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arXiv:2605.05106

Kink-kink correlations in nonlinear quenches across a quantum critical point

Lakshita Jindal, Kavita Jain

cond-mat.stat-mechquant-phSubmitted May 6, 2026• Announced May 7, 2026• 0 citations
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Abstract

When a quantum system exhibiting a second order phase transition is quenched across the critical point in large but finite time, the dynamics are not adiabatic in the critical region and the Kibble-Zurek (KZ) mechanism provides a framework to determine local observables such as the mean defect density. However, to find higher-point functions, one has to go beyond the KZ paradigm asshown in recent works on one-dimensional transverse field Ising model (TFIM) following a linear quench. It has been found that (i) besides the KZ scale, the quench dynamics depend on another length scale that arises due to the finite phase difference between the low energy modes, and (ii) contrary to the expectations based on the KZ mechanism, in general, the correlation functions do not decay exponentially with distance. Motivated by these results for the linear quench, we are interested in understanding if these properties are universal, and consider the 1D TFIM when the transverse field varies algebraically in the vicinity of the critical field. We focus on the equal-time,longitudinal kink-kink correlation function at the end of the quench from the paramagnetic to the ferromagnetic phase, and find that (i) the correlator depends only on the KZ length for superlinear quenches, otherwise an additional dephasing length is required to describe it, and (ii) the dephased correlator decays as a compressed exponential with an exponent that changes continuously with the quench exponent. Our results are obtained using an adiabatic perturbation theory, analytical arguments and exact numerical integration of the relevant equations.

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When a quantum system exhibiting a second order phase transition is quenched across the critical point in large but finite time, the dynamics are not adiabatic in the critical region and the Kibble-Zurek (KZ) mechanism provides a framework to determine local observables such as the mean defect density. However, to find higher-point functions, one has to go beyond the KZ paradigm asshown in recent works on one-dimensional transverse field Ising model (TFIM) following a linear quench. It has been found that (i) besides the KZ scale, the quench dynamics depend on another length scale that arises due to the finite phase difference between the low energy modes, and (ii) contrary to the expectations based on the KZ mechanism, in general, the correlation functions do not decay exponentially with distance. Motivated by these results for the linear quench, we are interested in understanding if these properties are universal, and consider the 1D TFIM when the transverse field varies algebraically in the vicinity of the critical field. We focus on the equal-time,longitudinal kink-kink correlation function at the end of the quench from the paramagnetic to the ferromagnetic phase, and find that (i) the correlator depends only on the KZ length for superlinear quenches, otherwise an additional dephasing length is required to describe it, and (ii) the dephased correlator decays as a compressed exponential with an exponent that changes continuously with the quench exponent. Our results are obtained using an adiabatic perturbation theory, analytical arguments and exact numerical integration of the relevant equations.
The transverse field in the quantum Ising chain is linearly ramped from the para- to the ferromagnetic phase across the quantum critical point at a rate characterized by a quench time ${\ensuremath{\tau}}_{Q}$. We calculate a connected kink-kink correlator in the final state at zero transverse field. The correlator is a sum of two terms: a negative (antibunching) Gaussian that depends on the Kibble-Zurek correlation length only and a positive term that depends on a second longer scale of length. The second length is made longer by dephasing of the state excited near the critical point during the following ramp across the ferromagnetic phase. This interpretation is corroborated by considering a linear ramp that is halted in the ferromagnetic phase for a finite waiting time and then continued at the same rate as before the halt. The extra time available for dephasing increases the second scale of length that asymptotically grows linearly with the waiting time. The dephasing also suppresses magnitude of the second term, making it negligible for waiting times much longer than ${\ensuremath{\tau}}_{Q}$. The same dephasing can be obtained with a smooth ramp that slows down in the ferromagnetic phase. Assuming sufficient dephasing, we also obtain higher order kink correlators and the ferromagnetic correlation function.
Linear quench of the transverse field drives the quantum Ising chain across a quantum critical point from the paramagnetic to the ferromagnetic phase. We focus on normal and anomalous quadratic correlators between fermionic kink creation and annihilation operators. They depend not only on the Kibble-Zurek (KZ) correlation length but also on a dephasing length scale, which differs from the KZ length by a logarithmic correction. Additional slowing down of the ramp in the ferromagnetic phase further increases the dephasing length and suppresses the anomalous correlator. The quadratic correlators enter Pfaffians that yield experimentally relevant kink correlation functions, the probability distribution of ferromagnetic domain sizes, and, closely related, emptiness formation probability. The latter takes the form of a Pfaffian of a block Toeplitz matrix that allows for some analytic asymptotes. Finally, we obtain further insight into the structure of the state at the end of the ramp by interpreting it as a paired state of fermionic kinks characterized by its pair wave function. All those quantities are sensitive to quantum coherence between eigenstates with different numbers of kinks, thus making them a convenient probe of the quantumness of a quantum simulator platform.
Quantum Ising model in one dimension is an exactly solvable example of a quantum phase transition. We investigate its behavior during a quench caused by a gradual turning off of the transverse bias field. The system is then driven at a fixed rate characterized by the quench time ${\ensuremath{\tau}}_{Q}$ across the critical point from a paramagnetic to ferromagnetic phase. In agreement with Kibble-Zurek mechanism (which recognizes that evolution is approximately adiabatic far away, but becomes approximately impulse sufficiently near the critical point), quantum state of the system after the transition exhibits a characteristic correlation length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ proportional to the square root of the quench time ${\ensuremath{\tau}}_{Q}$: $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}=\sqrt{{\ensuremath{\tau}}_{Q}}$. The inverse of this correlation length is known to determine average density of defects (e.g., kinks) after the transition. In this paper, we show that this same $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ controls the entropy of entanglement, e.g., entropy of a block of LLL spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large LLL, this entropy saturates at $\frac{1}{6}\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size $L\ensuremath{\approx}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, but---in the subsequent evolution in the ferromagnetic phase---a somewhat larger length scale $l=\sqrt{{\ensuremath{\tau}}_{Q}}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\tau}}_{Q}$ develops as a result of a dephasing process that can be regarded as a quantum analog of phase ordering, and the entropy saturates when $L\ensuremath{\approx}l$. We also study the spin-spin correlation using both analytic methods and real time simulations with the Vidal algorithm. We find that at an instant when quench is crossing the critical point, ferromagnetic correlations decay exponentially with the dynamical correlation length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, but (as for entropy of entanglement) in the following evolution length scale lll gradually develops. The correlation function becomes oscillatory at distances less than this scale. However, both the wavelength and the correlation length of these oscillations are still determined by $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.
We consider the finite-time quench dynamics in the quantum transverse field Ising model which exhibits a second order phase transition from a paramagnetic to a ferromagnetic phase, as the transverse magnetic field is decreased. These dynamics have been thoroughly investigated in previous studies when the critical point is crossed during the quench; here, we quench the system from deep in the paramagnetic phase to just above the critical field so that the system remains in the gapped phase throughout the quench duration. On linearly quenching the infinitely large system, we find that the behavior of mean longitudinal defect density and mean transverse magnetization at the end of the quench falls into three distinct scaling regimes as the quench time is increased. For sufficiently small quench times, these observables remain roughly constant, but for larger quench times, a crossover occurs from the Kibble-Zurek scaling law to the quadratic quench rate law when the Kibble-Zurek time is of the order of relaxation time at the final quench field. These results are shown analytically using power series and uniform asymptotic expansions of the exact solution of the model, and also compared with an adiabatic perturbation theory in the third regime. We find that the above mentioned scaling regimes hold for quenches within the ferromagnetic phase also, and provide a general scaling argument for crossover from the Kibble-Zurek regime to an adiabatic regime for slow quenches within a gapped phase.
According to the Kibble-Zurek mechanism, there is a universal power-law relationship between the defect density and the quench rate during a slow linear quench through a critical point. It is generally accepted that a fast quench results in a deviation from the Kibble-Zurek scaling law and leads to the formation of a saturated plateau in the defect density. By adjusting the quench rate from slow to very fast limits, we observe the varying quench dynamics and identify a presaturated regime that lies between the saturated and Kibble-Zurek regimes. This significant result is first elucidated through the adiabatic-impulse approximation, then verified by a rigorous analysis on the transverse Ising chain as well. As we approach the turning point from the saturated to presaturated regimes, we notice a change in scaling laws and, with an increase in the initial transverse field, a shrinking of the saturated regime until it disappears. During another turning point from the Kibble-Zurek to presaturated regimes, we observe an attenuation of the dephasing effect and a change in the behavior of the kink-kink correlation function from a Gaussian decay to an exponential decay. Finally, the coherent many-body oscillation after quench exhibits different behaviors in the three regimes and shows a significant change of scaling behavior between the S and PS regimes.
We analyze the quantum phase transitions taking place in a one-dimensional transverse-field Ising model with long-range couplings that decay algebraically with distance. We are interested in the Kibble-Zurek universal scaling laws emerging in nonequilibrium dynamics and in the potential for the unambiguous observation of such behavior in a realistic experimental setup based on trapped ions. To this end, we determine the phase diagram of the model and the critical exponents characterizing its quantum phase transitions by means of density-matrix renormalization group calculations and finite-size scaling theory, which allows us to obtain good estimates for different range of ferromagnetic and antiferromagnetic interactions. Beyond critical equilibrium properties, we tackle a nonequilibrium scenario in which quantum Kibble-Zurek scaling laws may be retrieved. Here, it is found that the predicted nonequilibrium universal behavior, i.e., the scaling laws as a function of the quench rate and critical exponents, can be observed in systems comprising an experimentally feasible number of spins. Finally, a scheme is introduced to simulate the algebraically decaying couplings accurately by means of a digital quantum simulation with trapped ions. Our results suggest that quantum Kibble-Zurek physics can be explored and observed in state-of-the-art experiments with trapped ions realizing long-range Ising models.