) = − Tr [ ρ ^ ≤ j ( t ) ln ρ ^ ≤ j ( t ) ] , (14) with ρ ^ ≤ j ( t ) \hat{\rho}_{\leq j}(t) ρ ^ ≤ j ( t ) 1 , … , j 1,\ldots,j 1 , … , j S b o n d ( j , t ) S_{\mathrm{bond}}(j,t) S bond ( j , t ) S b o n d ( j , t ) = s S_{\mathrm{bond}}(j,t) = s S bond ( j , t ) = s [67 ] . At each time, it lies at the first bond on either side of j ∗ j^* j ∗ S b o n d S_{\mathrm{bond}} S bond s s s s = 0.75 s = 0.75 s = 0.75 s = 1 s = 1 s = 1 Δ s = 0.025 \Delta s = 0.025 Δ s = 0.025 S b o n d ≲ 0.68 S_{\mathrm{bond}} \lesssim0.68 S bond ≲ 0.68 S b o n d ≲ 1.3 S_{\mathrm{bond}} \lesssim1.3 S bond ≲ 1.3 Figure 6 , and fitted the fronts over the whole simulated time window, t ≤ t max t \leq t_{\max} t ≤ t m a x
Figure 6. Entanglement light cone at criticality for fractional order q = 1.5 q = 1.5 q = 1.5 L = 150 L = 150 L = 150 S b o n d ( j , t ) S_{\mathrm{bond}}(j,t) S bond ( j , t ) S b o n d ( j , t ) = s S_{\mathrm{bond}}(j,t)=s S bond ( j , t ) = s s = 0.8 s=0.8 s = 0.8 s = 1 s=1 s = 1 ∣ j − j ∗ ∣ ∝ t 1 / z \lvert j-j^*\rvert\propto t^{1/z} ∣ j − j ∗ ∣ ∝ t 1/ z ∣ j − j ∗ ∣ ∝ t \lvert j-j^*\rvert\propto t ∣ j − j ∗ ∣ ∝ t
∣ j − j ∗ ∣ ∼ t 1 / z . (15) |j-j^*| \sim t^{1/z}. \tag*{(15)} ∣ j − j ∗ ∣ ∼ t 1/ z . (15) Larger z z z [67 ] . At q = 1.5 q = 1.5 q = 1.5 z = 0.82 ( 4 ) z = 0.82(4) z = 0.82 ( 4 ) Figure 6 for s = 0.8 s = 0.8 s = 0.8 s = 1 s = 1 s = 1 z ≃ 0.82 z \simeq0.82 z ≃ 0.82 Figure 6 also shows a strictly ballistic cone, ∣ j − j ∗ ∣ = v t |j-j^*| = vt ∣ j − j ∗ ∣ = v t t = 0 t = 0 t = 0 j ∗ j^* j ∗ v = ∣ j − j ∗ ∣ max / t max v = |j-j^*|_{\max}/t_{\max} v = ∣ j − j ∗ ∣ m a x / t m a x q = 1.5 q = 1.5 q = 1.5 q < 1 q < 1 q < 1 r − 2 r^{-2} r − 2 z z z [35 , 86 ] . Accordingly, Figure 2 shows the KZ exponents obtained with this z z z ν \nu ν q > 1 q>1 q > 1
Free-fermion description
In this section, we investigate to what extent the full many-body dynamics can be captured by the simplest free-fermion approximation. For this, we employ the Jordan-Wigner transformation for the spin operators and truncate the arising fermionic multibody interaction terms at quadratic order, which yields a free-fermion description. The resulting model is a fractional p p p q q q
Truncated Jordan-Wigner mapping and the fractional p-wave superconductor
The Jordan-Wigner transformation σ ^ i z = 1 − 2 c ^ i † c ^ i \hat{\sigma}_i^z=1-2\hat{c}_i^\dagger\hat{c}_i σ ^ i z = 1 − 2 c ^ i † c ^ i σ ^ i x = K ^ i ( c ^ i † + c ^ i ) \hat{\sigma}_i^x=\hat{\mathcal{K}}_i(\hat{c}_i^\dagger+\hat{c}_i) σ ^ i x = K ^ i ( c ^ i † + c ^ i ) K ^ i = ∏ j < i ( 1 − 2 c ^ j † c ^ j ) \hat{\mathcal{K}}_i=\prod_{j<i}(1-2\hat{c}_j^\dagger\hat{c}_j) K ^ i = ∏ j < i ( 1 − 2 c ^ j † c ^ j ) − ∑ i σ ^ i z = 2 ∑ i c ^ i † c ^ i + c o n s t -\sum_i\hat{\sigma}_i^z=2\sum_i\hat{c}_i^\dagger\hat{c}_i+\mathrm{const} − ∑ i σ ^ i z = 2 ∑ i c ^ i † c ^ i + const
σ ^ i x σ ^ i + r x = ( c ^ i † − c ^ i ) ∏ j = i + 1 i + r − 1 ( 1 − 2 c ^ j † c ^ j ) ( c ^ i + r † + c ^ i + r ) , (16) \hat{\sigma}_i^x\hat{\sigma}_{i+r}^x=(\hat{c}_i^\dagger-\hat{c}_i)\prod_{j=i+1}^{i+r-1}(1-2\hat{c}_j^\dagger\hat{c}_j)(\hat{c}_{i+r}^\dagger+\hat{c}_{i+r}), \tag*{(16)} σ ^ i x σ ^ i + r x = ( c ^ i † − c ^ i ) j = i + 1 ∏ i + r − 1 ( 1 − 2 c ^ j † c ^ j ) ( c ^ i + r † + c ^ i + r ) , (16) with the c ^ i \hat{c}_i c ^ i c ^ i † \hat{c}_i^\dagger c ^ i † K ^ i \hat{\mathcal{K}}_i K ^ i K ^ i = 1 \hat{\mathcal{K}}_i=1 K ^ i = 1 σ ^ i x σ ^ i + r x ≈ c ^ i † c ^ i + r + c ^ i † c ^ i + r † + h . c . \hat{\sigma}_i^x\hat{\sigma}_{i+r}^x\approx\hat{c}_i^\dagger\hat{c}_{i+r}+\hat{c}_i^\dagger\hat{c}_{i+r}^\dagger+\mathrm{h.c.} σ ^ i x σ ^ i + r x ≈ c ^ i † c ^ i + r + c ^ i † c ^ i + r † + h.c. q = 2 q=2 q = 2 [45 ] . Away from q = 2 q=2 q = 2 [76 ] , and the resulting long-range Kitaev chain reaches short-range exponents already for q > 1 q>1 q > 1 [90 ] , whereas Ising chains with power-law couplings r − ( 1 + q ) r^{-(1+q)} r − ( 1 + q ) q > 7 / 4 q>7/4 q > 7/4 [35 , 69 ] . After the truncation, the Hamiltonian takes the form of a p p p
H ^ F F ( t ) = 2 g ( t ) ∑ i c ^ i † c ^ i − ( 1 − g ( t ) ) ∑ i ∑ r J r , q ( c ^ i † c ^ i + r + c ^ i † c ^ i + r † + h . c . ) , (17) \hat{H}_{\mathrm{FF}}(t)=2g(t)\sum_i\hat{c}_i^\dagger\hat{c}_i-(1-g(t))\sum_i\sum_r J_{r,q}\left(\hat{c}_i^\dagger\hat{c}_{i+r}+\hat{c}_i^\dagger\hat{c}_{i+r}^\dagger+\mathrm{h.c.}\right), \tag*{(17)} H ^ FF ( t ) = 2 g ( t ) i ∑ c ^ i † c ^ i − ( 1 − g ( t )) i ∑ r ∑ J r , q ( c ^ i † c ^ i + r + c ^ i † c ^ i + r † + h.c. ) , (17) with hopping and pairing amplitudes given by the same fractional couplings J r , q J_{r,q} J r , q (2) . The system is driven by the same linear protocol as in Sec. II A, from the trivial phase at g = 1 g=1 g = 1 p p p g = 0 g = 0 g = 0 c ^ k = L − 1 / 2 e − i π / 4 ∑ j e i k j c ^ j \hat{c}_k = L^{-1/2}e^{-i\pi/4}\sum_j e^{ikj}\hat{c}_j c ^ k = L − 1/2 e − iπ /4 ∑ j e ik j c ^ j k k k − k -k − k k > 0 k > 0 k > 0
H ^ F F ( t ) = ∑ k > 0 ψ ^ k † H k ( t ) ψ ^ k , H k ( t ) = ξ k ( t ) τ z + Δ k ( t ) τ x (18) \hat{H}_{\mathrm{FF}}(t) = \sum_{k>0} \hat{\psi}_k^\dagger H_k(t)\hat{\psi}_k,\qquad H_k(t) = \xi_k(t)\tau^z+\Delta_k(t)\tau^x \tag*{(18)} H ^ FF ( t ) = k > 0 ∑ ψ ^ k † H k ( t ) ψ ^ k , H k ( t ) = ξ k ( t ) τ z + Δ k ( t ) τ x (18) with the Nambu spinor ψ ^ k : = ( c ^ k , c ^ − k † ) T \hat{\psi}_k := (\hat{c}_k,\hat{c}_{-k}^\dagger)^T ψ ^ k := ( c ^ k , c ^ − k † ) T τ x , y , z \tau^{x,y,z} τ x , y , z
ξ k ( t ) = 2 g ( t ) − ( 1 − g ( t ) ) j q ( k ) , \xi_k(t)=2g(t)-(1-g(t))j_q(k), ξ k ( t ) = 2 g ( t ) − ( 1 − g ( t )) j q ( k ) , Δ k ( t ) = ( 1 − g ( t ) ) d q ( k ) . (19) \Delta_k(t)=(1-g(t))d_q(k). \tag*{(19)} Δ k ( t ) = ( 1 − g ( t )) d q ( k ) . (19) The hopping and pairing dispersions admit the closed forms
j q ( k ) = 2 ∑ r ≥ 1 J r , q cos ( k r ) = J q − ∣ 2 sin k 2 ∣ q (20) j_q(k)=2\sum_{r\geq1}J_{r,q}\cos(kr)=J_q-\left|2\sin\frac{k}{2}\right|^q \tag*{(20)} j q ( k ) = 2 r ≥ 1 ∑ J r , q cos ( k r ) = J q − 2 sin 2 k q (20) d q ( k ) = 2 ∑ r ≥ 1 J r , q sin ( k r ) = − 2 ( q q / 2 ) Im [ 2 F 1 ( − q 2 , 1 ; q 2 + 1 ; e i k ) ] , (21) \begin{aligned}
d_q(k)=2\sum_{r\geq1}J_{r,q}\sin(kr) \\
&=-2\binom{q}{q/2}\operatorname{Im}\left[{}_2F_1\left(-\frac{q}{2},1;\frac{q}{2}+1;e^{ik}\right)\right], \tag*{(21)}
\end{aligned} d q ( k ) = 2 r ≥ 1 ∑ J r , q sin ( k r ) = − 2 ( q /2 q ) Im [ 2 F 1 ( − 2 q , 1 ; 2 q + 1 ; e ik ) ] , (21) with J q ≡ ( q q / 2 ) J_q\equiv\binom{q}{q/2} J q ≡ ( q /2 q ) j q j_q j q d q d_q d q k k k d q ( 0 ) = 0 d_q(0)=0 d q ( 0 ) = 0
Each TLS (18) is identical to a spin coupled to the effective field h ⃗ k = ( Δ k , 0 , ξ k ) \vec{h}_k=(\Delta_k,0,\xi_k) h k = ( Δ k , 0 , ξ k ) x z xz x z z z z
ϵ k = ∣ h ⃗ k ∣ = ξ k 2 + Δ k 2 , tan 2 ϑ k = Δ k ξ k . (22) \epsilon_k=|\vec{h}_k|=\sqrt{\xi_k^2+\Delta_k^2},\qquad\tan2\vartheta_k=\frac{\Delta_k}{\xi_k}. \tag*{(22)} ϵ k = ∣ h k ∣ = ξ k 2 + Δ k 2 , tan 2 ϑ k = ξ k Δ k . (22) As h ⃗ k \vec{h}_k h k x z xz x z y y y ϑ k \vartheta_k ϑ k γ ^ k = cos ϑ k c ^ k + sin ϑ k c ^ − k † \hat{\gamma}_k=\cos\vartheta_k\hat{c}_k+\sin\vartheta_k\hat{c}_{-k}^\dagger γ ^ k = cos ϑ k c ^ k + sin ϑ k c ^ − k † γ ^ − k = − sin ϑ k c ^ k + cos ϑ k c ^ − k † \hat{\gamma}_{-k}=-\sin\vartheta_k\hat{c}_k+\cos\vartheta_k\hat{c}_{-k}^\dagger γ ^ − k = − sin ϑ k c ^ k + cos ϑ k c ^ − k †
H ^ = ∑ k > 0 ϵ k ( γ ^ k † γ ^ k + γ ^ − k † γ ^ − k − 1 ) . (23) \hat{H}=\sum_{k>0}\epsilon_k\left(\hat{\gamma}_k^\dagger\hat{\gamma}_k+\hat{\gamma}_{-k}^\dagger\hat{\gamma}_{-k}-1\right). \tag*{(23)} H ^ = k > 0 ∑ ϵ k ( γ ^ k † γ ^ k + γ ^ − k † γ ^ − k − 1 ) . (23) The instantaneous ground state is the quasiparticle vacuum, γ ^ ± k ∣ Ω ⟩ = 0 \hat{\gamma}_{\pm k}|\Omega\rangle=0 γ ^ ± k ∣Ω ⟩ = 0 ∣ Ω ⟩ = ∏ k > 0 ( cos ϑ k − sin ϑ k c ^ k † c ^ − k † ) ∣ 0 ⟩ |\Omega\rangle=\prod_{k>0}(\cos\vartheta_k-\sin\vartheta_k\hat{c}_k^\dagger\hat{c}_{-k}^\dagger)|0\rangle ∣Ω ⟩ = ∏ k > 0 ( cos ϑ k − sin ϑ k c ^ k † c ^ − k † ) ∣0 ⟩ ∣ 0 ⟩ |0\rangle ∣0 ⟩ ∣ 0 ⟩ k |0\rangle_k ∣0 ⟩ k c ^ k † c ^ − k † ∣ 0 ⟩ k \hat{c}_k^\dagger\hat{c}_{-k}^\dagger|0\rangle_k c ^ k † c ^ − k † ∣0 ⟩ k
As d q ( 0 ) = 0 d_q(0)=0 d q ( 0 ) = 0 h ⃗ 0 \vec{h}_0 h 0 ξ 0 = 0 \xi_0=0 ξ 0 = 0 k = 0 k=0 k = 0
g c F F = J q 2 + J q . (24) g_c^{\mathrm{FF}}=\frac{J_q}{2+J_q}. \tag*{(24)} g c FF = 2 + J q J q . (24) As shown in App. B, for low momenta the hopping term scales as j q ( k ) ≃ J q − k q j_q(k)\simeq J_q-k^q j q ( k ) ≃ J q − k q 2 F 1 {}_2F_1 2 F 1 ζ = e i k = 1 \zeta=e^{ik}=1 ζ = e ik = 1 d q ( k ) = a q k + c q k q + O ( k min ( 3 , q + 2 ) ) d_q(k)=a_qk+c_qk^{q}+\mathcal{O}(k^{\min(3,q+2)}) d q ( k ) = a q k + c q k q + O ( k m i n ( 3 , q + 2 ) )
a q = q 2 ( q − 1 ) ( q q / 2 ) , a_q=\frac{q}{2(q-1)}\binom{q}{q/2}, a q = 2 ( q − 1 ) q ( q /2 q ) , c q = 2 ( q q / 2 ) Γ ( q 2 + 1 ) Γ ( − q ) Γ ( − q 2 ) sin ( π q 2 ) , (25) c_q=2\binom{q}{q/2}\frac{\Gamma\left(\frac{q}{2}+1\right)\Gamma(-q)}{\Gamma\left(-\frac{q}{2}\right)}\sin\left(\frac{\pi q}{2}\right), \tag*{(25)} c q = 2 ( q /2 q ) Γ ( − 2 q ) Γ ( 2 q + 1 ) Γ ( − q ) sin ( 2 π q ) , (25) and the smaller of the two exponents controls the limit,
d q ( k ) ≃ c ∗ k q ∗ , q ∗ = min ( q , 1 ) , (26) d_q(k)\simeq c_*k^{q_*},\qquad q_*=\min(q,1), \tag*{(26)} d q ( k ) ≃ c ∗ k q ∗ , q ∗ = min ( q , 1 ) , (26) where
c ∗ = { c q , 0 < q < 1 , a q = q 2 ( q − 1 ) ( q q / 2 ) , q > 1. (27) c_*=
\begin{cases}
c_q, & 0<q<1,\\
a_q=\dfrac{q}{2(q-1)}\binom{q}{q/2}, & q>1.
\end{cases} \tag*{(27)} c ∗ = ⎩ ⎨ ⎧ c q , a q = 2 ( q − 1 ) q ( q /2 q ) , 0 < q < 1 , q > 1. (27) The non-analytic term therefore dominates for 0 < q < 1 0<q<1 0 < q < 1 q > 1 q>1 q > 1 q = 1 q=1 q = 1 a q a_q a q d 1 ( k ) = 4 π sin k 2 ln cot k 4 ≃ 2 π k ln 4 k d_1(k)=\frac{4}{\pi}\sin\frac{k}{2}\ln\cot\frac{k}{4}\simeq\frac{2}{\pi}k\ln\frac{4}{k} d 1 ( k ) = π 4 sin 2 k ln cot 4 k ≃ π 2 k ln k 4 q > 1 q>1 q > 1 a q = 2 ∑ r r J r , q = V 0 / 2 a_q=2\sum_r rJ_{r,q}=V_0/2 a q = 2 ∑ r r J r , q = V 0 /2 (4) . At even integers q = 2 , 4 , … q=2,4,\ldots q = 2 , 4 , … c q = 0 c_q=0 c q = 0 ξ k ( g c F F ) ∝ k q \xi_k(g_c^{\mathrm{FF}})\propto k^q ξ k ( g c FF ) ∝ k q Δ k ( g c F F ) ∝ k q ∗ \Delta_k(g_c^{\mathrm{FF}})\propto k^{q_*} Δ k ( g c FF ) ∝ k q ∗ ϵ k \epsilon_k ϵ k
ϵ k ( g c F F ) ∼ { k q ∗ , q ≠ 1 , k ln ( 1 / k ) , q = 1. (28) \epsilon_k(g_c^{\mathrm{FF}})\sim
\begin{cases}
k^{q_*}, & q\ne1,\\
k\ln(1/k), & q=1.
\end{cases} \tag*{(28)} ϵ k ( g c FF ) ∼ { k q ∗ , k ln ( 1/ k ) , q = 1 , q = 1. (28) so that z = min ( q , 1 ) z=\min(q,1) z = min ( q , 1 ) q = 1 q=1 q = 1 ϵ 0 ( g ) ∼ ∣ g − g c F F ∣ \epsilon_0(g)\sim|g-g_c^{\mathrm{FF}}| ϵ 0 ( g ) ∼ ∣ g − g c FF ∣ z ν = 1 z\nu=1 z ν = 1 ν = 1 / z \nu=1/z ν = 1/ z
Excitation and kink statistics
Due to the truncation of the Jordan-Wigner strings away from q = 2 q=2 q = 2 g = 0 g=0 g = 0 (1) , and already contains an extensive number of kinks. Accordingly, the kink number operator of Eq. (6) differs from the excitation number operator at g = 0 g=0 g = 0 N ^ e x = ∑ k > 0 ( γ ^ k † γ ^ k + γ ^ − k † γ ^ − k ) \hat{N}_{\mathrm{ex}}=\sum_{k>0}(\hat{\gamma}_k^\dagger\hat{\gamma}_k+\hat{\gamma}_{-k}^\dagger\hat{\gamma}_{-k}) N ^ ex = ∑ k > 0 ( γ ^ k † γ ^ k + γ ^ − k † γ ^ − k ) γ ^ ± k \hat{\gamma}_{\pm k} γ ^ ± k g = 0 g=0 g = 0
Ramping through the critical point g c F F g_c^{\mathrm{FF}} g c FF (24) , the time-evolved state becomes ∣ Ψ k ( t ) ⟩ = u k ( t ) ∣ 0 ⟩ k + |\Psi_k(t)\rangle=u_k(t)|0\rangle_k+ ∣ Ψ k ( t )⟩ = u k ( t ) ∣0 ⟩ k + v k ( t ) c ^ k † c ^ − k † ∣ 0 ⟩ k v_k(t)\hat{c}^{\dagger}_k\hat{c}^{\dagger}_{-k}|0\rangle_k v k ( t ) c ^ k † c ^ − k † ∣0 ⟩ k ∣ u k ∣ 2 + ∣ v k ∣ 2 = 1 |u_k|^2+|v_k|^2=1 ∣ u k ∣ 2 + ∣ v k ∣ 2 = 1 u k ( 0 ) = 1 u_k(0)=1 u k ( 0 ) = 1 v k ( 0 ) = 0 v_k(0)=0 v k ( 0 ) = 0
i d d t ( u k v k ) = ( − ξ k ( t ) Δ k ( t ) Δ k ( t ) ξ k ( t ) ) ( u k v k ) . (29) i\frac{d}{dt}\begin{pmatrix}u_k\\v_k\end{pmatrix}=
\begin{pmatrix}-\xi_k(t)&\Delta_k(t)\\\Delta_k(t)&\xi_k(t)\end{pmatrix}
\begin{pmatrix}u_k\\v_k\end{pmatrix}. \tag*{(29)} i d t d ( u k v k ) = ( − ξ k ( t ) Δ k ( t ) Δ k ( t ) ξ k ( t ) ) ( u k v k ) . (29) As this differential equation takes the Landau-Zener form in the leading order of k k k ⟨ γ ^ k † γ ^ k ⟩ \langle\hat{\gamma}^{\dagger}_k\hat{\gamma}_k\rangle ⟨ γ ^ k † γ ^ k ⟩ ( k , − k ) (k,-k) ( k , − k )
p k = exp [ − 4 π τ Q d q 2 ( k ) ( 2 + J q ( k ) ) 3 ] ≃ exp ( − λ q τ Q k 2 q ∗ ) , (30) p_k=\exp\left[-\frac{4\pi\tau_Qd_q^2(k)}{(2+J_q(k))^3}\right]\simeq\exp\left(-\lambda_q\tau_Q k^{2q_*}\right), \tag*{(30)} p k = exp [ − ( 2 + J q ( k ) ) 3 4 π τ Q d q 2 ( k ) ] ≃ exp ( − λ q τ Q k 2 q ∗ ) , (30) with λ q = 4 π c ∗ 2 / ( 2 + J q ) 3 \lambda_q=4\pi c_*^2/(2+J_q)^3 λ q = 4 π c ∗ 2 / ( 2 + J q ) 3 k k k q ≠ 1 q\ne1 q = 1 (30) only applies to the modes for which ξ k \xi_k ξ k J q ( k ) > 0 J_q(k)>0 J q ( k ) > 0 k < k c k<k_c k < k c 2 sin ( k c / 2 ) = J q 1 / q 2\sin(k_c/2)=J_q^{1/q} 2 sin ( k c /2 ) = J q 1/ q k c < k ≤ π k_c<k\le\pi k c < k ≤ π g = 0 g=0 g = 0 ϵ k ( g = 0 ) = J q 2 ( k ) + d q 2 ( k ) \epsilon_k(g=0)=\sqrt{J_q^2(k)+d_q^2(k)} ϵ k ( g = 0 ) = J q 2 ( k ) + d q 2 ( k ) k = π k=\pi k = π 2 q − J q ≃ q ln 2 2^q-J_q\simeq q\ln2 2 q − J q ≃ q ln 2 q q q
p k ≃ d q 2 ( k ) 4 τ Q 2 ϵ k 6 ( g = 0 ) . (31) p_k\simeq\frac{d_q^2(k)}{4\tau_Q^2\epsilon_k^6(g=0)}. \tag*{(31)} p k ≃ 4 τ Q 2 ϵ k 6 ( g = 0 ) d q 2 ( k ) . (31) Approximating the sum over all momentum modes with an integral in the continuum limit, one finds
n e x = 2 L ∑ k > 0 p k ≃ N q τ Q − 1 / ( 2 q ∗ ) , N q = Γ ( 1 + 1 2 q ∗ ) π λ q 1 / ( 2 q ∗ ) , (32) n_{\rm ex}=\frac{2}{L}\sum_{k>0}p_k\simeq\mathcal{N}_q\tau_Q^{-1/(2q_*)},\qquad
\mathcal{N}_q=\frac{\Gamma\left(1+\frac{1}{2q_*}\right)}{\pi\lambda_q^{1/(2q_*)}}, \tag*{(32)} n ex = L 2 k > 0 ∑ p k ≃ N q τ Q − 1/ ( 2 q ∗ ) , N q = π λ q 1/ ( 2 q ∗ ) Γ ( 1 + 2 q ∗ 1 ) , (32) for q > 1 / 4 q>1/4 q > 1/4 q < 1 / 4 q<1/4 q < 1/4 τ Q ≫ ( q ln 2 ) − 2 \tau_Q\gg(q\ln2)^{-2} τ Q ≫ ( q ln 2 ) − 2
n e x ≈ B q τ Q − 2 , (33) n_{\rm ex}\approx B_q\tau_Q^{-2}, \tag*{(33)} n ex ≈ B q τ Q − 2 , (33) B q = 1 π ∫ k c π d q 2 ( k ) 4 ϵ k 6 ( g = 0 ) d k = I 0 4 π q − 4 ( 1 + 1.46 q ) + O ( q − 2 ) , B_q=\frac{1}{\pi}\int_{k_c}^{\pi}\frac{d_q^2(k)}{4\epsilon_k^6(g=0)}\,dk=\frac{I_0}{4\pi}q^{-4}(1+1.46q)+O(q^{-2}), B q = π 1 ∫ k c π 4 ϵ k 6 ( g = 0 ) d q 2 ( k ) d k = 4 π I 0 q − 4 ( 1 + 1.46 q ) + O ( q − 2 ) , with I 0 ≈ 2.10 I_0\approx2.10 I 0 ≈ 2.10 q > 1 / 4 q>1/4 q > 1/4 L L L q < 1 / 4 q<1/4 q < 1/4 k = π k=\pi k = π q = 1 q=1 q = 1 n e x ∝ τ Q − 1 / 2 / ln τ Q n_{\rm ex}\propto\tau_Q^{-1/2}/\ln\tau_Q n ex ∝ τ Q − 1/2 / ln τ Q 1 / 2 + 1 / ln τ Q + … 1/2+1/\ln\tau_Q+\ldots 1/2 + 1/ ln τ Q + … 1 / 2 1/2 1/2
in App. B. The logarithm makes the effective finite-time exponent 1 / 2 + 1 / ln τ Q + … 1/2+1/\ln\tau_Q+\ldots 1/2 + 1/ ln τ Q + … 1 / 2 1/2 1/2
Similarly to Sec. II B, for the full counting statistics of N ^ e x \hat{N}_{\rm ex} N ^ ex p k p_k p k k k k [12 , 29 , 32 ]
ln ⟨ e i θ N ^ e x ⟩ = ∑ k > 0 ln [ 1 + ( e 2 i θ − 1 ) p k ] = − L 2 n e x Li 1 + 1 2 q ∗ ( 1 − e 2 i θ ) , (34) \begin{aligned}
\ln\langle e^{i\theta\hat{N}_{\rm ex}}\rangle=\sum_{k>0}\ln\left[1+(e^{2i\theta}-1)p_k\right] \\
&=-\frac{L}{2}n_{\rm ex}\operatorname{Li}_{1+\frac{1}{2q_*}}(1-e^{2i\theta}), \tag*{(34)}
\end{aligned} ln ⟨ e i θ N ^ ex ⟩ = k > 0 ∑ ln [ 1 + ( e 2 i θ − 1 ) p k ] = − 2 L n ex Li 1 + 2 q ∗ 1 ( 1 − e 2 i θ ) , (34) with Li s ( x ) = ∑ k = 1 ∞ x k k s \operatorname{Li}_s(x)=\sum_{k=1}^{\infty}\frac{x^k}{k^s} Li s ( x ) = ∑ k = 1 ∞ k s x k ⟨ ⋯ ⟩ \langle\cdots\rangle ⟨ ⋯ ⟩ (29) . The last equality holds in the continuum limit in the leading order in $\tau_Q$ for $q>1/4$ and $q\ne1$, and reduces to the $\operatorname{Li}_{3/2}$ form familiar from the transverse-field Ising chain when $q_*=1$ [29 ] `. As a result, all excitation cumulants scale with the same exponent,
κ ℓ e x = C ℓ ( q ) τ Q − 1 / ( 2 q ∗ ) , C ℓ C 1 = 2 ℓ − 1 S ℓ ( 1 2 q ∗ ) , (35) \kappa_\ell^{\rm ex}=C_\ell(q)\tau_Q^{-1/(2q_*)},\qquad
\frac{C_\ell}{C_1}=2^{\ell-1}S_\ell\left(\frac{1}{2q_*}\right), \tag*{(35)} κ ℓ ex = C ℓ ( q ) τ Q − 1/ ( 2 q ∗ ) , C 1 C ℓ = 2 ℓ − 1 S ℓ ( 2 q ∗ 1 ) , (35) with C 1 = L N q C_1=LN_q C 1 = L N q S 2 ( w ) = 1 − 2 − w S_2(w)=1-2^{-w} S 2 ( w ) = 1 − 2 − w S 3 ( w ) = 1 − 3 ⋅ 2 − w + 2 ⋅ 3 − w S_3(w)=1-3\cdot2^{-w}+2\cdot3^{-w} S 3 ( w ) = 1 − 3 ⋅ 2 − w + 2 ⋅ 3 − w R ℓ = κ ℓ e x / κ 1 e x R_\ell=\kappa_\ell^{\rm ex}/\kappa_1^{\rm ex} R ℓ = κ ℓ ex / κ 1 ex q ∗ q_* q ∗ q ∗ = 1 q_*=1 q ∗ = 1 R 2 ≃ 0.586 R_2\simeq0.586 R 2 ≃ 0.586 R 3 ≃ 0.134 R_3\simeq0.134 R 3 ≃ 0.134 κ 2 e x = κ 1 e x \kappa_2^{\rm ex}=\kappa_1^{\rm ex} κ 2 ex = κ 1 ex R 2 = 2 S 2 ( 1 / ( 2 q ∗ ) ) R_2=2S_2(1/(2q_*)) R 2 = 2 S 2 ( 1/ ( 2 q ∗ )) q > 1 / 2 q>1/2 q > 1/2 q < 1 / 2 q<1/2 q < 1/2 κ 2 e x < 2 κ 1 e x \kappa_2^{\rm ex}<2\kappa_1^{\rm ex} κ 2 ex < 2 κ 1 ex ∣ κ 3 e x ∣ ≤ 2 κ 2 e x |\kappa_3^{\rm ex}|\le2\kappa_2^{\rm ex} ∣ κ 3 ex ∣ ≤ 2 κ 2 ex R 2 < 2 R_2<2 R 2 < 2 ∣ R 3 ∣ ≤ 2 R 2 |R_3|\le2R_2 ∣ R 3 ∣ ≤ 2 R 2
For q < 1 / 4 q<1/4 q < 1/4 p k ∝ τ Q − 2 p_k\propto\tau_Q^{-2} p k ∝ τ Q − 2
κ ℓ e x = 2 ℓ − 1 L B q τ Q − 2 + O ( τ Q − 4 ) . (36) \kappa_\ell^{\rm ex}=2^{\ell-1}LB_q\tau_Q^{-2}+O(\tau_Q^{-4}). \tag*{(36)} κ ℓ ex = 2 ℓ − 1 L B q τ Q − 2 + O ( τ Q − 4 ) . (36) Accordingly, one finds in the leading order that κ 2 e x / κ 1 e x = 2 \kappa_2^{\rm ex}/\kappa_1^{\rm ex}=2 κ 2 ex / κ 1 ex = 2 κ 3 e x / κ 1 e x = 4 \kappa_3^{\rm ex}/\kappa_1^{\rm ex}=4 κ 3 ex / κ 1 ex = 4 κ 2 e x / κ 1 e x = 2 [ 1 − J 0 / ( 4 I 0 q 4 τ Q 2 ) ] \kappa_2^{\rm ex}/\kappa_1^{\rm ex}=2[1-J_0/(4I_0q^4\tau_Q^2)] κ 2 ex / κ 1 ex = 2 [ 1 − J 0 / ( 4 I 0 q 4 τ Q 2 )] κ 3 e x / κ 1 e x = 4 [ 1 − 3 J 0 / ( 4 I 0 q 4 τ Q 2 ) ] \kappa_3^{\rm ex}/\kappa_1^{\rm ex}=4[1-3J_0/(4I_0q^4\tau_Q^2)] κ 3 ex / κ 1 ex = 4 [ 1 − 3 J 0 / ( 4 I 0 q 4 τ Q 2 )] I 0 ≃ 2.10 I_0\simeq2.10 I 0 ≃ 2.10
J 0 = ∫ π / 2 π ( π 2 − k ) [ ln 2 ( 2 sin k 2 ) + ( π 2 − k ) 2 ] − 2 d k ≃ 2.72 , J_0=\int_{\pi/2}^{\pi}\left(\frac{\pi}{2}-k\right)\left[\ln^2\left(2\sin\frac{k}{2}\right)+\left(\frac{\pi}{2}-k\right)^2\right]^{-2}dk\simeq2.72, J 0 = ∫ π /2 π ( 2 π − k ) [ ln 2 ( 2 sin 2 k ) + ( 2 π − k ) 2 ] − 2 d k ≃ 2.72 , keeping both ratios stay below these values. The number of excited ( k , − k ) (k,-k) ( k , − k ) N ^ e x \hat{N}_{\rm ex} N ^ ex L L L
P e x ( n ) → 1 2 π κ 2 e x exp [ − ( n − κ 1 e x ) 2 2 κ 2 e x ] (37) P_{\mathrm{ex}}(n) \to\frac{1}{\sqrt{2\pi\kappa_{2}^{\mathrm{ex}}}} \exp\left[-\frac{(n-\kappa_{1}^{\mathrm{ex}})^2}{2\kappa_{2}^{\mathrm{ex}}}\right] \tag*{(37)} P ex ( n ) → 2 π κ 2 ex 1 exp [ − 2 κ 2 ex ( n − κ 1 ex ) 2 ] (37) which also describes the kink statistics of Sec. II B for large enough mean values.
Regarding the kink number statistics within the free-fermion setting, with periodic boundary conditions, the kink number operator of Eq. (6) is exactly quadratic (App. E),
N ^ = ∑ k > 0 n ^ k , n ^ k = 1 + ψ ^ k † ( − cos k τ z + sin k τ x ) ψ ^ k , (38) \hat{N}=\sum_{k>0}\hat{n}_k,\qquad\hat{n}_k=1+\hat{\psi}_k^\dagger(-\cos k\,\tau^z+\sin k\,\tau^x)\hat{\psi}_k, \tag*{(38)} N ^ = k > 0 ∑ n ^ k , n ^ k = 1 + ψ ^ k † ( − cos k τ z + sin k τ x ) ψ ^ k , (38) where n ^ k \hat{n}_k n ^ k k k k ( k , − k ) (k,-k) ( k , − k )
p ~ k = 1 2 ⟨ Ψ k ∣ n ^ k ∣ Ψ k ⟩ = ∣ cos k 2 u k + sin k 2 v k ∣ 2 , (39) \tilde{p}_k=\frac{1}{2}\langle\Psi_k|\hat{n}_k|\Psi_k\rangle=\left|\cos\frac{k}{2}u_k+\sin\frac{k}{2}v_k\right|^2, \tag*{(39)} p ~ k = 2 1 ⟨ Ψ k ∣ n ^ k ∣ Ψ k ⟩ = cos 2 k u k + sin 2 k v k 2 , (39) and it is related to p k p_k p k δ k \delta_k δ k g = 0 g=0 g = 0
sin 2 δ k = 1 2 [ 1 − j q ( k ) cos k + d q ( k ) sin k ϵ k ( g = 0 ) ] , (40) \sin^2\delta_k=\frac{1}{2}\left[1-\frac{j_q(k)\cos k+d_q(k)\sin k}{\epsilon_k(g=0)}\right], \tag*{(40)} sin 2 δ k = 2 1 [ 1 − ϵ k ( g = 0 ) j q ( k ) cos k + d q ( k ) sin k ] , (40) which vanishes at q = 2 q=2 q = 2 j 2 ( k ) = 2 cos k j_2(k)=2\cos k j 2 ( k ) = 2 cos k d 2 ( k ) = 2 sin k d_2(k)=2\sin k d 2 ( k ) = 2 sin k p k p_k p k Φ k \Phi_k Φ k g = 0 g=0 g = 0
p ~ k = sin 2 δ k + p k cos 2 δ k − p k ( 1 − p k ) sin 2 δ k cos Φ k . (41) \tilde{p}_k=\sin^2\delta_k+p_k\cos2\delta_k-\sqrt{p_k(1-p_k)}\sin2\delta_k\cos\Phi_k. \tag*{(41)} p ~ k = sin 2 δ k + p k cos 2 δ k − p k ( 1 − p k ) sin 2 δ k cos Φ k . (41) Here the first term gives the ground-state kinks of the free-fermion model at g = 0 g=0 g = 0 κ 1 G S = 2 ∑ k > 0 sin 2 δ k ∝ L \kappa_1^{\mathrm{GS}}=2\sum_{k>0}\sin^2\delta_k\propto L κ 1 GS = 2 ∑ k > 0 sin 2 δ k ∝ L Φ k \Phi_k Φ k k k k q > 1 / 4 q>1/4 q > 1/4 δ k → 0 \delta_k\to0 δ k → 0 p ~ k ≃ p k \tilde{p}_k\simeq p_k p ~ k ≃ p k q < 1 / 4 q<1/4 q < 1/4 k = π k=\pi k = π p ~ k − sin 2 δ k ≃ w k p k \tilde{p}_k-\sin^2\delta_k\simeq w_kp_k p ~ k − sin 2 δ k ≃ w k p k w k w_k w k w π ≃ 3.32 w_\pi\simeq3.32 w π ≃ 3.32 k = π k=\pi k = π q → 0 q\to0 q → 0 p k sin 2 δ k p_k\sin^2\delta_k p k sin 2 δ k
ln P ~ ( θ ) = ln ⟨ e i θ N ^ ⟩ ≃ ∑ k > 0 ln [ 1 + ( e 2 i θ − 1 ) sin 2 δ k ] + ∑ k > 0 ln [ 1 + ( e 2 i θ − 1 ) p ~ k ] . (42) \ln\tilde{P}(\theta)=\ln\langle e^{i\theta\hat{N}}\rangle\simeq\sum_{k>0}\ln\left[1+(e^{2i\theta}-1)\sin^2\delta_k\right]
+\sum_{k>0}\ln\left[1+(e^{2i\theta}-1)\tilde{p}_k\right]. \tag*{(42)} ln P ~ ( θ ) = ln ⟨ e i θ N ^ ⟩ ≃ k > 0 ∑ ln [ 1 + ( e 2 i θ − 1 ) sin 2 δ k ] + k > 0 ∑ ln [ 1 + ( e 2 i θ − 1 ) p ~ k ] . (42) The first sum gives the τ Q \tau_Q τ Q κ ℓ G S \kappa_\ell^{\mathrm{GS}} κ ℓ GS g = 0 g=0 g = 0 q > 1 / 4 q>1/4 q > 1/4 (34) , i.e., the kink cumulants measured from the ground state coincide at leading order with those of the excitations, κ ℓ − κ ℓ G S ≃ κ ℓ e x \kappa_\ell-\kappa_\ell^{\mathrm{GS}}\simeq\kappa_\ell^{\mathrm{ex}} κ ℓ − κ ℓ GS ≃ κ ℓ ex q < 1 / 4 q<1/4 q < 1/4 κ 1 − κ 1 G S ≃ ( B ~ q / B q ) κ 1 e x \kappa_1-\kappa_1^{\mathrm{GS}}\simeq(\tilde{B}_q/B_q)\kappa_1^{\mathrm{ex}} κ 1 − κ 1 GS ≃ ( B ~ q / B q ) κ 1 ex B ~ q = 1 π ∫ 0 π w k d q 2 ( k ) 4 ϵ k 6 ( g = 0 ) d k \tilde{B}_q=\frac{1}{\pi}\int_0^\pi w_k\,\frac{d_q^2(k)}{4\epsilon_k^6(g=0)}\,dk B ~ q = π 1 ∫ 0 π w k 4 ϵ k 6 ( g = 0 ) d q 2 ( k ) d k R ~ ℓ ≡ ( κ ℓ − κ ℓ G S ) / ( κ 1 − κ 1 G S ) \tilde{R}_\ell\equiv(\kappa_\ell-\kappa_\ell^{\mathrm{GS}})/(\kappa_1-\kappa_1^{\mathrm{GS}}) R ~ ℓ ≡ ( κ ℓ − κ ℓ GS ) / ( κ 1 − κ 1 GS ) R ~ 2 ≃ 1.89 \tilde{R}_2\simeq1.89 R ~ 2 ≃ 1.89 R ~ 3 ≃ 3.36 \tilde{R}_3\simeq3.36 R ~ 3 ≃ 3.36 q → 0 q\to0 q → 0 R ~ 2 ≃ 1.93 \tilde{R}_2\simeq1.93 R ~ 2 ≃ 1.93 R ~ 3 ≃ 3.57 \tilde{R}_3\simeq3.57 R ~ 3 ≃ 3.57 q = 1 / 4 q=1/4 q = 1/4
The kink number cumulant ratios thus exhibit the same behavior as the excitations for q > 1 / 4 q>1/4 q > 1/4 q < 1 / 4 q<1/4 q < 1/4 R ~ 2 \tilde{R}_2 R ~ 2 R ~ 3 \tilde{R}_3 R ~ 3 q q q ( κ 2 − κ 2 G S ) / ( κ 1 − κ 1 G S ) ≃ R ~ 2 − J ~ 0 / ( 2 I ~ 0 q 4 τ Q 2 ) (\kappa_2-\kappa_2^{\mathrm{GS}})/(\kappa_1-\kappa_1^{\mathrm{GS}})\simeq\tilde{R}_2-\tilde{J}_0/(2\tilde{I}_0q^4\tau_Q^2) ( κ 2 − κ 2 GS ) / ( κ 1 − κ 1 GS ) ≃ R ~ 2 − J ~ 0 / ( 2 I ~ 0 q 4 τ Q 2 ) ( κ 3 − κ 3 G S ) / ( κ 1 − κ 1 G S ) ≃ R ~ 3 − 3 J ~ 0 / ( I ~ 0 q 4 τ Q 2 ) (\kappa_3-\kappa_3^{\mathrm{GS}})/(\kappa_1-\kappa_1^{\mathrm{GS}})\simeq\tilde{R}_3-3\tilde{J}_0/(\tilde{I}_0q^4\tau_Q^2) ( κ 3 − κ 3 GS ) / ( κ 1 − κ 1 GS ) ≃ R ~ 3 − 3 J ~ 0 / ( I ~ 0 q 4 τ Q 2 ) I ~ 0 ≃ 4.89 \tilde{I}_0\simeq4.89 I ~ 0 ≃ 4.89 (106) and J ~ 0 = ∫ π / 3 π w k 2 ( π − k 2 ) 4 [ ln 2 ( 2 sin k 2 ) + ( π − k 2 ) 2 ] − 6 d k ≃ 15.1 \tilde{J}_0=\int_{\pi/3}^{\pi}w_k^2\left(\frac{\pi-k}{2}\right)^4\left[\ln^2\left(2\sin\frac{k}{2}\right)+\left(\frac{\pi-k}{2}\right)^2\right]^{-6}dk\simeq15.1 J ~ 0 = ∫ π /3 π w k 2 ( 2 π − k ) 4 [ ln 2 ( 2 sin 2 k ) + ( 2 π − k ) 2 ] − 6 d k ≃ 15.1
The mean kink numbers with the free-fermion numerics (exact evolution of the TLSs with L = 150 L=150 L = 150 Figure 1 , showing a progressively improving agreement with the TDVP results as q q q q = 2 q=2 q = 2 q ≳ 2 q\gtrsim2 q ≳ 2
Figure 1. Mean kink number, κ 1 \kappa_1 κ 1 q = 0.1 q=0.1 q = 0.1 q = 0.2 q=0.2 q = 0.2 q = 1.5 q=1.5 q = 1.5 q = 2.25 q=2.25 q = 2.25 g = 0 g=0 g = 0 ∝ τ Q − 1 / ( 2 q ∗ ) \propto\tau_Q^{-1/(2q_*)} ∝ τ Q − 1/ ( 2 q ∗ ) (35) , with q ∗ = min ( q , 1 ) q_*=\min(q,1) q ∗ = min ( q , 1 ) ∝ τ Q − 2 \propto\tau_Q^{-2} ∝ τ Q − 2 k > k c k>k_c k > k c (33) , which dominates slow ramps for q < 1 / 4 q<1/4 q < 1/4 q q q q = 0.1 q=0.1 q = 0.1 q = 0.2 q=0.2 q = 0.2 q = 1.5 q=1.5 q = 1.5 q = 2.25 q=2.25 q = 2.25
Figure 4 compares the second cumulant of the TDVP simulations with the same free-fermion results on the open chain and on the ring and with the analytical predictions. Somewhat differently from the case of the mean, the free-fermion results start matching the TDVP ones slightly earlier, for q ≳ 1.25 q\gtrsim1.25 q ≳ 1.25 § 1 .
Finally, we show the comparison between the cumulant ratios averaged over the middle half of the considered driving times as a function of the fractional order q q q Figure 8 . As for the variance, the free-fermion kink ratios measured from the ground state on the open chain and the analytical ratios R ℓ R_\ell R ℓ (35) on the ring converge to the TDVP results for q ≳ 1.25 q\gtrsim1.25 q ≳ 1.25 q ≲ 0.3 q\lesssim0.3 q ≲ 0.3 κ 2 / κ 1 \kappa_2/\kappa_1 κ 2 / κ 1 (4) . A related departure from independently formed defects was observed in Rydberg chains [54 ] . As an additional check for small system sizes, we evolved the full spin Hamiltonian, Eq. (1) , exactly on open chains of L = 12 L=12 L = 12 q = 0.2 q=0.2 q = 0.2 τ Q = 50 \tau_Q=50 τ Q = 50 κ 3 / κ 2 \kappa_3/\kappa_2 κ 3 / κ 2 κ 2 / κ 1 ≃ 2.0 \kappa_2/\kappa_1\simeq2.0 κ 2 / κ 1 ≃ 2.0 q = 0.1 q=0.1 q = 0.1 τ Q = 100 \tau_Q=100 τ Q = 100 L = 16 L=16 L = 16 κ 2 / κ 1 ≃ 2.2 \kappa_{2}/\kappa_{1} \simeq2.2 κ 2 / κ 1 ≃ 2.2 2.4 2.4 2.4 κ 3 / κ 2 ≃ 2.3 \kappa_{3}/\kappa_{2} \simeq2.3 κ 3 / κ 2 ≃ 2.3 2.6 2.6 2.6 L = 16 L = 16 L = 16 20 20 20
Figure 8. Cumulant ratios κ 2 / κ 1 \kappa_{2}/\kappa_{1} κ 2 / κ 1 κ 3 / κ 1 \kappa_{3}/\kappa_{1} κ 3 / κ 1 q q q κ 2 \kappa_{2} κ 2 κ 3 \kappa_{3} κ 3 (35) alone (black and blue dashed lines) start converging to the TDVP ones (orange circles for κ 2 \kappa_{2} κ 2 κ 3 \kappa_{3} κ 3 q ≳ 1.25 q \gtrsim1.25 q ≳ 1.25 q ≲ 0.3 q \lesssim0.3 q ≲ 0.3 κ 2 / κ 1 \kappa_{2}/\kappa_{1} κ 2 / κ 1
Kink-kink correlations
In this subsection, we investigate the spatial structure of the kinks, which provides further support for the Gaussian form of the histograms in Sec. II B. To this end, we extract the connected correlation function between the middle bond and the bond at distance r r r
C ( r ) = ⟨ n ^ L / 2 n ^ L / 2 + r ⟩ − ⟨ n ^ L / 2 ⟩ ⟨ n ^ L / 2 + r ⟩ , (43) C(r) = \langle\hat{n}_{L/2}\hat{n}_{L/2+r}\rangle- \langle\hat{n}_{L/2}\rangle\langle\hat{n}_{L/2+r}\rangle, \tag*{(43)} C ( r ) = ⟨ n ^ L /2 n ^ L /2 + r ⟩ − ⟨ n ^ L /2 ⟩ ⟨ n ^ L /2 + r ⟩ , (43) with the kink projectors n ^ j = ( 1 − σ ^ j x σ ^ j + 1 x ) / 2 \hat{n}_{j} = (1-\hat{\sigma}^{x}_{j}\hat{\sigma}^{x}_{j+1})/2 n ^ j = ( 1 − σ ^ j x σ ^ j + 1 x ) /2 (45) and r = − ( L / 2 − 1 ) , … , L / 2 − 1 r = -(L/2 - 1),\ldots,L/2 - 1 r = − ( L /2 − 1 ) , … , L /2 − 1 ∣ r ∣ \lvert r\rvert ∣ r ∣
As for the cumulants, we also compare the results with those obtained within the free-fermion approximation, i.e., with the Hamiltonian of Eq. (17) with all couplings r ≤ L − 1 r \leq L - 1 r ≤ L − 1
n ^ j = 1 2 [ 1 − ( c ^ j † − c ^ j ) ( c ^ j + 1 † + c ^ j + 1 ) ] , (44) \hat{n}_{j} = \frac{1}{2}\left[1-(\hat{c}^{\dagger}_{j}-\hat{c}_{j})(\hat{c}^{\dagger}_{j+1}+\hat{c}_{j+1})\right], \tag*{(44)} n ^ j = 2 1 [ 1 − ( c ^ j † − c ^ j ) ( c ^ j + 1 † + c ^ j + 1 ) ] , (44) the correlation function is affected by the free-fermion approximation solely through the time-evolved wave function and can be expressed by the two-point functions ⟨ c ^ i † c ^ j ⟩ \langle\hat{c}^{\dagger}_{i}\hat{c}_{j}\rangle ⟨ c ^ i † c ^ j ⟩ ⟨ c ^ i c ^ j ⟩ \langle\hat{c}_{i}\hat{c}_{j}\rangle ⟨ c ^ i c ^ j ⟩ ∣ ∑ r ≠ 0 C ( r ) ∣ ≤ C ( 0 ) \left|\sum_{r\ne0} C(r)\right| \leq C(0) ∑ r = 0 C ( r ) ≤ C ( 0 ) L L L
Figure 7 shows that for all fractional orders the correlations decay within a few tens of bonds around the middle of the chain. For q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25 q = 0.2 q = 0.2 q = 0.2 q = 0.5 q = 0.5 q = 0.5 Figure 8 and with the binding of domain walls into pairs, Eq. (4) . As κ 2 > 2 κ 1 ≥ 2 ∑ j n j ( 1 − n j ) \kappa_{2} > 2\kappa_{1} \geq2\sum_{j} n_{j}(1-n_{j}) κ 2 > 2 κ 1 ≥ 2 ∑ j n j ( 1 − n j ) q ≲ 0.2 q \lesssim0.2 q ≲ 0.2 (117) , and this holds for the middle bond itself: at q = 0.2 q = 0.2 q = 0.2 ∑ r ≠ 0 C ( r ) ≃ 1.5 \sum_{r\ne0} C(r) \simeq1.5 ∑ r = 0 C ( r ) ≃ 1.5 2.3 C ( 0 ) 2.3\,C(0) 2.3 C ( 0 ) τ Q ≥ 30 \tau_{Q} \geq30 τ Q ≥ 30 0.8 0.8 0.8 0.9 C ( 0 ) 0.9\,C(0) 0.9 C ( 0 ) ∣ r ∣ − 2 \lvert r\rvert^{-2} ∣ r ∣ − 2 ∣ r ∣ − 1 \lvert r\rvert^{-1} ∣ r ∣ − 1 § 2.13 .
Figure 7. Connected kink-kink correlation function of Eq. (43) , normalized by the local variance ⟨ n ^ L / 2 2 ⟩ \langle\hat{n}_{L/2}^{2}\rangle ⟨ n ^ L /2 2 ⟩ q = 0.2 q = 0.2 q = 0.2 q = 0.5 q = 0.5 q = 0.5 q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25 r = 0 r = 0 r = 0 ∣ r ∣ − 1 \lvert r\rvert^{-1} ∣ r ∣ − 1 ∣ r ∣ − 2 \lvert r\rvert^{-2} ∣ r ∣ − 2 q ≤ 0.5 q \leq0.5 q ≤ 0.5 q ≥ 1.5 q \geq1.5 q ≥ 1.5
Summary and outlook
Long-range interactions provide a continuous parameter along which critical exponents vary. In the fractional Ising chain a single derivative order fixes the couplings at every scale, from the nearest-neighbor bond to the far tail, which makes the chain a testbed for the Kibble-Zurek mechanism in nonintegrable systems across the range of interactions.
We have characterized the kink number statistics generated by driving the fractional Ising chain across its quantum critical point. In particular, we have reported the full counting statistics of kinks and the scaling of their cumulants with the driving time, obtained from TDVP simulations within the MPS and MPO formalism, and benchmarked throughout against the exactly solvable fractional p p p
First, we tested the KZ scaling of the mean kink number with the quench time, using the correlation-length and dynamical critical exponents extracted from the finite-size scaling of the gap minimum obtained by DMRG and, for z z z q ≲ 1 q \lesssim1 q ≲ 1 q ≳ 1 q \gtrsim1 q ≳ 1 L = 150 L = 150 L = 150 q ≳ 1.25 q \gtrsim1.25 q ≳ 1.25
We also compared our results with a free-fermion p p p q ≤ 1 q \le1 q ≤ 1 q = 1 / 4 q = 1/4 q = 1/4 τ Q − 2 \tau_Q^{-2} τ Q − 2 q ≳ 1 q \gtrsim1 q ≳ 1 q ≳ 2 q \gtrsim2 q ≳ 2 q = 1.5 q = 1.5 q = 1.5 q q q q ≳ 1.25 q \gtrsim1.25 q ≳ 1.25 q ≲ 0.3 q \lesssim0.3 q ≲ 0.3
Fractional couplings are not an exotic choice. Lévy statistics are as generic as Gaussian ones, and fractional operators are their continuum description in disordered and multiscale media [55 , 56 , 57 ] . As quantum simulators move toward realistic materials, how defects form under such couplings becomes a practical question. The Kibble-Zurek argument is at heart causal. Regions farther apart than the frozen correlation length order independently because no signal connects them before freeze-out. Fractional couplings weaken this premise, since correlations spread along Lévy light cones that widen as q q q [67 ] . The breakdown of KZ scaling found here lies in the same regime. Whether it reflects this altered causal structure, the binding of defects, or both is open, and deciding it requires a theory of defect formation with an algebraic rather than linear causal horizon.
At small q q q (4) continuously, from growth as m 1 − q m^{1-q} m 1 − q q = 1 q = 1 q = 1 1 < q < 2 1 < q < 2 1 < q < 2 [18 , 72 , 73 ] .
Cumulant ratios take universal values when defects form independently and depart from them when defects interact. Counting statistics can therefore probe critical points that lack an equilibrium theory, using the snapshot measurements already available on Rydberg arrays, trapped ions and quantum annealers [15 , 50 , 53 , 54 ] . Fractional derivatives extend directly to higher dimensions and, in time, to memory effects, which opens defect formation in multiscale quantum media to the same analysis.
Acknowledgments
AG and AdC acknowledge the Luxembourg National Research Fund under Grant No. C22/MS/17132054/AQCQNET and AG acknowledges the Luxembourg National Research Fund under Grant No. CC25/MS/19559370/FastQOPT. This material is based upon work supported by the U.S. National Science Foundation under Grant No. PHY-2515059. LDC thanks Maksym Serbyn for hosting him during the sabbatical in which he completed this work. Claude (Anthropic) was used to assist with checking calculations and references and with editing the text; the authors take full responsibility for the content.
Characteristic-function reconstruction of the defect distribution
For the MPS simulations of the fractional Ising chain, it is convenient to characterize the full counting statistics of domain-wall defects through the characteristic function rather than through high moments alone, which become increasingly noisy once the corresponding MPOs are contracted. To this end, we define the defect-number operator
N ^ = ∑ i = 1 L − 1 n ^ i , n ^ i = 1 − σ ^ i x σ ^ i + 1 x 2 , (45) \hat{N}=\sum_{i=1}^{L-1}\hat{n}_i,\qquad\hat{n}_i=\frac{1-\hat{\sigma}_i^x\hat{\sigma}_{i+1}^x}{2}, \tag*{(45)} N ^ = i = 1 ∑ L − 1 n ^ i , n ^ i = 2 1 − σ ^ i x σ ^ i + 1 x , (45) where n ^ i \hat{n}_i n ^ i ( i , i + 1 ) (i,i+1) ( i , i + 1 ) n ^ i \hat{n}_i n ^ i 0 , 1 0,1 0 , 1 n ^ i 2 = n ^ i \hat{n}_i^2=\hat{n}_i n ^ i 2 = n ^ i [ n ^ i , n ^ j ] = 0 [\hat{n}_i,\hat{n}_j]=0 [ n ^ i , n ^ j ] = 0 σ ^ x \hat{\sigma}^x σ ^ x
P ~ ( θ ) ≡ ⟨ Ψ ( τ Q ) ∣ e i θ N ^ ∣ Ψ ( τ Q ) ⟩ , (46) \widetilde{P}(\theta)\equiv\langle\Psi(\tau_Q)|e^{i\theta\hat{N}}|\Psi(\tau_Q)\rangle, \tag*{(46)} P ( θ ) ≡ ⟨ Ψ ( τ Q ) ∣ e i θ N ^ ∣Ψ ( τ Q )⟩ , (46) factorizes exactly as
P ~ ( θ ) = ⟨ Ψ ( τ Q ) ∣ ∏ i = 1 L − 1 e i θ n ^ i ∣ Ψ ( τ Q ) ⟩ . (47) \widetilde{P}(\theta)=\langle\Psi(\tau_Q)|\prod_{i=1}^{L-1}e^{i\theta\hat{n}_i}|\Psi(\tau_Q)\rangle. \tag*{(47)} P ( θ ) = ⟨ Ψ ( τ Q ) ∣ i = 1 ∏ L − 1 e i θ n ^ i ∣Ψ ( τ Q )⟩ . (47) This factorization makes the evaluation efficient on an MPS: rather than constructing N ^ ℓ \hat{N}^{\ell} N ^ ℓ ( i , i + 1 ) (i,i+1) ( i , i + 1 )
U ^ i ( θ ) ≡ e i θ n ^ i = I + ( e i θ − 1 ) n ^ i . (48) \hat{U}_i(\theta)\equiv e^{i\theta\hat{n}_i}=\mathbb{I}+(e^{i\theta}-1)\hat{n}_i. \tag*{(48)} U ^ i ( θ ) ≡ e i θ n ^ i = I + ( e i θ − 1 ) n ^ i . (48) In the two-spin σ ^ x \hat{\sigma}^x σ ^ x { ∣ ↑ ↑ ⟩ , ∣ ↑ ↓ ⟩ , ∣ ↓ ↑ ⟩ , ∣ ↓ ↓ ⟩ } \{|\uparrow\uparrow\rangle,|\uparrow\downarrow\rangle,|\downarrow\uparrow\rangle,|\downarrow\downarrow\rangle\} { ∣ ↑↑ ⟩ , ∣ ↑↓ ⟩ , ∣ ↓↑ ⟩ , ∣ ↓↓ ⟩}
U ^ i ( θ ) = diag ( 1 , e i θ , e i θ , 1 ) , (49) \hat{U}_i(\theta)=\operatorname{diag}(1,e^{i\theta},e^{i\theta},1), \tag*{(49)} U ^ i ( θ ) = diag ( 1 , e i θ , e i θ , 1 ) , (49) showing explicitly that aligned neighboring spins contribute phase 1 1 1 e i θ e^{i\theta} e i θ
For a many-body state ∣ Ψ ( τ Q ) ⟩ |\Psi(\tau_Q)\rangle ∣Ψ ( τ Q )⟩
P ~ ( θ ) = ⟨ Ψ ( τ Q ) ∣ ∏ i = 1 L − 1 U ^ i ( θ ) ∣ Ψ ( τ Q ) ⟩ ⟨ Ψ ( τ Q ) ∣ Ψ ( τ Q ) ⟩ . (50) \widetilde{P}(\theta)=\frac{\langle\Psi(\tau_Q)|\prod_{i=1}^{L-1}\hat{U}_i(\theta)|\Psi(\tau_Q)\rangle}{\langle\Psi(\tau_Q)|\Psi(\tau_Q)\rangle}. \tag*{(50)} P ( θ ) = ⟨ Ψ ( τ Q ) ∣Ψ ( τ Q )⟩ ⟨ Ψ ( τ Q ) ∣ ∏ i = 1 L − 1 U ^ i ( θ ) ∣Ψ ( τ Q )⟩ . (50) Because all U ^ i ( θ ) \hat{U}_i(\theta) U ^ i ( θ ) e i θ n ^ i = e i θ / 2 [ cos θ 2 − i sin θ 2 σ ^ i x σ ^ i + 1 x ] e^{i\theta\hat{n}_i}=e^{i\theta/2}\left[\cos\frac{\theta}{2}-i\sin\frac{\theta}{2}\hat{\sigma}_i^x\hat{\sigma}_{i+1}^x\right] e i θ n ^ i = e i θ /2 [ cos 2 θ − i sin 2 θ σ ^ i x σ ^ i + 1 x ] 2 2 2 (50) scales as that of the norm, O ( L χ 3 ) \mathcal{O}(L\chi^3) O ( L χ 3 ) θ \theta θ 10 − 12 10^{-12} 1 0 − 12 P ~ ( θ ) \widetilde{P}(\theta) P ( θ )
θ m = 2 π m M , m = 0 , 1 , … , M − 1 , (51) \theta_m=\frac{2\pi m}{M},\qquad m=0,1,\ldots,M-1, \tag*{(51)} θ m = M 2 π m , m = 0 , 1 , … , M − 1 , (51) with M ≥ L M\geq L M ≥ L N ^ \hat{N} N ^ N ^ ∈ { 0 , 1 , … , L − 1 } \hat{N}\in\{0,1,\ldots,L-1\} N ^ ∈ { 0 , 1 , … , L − 1 } P ( n ) = Prob ( N ^ = n ) P(n)=\operatorname{Prob}(\hat{N}=n) P ( n ) = Prob ( N ^ = n )
P ( n ) = 1 M ∑ m = 0 M − 1 e − i θ m n P ~ ( θ m ) , (52) P(n)=\frac{1}{M}\sum_{m=0}^{M-1}e^{-i\theta_m n}\widetilde{P}(\theta_m), \tag*{(52)} P ( n ) = M 1 m = 0 ∑ M − 1 e − i θ m n P ( θ m ) , (52) for n = 0 , 1 , … , L − 1 n=0,1,\ldots,L-1 n = 0 , 1 , … , L − 1
Low-momentum expansion of the dispersions
Here we derive the small-k k k j q ( k ) j_q(k) j q ( k ) d q ( k ) d_q(k) d q ( k ) q = 1 q=1 q = 1 c ∗ c_* c ∗ (27) . The cosine sum closes in terms of the generating identity of the centered generalized difference,
∑ r = − ∞ ∞ ( − 1 ) r ( q q / 2 + r ) e i k r = ∣ 2 sin k 2 ∣ q , (53) \sum_{r=-\infty}^{\infty}(-1)^r\binom{q}{q/2+r}e^{ikr}=\left|2\sin\frac{k}{2}\right|^q, \tag*{(53)} r = − ∞ ∑ ∞ ( − 1 ) r ( q /2 + r q ) e ik r = 2 sin 2 k q , (53) so that, separating the r = 0 r=0 r = 0 (20) . Expanding the sine for k → 0 k\to0 k → 0
j q ( k ) = J q − k q [ 1 − q 24 k 2 + O ( k 4 ) ] . (54) j_q(k)=J_q-k^q\left[1-\frac{q}{24}k^2+\mathcal{O}(k^4)\right]. \tag*{(54)} j q ( k ) = J q − k q [ 1 − 24 q k 2 + O ( k 4 ) ] . (54) No non-analytic contribution beyond k q k^q k q (53) holds identically on the Brillouin zone. The sine sum does not terminate and is expressed through the ordinary hypergeometric function, Eq. (20) . The limit k → 0 k\to0 k → 0 ζ = e i k → 1 \zeta=e^{ik}\to1 ζ = e ik → 1 2 F 1 {}_2F_1 2 F 1
2 F 1 ( a , b ; c ; ζ ) = Γ ( c ) Γ ( c − a − b ) Γ ( c − a ) Γ ( c − b ) 2 F 1 ( a , b ; a + b − c + 1 ; 1 − ζ ) + Γ ( c ) Γ ( a + b − c ) Γ ( a ) Γ ( b ) ( 1 − ζ ) c − a − b 2 F 1 ( c − a , c − b ; c − a − b + 1 ; 1 − ζ ) (55) {}_2F_1(a,b;c;\zeta)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}{}_2F_1(a,b;a+b-c+1;1-\zeta)
+\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}(1-\zeta)^{c-a-b}{}_2F_1(c-a,c-b;c-a-b+1;1-\zeta) \tag*{(55)} 2 F 1 ( a , b ; c ; ζ ) = Γ ( c − a ) Γ ( c − b ) Γ ( c ) Γ ( c − a − b ) 2 F 1 ( a , b ; a + b − c + 1 ; 1 − ζ ) + Γ ( a ) Γ ( b ) Γ ( c ) Γ ( a + b − c ) ( 1 − ζ ) c − a − b 2 F 1 ( c − a , c − b ; c − a − b + 1 ; 1 − ζ ) (55) with ( a , b , c ) = ( − q / 2 , 1 , q / 2 + 1 ) (a,b,c)=(-q/2,1,q/2+1) ( a , b , c ) = ( − q /2 , 1 , q /2 + 1 ) c − a − b = q c-a-b=q c − a − b = q ( 1 − ζ ) q (1-\zeta)^q ( 1 − ζ ) q 1 − ζ 1-\zeta 1 − ζ 1 − e i k = − i k + k 2 / 2 + O ( k 3 ) 1-e^{ik}=-ik+k^2/2+\mathcal{O}(k^3) 1 − e ik = − ik + k 2 /2 + O ( k 3 )
d q ( k ) = a q k + c q k q + O ( k min ( 3 , q + 2 ) ) , (56) d_q(k)=a_q k+c_q k^q+\mathcal{O}\left(k^{\min(3,q+2)}\right), \tag*{(56)} d q ( k ) = a q k + c q k q + O ( k m i n ( 3 , q + 2 ) ) , (56) with the amplitudes a q a_q a q c q c_q c q (25) . Whichever exponent is the smaller controls the limit, giving d q ( k ) ≃ c ∗ k q ∗ d_q(k)\simeq c_*k^{q_*} d q ( k ) ≃ c ∗ k q ∗ q ∗ = min ( q , 1 ) q_*=\min(q,1) q ∗ = min ( q , 1 ) c ∗ c_* c ∗ (27) .
Two special cases follow directly. At even integers q = 2 , 4 , … q=2,4,\ldots q = 2 , 4 , … c q = 0 c_q=0 c q = 0 1 / Γ ( − q / 2 ) = 0 1/\Gamma(-q/2)=0 1/Γ ( − q /2 ) = 0 sin ( π q / 2 ) \sin(\pi q/2) sin ( π q /2 ) Γ ( − q ) \Gamma(-q) Γ ( − q ) q = 2 q = 2 q = 2 d 2 ( k ) = 2 sin k d_2(k) = 2\sin k d 2 ( k ) = 2 sin k ln k \ln k ln k q = 1 q = 1 q = 1
d 1 ( k ) = 4 π sin k 2 ln cot k 4 ≃ 2 π k ln 4 k , (57) d_1(k) = \frac{4}{\pi}\sin\frac{k}{2}\ln\cot\frac{k}{4} \simeq\frac{2}{\pi}k\ln\frac{4}{k}, \tag*{(57)} d 1 ( k ) = π 4 sin 2 k ln cot 4 k ≃ π 2 k ln k 4 , (57) consistent with the divergence of a q a_q a q q → 1 q \to1 q → 1 a q a_q a q Γ ( − q ) \Gamma(-q) Γ ( − q ) c q c_q c q
At q = 1 q = 1 q = 1 (30) therefore becomes
p k ≃ exp [ − β k 2 ln 2 ( 4 / k ) ] , β = 4 π c 0 2 τ Q ( 2 + J q ) 3 , (58) p_k \simeq\exp\left[-\beta k^2\ln^2(4/k)\right], \qquad\beta= \frac{4\pi c_0^2\tau_Q}{(2+J_q)^3}, \tag*{(58)} p k ≃ exp [ − β k 2 ln 2 ( 4/ k ) ] , β = ( 2 + J q ) 3 4 π c 0 2 τ Q , (58) with c 0 = 2 / π c_0 = 2/\pi c 0 = 2/ π (57) . Writing L = ln τ Q \mathcal{L} = \ln\tau_Q L = ln τ Q (32) is dominated by k ≲ k ∗ ∼ τ Q − 1 / 2 / L k \lesssim k_* \sim\tau_Q^{-1/2}/\mathcal{L} k ≲ k ∗ ∼ τ Q − 1/2 / L k ∗ k_* k ∗
n e x ∣ q = 1 = c τ Q − 1 / 2 L − 2 c τ Q − 1 / 2 ln L L 2 + O ( τ Q − 1 / 2 L 2 ) , (59) n_{\mathrm{ex}}\big|_{q=1} = c\frac{\tau_Q^{-1/2}}{\mathcal{L}} - 2c\frac{\tau_Q^{-1/2}\ln\mathcal{L}}{\mathcal{L}^2} + \mathcal{O}\left(\frac{\tau_Q^{-1/2}}{\mathcal{L}^2}\right), \tag*{(59)} n ex q = 1 = c L τ Q − 1/2 − 2 c L 2 τ Q − 1/2 ln L + O ( L 2 τ Q − 1/2 ) , (59) with C = ( 2 + J q ) 3 / 2 / ( 2 π c 0 ) ≃ 1.481 C = (2+J_q)^{3/2}/(2\pi c_0) \simeq1.481 C = ( 2 + J q ) 3/2 / ( 2 π c 0 ) ≃ 1.481 τ Q − 1 / 2 \tau_Q^{-1/2} τ Q − 1/2 1 / 2 1/2 1/2
Finally, at g = g c F F g = g_c^{\mathrm{FF}} g = g c FF (24) the kinetic term reduces to
ξ k ( g c F F ) = ( 1 − g c F F ) ∣ 2 sin k 2 ∣ q ≃ 2 2 + J q k q , (60) \xi_k(g_c^{\mathrm{FF}}) = \left(1-g_c^{\mathrm{FF}}\right)\left|2\sin\frac{k}{2}\right|^q \simeq\frac{2}{2+J_q}k^q, \tag*{(60)} ξ k ( g c FF ) = ( 1 − g c FF ) 2 sin 2 k q ≃ 2 + J q 2 k q , (60) with the constant part canceling by the definition of g c F F g_c^{\mathrm{FF}} g c FF Δ k ( g c F F ) ≃ 2 c ∗ k q ∗ / ( 2 + J q ) \Delta_k(g_c^{\mathrm{FF}}) \simeq2c_*k^{q_*}/(2+J_q) Δ k ( g c FF ) ≃ 2 c ∗ k q ∗ / ( 2 + J q )
ϵ k 2 ( g c F F ) = ξ k 2 + Δ k 2 ∝ k 2 q + k 2 q ∗ , (61) \epsilon_k^2(g_c^{\mathrm{FF}}) = \xi_k^2+\Delta_k^2 \propto k^{2q}+k^{2q_*}, \tag*{(61)} ϵ k 2 ( g c FF ) = ξ k 2 + Δ k 2 ∝ k 2 q + k 2 q ∗ , (61) so that ϵ k ( g c F F ) ∝ k q ∗ \epsilon_k(g_c^{\mathrm{FF}}) \propto k^{q_*} ϵ k ( g c FF ) ∝ k q ∗ z = min ( q , 1 ) z = \min(q,1) z = min ( q , 1 ) q = 1 q = 1 q = 1
As a check of the normalization, at q = 2 q = 2 q = 2 J 2 = 2 J_2 = 2 J 2 = 2 g c F F = 1 / 2 g_c^{\mathrm{FF}} = 1/2 g c FF = 1/2 c ∗ = a 2 = 2 c_* = a_2 = 2 c ∗ = a 2 = 2 d 2 ( k ) = 2 sin k d_2(k) = 2\sin k d 2 ( k ) = 2 sin k (30) then gives λ 2 = 4 π c ∗ 2 / ( 2 + J 2 ) 3 = π / 4 \lambda_2 = 4\pi c_*^2/(2+J_2)^3 = \pi/4 λ 2 = 4 π c ∗ 2 / ( 2 + J 2 ) 3 = π /4 p k = e − π τ Q k 2 / 4 p_k = e^{-\pi\tau_Q k^2/4} p k = e − π τ Q k 2 /4 n e x = 1 / ( π τ Q ) n_{\mathrm{ex}} = 1/(\pi\sqrt{\tau_Q}) n ex = 1/ ( π τ Q ) (30) against direct integration of the Bogoliubov-de Gennes equations (29) , obtaining agreement to within two percent over 0.5 ≤ q ≤ 3 0.5 \le q \le3 0.5 ≤ q ≤ 3
Finite-size threshold for the low-momentum excitations
In this section, we show how the time scale for the breakdown of the continuum approximation in the low-momentum regime changes with the fractional order.
Adiabatic perturbation theory for high-momentum modes at small q
The continuum limit of Eq. (32) requires a sufficient number of modes below the Landau-Zener momentum k ∗ = ( λ q τ Q ) − 1 / ( 2 q ∗ ) k_* = (\lambda_q\tau_Q)^{-1/(2q_*)} k ∗ = ( λ q τ Q ) − 1/ ( 2 q ∗ ) λ q = 4 π c ∗ 2 / ( 2 + J q ) 3 \lambda_q = 4\pi c_*^2/(2+J_q)^3 λ q = 4 π c ∗ 2 / ( 2 + J q ) 3 q ∗ = min ( q , 1 ) q_* = \min(q,1) q ∗ = min ( q , 1 ) k ∗ k_* k ∗ π / L \pi/L π / L
τ F S = 1 λ q ( L π ) 2 q ∗ , (62) \tau_{\mathrm{FS}} = \frac{1}{\lambda_q}\left(\frac{L}{\pi}\right)^{2q_*}, \tag*{(62)} τ FS = λ q 1 ( π L ) 2 q ∗ , (62) beyond which only the pair k = π / L k = \pi/L k = π / L τ a d \tau_{\mathrm{ad}} τ ad τ F S \tau_{\mathrm{FS}} τ FS N e x = L N q τ Q − 1 / ( 2 q ∗ ) ≲ 2 N_{\mathrm{ex}} = L\mathcal{N}_q\tau_Q^{-1/(2q_*)} \lesssim2 N ex = L N q τ Q − 1/ ( 2 q ∗ ) ≲ 2 N q = Γ ( 1 + 1 / ( 2 q ∗ ) ) / ( π λ q 1 / ( 2 q ∗ ) ) \mathcal{N}_q = \Gamma(1+1/(2q_*))/(\pi\lambda_q^{1/(2q_*)}) N q = Γ ( 1 + 1/ ( 2 q ∗ )) / ( π λ q 1/ ( 2 q ∗ ) )
τ a d = 1 λ q [ L Γ ( 1 + 1 2 q ∗ ) 2 π ] 2 q ∗ . (63) \tau_{\mathrm{ad}} = \frac{1}{\lambda_q}\left[\frac{L\Gamma\left(1+\frac{1}{2q_*}\right)}{2\pi}\right]^{2q_*}. \tag*{(63)} τ ad = λ q 1 2 π L Γ ( 1 + 2 q ∗ 1 ) 2 q ∗ . (63) Both time scales grow as ( L / π ) 2 q ∗ / λ q (L/\pi)^{2q_*}/\lambda_q ( L / π ) 2 q ∗ / λ q
τ a d τ F S = [ Γ ( 1 + 1 2 q ∗ ) 2 ] 2 q ∗ . (64) \frac{\tau_{\mathrm{ad}}}{\tau_{\mathrm{FS}}} = \left[\frac{\Gamma\left(1+\frac{1}{2q_*}\right)}{2}\right]^{2q_*}. \tag*{(64)} τ FS τ ad = 2 Γ ( 1 + 2 q ∗ 1 ) 2 q ∗ . (64) This function decreases monotonically, becomes unity at q = 1 / 4 q = 1/4 q = 1/4 π / 16 \pi/16 π /16 q ≥ 1 q \ge1 q ≥ 1 τ a d ≃ 0.2 τ F S \tau_{\mathrm{ad}} \simeq0.2\tau_{\mathrm{FS}} τ ad ≃ 0.2 τ FS q > 1 / 4 q > 1/4 q > 1/4 τ a d < τ F S \tau_{\mathrm{ad}} < \tau_{\mathrm{FS}} τ ad < τ FS q < 1 / 4 q < 1/4 q < 1/4 τ F S < τ a d \tau_{\mathrm{FS}} < \tau_{\mathrm{ad}} τ FS < τ ad L L L q ≳ 0.1 q \gtrsim0.1 q ≳ 0.1 16 / π ≃ 5 16/\pi\simeq5 16/ π ≃ 5 L = 150 L = 150 L = 150 (62) and (63) give τ F S ≃ 1.5 × 10 3 \tau_{\mathrm{FS}} \simeq1.5\times10^3 τ FS ≃ 1.5 × 1 0 3 τ a d ≃ 2.9 × 10 2 \tau_{\mathrm{ad}} \simeq2.9\times10^2 τ ad ≃ 2.9 × 1 0 2 q = 1.5 q = 1.5 q = 1.5 τ F S ≃ 3.4 × 10 3 \tau_{\mathrm{FS}} \simeq3.4\times10^3 τ FS ≃ 3.4 × 1 0 3 τ a d ≃ 6.6 × 10 2 \tau_{\mathrm{ad}} \simeq6.6\times10^2 τ ad ≃ 6.6 × 1 0 2 q = 2.25 q = 2.25 q = 2.25 τ F S ≃ 90 \tau_{\mathrm{FS}} \simeq90 τ FS ≃ 90 τ a d ≃ 60 \tau_{\mathrm{ad}} \simeq60 τ ad ≃ 60 0.1 ≤ q ≤ 0.5 0.1 \le q \le0.5 0.1 ≤ q ≤ 0.5 L = 150 L = 150 L = 150 q < 1 / 4 q < 1/4 q < 1/4 k = π k = \pi k = π k k k
Appendix D: Adiabatic perturbation theory for high-momentum modes at small q q q
In this section, we derive the leading-order formula, p k ≃ q ˙ k 2 / ( 4 τ Q 2 ϵ k 6 ( g = 0 ) ) p_k \simeq\dot{q}_k^2/(4\tau_Q^2\epsilon_k^6(g=0)) p k ≃ q ˙ k 2 / ( 4 τ Q 2 ϵ k 6 ( g = 0 )) ξ k \xi_k ξ k (18) does not change sign during the ramp. As ξ k = 2 g − ( 1 − g ) J q ( k ) \xi_k = 2g-(1-g)J_q(k) ξ k = 2 g − ( 1 − g ) J q ( k ) g = 1 g=1 g = 1 − j q ( k ) -j_q(k) − j q ( k ) g = 0 g=0 g = 0 j q ( k ) = J q − ∣ 2 sin ( k / 2 ) ∣ q > 0 j_q(k)=J_q-\lvert2\sin(k/2)\rvert^q>0 j q ( k ) = J q − ∣ 2 sin ( k /2 ) ∣ q > 0 k c < k < π k_c<k<\pi k c < k < π
2 sin k c 2 = J q 1 / q , (65) 2\sin\frac{k_c}{2}=J_q^{1/q}, \tag*{(65)} 2 sin 2 k c = J q 1/ q , (65) no avoided crossing emerges, and the Landau-Zener form of p k = exp [ − 4 π τ Q d q 2 ( k ) ( 2 + j q ( k ) ) 3 ] p_k=\exp\left[-\frac{4\pi\tau_Q d_q^2(k)}{(2+j_q(k))^3}\right] p k = exp [ − ( 2 + j q ( k ) ) 3 4 π τ Q d q 2 ( k ) ] k = π k=\pi k = π g = 0 g=0 g = 0 d q ( π ) = 0 d_q(\pi)=0 d q ( π ) = 0 ϵ π ( g = 0 ) = ∣ j q ( π ) ∣ = 2 q − J q ≃ q ln 2 \epsilon_\pi(g=0)=\lvert j_q(\pi)\rvert=2^q-J_q\simeq q\ln2 ϵ π ( g = 0 ) = ∣ j q ( π )∣ = 2 q − J q ≃ q ln 2 q q q ( k , − k ) (k,-k) ( k , − k ) ψ ^ k † H k ( t ) ψ ^ k \hat{\psi}_k^\dagger H_k(t)\hat{\psi}_k ψ ^ k † H k ( t ) ψ ^ k ψ ^ k = ( c ^ k , c ^ − k † ) T \hat{\psi}_k=(\hat{c}_k,\hat{c}_{-k}^\dagger)^T ψ ^ k = ( c ^ k , c ^ − k † ) T H k ( t ) = ξ k ( t ) τ z + Δ k ( t ) τ x H_k(t)=\xi_k(t)\tau^z+\Delta_k(t)\tau^x H k ( t ) = ξ k ( t ) τ z + Δ k ( t ) τ x τ x , z \tau^{x,z} τ x , z k k k ∣ 0 ⟩ k \lvert0\rangle_k ∣ 0 ⟩ k ∣ 0 ⟩ k \lvert0\rangle_k ∣ 0 ⟩ k ∣ 2 ⟩ k ≡ c ^ k † c ^ − k † ∣ 0 ⟩ k \lvert2\rangle_k\equiv\hat{c}_k^\dagger\hat{c}_{-k}^\dagger\lvert0\rangle_k ∣ 2 ⟩ k ≡ c ^ k † c ^ − k † ∣ 0 ⟩ k k k k
∣ Ω k ( t ) ⟩ = cos ϑ k ∣ 0 ⟩ k − sin ϑ k ∣ 2 ⟩ k , \lvert\Omega_k(t)\rangle=\cos\vartheta_k\lvert0\rangle_k-\sin\vartheta_k\lvert2\rangle_k, ∣ Ω k ( t )⟩ = cos ϑ k ∣ 0 ⟩ k − sin ϑ k ∣ 2 ⟩ k , ∣ Ω k + ( t ) ⟩ = sin ϑ k ∣ 0 ⟩ k + cos ϑ k ∣ 2 ⟩ k , (66) \lvert\Omega_k^+(t)\rangle=\sin\vartheta_k\lvert0\rangle_k+\cos\vartheta_k\lvert2\rangle_k, \tag*{(66)} ∣ Ω k + ( t )⟩ = sin ϑ k ∣ 0 ⟩ k + cos ϑ k ∣ 2 ⟩ k , (66) with the energies ∓ ϵ k \mp\epsilon_k ∓ ϵ k ϑ k ( t ) \vartheta_k(t) ϑ k ( t ) (22) . The time-evolved state of Eq. (29) is written as
∣ Ψ k ( t ) ⟩ = a k ( t ) e i φ k ( t ) ∣ Ω k ( t ) ⟩ + b k ( t ) e − i φ k ( t ) ∣ Ω k + ( t ) ⟩ , (67) \lvert\Psi_k(t)\rangle=a_k(t)e^{i\varphi_k(t)}\lvert\Omega_k(t)\rangle+b_k(t)e^{-i\varphi_k(t)}\lvert\Omega_k^+(t)\rangle, \tag*{(67)} ∣ Ψ k ( t )⟩ = a k ( t ) e i φ k ( t ) ∣ Ω k ( t )⟩ + b k ( t ) e − i φ k ( t ) ∣ Ω k + ( t )⟩ , (67) with φ k ( t ) = ∫ 0 t ϵ k ( t ′ ) d t ′ \varphi_k(t)=\int_0^t\epsilon_k(t')\,\mathrm{d}t' φ k ( t ) = ∫ 0 t ϵ k ( t ′ ) d t ′ a k ( 0 ) = 1 a_k(0)=1 a k ( 0 ) = 1 b k ( 0 ) = 0 b_k(0)=0 b k ( 0 ) = 0 g = 1 g=1 g = 1 ϑ k = 0 \vartheta_k=0 ϑ k = 0 ∣ Ω k ⟩ = ∣ 0 ⟩ k \lvert\Omega_k\rangle=\lvert0\rangle_k ∣ Ω k ⟩ = ∣ 0 ⟩ k ∂ t ∣ Ω k ⟩ = − ϑ ˙ k ∣ Ω k + ⟩ \partial_t\lvert\Omega_k\rangle=-\dot{\vartheta}_k\lvert\Omega_k^+\rangle ∂ t ∣ Ω k ⟩ = − ϑ ˙ k ∣ Ω k + ⟩ ∂ t ∣ Ω k + ⟩ = ϑ ˙ k ∣ Ω k ⟩ \partial_t\lvert\Omega_k^+\rangle=\dot{\vartheta}_k\lvert\Omega_k\rangle ∂ t ∣ Ω k + ⟩ = ϑ ˙ k ∣ Ω k ⟩
a ˙ k = − ϑ ˙ k e − 2 i φ k b k , b ˙ k = ϑ ˙ k e 2 i φ k a k . (68) \dot{a}_k=-\dot{\vartheta}_k e^{-2i\varphi_k}b_k,\qquad\dot{b}_k=\dot{\vartheta}_k e^{2i\varphi_k}a_k. \tag*{(68)} a ˙ k = − ϑ ˙ k e − 2 i φ k b k , b ˙ k = ϑ ˙ k e 2 i φ k a k . (68) To the leading order in τ Q \tau_Q τ Q ϑ ˙ k ∝ τ Q − 1 \dot{\vartheta}_k\propto\tau_Q^{-1} ϑ ˙ k ∝ τ Q − 1 a k ≃ 1 a_k\simeq1 a k ≃ 1 e 2 i φ k = ( 2 i ϵ k ) − 1 d e 2 i φ k / d t e^{2i\varphi_k}=(2i\epsilon_k)^{-1}\mathrm{d}e^{2i\varphi_k}/\mathrm{d}t e 2 i φ k = ( 2 i ϵ k ) − 1 d e 2 i φ k / d t
b k ( τ Q ) = ∫ 0 τ Q ϑ ˙ k e 2 i φ k d t = [ ϑ ˙ k 2 i ϵ k e 2 i φ k ] 0 τ Q − ∫ 0 τ Q d d t ( ϑ ˙ k 2 i ϵ k ) e 2 i φ k d t . (69) \begin{aligned}
b_k(\tau_Q)=\int_0^{\tau_Q}\dot{\vartheta}_k e^{2i\varphi_k}\,\mathrm{d}t \\
&=\left[\frac{\dot{\vartheta}_k}{2i\epsilon_k}e^{2i\varphi_k}\right]_0^{\tau_Q}-\int_0^{\tau_Q}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\dot{\vartheta}_k}{2i\epsilon_k}\right)e^{2i\varphi_k}\,\mathrm{d}t. \tag*{(69)}
\end{aligned} b k ( τ Q ) = ∫ 0 τ Q ϑ ˙ k e 2 i φ k d t = [ 2 i ϵ k ϑ ˙ k e 2 i φ k ] 0 τ Q − ∫ 0 τ Q d t d ( 2 i ϵ k ϑ ˙ k ) e 2 i φ k d t . (69) where the remaining integral is of order τ Q − 2 \tau_Q^{-2} τ Q − 2 tan 2 ϑ k = ( 1 − g ) d q ( k ) / [ 2 g − ( 1 − g ) j q ( k ) ] \tan2\vartheta_k=(1-g)d_q(k)/[2g-(1-g)j_q(k)] tan 2 ϑ k = ( 1 − g ) d q ( k ) / [ 2 g − ( 1 − g ) j q ( k )] g ˙ = − 1 / τ Q \dot{g}=-1/\tau_Q g ˙ = − 1/ τ Q ξ k = 2 \xi_k=2 ξ k = 2 Δ k = 0 \Delta_k=0 Δ k = 0 d ( tan 2 ϑ k ) / d g = − d q ( k ) / 2 \mathrm{d}(\tan2\vartheta_k)/\mathrm{d}g=-d_q(k)/2 d ( tan 2 ϑ k ) / d g = − d q ( k ) /2 g = 1 g=1 g = 1 d ( tan 2 ϑ k ) / d g = − 2 d q ( k ) / j q 2 ( k ) \mathrm{d}(\tan2\vartheta_k)/\mathrm{d}g=-2d_q(k)/j_q^2(k) d ( tan 2 ϑ k ) / d g = − 2 d q ( k ) / j q 2 ( k ) cos 2 2 ϑ k = j q 2 ( k ) / ϵ k 2 ( g = 0 ) \cos^2 2\vartheta_k=j_q^2(k)/\epsilon_k^2(g=0) cos 2 2 ϑ k = j q 2 ( k ) / ϵ k 2 ( g = 0 ) g = 0 g=0 g = 0
ϑ ˙ k ∣ g = 1 = d q ( k ) 4 τ Q , ϑ ˙ k ∣ g = 0 = d q ( k ) τ Q ϵ k 2 ( g = 0 ) , (70) \left.\dot{\vartheta}_k\right|_{g=1}=\frac{d_q(k)}{4\tau_Q},\qquad\left.\dot{\vartheta}_k\right|_{g=0}=\frac{d_q(k)}{\tau_Q\epsilon_k^2(g=0)}, \tag*{(70)} ϑ ˙ k g = 1 = 4 τ Q d q ( k ) , ϑ ˙ k g = 0 = τ Q ϵ k 2 ( g = 0 ) d q ( k ) , (70) which yields for the excitation probability
p k = ∣ b k ( τ Q ) ∣ 2 ≃ d q 2 ( k ) 4 τ Q 2 ∣ e 2 i φ k ( τ Q ) ϵ k 3 ( g = 0 ) − 1 8 ∣ 2 . (71) p_k=\lvert b_k(\tau_Q)\rvert^2\simeq\frac{d_q^2(k)}{4\tau_Q^2}\left\lvert\frac{e^{2i\varphi_k(\tau_Q)}}{\epsilon_k^3(g=0)}-\frac{1}{8}\right\rvert^2. \tag*{(71)} p k = ∣ b k ( τ Q ) ∣ 2 ≃ 4 τ Q 2 d q 2 ( k ) ϵ k 3 ( g = 0 ) e 2 i φ k ( τ Q ) − 8 1 2 . (71) As for small q q q ϵ k ( g = 0 ) ∝ q \epsilon_k(g=0)\propto q ϵ k ( g = 0 ) ∝ q
p k ≃ d q 2 ( k ) 4 τ Q 2 ϵ k 6 ( g = 0 ) . (72) p_k\simeq\frac{d_q^2(k)}{4\tau_Q^2\epsilon_k^6(g=0)}. \tag*{(72)} p k ≃ 4 τ Q 2 ϵ k 6 ( g = 0 ) d q 2 ( k ) . (72) Neglecting the remaining integral in Eq. (69) requires the gap to vary slowly on the time scale 1 / ϵ k ( g = 0 ) 1/\epsilon_k(g=0) 1/ ϵ k ( g = 0 ) τ Q ϵ k 2 ( g = 0 ) ≫ 1 \tau_Q\epsilon_k^2(g=0)\gg1 τ Q ϵ k 2 ( g = 0 ) ≫ 1 k = π k=\pi k = π τ Q ≫ ( 2 q − J q ) − 2 ≃ ( q ln 2 ) − 2 \tau_Q\gg(2^q-J_q)^{-2}\simeq(q\ln2)^{-2} τ Q ≫ ( 2 q − J q ) − 2 ≃ ( q ln 2 ) − 2 q q q (72) over the modes of Eq. (65) in the continuum limit, modes above the threshold k c < k < π k_c<k<\pi k c < k < π
N n c = L B q τ Q − 2 , B q = 1 π ∫ k c π d q 2 ( k ) 4 ϵ k 6 ( g = 0 ) d k . (73) N_{\mathrm{nc}}=LB_q\tau_Q^{-2},\qquad B_q=\frac{1}{\pi}\int_{k_c}^{\pi}\frac{d_q^2(k)}{4\epsilon_k^6(g=0)}\,\mathrm{d}k. \tag*{(73)} N nc = L B q τ Q − 2 , B q = π 1 ∫ k c π 4 ϵ k 6 ( g = 0 ) d q 2 ( k ) d k . (73) First assuming that the continuum approximation can be applied for the low-momentum excitations, N e x = L π ∫ 0 ∞ e − λ q τ Q k 2 q ∗ d k = L Γ ( 1 + 1 2 q ∗ ) / [ π ( λ q τ Q ) 1 / ( 2 q ∗ ) ] N_{\mathrm{ex}}=\frac{L}{\pi}\int_0^\infty e^{-\lambda_q\tau_Q k^{2q^*}}\,\mathrm{d}k= L\Gamma\left(1+\frac{1}{2q^*}\right)/[\pi(\lambda_q\tau_Q)^{1/(2q^*)}] N ex = π L ∫ 0 ∞ e − λ q τ Q k 2 q ∗ d k = L Γ ( 1 + 2 q ∗ 1 ) / [ π ( λ q τ Q ) 1/ ( 2 q ∗ ) ]
N n c N e x = π B q λ q 1 / ( 2 q ∗ ) Γ ( 1 + 1 2 q ∗ ) 1 τ Q 2 − 1 2 q ∗ . (74) \frac{N_{\mathrm{nc}}}{N_{\mathrm{ex}}}=\frac{\pi B_q\lambda_q^{1/(2q^*)}}{\Gamma\left(1+\frac{1}{2q^*}\right)}\frac{1}{\tau_Q^{2-\frac{1}{2q^*}}}. \tag*{(74)} N ex N nc = Γ ( 1 + 2 q ∗ 1 ) π B q λ q 1/ ( 2 q ∗ ) τ Q 2 − 2 q ∗ 1 1 . (74) As it grows with τ Q \tau_Q τ Q q < 1 / 4 q<1/4 q < 1/4 q > 1 / 4 q>1/4 q > 1/4 q > 1 / 4 q>1/4 q > 1/4 q < 1 / 4 q<1/4 q < 1/4 k = π k=\pi k = π q > 1 / 4 q>1/4 q > 1/4 τ a d ≪ τ F S \tau_{\mathrm{ad}}\ll\tau_{\mathrm{FS}} τ ad ≪ τ FS τ a d \tau_{\mathrm{ad}} τ ad § 2.1.1 .
In the complementary regime, where τ Q ≫ τ F S \tau_Q\gg\tau_{\mathrm{FS}} τ Q ≫ τ FS N e x ≃ 2 e − τ Q / τ F S N_{\mathrm{ex}}\simeq2e^{-\tau_Q/\tau_{\mathrm{FS}}} N ex ≃ 2 e − τ Q / τ FS N n c N_{\mathrm{nc}} N nc (73) remains unchanged, as the excitation probability of Eq. (72) varies with k k k x = τ Q / τ F S x=\tau_Q/\tau_{\mathrm{FS}} x = τ Q / τ FS
N n c N e x = L B q 2 τ Q 2 e τ Q / τ F S = L B q 2 τ F S 2 e x x 2 , (75) \frac{N_{\mathrm{nc}}}{N_{\mathrm{ex}}}=\frac{LB_q}{2\tau_Q^2}e^{\tau_Q/\tau_{\mathrm{FS}}}=\frac{LB_q}{2\tau_{\mathrm{FS}}^2}\frac{e^x}{x^2}, \tag*{(75)} N ex N nc = 2 τ Q 2 L B q e τ Q / τ FS = 2 τ FS 2 L B q x 2 e x , (75) which grows exponentially with τ Q \tau_Q τ Q q q q k = π k=\pi k = π q q q x x x x ∗ x_* x ∗ e x ∗ / x ∗ 2 = 2 τ F S 2 / ( L B q ) e^{x_*}/x_*^2=2\tau_{\mathrm{FS}}^2/(LB_q) e x ∗ / x ∗ 2 = 2 τ FS 2 / ( L B q ) q q q J r , q ≃ q / ( 2 r ) J_{r,q} \simeq q/(2r) J r , q ≃ q / ( 2 r ) j q ( k ) ≃ − q ln ( 2 sin k / 2 ) j_q(k) \simeq-q\ln(2\sin k/2) j q ( k ) ≃ − q ln ( 2 sin k /2 ) d q ( k ) ≃ q ( π − k ) / 2 d_q(k) \simeq q(\pi-k)/2 d q ( k ) ≃ q ( π − k ) /2 J q ≃ 1 J_q \simeq1 J q ≃ 1 c q ≃ π q / 2 c_q \simeq\pi q/2 c q ≃ π q /2 k c ≃ π / 3 k_c \simeq\pi/3 k c ≃ π /3 λ q ≃ π 3 q 2 / 27 \lambda_q \simeq\pi^3q^2/27 λ q ≃ π 3 q 2 /27 τ F S ≃ 27 / ( π 3 q 2 ) ( L / π ) 2 q \tau_{\mathrm{FS}} \simeq27/(\pi^3q^2)(L/\pi)^{2q} τ FS ≃ 27/ ( π 3 q 2 ) ( L / π ) 2 q
B q ≃ I 0 4 π q 4 , I 0 = ∫ π / 3 π ( ( π − k ) / 2 ) 2 [ ln 2 ( 2 sin ( k / 2 ) ) + ( ( π − k ) / 2 ) 2 ] 3 d k ≃ 2.10. (76) B_q \simeq\frac{I_0}{4\pi q^4}, \qquad I_0=\int_{\pi/3}^{\pi}\frac{((\pi-k)/2)^2}{\left[\ln^2(2\sin(k/2))+((\pi-k)/2)^2\right]^3}\,\mathrm{d}k\simeq2.10. \tag*{(76)} B q ≃ 4 π q 4 I 0 , I 0 = ∫ π /3 π [ ln 2 ( 2 sin ( k /2 )) + (( π − k ) /2 ) 2 ] 3 (( π − k ) /2 ) 2 d k ≃ 2.10. (76) Reassuringly, in this regime τ Q ≫ τ F S \tau_Q\gg\tau_{\mathrm{FS}} τ Q ≫ τ FS τ Q ≫ q − 2 \tau_Q\gg q^{-2} τ Q ≫ q − 2 q − 4 q^{-4} q − 4 B q B_q B q τ F S − 2 ∝ q 4 \tau_{\mathrm{FS}}^{-2}\propto q^4 τ FS − 2 ∝ q 4 (75) becomes
N n c N e x ≃ π 5 I 0 5832 π 4 q L 1 − 4 q x 5 x 2 , π 5 I 0 5832 ≃ 0.11. (77) \frac{N_{\mathrm{nc}}}{N_{\mathrm{ex}}}\simeq\frac{\pi^5 I_0}{5832}\pi^{4q}L^{1-4q}\frac{x^5}{x^2},\qquad\frac{\pi^5 I_0}{5832}\simeq0.11. \tag*{(77)} N ex N nc ≃ 5832 π 5 I 0 π 4 q L 1 − 4 q x 2 x 5 , 5832 π 5 I 0 ≃ 0.11. (77) The modes near k = π k=\pi k = π q ≲ ln ( 0.11 L x / x 2 ) / [ 4 ln ( L / π ) ] q\lesssim\ln(0.11L^{x}/x^2)/[4\ln(L/\pi)] q ≲ ln ( 0.11 L x / x 2 ) / [ 4 ln ( L / π )] x ≃ 1 x\simeq1 x ≃ 1 q ≲ ln ( 0.30 L ) / [ 4 ln ( L / π ) ] q\lesssim\ln(0.30L)/[4\ln(L/\pi)] q ≲ ln ( 0.30 L ) / [ 4 ln ( L / π )] q ≲ 0.246 q\lesssim0.246 q ≲ 0.246 L = 150 L=150 L = 150 q = 1 / 4 q=1/4 q = 1/4 L → ∞ L\to\infty L → ∞ (74) .
Kink number cumulants and their relation to excitations
In this section, we provide the analytical details for the kink number cumulants and show that, at leading order in τ Q \tau_Q τ Q k = π k=\pi k = π
N ^ = 1 2 ∑ i = 1 L ( 1 − σ ^ i x σ ^ i + 1 x ) , σ ^ L + 1 x = σ ^ 1 x . (78) \hat N=\frac{1}{2}\sum_{i=1}^{L}\left(1-\hat\sigma_i^x\hat\sigma_{i+1}^x\right),\qquad\hat\sigma_{L+1}^x=\hat\sigma_1^x. \tag*{(78)} N ^ = 2 1 i = 1 ∑ L ( 1 − σ ^ i x σ ^ i + 1 x ) , σ ^ L + 1 x = σ ^ 1 x . (78) The Jordan-Wigner transformation
σ ^ i z = 1 − 2 c ^ i † c ^ i , σ ^ i x = ∏ j < i ( 1 − 2 c ^ j † c ^ j ) ( c ^ i † + c ^ i ) , (79) \hat\sigma_i^z=1-2\hat c_i^\dagger\hat c_i,\qquad\hat\sigma_i^x=\prod_{j<i}(1-2\hat c_j^\dagger\hat c_j)(\hat c_i^\dagger+\hat c_i), \tag*{(79)} σ ^ i z = 1 − 2 c ^ i † c ^ i , σ ^ i x = j < i ∏ ( 1 − 2 c ^ j † c ^ j ) ( c ^ i † + c ^ i ) , (79) together with ( c ^ i † + c ^ i ) ( 1 − 2 c ^ i † c ^ i ) = c ^ i † − c ^ i (\hat c_i^\dagger+\hat c_i)(1-2\hat c_i^\dagger\hat c_i)=\hat c_i^\dagger-\hat c_i ( c ^ i † + c ^ i ) ( 1 − 2 c ^ i † c ^ i ) = c ^ i † − c ^ i i = 1 , … , L − 1 i=1,\ldots,L-1 i = 1 , … , L − 1
σ ^ i x σ ^ i + 1 x = c ^ i † c ^ i + 1 + c ^ i + 1 † c ^ i + c ^ i † c ^ i + 1 † + c ^ i + 1 c ^ i . (80) \hat\sigma_i^x\hat\sigma_{i+1}^x=\hat c_i^\dagger\hat c_{i+1}+\hat c_{i+1}^\dagger\hat c_i+\hat c_i^\dagger\hat c_{i+1}^\dagger+\hat c_{i+1}\hat c_i. \tag*{(80)} σ ^ i x σ ^ i + 1 x = c ^ i † c ^ i + 1 + c ^ i + 1 † c ^ i + c ^ i † c ^ i + 1 † + c ^ i + 1 c ^ i . (80) For the closing bond, which carries the string of all other sites, using ( 1 − 2 c ^ L † c ^ L ) ( c ^ L † + c ^ L ) = c ^ L † − c ^ L (1-2\hat c_L^\dagger\hat c_L)(\hat c_L^\dagger+\hat c_L)=\hat c_L^\dagger-\hat c_L ( 1 − 2 c ^ L † c ^ L ) ( c ^ L † + c ^ L ) = c ^ L † − c ^ L
σ ^ L x σ ^ 1 x = − P ^ ( c ^ L † − c ^ L ) ( c ^ 1 † + c ^ 1 ) , P ^ = ∏ j = 1 L ( 1 − 2 c ^ j † c ^ j ) . (81) \hat\sigma_L^x\hat\sigma_1^x=-\hat P(\hat c_L^\dagger-\hat c_L)(\hat c_1^\dagger+\hat c_1),\qquad\hat P=\prod_{j=1}^{L}(1-2\hat c_j^\dagger\hat c_j). \tag*{(81)} σ ^ L x σ ^ 1 x = − P ^ ( c ^ L † − c ^ L ) ( c ^ 1 † + c ^ 1 ) , P ^ = j = 1 ∏ L ( 1 − 2 c ^ j † c ^ j ) . (81) where P ^ \hat P P ^ P ^ \hat P P ^ P ^ = 1 \hat P=1 P ^ = 1 c ^ L + 1 = − c ^ 1 \hat c_{L+1}=-\hat c_1 c ^ L + 1 = − c ^ 1
N ^ = L 2 − 1 2 ∑ i = 1 L ( c ^ i † c ^ i + 1 + c ^ i + 1 † c ^ i + c ^ i † c ^ i + 1 † + c ^ i + 1 c ^ i ) . (82) \hat N=\frac{L}{2}-\frac{1}{2}\sum_{i=1}^{L}\left(\hat c_i^\dagger\hat c_{i+1}+\hat c_{i+1}^\dagger\hat c_i+\hat c_i^\dagger\hat c_{i+1}^\dagger+\hat c_{i+1}\hat c_i\right). \tag*{(82)} N ^ = 2 L − 2 1 i = 1 ∑ L ( c ^ i † c ^ i + 1 + c ^ i + 1 † c ^ i + c ^ i † c ^ i + 1 † + c ^ i + 1 c ^ i ) . (82) The fermionic operators are decomposed in Fourier space as
c ^ j = e i π / 4 L ∑ k e − i j k c ^ k , k = π ( 2 m − 1 ) L , (83) \hat c_j=\frac{e^{i\pi/4}}{\sqrt L}\sum_k e^{-ijk}\hat c_k,\qquad k=\frac{\pi(2m-1)}{L}, \tag*{(83)} c ^ j = L e iπ /4 k ∑ e − ij k c ^ k , k = L π ( 2 m − 1 ) , (83) with m = − L / 2 + 1 , … , L / 2 m=-L/2+1,\ldots,L/2 m = − L /2 + 1 , … , L /2 (83) into the four terms of Eq. (82) , using 1 L ∑ j = 1 L e i j ( k − k ′ ) = δ k k ′ \frac{1}{L}\sum_{j=1}^{L}e^{ij(k-k')}=\delta_{kk'} L 1 ∑ j = 1 L e ij ( k − k ′ ) = δ k k ′ k k k − k -k − k k > 0 k>0 k > 0 c ^ − k † c ^ k † = − c ^ k † c ^ − k † \hat c_{-k}^\dagger\hat c_k^\dagger=-\hat c_k^\dagger\hat c_{-k}^\dagger c ^ − k † c ^ k † = − c ^ k † c ^ − k †
∑ i = 1 L σ ^ i x σ ^ i + 1 x = ∑ k > 0 [ 2 cos k ( c ^ k † c ^ k + c ^ − k † c ^ − k ) − 2 sin k ( c ^ k † c ^ − k † + c ^ − k c ^ k ) ] . (84) \sum_{i=1}^{L}\hat\sigma_i^x\hat\sigma_{i+1}^x=\sum_{k>0}\left[2\cos k\left(\hat c_k^\dagger\hat c_k+\hat c_{-k}^\dagger\hat c_{-k}\right)-2\sin k\left(\hat c_k^\dagger\hat c_{-k}^\dagger+\hat c_{-k}\hat c_k\right)\right]. \tag*{(84)} i = 1 ∑ L σ ^ i x σ ^ i + 1 x = k > 0 ∑ [ 2 cos k ( c ^ k † c ^ k + c ^ − k † c ^ − k ) − 2 sin k ( c ^ k † c ^ − k † + c ^ − k c ^ k ) ] . (84) With ψ ^ k = ( c ^ k , c ^ − k † ) T \hat\psi_k=(\hat c_k,\hat c_{-k}^\dagger)^T ψ ^ k = ( c ^ k , c ^ − k † ) T
N ^ = L 2 + 1 2 ∑ k > 0 ψ ^ k † ( − 2 cos k τ z + 2 sin k τ x ) ψ ^ k = ∑ k > 0 n ^ k , (85) \hat N=\frac{L}{2}+\frac{1}{2}\sum_{k>0}\hat\psi_k^\dagger(-2\cos k\,\tau^z+2\sin k\,\tau^x)\hat\psi_k=\sum_{k>0}\hat n_k, \tag*{(85)} N ^ = 2 L + 2 1 k > 0 ∑ ψ ^ k † ( − 2 cos k τ z + 2 sin k τ x ) ψ ^ k = k > 0 ∑ n ^ k , (85) with n ^ k = 1 + ψ ^ k † ( − cos k τ z + sin k τ x ) ψ ^ k \hat n_k=1+\hat\psi_k^\dagger(-\cos k\,\tau^z+\sin k\,\tau^x)\hat\psi_k n ^ k = 1 + ψ ^ k † ( − cos k τ z + sin k τ x ) ψ ^ k τ x , z \tau^{x,z} τ x , z q = 2 q=2 q = 2 J r , 2 = δ r , 1 J_{r,2}=\delta_{r,1} J r , 2 = δ r , 1 j 2 ( k ) = 2 cos k j_2(k)=2\cos k j 2 ( k ) = 2 cos k d 2 ( k ) = 2 sin k d_2(k)=2\sin k d 2 ( k ) = 2 sin k (85) reduces to the transverse-field Ising result, N ^ = L / 2 + 1 2 H ^ F F ( g = 0 ) \hat N=L/2+\frac{1}{2}\hat H_{\mathrm{FF}}(g=0) N ^ = L /2 + 2 1 H ^ FF ( g = 0 ) ∣ 0 ⟩ k |0\rangle_k ∣0 ⟩ k ∣ 2 ⟩ k ≡ c ^ k † c ^ − k † ∣ 0 ⟩ k |2\rangle_k\equiv\hat c_k^\dagger\hat c_{-k}^\dagger|0\rangle_k ∣2 ⟩ k ≡ c ^ k † c ^ − k † ∣0 ⟩ k ∣ u k ⟩ = u k ∣ 0 ⟩ k + v k ∣ 2 ⟩ k |u_k\rangle=u_k|0\rangle_k+v_k|2\rangle_k ∣ u k ⟩ = u k ∣0 ⟩ k + v k ∣2 ⟩ k (29) with initial condition u k ( 0 ) = 1 u_k(0)=1 u k ( 0 ) = 1 v k ( 0 ) = 0 v_k(0)=0 v k ( 0 ) = 0 n ^ k \hat n_k n ^ k k k k
p ~ k = 1 2 ⟨ ψ k ∣ n ^ k ∣ ψ k ⟩ = ∣ cos k 2 u k + sin k 2 v k ∣ 2 . (86) \tilde p_k=\frac{1}{2}\langle\psi_k|\hat n_k|\psi_k\rangle=\left|\cos\frac{k}{2}u_k+\sin\frac{k}{2}v_k\right|^2. \tag*{(86)} p ~ k = 2 1 ⟨ ψ k ∣ n ^ k ∣ ψ k ⟩ = cos 2 k u k + sin 2 k v k 2 . (86) For relating it to the excitation probability, we use again the instantaneous basis of H k H_k H k g = 0 g=0 g = 0 (66) ,
∣ Ω k ⟩ = cos ϑ k ∣ 0 ⟩ k − sin ϑ k ∣ 2 ⟩ k , ∣ Ω k ∗ ⟩ = sin ϑ k ∣ 0 ⟩ k + cos ϑ k ∣ 2 ⟩ k . (87) |\Omega_k\rangle=\cos\vartheta_k|0\rangle_k-\sin\vartheta_k|2\rangle_k,\qquad|\Omega_k^*\rangle=\sin\vartheta_k|0\rangle_k+\cos\vartheta_k|2\rangle_k. \tag*{(87)} ∣ Ω k ⟩ = cos ϑ k ∣0 ⟩ k − sin ϑ k ∣2 ⟩ k , ∣ Ω k ∗ ⟩ = sin ϑ k ∣0 ⟩ k + cos ϑ k ∣2 ⟩ k . (87) where, by Eqs. (19) and (22) with ξ k = − j q ( k ) \xi_k=-j_q(k) ξ k = − j q ( k ) Δ k = d q ( k ) \Delta_k=d_q(k) Δ k = d q ( k ) g = 0 g=0 g = 0 ϑ k \vartheta_k ϑ k tan 2 ϑ k = − d q ( k ) / j q ( k ) \tan2\vartheta_k=-d_q(k)/j_q(k) tan 2 ϑ k = − d q ( k ) / j q ( k ) cos 2 ϑ k = − j q ( k ) / ϵ k ( g = 0 ) \cos2\vartheta_k=-j_q(k)/\epsilon_k(g=0) cos 2 ϑ k = − j q ( k ) / ϵ k ( g = 0 ) sin 2 ϑ k = d q ( k ) / ϵ k ( g = 0 ) \sin2\vartheta_k=d_q(k)/\epsilon_k(g=0) sin 2 ϑ k = d q ( k ) / ϵ k ( g = 0 ) ϵ k ( g = 0 ) = j q 2 ( k ) + d q 2 ( k ) \epsilon_k(g=0)=\sqrt{j_q^2(k)+d_q^2(k)} ϵ k ( g = 0 ) = j q 2 ( k ) + d q 2 ( k )
∣ Ψ k ⟩ = α k ∣ Ω k ⟩ + β k ∣ Ω k + ⟩ . (88) \lvert\Psi_k\rangle=\alpha_k\lvert\Omega_k\rangle+\beta_k\lvert\Omega_k^+\rangle. \tag*{(88)} ∣ Ψ k ⟩ = α k ∣ Ω k ⟩ + β k ∣ Ω k + ⟩ . (88) Comparing with ∣ Ψ k ⟩ = u k ∣ 0 ⟩ k + v k ∣ 2 ⟩ k \lvert\Psi_k\rangle=u_k\lvert0\rangle_k+v_k\lvert2\rangle_k ∣ Ψ k ⟩ = u k ∣ 0 ⟩ k + v k ∣ 2 ⟩ k
u k = α k cos ϑ k + β k sin ϑ k , v k = − α k sin ϑ k + β k cos ϑ k . (89) u_k=\alpha_k\cos\vartheta_k+\beta_k\sin\vartheta_k,\qquad v_k=-\alpha_k\sin\vartheta_k+\beta_k\cos\vartheta_k. \tag*{(89)} u k = α k cos ϑ k + β k sin ϑ k , v k = − α k sin ϑ k + β k cos ϑ k . (89) Inserting Eq. (89) into Eq. (86) and using the angle addition formulas gives
cos k 2 u k + sin k 2 v k = α k cos ( ϑ k + k 2 ) + β k sin ( ϑ k + k 2 ) . (90) \cos\frac{k}{2}u_k+\sin\frac{k}{2}v_k=\alpha_k\cos\left(\vartheta_k+\frac{k}{2}\right)+\beta_k\sin\left(\vartheta_k+\frac{k}{2}\right). \tag*{(90)} cos 2 k u k + sin 2 k v k = α k cos ( ϑ k + 2 k ) + β k sin ( ϑ k + 2 k ) . (90) Measuring ϑ k \vartheta_k ϑ k ( π − k ) / 2 (\pi-k)/2 ( π − k ) /2
δ k = ϑ k − π − k 2 , cos 2 δ k = j q ( k ) cos k + d q ( k ) sin k ϵ k ( g = 0 ) , (91) \delta_k=\vartheta_k-\frac{\pi-k}{2},\qquad\cos2\delta_k=\frac{j_q(k)\cos k+d_q(k)\sin k}{\epsilon_k(g=0)}, \tag*{(91)} δ k = ϑ k − 2 π − k , cos 2 δ k = ϵ k ( g = 0 ) j q ( k ) cos k + d q ( k ) sin k , (91) one has cos ( ϑ k + k / 2 ) = − sin δ k \cos(\vartheta_k+k/2)=-\sin\delta_k cos ( ϑ k + k /2 ) = − sin δ k sin ( ϑ k + k / 2 ) = cos δ k \sin(\vartheta_k+k/2)=\cos\delta_k sin ( ϑ k + k /2 ) = cos δ k
p ~ k = ∣ β k cos δ k − α k sin δ k ∣ 2 . (92) \widetilde p_k=\left\lvert\beta_k\cos\delta_k-\alpha_k\sin\delta_k\right\rvert^2. \tag*{(92)} p k = ∣ β k cos δ k − α k sin δ k ∣ 2 . (92) The squared amplitude of the excited state is the excitation probability of Sec. IV B, ∣ β k ∣ 2 = p k \lvert\beta_k\rvert^2=p_k ∣ β k ∣ 2 = p k ∣ α k ∣ 2 = 1 − p k \lvert\alpha_k\rvert^2=1-p_k ∣ α k ∣ 2 = 1 − p k α k ∗ β k = p k ( 1 − p k ) e i Φ k \alpha_k^*\beta_k=\sqrt{p_k(1-p_k)}e^{i\Phi_k} α k ∗ β k = p k ( 1 − p k ) e i Φ k Φ k \Phi_k Φ k α k \alpha_k α k β k \beta_k β k (92) then relates the kink probability to the excitation probability,
p ~ k = sin 2 δ k + p k cos 2 δ k − p k ( 1 − p k ) sin 2 δ k cos Φ k . (93) \widetilde p_k=\sin^2\delta_k+p_k\cos2\delta_k-\sqrt{p_k(1-p_k)}\sin2\delta_k\cos\Phi_k. \tag*{(93)} p k = sin 2 δ k + p k cos 2 δ k − p k ( 1 − p k ) sin 2 δ k cos Φ k . (93) For small momenta the hopping and pairing dispersions behave as j q ( k ) ≃ J q − k q ≃ J q > 0 j_q(k)\simeq J_q-k^{q}\simeq J_q>0 j q ( k ) ≃ J q − k q ≃ J q > 0 d q ( k ) ≃ c ∗ k q ∗ > 0 d_q(k)\simeq c_*k^{q^*}>0 d q ( k ) ≃ c ∗ k q ∗ > 0 cos 2 ϑ k = − j q ( k ) / ϵ k ( g = 0 ) < 0 \cos2\vartheta_k=-j_q(k)/\epsilon_k(g=0)<0 cos 2 ϑ k = − j q ( k ) / ϵ k ( g = 0 ) < 0 sin 2 ϑ k = d q ( k ) / ϵ k ( g = 0 ) > 0 \sin2\vartheta_k=d_q(k)/\epsilon_k(g=0)>0 sin 2 ϑ k = d q ( k ) / ϵ k ( g = 0 ) > 0 2 ϑ k 2\vartheta_k 2 ϑ k tan 2 ϑ k = − d q ( k ) / j q ( k ) \tan2\vartheta_k=-d_q(k)/j_q(k) tan 2 ϑ k = − d q ( k ) / j q ( k )
2 ϑ k = π − arctan d q ( k ) j q ( k ) ≃ π − c ∗ J q k q ∗ . (94) 2\vartheta_k=\pi-\arctan\frac{d_q(k)}{j_q(k)}\simeq\pi-\frac{c_*}{J_q}k^{q^*}. \tag*{(94)} 2 ϑ k = π − arctan j q ( k ) d q ( k ) ≃ π − J q c ∗ k q ∗ . (94) Using that 2 δ k = 2 ϑ k − π + k 2\delta_k=2\vartheta_k-\pi+k 2 δ k = 2 ϑ k − π + k
2 δ k ≃ k − c ∗ J q k q ∗ → k → 0 0. (95) 2\delta_k\simeq k-\frac{c_*}{J_q}k^{q^*}\xrightarrow{k\to0}0. \tag*{(95)} 2 δ k ≃ k − J q c ∗ k q ∗ k → 0 0. (95) As a result, for the low-energy, Landau-Zener type excitations only the second term of Eq. (93) contributes to the kinks above the ground state,
p ~ k ≃ p k + O ( k q ∗ ) , (96) \widetilde p_k\simeq p_k+O(k^{q^*}), \tag*{(96)} p k ≃ p k + O ( k q ∗ ) , (96) where the correction stems from the interference term of Eq. (93) , whose phase Φ k \Phi_k Φ k k k k
it averages out in the sum over modes. For the remaining modes with avoided crossings, where p k p_k p k (93) instead gives p ~ k = sin 2 δ k \widetilde p_k=\sin^2\delta_k p k = sin 2 δ k g = 0 g=0 g = 0 k = π k=\pi k = π p k sin 2 δ k ≃ 0 p_k\sin^2\delta_k\simeq0 p k sin 2 δ k ≃ 0 k k k p k p_k p k
ln ⟨ e i θ N ^ ⟩ ≃ ∑ k > 0 ln [ 1 + ( e 2 i θ − 1 ) sin 2 δ k ] + ∑ k > 0 ln [ 1 + ( e 2 i θ − 1 ) p k ] . (97) \ln\langle e^{i\theta\hat N}\rangle\simeq\sum_{k>0}\ln\left[1+\left(e^{2i\theta}-1\right)\sin^2\delta_k\right]+\sum_{k>0}\ln\left[1+\left(e^{2i\theta}-1\right)p_k\right]. \tag*{(97)} ln ⟨ e i θ N ^ ⟩ ≃ k > 0 ∑ ln [ 1 + ( e 2 i θ − 1 ) sin 2 δ k ] + k > 0 ∑ ln [ 1 + ( e 2 i θ − 1 ) p k ] . (97) Here the first sum gives the τ Q \tau_Q τ Q g = 0 g=0 g = 0 (34) . As a result, kinks relative to the ground state follow the same statistics as those of the excitations as long as they are limited to the low-energy region, i.e. for fractional orders q > 1 / 4 q>1/4 q > 1/4
Next we consider the modes without an avoided crossing close to k = π k=\pi k = π q q q j q ( k ) < 0 j_q(k)<0 j q ( k ) < 0 ξ k = 2 g − ( 1 − g ) j q ( k ) \xi_k=2g-(1-g)j_q(k) ξ k = 2 g − ( 1 − g ) j q ( k ) H k ( t ) H_k(t) H k ( t ) ∣ Ψ k ⟩ = a k e i φ k ∣ Ω k ⟩ + b k e − i φ k ∣ Ω k + ⟩ \lvert\Psi_k\rangle=a_ke^{i\varphi_k}\lvert\Omega_k\rangle+b_ke^{-i\varphi_k}\lvert\Omega_k^+\rangle ∣ Ψ k ⟩ = a k e i φ k ∣ Ω k ⟩ + b k e − i φ k ∣ Ω k + ⟩ φ k ( t ) = ∫ 0 t ϵ k ( t ′ ) d t ′ \varphi_k(t)=\int_0^t\epsilon_k(t')\,dt' φ k ( t ) = ∫ 0 t ϵ k ( t ′ ) d t ′ a k ( 0 ) = 1 a_k(0)=1 a k ( 0 ) = 1 b k ( 0 ) = 0 b_k(0)=0 b k ( 0 ) = 0 a ˙ k ( 0 ) = 0 \dot a_k(0)=0 a ˙ k ( 0 ) = 0 b ˙ k ( 0 ) = ϑ ˙ k e 2 i φ k a k \dot b_k(0)=\dot\vartheta_ke^{2i\varphi_k}a_k b ˙ k ( 0 ) = ϑ ˙ k e 2 i φ k a k a k ≃ 1 a_k\simeq1 a k ≃ 1
b k ( τ Q ) = ∫ 0 τ Q ϑ ˙ k e 2 i φ k d t = [ ( − i ϑ ˙ k 2 ϵ k + 1 4 ϵ k d d t ϑ ˙ k ϵ k ) e 2 i φ k ] 0 τ Q + O ( τ Q − 3 ) . (98) \begin{aligned}
b_k(\tau_Q)=\int_0^{\tau_Q}\dot\vartheta_k e^{2i\varphi_k}\,dt \\
&=\left[\left(-i\frac{\dot\vartheta_k}{2\epsilon_k}+\frac{1}{4\epsilon_k}\frac{d}{dt}\frac{\dot\vartheta_k}{\epsilon_k}\right)e^{2i\varphi_k}\right]_0^{\tau_Q}+O(\tau_Q^{-3}). \tag*{(98)}
\end{aligned} b k ( τ Q ) = ∫ 0 τ Q ϑ ˙ k e 2 i φ k d t = [ ( − i 2 ϵ k ϑ ˙ k + 4 ϵ k 1 d t d ϵ k ϑ ˙ k ) e 2 i φ k ] 0 τ Q + O ( τ Q − 3 ) . (98) As each further integration by parts brings an additional factor 1 / ( τ Q ϵ k ) 1/(\tau_Q\epsilon_k) 1/ ( τ Q ϵ k ) τ Q ϵ k 2 ≫ 1 \tau_Q\epsilon_k^2\gg1 τ Q ϵ k 2 ≫ 1 τ Q ≫ ( q ln 2 ) − 2 \tau_Q\gg(q\ln2)^{-2} τ Q ≫ ( q ln 2 ) − 2 q q q Δ k ′ ξ k − Δ k ξ k ′ = − 2 d q ( k ) \Delta_k'\xi_k-\Delta_k\xi_k'=-2d_q(k) Δ k ′ ξ k − Δ k ξ k ′ = − 2 d q ( k ) g g g ( … ) ′ = d ( … ) / d g (\ldots)'=d(\ldots)/dg ( … ) ′ = d ( … ) / d g ϑ ˙ k = d q ( k ) / ( τ Q ϵ k 2 ) \dot\vartheta_k=d_q(k)/(\tau_Q\epsilon_k^2) ϑ ˙ k = d q ( k ) / ( τ Q ϵ k 2 ) e 2 i φ k ( τ Q ) e^{2i\varphi_k(\tau_Q)} e 2 i φ k ( τ Q ) (88) at the end of the ramp, one has α k = a k ( τ Q ) e i φ k ( τ Q ) \alpha_k=a_k(\tau_Q)e^{i\varphi_k(\tau_Q)} α k = a k ( τ Q ) e i φ k ( τ Q ) β k = b k ( τ Q ) e − i φ k ( τ Q ) \beta_k=b_k(\tau_Q)e^{-i\varphi_k(\tau_Q)} β k = b k ( τ Q ) e − i φ k ( τ Q ) g = 0 g=0 g = 0 α k ≃ e i φ k ( τ Q ) \alpha_k\simeq e^{i\varphi_k(\tau_Q)} α k ≃ e i φ k ( τ Q )
β k ≃ e i φ k ( τ Q ) ( − i X k + Y k ) , X k = d q ( k ) 2 τ Q ϵ k 3 , \beta_k\simeq e^{i\varphi_k(\tau_Q)}(-iX_k+Y_k),\qquad X_k=\frac{d_q(k)}{2\tau_Q\epsilon_k^3}, β k ≃ e i φ k ( τ Q ) ( − i X k + Y k ) , X k = 2 τ Q ϵ k 3 d q ( k ) , Y k = − 3 d q ( k ) [ 2 j q ( k ) + ϵ k 2 ] 4 τ Q 2 ϵ k 6 . (99) Y_k=-\frac{3d_q(k)\left[2j_q(k)+\epsilon_k^2\right]}{4\tau_Q^2\epsilon_k^6}. \tag*{(99)} Y k = − 4 τ Q 2 ϵ k 6 3 d q ( k ) [ 2 j q ( k ) + ϵ k 2 ] . (99) where ϵ k ≡ ϵ k ( g = 0 ) \epsilon_k \equiv\epsilon_k(g=0) ϵ k ≡ ϵ k ( g = 0 ) p k ≃ X k 2 = d q 2 ( k ) / ( 4 τ Q 2 ϵ k 6 ) p_k \simeq X_k^2 = d_q^2(k)/(4\tau_Q^2\epsilon_k^6) p k ≃ X k 2 = d q 2 ( k ) / ( 4 τ Q 2 ϵ k 6 ) p ~ k = ∣ β k cos δ k − α k sin δ k ∣ 2 \tilde p_k = |\beta_k\cos\delta_k-\alpha_k\sin\delta_k|^2 p ~ k = ∣ β k cos δ k − α k sin δ k ∣ 2 sin 2 δ k = [ j q ( k ) sin k − d q ( k ) cos k ] / ϵ k \sin2\delta_k=[j_q(k)\sin k-d_q(k)\cos k]/\epsilon_k sin 2 δ k = [ j q ( k ) sin k − d q ( k ) cos k ] / ϵ k
p ~ k ≃ sin 2 δ k + w k p k (100) \tilde p_k \simeq\sin^2\delta_k+w_kp_k \tag*{(100)} p ~ k ≃ sin 2 δ k + w k p k (100) w k = cos 2 δ k + 3 [ 2 j q ( k ) + ϵ k 2 ] [ j q ( k ) sin k − d q ( k ) cos k ] d q ( k ) ϵ k . w_k=\cos^2\delta_k+\frac{3[2j_q(k)+\epsilon_k^2][j_q(k)\sin k-d_q(k)\cos k]}{d_q(k)\epsilon_k}. w k = cos 2 δ k + d q ( k ) ϵ k 3 [ 2 j q ( k ) + ϵ k 2 ] [ j q ( k ) sin k − d q ( k ) cos k ] . In contrast to small momenta, the second-order amplitude Y k Y_k Y k k = π − κ k=\pi-\kappa k = π − κ d q ( k ) ≃ D q κ d_q(k)\simeq D_q\kappa d q ( k ) ≃ D q κ D q = − d q ′ ( π ) D_q=-d'_q(\pi) D q = − d q ′ ( π ) j q ( k ) ≃ − ϵ π j_q(k)\simeq-\epsilon_\pi j q ( k ) ≃ − ϵ π ϵ π = 2 q − J q \epsilon_\pi=2^q-J_q ϵ π = 2 q − J q 2 δ k ≃ κ ( D q / ϵ π − 1 ) 2\delta_k\simeq\kappa(D_q/\epsilon_\pi-1) 2 δ k ≃ κ ( D q / ϵ π − 1 ) X k ≃ D q κ / ( 2 τ Q ϵ π 3 ) X_k\simeq D_q\kappa/(2\tau_Q\epsilon_\pi^3) X k ≃ D q κ / ( 2 τ Q ϵ π 3 ) κ \kappa κ Y k Y_k Y k sin δ k \sin\delta_k sin δ k κ 2 / τ Q 2 \kappa^2/\tau_Q^2 κ 2 / τ Q 2 p k p_k p k
w π = 1 + 3 ( ϵ π − D q ) ( 2 − ϵ π ) D q (101) w_\pi=1+\frac{3(\epsilon_\pi-D_q)(2-\epsilon_\pi)}{D_q} \tag*{(101)} w π = 1 + D q 3 ( ϵ π − D q ) ( 2 − ϵ π ) (101) which reaches 1 + 6 ( 2 ln 2 − 1 ) ≃ 3.32 1+6(2\ln2-1)\simeq3.32 1 + 6 ( 2 ln 2 − 1 ) ≃ 3.32 q → 0 q\to0 q → 0 D q / ϵ π → 1 / ( 2 ln 2 ) D_q/\epsilon_\pi\to1/(2\ln2) D q / ϵ π → 1/ ( 2 ln 2 ) q = 2 q=2 q = 2 D 2 = ϵ π = 2 D_2=\epsilon_\pi=2 D 2 = ϵ π = 2 k = π k=\pi k = π τ Q − 2 \tau_Q^{-2} τ Q − 2 w k w_k w k
For the higher cumulants, the overlap with the ground-state kinks no longer drops out. The kink number of each pair of modes takes the values 0 and 2 with probabilities 1 − p ~ k 1-\tilde p_k 1 − p ~ k p ~ k \tilde p_k p ~ k 2 p ~ k 2\tilde p_k 2 p ~ k 4 p ~ k ( 1 − p ~ k ) 4\tilde p_k(1-\tilde p_k) 4 p ~ k ( 1 − p ~ k ) 8 p ~ k ( 1 − p ~ k ) ( 1 − 2 p ~ k ) 8\tilde p_k(1-\tilde p_k)(1-2\tilde p_k) 8 p ~ k ( 1 − p ~ k ) ( 1 − 2 p ~ k ) p ~ k − sin 2 δ k = w k p k \tilde p_k-\sin^2\delta_k=w_kp_k p ~ k − sin 2 δ k = w k p k sin 2 δ k \sin^2\delta_k sin 2 δ k
κ 1 − κ 1 G S ≃ 2 ∑ k > 0 w k p k , κ 2 − κ 2 G S ≃ 4 ∑ k > 0 w k p k cos 2 δ k , κ 3 − κ 3 G S ≃ 8 ∑ k > 0 w k p k ( 1 − 3 2 sin 2 2 δ k ) , (102) \begin{aligned}
\kappa_1-\kappa_1^{\mathrm{GS}}&\simeq2\sum_{k>0}w_kp_k, \qquad
\kappa_2-\kappa_2^{\mathrm{GS}}\simeq4\sum_{k>0}w_kp_k\cos2\delta_k,\\
\kappa_3-\kappa_3^{\mathrm{GS}}&\simeq8\sum_{k>0}w_kp_k\left(1-\frac{3}{2}\sin^22\delta_k\right), \tag*{(102)}
\end{aligned} κ 1 − κ 1 GS κ 3 − κ 3 GS ≃ 2 k > 0 ∑ w k p k , κ 2 − κ 2 GS ≃ 4 k > 0 ∑ w k p k cos 2 δ k , ≃ 8 k > 0 ∑ w k p k ( 1 − 2 3 sin 2 2 δ k ) , (102) which corresponds to replacing the second sum of Eq. (97) by ∑ k > 0 ( e 2 i θ − 1 ) w k p k / [ 1 + ( e 2 i θ − 1 ) sin 2 δ k ] \sum_{k>0}(e^{2i\theta}-1)w_kp_k/[1+(e^{2i\theta}-1)\sin^2\delta_k] ∑ k > 0 ( e 2 i θ − 1 ) w k p k / [ 1 + ( e 2 i θ − 1 ) sin 2 δ k ] δ k \delta_k δ k k = π k=\pi k = π sin 2 δ k ≃ 0.07 \sin^2\delta_k\simeq0.07 sin 2 δ k ≃ 0.07 k c k_c k c (91) with j q ( k c ) = 0 j_q(k_c)=0 j q ( k c ) = 0 k c → π / 3 k_c\to\pi/3 k c → π /3 q q q sin 2 δ k c = ( 1 − sin k c ) / 2 ≃ ( 1 − 3 / 2 ) / 2 \sin^2\delta_{k_c}=(1-\sin k_c)/2\simeq(1-\sqrt{3}/2)/2 sin 2 δ k c = ( 1 − sin k c ) /2 ≃ ( 1 − 3 /2 ) /2 R ~ ℓ \tilde R_\ell R ~ ℓ R ~ 2 ≃ 1.89 \tilde R_2\simeq1.89 R ~ 2 ≃ 1.89 R ~ 3 ≃ 3.36 \tilde R_3\simeq3.36 R ~ 3 ≃ 3.36 q → 0 q\to0 q → 0 R ~ 2 ≃ 1.93 \tilde R_2\simeq1.93 R ~ 2 ≃ 1.93 R ~ 3 ≃ 3.57 \tilde R_3\simeq3.57 R ~ 3 ≃ 3.57 q = 1 / 4 q=1/4 q = 1/4 w k p k w_kp_k w k p k − 2 ∑ k w k 2 p k 2 / ∑ k w k p k ≃ − J ~ 0 / ( 2 I ~ 0 q 4 τ Q 2 ) -2\sum_k w_k^2p_k^2/\sum_k w_kp_k\simeq-\tilde J_0/(2\tilde I_0q^4\tau_Q^2) − 2 ∑ k w k 2 p k 2 / ∑ k w k p k ≃ − J ~ 0 / ( 2 I ~ 0 q 4 τ Q 2 ) − 12 ∑ k w k 2 p k 2 cos 2 δ k / ∑ k w k p k ≃ − 3 J ~ 0 / ( I ~ 0 q 4 τ Q 2 ) -12\sum_k w_k^2p_k^2\cos2\delta_k/\sum_k w_kp_k\simeq-3\tilde J_0/(\tilde I_0q^4\tau_Q^2) − 12 ∑ k w k 2 p k 2 cos 2 δ k / ∑ k w k p k ≃ − 3 J ~ 0 / ( I ~ 0 q 4 τ Q 2 ) I ~ 0 \tilde I_0 I ~ 0 (106) and J ~ 0 \tilde J_0 J ~ 0
Summing over the modes in the continuum limit, 2 ∑ k > 0 → L / π ∫ d k 2\sum_{k>0}\to L/\pi\int dk 2 ∑ k > 0 → L / π ∫ d k N L Z = L N q τ Q − 1 / ( 2 q ∗ ) N_{\mathrm{LZ}}=LN_q\tau_Q^{-1/(2q_*)} N LZ = L N q τ Q − 1/ ( 2 q ∗ ) N q = Γ ( 1 + 1 / ( 2 q ∗ ) ) / ( π λ q 1 / ( 2 q ∗ ) ) N_q=\Gamma(1+1/(2q_*))/(\pi\lambda_q^{1/(2q_*)}) N q = Γ ( 1 + 1/ ( 2 q ∗ )) / ( π λ q 1/ ( 2 q ∗ ) )
The modes without an avoided crossing, k c < k ≤ π k_c<k\le\pi k c < k ≤ π 2 sin ( k c / 2 ) = J q 1 / q 2\sin(k_c/2)=J_q^{1/q} 2 sin ( k c /2 ) = J q 1/ q τ Q ϵ π 2 ≫ 1 \tau_Q\epsilon_\pi^2\gg1 τ Q ϵ π 2 ≫ 1
N n c = L π ∫ k c π w k p k d k = L B ~ q τ Q − 2 , B ~ q = 1 π ∫ k c π w k d q 2 ( k ) 4 ϵ k 6 d k . (103) N_{\mathrm{nc}}=\frac{L}{\pi}\int_{k_c}^{\pi}w_kp_k\,dk=L\tilde B_q\tau_Q^{-2},\qquad\tilde B_q=\frac{1}{\pi}\int_{k_c}^{\pi}\frac{w_kd_q^2(k)}{4\epsilon_k^6}\,dk. \tag*{(103)} N nc = π L ∫ k c π w k p k d k = L B ~ q τ Q − 2 , B ~ q = π 1 ∫ k c π 4 ϵ k 6 w k d q 2 ( k ) d k . (103) Thus the competition between them is identical to that of the excitations, Eq. (74) , up to a different, τ Q \tau_Q τ Q
N n c N L Z = B ~ q N q τ Q 1 2 q ∗ − 2 , (104) \frac{N_{\mathrm{nc}}}{N_{\mathrm{LZ}}}=\frac{\tilde B_q}{N_q}\tau_Q^{\frac{1}{2q_*}-2}, \tag*{(104)} N LZ N nc = N q B ~ q τ Q 2 q ∗ 1 − 2 , (104) growing with τ Q \tau_Q τ Q q < 1 / 4 q<1/4 q < 1/4 q > 1 / 4 q>1/4 q > 1/4 w k w_k w k τ Q ≫ ( q ln 2 ) − 2 \tau_Q\gg(q\ln2)^{-2} τ Q ≫ ( q ln 2 ) − 2 k = π k=\pi k = π q = 1 / 4 q=1/4 q = 1/4 q > 1 / 4 q>1/4 q > 1/4 τ a d < τ F S \tau_{\mathrm{ad}}<\tau_{\mathrm{FS}} τ ad < τ FS
Finally, we also provide a cross-check in the complementary regime, where the slowest ramps resolve individual momenta, i.e., for τ Q ≳ τ F S = λ q − 1 ( L / π ) 2 q ∗ \tau_Q\gtrsim\tau_{\mathrm{FS}}=\lambda_q^{-1}(L/\pi)^{2q_*} τ Q ≳ τ FS = λ q − 1 ( L / π ) 2 q ∗ ( λ q τ Q ) − 1 / ( 2 q ∗ ) (\lambda_q\tau_Q)^{-1/(2q_*)} ( λ q τ Q ) − 1/ ( 2 q ∗ ) π / L \pi/L π / L N L Z ≃ 2 e − τ Q / τ F S N_{\mathrm{LZ}}\simeq2e^{-\tau_Q/\tau_{\mathrm{FS}}} N LZ ≃ 2 e − τ Q / τ FS δ k \delta_k δ k k → 0 k\to0 k → 0 k = π k=\pi k = π (75) , up to the prefactor B ~ q \tilde B_q B ~ q
N n c N L Z = L B ~ q 2 τ F S 2 e x x 2 , x = τ Q / τ F S . (105) \frac{N_{\mathrm{nc}}}{N_{\mathrm{LZ}}}=\frac{L\tilde B_q}{2\tau_{\mathrm{FS}}^2}\frac{e^x}{x^2},\qquad x=\tau_Q/\tau_{\mathrm{FS}}. \tag*{(105)} N LZ N nc = 2 τ FS 2 L B ~ q x 2 e x , x = τ Q / τ FS . (105) The exponential growth in x x x q q q x ∗ x_* x ∗ e x ∗ / x ∗ 2 = 2 τ F S 2 / ( L B ~ q ) e^{x_*}/x_*^2=2\tau_{\mathrm{FS}}^2/(L\tilde B_q) e x ∗ / x ∗ 2 = 2 τ FS 2 / ( L B ~ q ) q q q q q q j q ( k ) ≃ − q ln ( 2 sin k 2 ) j_q(k)\simeq-q\ln(2\sin\frac{k}{2}) j q ( k ) ≃ − q ln ( 2 sin 2 k ) d q ( k ) ≃ q 2 ( π − k ) d_q(k)\simeq\frac{q}{2}(\pi-k) d q ( k ) ≃ 2 q ( π − k ) J r , q ≃ q / ( 2 r ) J_{r,q}\simeq q/(2r) J r , q ≃ q / ( 2 r ) w k w_k w k d q d_q d q q q q
B ~ q ≃ I ~ 0 4 π q 4 , I ~ 0 = ∫ π / 3 π w k ( π − k 2 ) 2 [ ln 2 ( 2 sin k 2 ) + ( π − k 2 ) 2 ] 3 d k ≃ 4.89. (106) \tilde B_q\simeq\frac{\tilde I_0}{4\pi q^4},\qquad\tilde I_0=\int_{\pi/3}^{\pi}\frac{w_k\left(\frac{\pi-k}{2}\right)^2}{\left[\ln^2\left(2\sin\frac{k}{2}\right)+\left(\frac{\pi-k}{2}\right)^2\right]^3}\,dk\simeq4.89. \tag*{(106)} B ~ q ≃ 4 π q 4 I ~ 0 , I ~ 0 = ∫ π /3 π [ ln 2 ( 2 sin 2 k ) + ( 2 π − k ) 2 ] 3 w k ( 2 π − k ) 2 d k ≃ 4.89. (106) Together with τ F S ≃ 27 / ( π 3 q 2 ) ( L / π ) 2 q \tau_{\mathrm{FS}}\simeq27/(\pi^3q^2)(L/\pi)^{2q} τ FS ≃ 27/ ( π 3 q 2 ) ( L / π ) 2 q τ Q ≫ τ F S \tau_Q\gg\tau_{\mathrm{FS}} τ Q ≫ τ FS q − 2 q^{-2} q − 2 q − 4 q^{-4} q − 4
N n c N L Z = π 5 I ~ 0 5832 π q L 1 − 4 q e x x 2 , π 5 I ~ 0 5832 ≃ 0.26 , (107) \frac{N_{\mathrm{nc}}}{N_{\mathrm{LZ}}}=\frac{\pi^5\tilde I_0}{5832}\pi^qL^{1-4q}\frac{e^x}{x^2},\qquad\frac{\pi^5\tilde I_0}{5832}\simeq0.26, \tag*{(107)} N LZ N nc = 5832 π 5 I ~ 0 π q L 1 − 4 q x 2 e x , 5832 π 5 I ~ 0 ≃ 0.26 , (107) so that the kinks near k = π k=\pi k = π q < ln ( 0.26 L e x / x 2 ) / [ 4 ln ( L / π ) ] q<\ln(0.26Le^x/x^2)/[4\ln(L/\pi)] q < ln ( 0.26 L e x / x 2 ) / [ 4 ln ( L / π )] τ Q ≫ τ F S \tau_Q\gg\tau_{\mathrm{FS}} τ Q ≫ τ FS x ≫ 1 x\gg1 x ≫ 1 q < ln ( 0.70 L ) / [ 4 ln ( L / π ) ] q<\ln(0.70L)/[4\ln(L/\pi)] q < ln ( 0.70 L ) / [ 4 ln ( L / π )] L L L q = 1 / 4 q=1/4 q = 1/4
Free-fermion bounds on the kink-number cumulants
In this section we show that for every Gaussian state of fermions the cumulants of the kink number N ^ \hat{N} N ^ κ 2 < 2 κ 1 \kappa_2 < 2\kappa_1 κ 2 < 2 κ 1 ∣ κ 3 ∣ ≤ 2 κ 2 |\kappa_3| \le2\kappa_2 ∣ κ 3 ∣ ≤ 2 κ 2 κ 1 > 0 \kappa_1 > 0 κ 1 > 0 (16) , which for r = 1 r=1 r = 1 η ^ j = c ^ j † + c ^ j \hat{\eta}_j=\hat{c}^{\dagger}_j+\hat{c}_j η ^ j = c ^ j † + c ^ j φ ^ j = i ( c ^ j † − c ^ j ) \hat{\varphi}_j=i(\hat{c}^{\dagger}_j-\hat{c}_j) φ ^ j = i ( c ^ j † − c ^ j )
σ ^ j x σ ^ j + 1 x = ( c ^ j † − c ^ j ) ( c ^ j + 1 † + c ^ j + 1 ) = − i φ ^ j η ^ j + 1 . (108) \hat{\sigma}^x_j\hat{\sigma}^x_{j+1}=(\hat{c}^{\dagger}_j-\hat{c}_j)(\hat{c}^{\dagger}_{j+1}+\hat{c}_{j+1})=-i\hat{\varphi}_j\hat{\eta}_{j+1}. \tag*{(108)} σ ^ j x σ ^ j + 1 x = ( c ^ j † − c ^ j ) ( c ^ j + 1 † + c ^ j + 1 ) = − i φ ^ j η ^ j + 1 . (108) Introducing the bond fermions
f ^ j = 1 2 ( φ ^ j + i η ^ j + 1 ) , j = 1 , … , L − 1 , (109) \hat{f}_j=\frac{1}{2}(\hat{\varphi}_j+i\hat{\eta}_{j+1}),\qquad j=1,\ldots,L-1, \tag*{(109)} f ^ j = 2 1 ( φ ^ j + i η ^ j + 1 ) , j = 1 , … , L − 1 , (109) the j j j (6) , can be expressed as
1 2 ( 1 − σ ^ j x σ ^ j + 1 x ) = f ^ j † f ^ j , N ^ = ∑ j = 1 L − 1 f ^ j † f ^ j . (110) \frac{1}{2}(1-\hat{\sigma}^x_j\hat{\sigma}^x_{j+1})=\hat{f}^{\dagger}_j\hat{f}_j,\qquad\hat{N}=\sum_{j=1}^{L-1}\hat{f}^{\dagger}_j\hat{f}_j. \tag*{(110)} 2 1 ( 1 − σ ^ j x σ ^ j + 1 x ) = f ^ j † f ^ j , N ^ = j = 1 ∑ L − 1 f ^ j † f ^ j . (110) As the f ^ j \hat{f}_j f ^ j c ^ j \hat{c}_j c ^ j c ^ j \hat{c}_j c ^ j f ^ j \hat{f}_j f ^ j ( L − 1 ) × ( L − 1 ) (L-1)\times(L-1) ( L − 1 ) × ( L − 1 ) G j l = ⟨ f ^ l † f ^ j ⟩ G_{jl}=\langle\hat{f}^{\dagger}_l\hat{f}_j\rangle G j l = ⟨ f ^ l † f ^ j ⟩ F j l = ⟨ f ^ l f ^ j ⟩ F_{jl}=\langle\hat{f}_l\hat{f}_j\rangle F j l = ⟨ f ^ l f ^ j ⟩ 0 ≤ G ≤ 1 0\le G\le1 0 ≤ G ≤ 1 f ^ j \hat{f}_j f ^ j f ^ j † \hat{f}^{\dagger}_j f ^ j † [91 ] , one finds the following inequality
F F † ≤ G − G 2 . (111) FF^{\dagger}\le G-G^2. \tag*{(111)} F F † ≤ G − G 2 . (111) Within this setup Wick’s theorem gives the first three cumulants as
κ 1 = Tr G , κ 2 = Tr ( G − G 2 ) + Tr ( F F † ) , (112) \kappa_1=\operatorname{Tr}G,\qquad\kappa_2=\operatorname{Tr}(G-G^2)+\operatorname{Tr}(FF^{\dagger}), \tag*{(112)} κ 1 = Tr G , κ 2 = Tr ( G − G 2 ) + Tr ( F F † ) , (112) κ 3 = Tr ( G − 3 G 2 + 2 G 3 ) + 3 Tr [ ( 1 − G ) F F † ] − 3 Tr ( F G T F † ) . (113) \kappa_3=\operatorname{Tr}(G-3G^2+2G^3)+3\operatorname{Tr}[(1-G)FF^{\dagger}]-3\operatorname{Tr}(FG^TF^{\dagger}). \tag*{(113)} κ 3 = Tr ( G − 3 G 2 + 2 G 3 ) + 3 Tr [( 1 − G ) F F † ] − 3 Tr ( F G T F † ) . (113) Inserting Eq. (111) into κ 2 \kappa_2 κ 2 Tr G 2 ≥ ( Tr G ) 2 / ( L − 1 ) \operatorname{Tr}G^2\ge(\operatorname{Tr}G)^2/(L-1) Tr G 2 ≥ ( Tr G ) 2 / ( L − 1 )
κ 2 ≤ 2 Tr ( G − G 2 ) ≤ 2 κ 1 ( 1 − κ 1 L − 1 ) < 2 κ 1 . (114) \kappa_2\le2\operatorname{Tr}(G-G^2)\le2\kappa_1\left(1-\frac{\kappa_1}{L-1}\right)<2\kappa_1. \tag*{(114)} κ 2 ≤ 2 Tr ( G − G 2 ) ≤ 2 κ 1 ( 1 − L − 1 κ 1 ) < 2 κ 1 . (114) For the third cumulant, Eq. (111) together with 0 ≤ G ≤ 1 0\le G\le1 0 ≤ G ≤ 1 0 ≤ Tr ( G F F † ) ≤ Tr ( F F † ) 0\le\operatorname{Tr}(GFF^{\dagger})\le\operatorname{Tr}(FF^{\dagger}) 0 ≤ Tr ( GF F † ) ≤ Tr ( F F † ) 0 ≤ Tr ( F G T F † ) ≤ Tr ( F F † ) 0\le\operatorname{Tr}(FG^TF^{\dagger})\le\operatorname{Tr}(FF^{\dagger}) 0 ≤ Tr ( F G T F † ) ≤ Tr ( F F † ) 2 κ 2 − κ 3 ≥ 2 Tr G 2 ( 1 − G ) ≥ 0 2\kappa_2-\kappa_3\ge2\operatorname{Tr}G^2(1-G)\ge0 2 κ 2 − κ 3 ≥ 2 Tr G 2 ( 1 − G ) ≥ 0 2 κ 2 + κ 3 ≥ 2 Tr G ( 1 − G ) 2 ≥ 0 2\kappa_2+\kappa_3\ge2\operatorname{Tr}G(1-G)^2\ge0 2 κ 2 + κ 3 ≥ 2 Tr G ( 1 − G ) 2 ≥ 0 ∣ κ 3 ∣ ≤ 2 κ 2 |\kappa_3|\le2\kappa_2 ∣ κ 3 ∣ ≤ 2 κ 2 F = 0 F=0 F = 0 κ 2 ≤ κ 1 \kappa_2\le\kappa_1 κ 2 ≤ κ 1 ∣ κ 3 ∣ ≤ κ 2 |\kappa_3|\le\kappa_2 ∣ κ 3 ∣ ≤ κ 2
As Eq. (111) holds for any fermionic state and Eqs. (112) and (F6) only require Wick’s theorem, the bounds apply to pure and mixed Gaussian states for any couplings, boundary conditions and driving protocol, and in particular to the free-fermion approximation of Sec. IV A. As the derivation only uses that N ^ \hat{N} N ^ N ^ e x \hat{N}_{\mathrm{ex}} N ^ ex L L L L − 1 L-1 L − 1 N ^ \hat{N} N ^ ℓ \ell ℓ ℓ \ell ℓ ℓ = 2 , 3 \ell=2,3 ℓ = 2 , 3
Free-fermion bounds on the kink-kink correlations and exact Gaussian limit
In this section we extend the bounds of Appendix F to the connected kink-kink correlations C i j = ⟨ n ^ i n ^ j ⟩ − ⟨ n ^ i ⟩ ⟨ n ^ j ⟩ C_{ij}=\langle\hat{n}_i\hat{n}_j\rangle-\langle\hat{n}_i\rangle\langle\hat{n}_j\rangle C ij = ⟨ n ^ i n ^ j ⟩ − ⟨ n ^ i ⟩ ⟨ n ^ j ⟩ n ^ j = f ^ j † f ^ j \hat{n}_j=\hat{f}^{\dagger}_j\hat{f}_j n ^ j = f ^ j † f ^ j (110) , which enter the correlation function of Eq. (43) as C ( r ) = C L / 2 , L / 2 + r C(r)=C_{L/2,L/2+r} C ( r ) = C L /2 , L /2 + r r = − L / 2 + 1 , … , L / 2 − 1 r=-L/2+1,\ldots,L/2-1 r = − L /2 + 1 , … , L /2 − 1
C i j = δ i j G i i − ∣ G i j ∣ 2 + ∣ F i j ∣ 2 . (115) C_{ij}=\delta_{ij}G_{ii}-|G_{ij}|^2+|F_{ij}|^2. \tag*{(115)} C ij = δ ij G ii − ∣ G ij ∣ 2 + ∣ F ij ∣ 2 . (115) By this, the local and total variances read C i i = n i ( 1 − n i ) C_{ii}=n_i(1-n_i) C ii = n i ( 1 − n i ) n i = G i i n_i=G_{ii} n i = G ii κ 2 = ∑ i , j C i j \kappa_2=\sum_{i,j}C_{ij} κ 2 = ∑ i , j C ij (112) . For a single pair of bonds i ≠ j i\ne j i = j ∣ F i j ∣ 2 ≤ n i ( 1 − n j ) |F_{ij}|^2\le n_i(1-n_j) ∣ F ij ∣ 2 ≤ n i ( 1 − n j ) ∣ G i j ∣ 2 ≤ min [ n i n j , ( 1 − n i ) ( 1 − n j ) ] |G_{ij}|^2\le\min[n_in_j,(1-n_i)(1-n_j)] ∣ G ij ∣ 2 ≤ min [ n i n j , ( 1 − n i ) ( 1 − n j )]
− min [ n i n j , ( 1 − n i ) ( 1 − n j ) ] ≤ C i j ≤ min [ n i ( 1 − n j ) , n j ( 1 − n i ) ] . (116) -\min[n_in_j,(1-n_i)(1-n_j)]\le C_{ij}\le\min[n_i(1-n_j),n_j(1-n_i)]. \tag*{(116)} − min [ n i n j , ( 1 − n i ) ( 1 − n j )] ≤ C ij ≤ min [ n i ( 1 − n j ) , n j ( 1 − n i )] . (116) These bounds, however, hold for any two commuting projectors in any state, since max ( 0 , n i + n j − 1 ) ≤ ⟨ n ^ i n ^ j ⟩ ≤ min ( n i , n j ) \max(0,n_i+n_j-1)\le\langle\hat{n}_i\hat{n}_j\rangle\le\min(n_i,n_j) max ( 0 , n i + n j − 1 ) ≤ ⟨ n ^ i n ^ j ⟩ ≤ min ( n i , n j )
The key part where the free-fermion structure becomes essential is the total correlation of a single bond. Summing Eq. (115) over j j j ∑ j C i j = ( G − G 2 ) i i + ( F F † ) i i \sum_j C_{ij}=(G-G^2)_{ii}+(FF^{\dagger})_{ii} ∑ j C ij = ( G − G 2 ) ii + ( F F † ) ii (111) also holds for the diagonal elements, ( F F † ) i i ≤ ( G − G 2 ) i i (FF^{\dagger})_{ii}\le(G-G^2)_{ii} ( F F † ) ii ≤ ( G − G 2 ) ii ( G 2 ) i i = ∑ j ∣ G i j ∣ 2 ≥ n i 2 (G^2)_{ii}=\sum_j|G_{ij}|^2\ge n_i^2 ( G 2 ) ii = ∑ j ∣ G ij ∣ 2 ≥ n i 2 0 ≤ ∑ j C i j ≤ 2 n i ( 1 − n i ) 0\le\sum_j C_{ij}\le2n_i(1-n_i) 0 ≤ ∑ j C ij ≤ 2 n i ( 1 − n i )
∣ ∑ j ≠ i C i j ∣ ≤ C i i = n i ( 1 − n i ) . (117) \left|\sum_{j\ne i}C_{ij}\right|\le C_{ii}=n_i(1-n_i). \tag*{(117)} j = i ∑ C ij ≤ C ii = n i ( 1 − n i ) . (117) Thus the total correlation of a bond with all other bonds cannot exceed its own variance, and summing Eq. (117) over i i i (114) . Far from the edges of the chain, where the kink density is approximately constant, Eq. (G3) reads ∑ r ≠ 0 C ( r ) ≤ C ( 0 ) \sum_{r\ne0} C(r) \le C(0) ∑ r = 0 C ( r ) ≤ C ( 0 ) (43) . As a result, the kink correlations of any free-fermion state are summable for every system size, so that the variance grows at most linearly with L L L [92 , 93 ] . Regarding the higher-order cumulants, as N ^ \hat{N} N ^ ∣ κ ℓ ∣ ≤ c ℓ L |\kappa_\ell| \le c_\ell L ∣ κ ℓ ∣ ≤ c ℓ L κ 2 \kappa_2 κ 2 L L L κ ℓ / κ 2 ℓ / 2 ∼ L 1 − ℓ / 2 \kappa_\ell/\kappa_2^{\ell/2} \sim L^{1-\ell/2} κ ℓ / κ 2 ℓ /2 ∼ L 1 − ℓ /2 ℓ ≥ 3 \ell\ge3 ℓ ≥ 3 P ( n ) P(n) P ( n ) L → ∞ L \to\infty L → ∞
Finite-size scaling of the gap
In this appendix, we provide further numerical details on the fitting parameters and their errors in the finite-size scaling of the minimal energy gap and its position g c ( L ) g_c(L) g c ( L ) Δ min \Delta_{\min} Δ m i n Δ = E 2 − E 0 \Delta= E_2-E_0 Δ = E 2 − E 0 g c F F g_c^{\mathrm{FF}} g c FF (24) and then located by a golden-section search, in which the bracketing interval is reduced by the factor ( 5 − 1 ) / 2 (\sqrt{5}-1)/2 ( 5 − 1 ) /2 2 × 10 − 4 2 \times10^{-4} 2 × 1 0 − 4 ± 10 % \pm10\% ± 10% g c ( L ) g_c(L) g c ( L ) Δ ( g c , L ) \Delta(g_c,L) Δ ( g c , L ) L = 20 L = 20 L = 20 L = 200 L = 200 L = 200
The scaling forms of Eqs. (9) and (10) are fitted by nonlinear least squares with all parameters free. The correction terms are retained only for ω , ω ′ ≥ 0.2 \omega,\omega' \ge0.2 ω , ω ′ ≥ 0.2 g c ( L ) g_c(L) g c ( L ) [94 ] n ln ( R S S / n ) + 2 k n\ln(\mathrm{RSS}/n) + 2k n ln ( RSS / n ) + 2 k n n n k k k [94 ] .
The uncertainties of z z z ν \nu ν ν / ( z ν + 1 ) \nu/(z\nu+1) ν / ( z ν + 1 ) n n n [95 , 96 ] σ = [ n − 1 n ∑ i = 1 n ( θ i − θ ˉ ) 2 ] 1 / 2 \sigma= \left[\frac{n-1}{n}\sum_{i=1}^{n}(\theta_i-\bar{\theta})^2\right]^{1/2} σ = [ n n − 1 ∑ i = 1 n ( θ i − θ ˉ ) 2 ] 1/2 θ i \theta_i θ i i i i θ ˉ \bar{\theta} θ ˉ θ i \theta_i θ i
A dash in Tables II and III marks a correction term that is not included, either because it is rejected by the above criteria or because it stays below 1% of the leading term for all sizes, and the data then follow the leading power law alone.
The minimum gap and its position are shown for selected fractional orders in Figure 9 , and the fitted parameters are listed in Tables II and III.
Further results on the cumulants
In this section we provide further results on the kink number cumulants. We first show the third cumulant and the cumulant ratios as functions of the driving time, and then the first three cumulants multiplied by the power of τ Q \tau_Q τ Q
The third cumulant is shown in Figure 10 for q = 0.1 q = 0.1 q = 0.1 q = 0.2 q = 0.2 q = 0.2 q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25 q ≳ 1.25 q \gtrsim1.25 q ≳ 1.25 q = 0.1 q = 0.1 q = 0.1 q = 0.2 q = 0.2 q = 0.2 q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25
The cumulant ratios as functions of τ Q \tau_Q τ Q Figure 11 for the same fractional orders. Although the mean kink numbers of the TDVP simulations and of free fermions on the ring differ for q ≲ 2 q \lesssim2 q ≲ 2 q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25 q = 0.1 q = 0.1 q = 0.1 q = 0.2 q = 0.2 q = 0.2 κ 2 / κ 1 \kappa_2/\kappa_1 κ 2 / κ 1 R ~ 2 ≃ 1.91 \widetilde{R}_2 \simeq1.91 R 2 ≃ 1.91 1.92 1.92 1.92 Figure 8 of the main text.
To better characterize the range of driving times where the cumulants follow a power law, we multiply them by τ Q α \tau_Q^{\alpha} τ Q α α \alpha α q > 1 / 4 q > 1/4 q > 1/4 α = 1 / ( 2 q ∗ ) \alpha= 1/(2q_*) α = 1/ ( 2 q ∗ ) (35) , while for q < 1 / 4 q < 1/4 q < 1/4 α = 2 \alpha= 2 α = 2 (33) . An expected power law then appears as a horizontal line, at the free-fermion values C ℓ C_\ell C ℓ q > 1 / 4 q > 1/4 q > 1/4 R ~ ℓ L ~ B ~ q \widetilde{R}_\ell\widetilde{L}\widetilde{B}_q R ℓ L B q R ~ 1 = 1 \widetilde{R}_1 = 1 R 1 = 1 q < 1 / 4 q < 1/4 q < 1/4
For the mean, Figure 12 , the free-fermion kinks and excitations at q = 2.25 q = 2.25 q = 2.25 τ Q \tau_Q τ Q τ F S \tau_{\mathrm{FS}} τ FS § 2.1.1 ). The TDVP data lie slightly below it and decrease slowly with τ Q \tau_Q τ Q q = 1.5 q = 1.5 q = 1.5 τ Q \tau_Q τ Q τ Q \tau_Q τ Q τ Q − 1 / 2 \tau_Q^{-1/2} τ Q − 1/2 q < 1 / 4 q < 1/4 q < 1/4 k > k c k > k_c k > k c q = 0.2 q = 0.2 q = 0.2 τ Q ≳ 80 \tau_Q \gtrsim80 τ Q ≳ 80 L B ~ q L\widetilde{B}_q L B q q = 0.1 q = 0.1 q = 0.1 τ Q ≫ ( q ln 2 ) − 2 ≈ 200 \tau_Q \gg(q\ln2)^{-2} \approx200 τ Q ≫ ( q ln 2 ) − 2 ≈ 200 q = 0.2 q = 0.2 q = 0.2 τ Q − 2 \tau_Q^{-2} τ Q − 2 q = 0.1 q = 0.1 q = 0.1 τ Q − 2 \tau_Q^{-2} τ Q − 2 Figure 1 and Figure 2 .
Figure 12. Mean kink number multiplied by τ Q α \tau_Q^\alpha τ Q α q = 0.1 q=0.1 q = 0.1 q = 0.2 q=0.2 q = 0.2 q = 1.5 q=1.5 q = 1.5 q = 2.25 q=2.25 q = 2.25 α = 2 \alpha=2 α = 2 q < 1 / 4 q<1/4 q < 1/4 α = 1 / ( 2 q ∗ ) \alpha=1/(2q_*) α = 1/ ( 2 q ∗ ) g = 0 g=0 g = 0 (35) and the contribution L B ˉ q τ Q 2 L\bar B_q\tau_Q^2 L B ˉ q τ Q 2 k > k c k>k_c k > k c
Table 2. Parameters of the fit of the minimum gap, Eq. (9) , over the system sizes L L L
The variance in Figure 13 behaves similarly to the mean at q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25 τ Q \tau_Q τ Q q = 0.2 q = 0.2 q = 0.2 R ~ 2 L ~ B ~ q \widetilde{R}_2\widetilde{L}\widetilde{B}_q R 2 L B q k > k c k > k_c k > k c τ Q ≳ 100 \tau_Q \gtrsim100 τ Q ≳ 100 q = 0.1 q = 0.1 q = 0.1
Figure 13. Variance of the kink number multiplied by τ Q α \tau_Q^\alpha τ Q α q = 0.1 q = 0.1 q = 0.1 q = 0.2 q = 0.2 q = 0.2 q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25 α \alpha α Figure 12 . The black dotted lines show the leading-order prediction C 2 τ Q − 1 / ( 2 q ∗ ) C_2\tau_Q^{-1/(2q_*)} C 2 τ Q − 1/ ( 2 q ∗ ) (35) and the red dashed lines the contribution R ~ 2 L B ~ q τ Q − 2 \widetilde{R}_2L\widetilde{B}_q\tau_Q^{-2} R 2 L B q τ Q − 2 k > k c k > k_c k > k c
For the third cumulant, Figure 14 , the TDVP data at
Figure 14. Third cumulant of the kink number multiplied by τ Q α \tau_Q^\alpha τ Q α q = 0.1 q = 0.1 q = 0.1 q = 0.2 q = 0.2 q = 0.2 q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25 α \alpha α Figure 12 . The black dotted lines show the leading-order prediction C 3 τ Q − 1 / ( 2 q ∗ ) C_3\tau_Q^{-1/(2q_*)} C 3 τ Q − 1/ ( 2 q ∗ ) (35) and the red dashed lines the contribution R ~ 3 L B ~ q τ Q − 2 \widetilde{R}_3L\widetilde{B}_q\tau_Q^{-2} R 3 L B q τ Q − 2 k > k c k > k_c k > k c
TABLE III. Parameters of the fit of the position of the gap minimum, Eq. (10) , over the system sizes of Table II and the resulting KZ exponent ν / ( z ν + 1 ) \nu/(z\nu+ 1) ν / ( z ν + 1 ) g c g_c g c (1) , while a ′ a' a ′ b ′ b' b ′ g c ( L ) / [ 1 − g c ( L ) ] g_c(L)/[1-g_c(L)] g c ( L ) / [ 1 − g c ( L )]
Table 3. Parameters of the fit of the position of the gap minimum, Eq. (10), over the system sizes of Table II and the resulting KZ exponent ν / ( z ν + 1 ) \nu/(z\nu+ 1) ν / ( z ν + 1 ) g c g_c g c a ′ a' a ′ b ′ b' b ′ g c ( L ) / [ 1 − g c ( L ) ] g_c(L)/[1-g_c(L)] g c ( L ) / [ 1 − g c ( L )]
q = 0.2 q=0.2 q = 0.2 τ Q ≳ 100 \tau_Q \gtrsim100 τ Q ≳ 100 κ 3 \kappa_3 κ 3 τ Q − 2 \tau_Q^{-2} τ Q − 2 k > k c k>k_c k > k c q = 0.1 q=0.1 q = 0.1 k > k c k>k_c k > k c τ Q ≳ 150 \tau_Q \gtrsim150 τ Q ≳ 150 q = 1.5 q=1.5 q = 1.5 q = 2.25 q=2.25 q = 2.25
Kink density profiles
In this section, we show the spatial distribution of the kinks at the end of the ramp, ⟨ n ^ j ⟩ \langle\hat{n}_j\rangle ⟨ n ^ j ⟩ n ^ j \hat{n}_j n ^ j (45) , which sum to the mean kink number,
κ 1 = ∑ j ⟨ n ^ j ⟩ \kappa_1=\sum_j\langle\hat{n}_j\rangle κ 1 = ∑ j ⟨ n ^ j ⟩ Figure 15 compares the TDVP profiles with those of the free-fermion approximation on the same open chain for the four fractional orders of the main text.
Figure 15. Kink density ⟨ n ^ j ⟩ \langle\hat{n}_j\rangle ⟨ n ^ j ⟩ j j j q = 0.1 q = 0.1 q = 0.1 q = 0.2 q = 0.2 q = 0.2 q = 1.5 q = 1.5 q = 1.5 q = 2.25 q = 2.25 q = 2.25
For q = 1.5 q=1.5 q = 1.5 q = 2.25 q=2.25 q = 2.25 τ Q \tau_Q τ Q L = 150 L=150 L = 150 q = 1.5 q=1.5 q = 1.5 q = 2.25 q=2.25 q = 2.25
For q = 0.1 q=0.1 q = 0.1 q = 0.2 q=0.2 q = 0.2 q = 0.2 q=0.2 q = 0.2 τ Q = 200 \tau_Q=200 τ Q = 200 q = 0.2 q=0.2 q = 0.2
The profile fixes the mean, while the variance in excess of the local contributions, κ 2 − ∑ j ⟨ n ^ j ⟩ ( 1 − ⟨ n ^ j ⟩ ) = ∑ i ≠ j C i j \kappa_2-\sum_j\langle\hat n_j\rangle(1-\langle\hat n_j\rangle)=\sum_{i\ne j}C_{ij} κ 2 − ∑ j ⟨ n ^ j ⟩ ( 1 − ⟨ n ^ j ⟩) = ∑ i = j C ij q q q
References 
