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arXiv:1806.10646 · Physical Review Letters

Universal Statistics of Topological Defects Formed in a Quantum Phase Transition

Adolfo del Campo

Published Nov 14, 2018• 97 citations
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Abstract

When a quantum phase transition is crossed in finite time, critical slowing down leads to the breakdown of adiabatic dynamics and the formation of topological defects. The average density of defects scales with the quench rate following a universal power law predicted by the Kibble-Zurek mechanism. We analyze the full counting statistics of kinks and report the exact kink number distribution in the transverse-field quantum Ising model. Kink statistics is described by the Poisson binomial distribution with all cumulants exhibiting a universal power law scaling with the quench rate. In the absence of finite-size effects, the distribution approaches a normal one, a feature that is expected to apply broadly in systems described by the Kibble-Zurek mechanism.

DOI
10.1103/physrevlett.121.200601

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When a quantum phase transition is crossed in finite time, critical slowing down leads to the breakdown of adiabatic dynamics and the formation of topological defects. The average density of defects scales with the quench rate following a universal power law predicted by the Kibble-Zurek mechanism. We analyze the full counting statistics of kinks and report the exact kink number distribution in the transverse-field quantum Ising model. Kink statistics is described by the Poisson binomial distribution with all cumulants exhibiting a universal power law scaling with the quench rate. In the absence of finite-size effects, the distribution approaches a normal one, a feature that is expected to apply broadly in systems described by the Kibble-Zurek mechanism.
Abstract When a quantum phase transition is crossed within a finite time, critical slowing down disrupts adiabatic dynamics, resulting in the formation of topological defects. The average density of these defects scales with the quench rate, adhering to a universal power law as predicted by the Kibble–Zurek mechanism (KZM). In this study, we aim to investigate the counting statistics of kink density in the 1D transverse-field quantum Ising model. We demonstrate on multiple quantum processing units up to 100 qubits, that higher-order cumulants follow a universal power law scaling as a function of the quench time. We also show the breakdown of the KZM for short quenches for finite-size systems. Tensor network simulations corroborate our quantum simulation results for bigger systems not in the asymptotic limit.
In a scenario of spontaneous symmetry breaking in finite time, topological defects are generated at a density that scales with the driving time according to the Kibble-Zurek mechanism (KZM). Signatures of universality beyond the KZM have recently been unveiled: The number distribution of topological defects has been shown to follow a binomial distribution, in which all cumulants inherit the universal power-law scaling with the quench rate, with cumulant rations being constant. In this work, we analyze the role of boundary conditions in the statistics of topological defects. In particular, we consider a lattice system with nearest-neighbor interactions subject to soft antiperiodic, open, and periodic boundary conditions implemented by an energy penalty term. We show that for fast and moderate quenches, the cumulants of the kink number distribution present a universal scaling with the quench rate that is independent of the boundary conditions except for an additive term, which becomes prominent in the limit of slow quenches, leading to the breaking of power-law behavior. We test our theoretical predictions with a one-dimensional scalar theory on a lattice.
We consider the number distribution of topological defects resulting from the finite-time crossing of a continuous phase transition and identify signatures of universality beyond the mean value, predicted by the Kibble-Zurek mechanism. Statistics of defects follows a binomial distribution with N Bernouilli trials associated with the probability of forming a topological defect at the locations where multiple domains merge. All cumulants of the distribution are predicted to exhibit a common universal power-law scaling with the quench time in which the transition is crossed. Knowledge of the distribution is used to discuss the onset of adiabatic dynamics and bound rare events associated with large deviations.
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.
The Kibble-Zurek mechanism (KZM) predicts that the average number of topological defects generated upon crossing a continuous or quantum phase transition obeys a universal scaling law with the quench time. Fluctuations in the defect number near equilibrium are approximately of Gaussian form, in agreement with the central limit theorem. Using large deviations theory, we characterize the universality of fluctuations beyond the KZM and report the exact form of the rate function in the transverse-field quantum Ising model. In addition, we characterize the scaling of large deviations in an arbitrary continuous phase transition, building on recent evidence establishing the universality of the defect number distribution.
We analyze mechanisms for universal out-of-equilibrium dynamics near criticality by exploring the effect of randomized quantum resetting (QR) under a finite-time quench across a quantum phase transition. Using the transverse-field Ising chain as a generic model and exploiting its exact solution, QR is found to cause a crossover of the scaling of the topological defect density with the time scale τττ of the quench, from Kibble-Zurek to anti-Kibble-Zurek scaling as τττ increases. This reflects a competition between non-adiabatic quench-driven excitations and QR, giving rise to local minima of the defect densities at optimal annealing times. These times and the corresponding local minima are shown to scale as universal power laws with the rate of QR. Additional results for the scaling of the mean excess energy suggest that a system driven across a quantum critical point exhibits the same scaling behavior under a linear quench with QR as with uncorrelated noise.