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arXiv:2412.20186 · Physical review. B./Physical review. B

Quantum Kibble-Zurek mechanism: The role of boundary conditions, endpoints, and kink types

Jose Soto Garcia, Natalia Chepiga

Published Jan 26, 2026• 2 citations
arXivHTMLPDF

Abstract

Quantum phase transitions are characterized by the universal scaling laws in the critical region surrounding the transitions. This universality is also manifested in the critical real-time dynamics through the quantum Kibble-Zurek mechanism. In recent experiments on a Rydberg atom quantum simulator, the Kibble-Zurek mechanism was used to probe the nature of quantum phase transitions. In this paper, we analyze the caveats associated with this method and develop strategies to improve its accuracy. Focusing on two minimal models—transverse-field Ising and quantum three-state Potts, both in one dimension—we study the effect of boundary conditions, the location of the endpoints, and some subtleties in the definition of the kink operators. In particular, we show that the critical scaling of the most intuitive types of kinks is extremely sensitive to the correct choice of endpoint, while more advanced types of kinks exhibit remarkably robust universal scaling. Furthermore, we show that when kinks are tracked over the entire chain, fixed boundary conditions improve the accuracy of the scaling. Surprisingly, the Kibble-Zurek critical scaling appears to be equally accurate whether the fixed boundary conditions are chosen to be symmetric or antisymmetric. We also show that the density of kinks extracted in the central part of long chains obeys the predicted universal scaling for all types of boundary conditions. Finally, we test our kink definition for the Ising transition on the period-2 phase of the Rydberg model and show that it is more robust against the endpoint than the standard definition.

DOI
10.1103/s1pr-8vlp

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Quantum phase transitions are characterized by the universal scaling laws in the critical region surrounding the transitions. This universality is also manifested in the critical real-time dynamics through the quantum Kibble-Zurek mechanism. In recent experiments on a Rydberg atom quantum simulator, the Kibble-Zurek mechanism was used to probe the nature of quantum phase transitions. In this paper, we analyze the caveats associated with this method and develop strategies to improve its accuracy. Focusing on two minimal models—transverse-field Ising and quantum three-state Potts, both in one dimension—we study the effect of boundary conditions, the location of the endpoints, and some subtleties in the definition of the kink operators. In particular, we show that the critical scaling of the most intuitive types of kinks is extremely sensitive to the correct choice of endpoint, while more advanced types of kinks exhibit remarkably robust universal scaling. Furthermore, we show that when kinks are tracked over the entire chain, fixed boundary conditions improve the accuracy of the scaling. Surprisingly, the Kibble-Zurek critical scaling appears to be equally accurate whether the fixed boundary conditions are chosen to be symmetric or antisymmetric. We also show that the density of kinks extracted in the central part of long chains obeys the predicted universal scaling for all types of boundary conditions. Finally, we test our kink definition for the Ising transition on the period-2 phase of the Rydberg model and show that it is more robust against the endpoint than the standard definition.
Quantum phase transitions (QPTs) involve transformations between different states of matter that are driven by quantum fluctuations. These fluctuations play a dominant role in the quantum critical region surrounding the transition point, where the dynamics are governed by the universal properties associated with the QPT. The resulting quantum criticality has been explored by probing linear response for systems near thermal equilibrium. While time dependent phenomena associated with classical phase transitions have been studied in various scientific fields, understanding critical real-time dynamics in isolated, non-equilibrium quantum systems is of fundamental importance both for exploring novel approaches to quantum information processing and realizing exotic new phases of matter. Here, we use a Rydberg atom quantum simulator with programmable interactions to study the quantum critical dynamics associated with several distinct QPTs. By studying the growth of spatial correlations while crossing the QPT at variable speeds, we experimentally verify the quantum Kibble-Zurek mechanism (QKZM) for an Ising-type QPT, explore scaling universality, and observe corrections beyond simple QKZM predictions. This approach is subsequently used to investigate novel QPTs associated with chiral clock model providing new insights into exotic systems, and opening the door for precision studies of critical phenomena and applications to quantum optimization.
Abstract The Kibble–Zurek mechanism (KZM) describes the dynamics across a phase transition leading to the formation of topological defects, such as vortices in superfluids and domain walls in spin systems. Here, we experimentally probe the distribution of kink pairs in a one-dimensional quantum Ising chain driven across the paramagnet-ferromagnet quantum phase transition, using a single trapped ion as a quantum simulator in momentum space. The number of kink pairs after the transition follows a Poisson binomial distribution, in which all cumulants scale with a universal power law as a function of the quench time in which the transition is crossed. We experimentally verified this scaling for the first cumulants and report deviations due to noise-induced dephasing of the trapped ion. Our results establish that the universal character of the critical dynamics can be extended beyond KZM, which accounts for the mean kink number, to characterize the full probability distribution of topological defects.
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.
The quantum Kibble-Zurek mechanism (QKZM) predicts universal dynamical behavior near the quantum phase transitions (QPTs). It is now well understood for the one-dimensional quantum matter. Higher-dimensional systems, however, remain a challenge, complicated by the fundamentally different character of the associated QPTs and their underlying conformal field theories. In this work, we take the first steps toward theoretical exploration of the QKZM in two dimensions for interacting quantum matter. We study the dynamical crossing of the QPT in the paradigmatic Ising model by a joint effort of modern state-of-the-art numerical methods, including artificial neural networks and tensor networks. As a central result, we quantify universal QKZM behavior close to the QPT. We also note that, upon traversing further into the ferromagnetic regime, deviations from the QKZM prediction appear. We explain the observed behavior by proposing an extended QKZM taking into account spectral information as well as phase ordering. Our work provides a testing platform for higher-dimensional quantum simulators.
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 experimental realization of the quantum Kibble-Zurek mechanism in arrays of trapped Rydberg atoms has brought the problem of commensurate-incommensurate transition back into the focus of active research. Relying on equilibrium simulations of finite intervals, direct chiral transitions at the boundary of the period-3 and period-4 phases have been predicted. Here, we study how these chiral transitions can be diagnosed experimentally with critical dynamics. We demonstrate that chiral transitions can be distinguished from the floating phases by comparing Kibble-Zurek dynamics on arrays with different numbers of atoms. Furthermore, by sweeping in the opposite direction and keeping track of the order parameter, we identify the location of conformal points. Finally, combining forward and backward sweeps, we extract all critical exponents characterizing the transition.