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Scientific Reports

Quench in the 1D Bose-Hubbard model: Topological defects and excitations from the Kosterlitz-Thouless phase transition dynamics

Jacek Dziarmaga, Wojciech Hubert Zurek

Published Aug 5, 2014• 31 citations
Publisher

Abstract

Kibble-Zurek mechanism (KZM) uses critical scaling to predict density of topological defects and other excitations created in second order phase transitions. We point out that simply inserting asymptotic critical exponents deduced from the immediate vicinity of the critical point to obtain predictions can lead to results that are inconsistent with a more careful KZM analysis based on causality - on the comparison of the relaxation time of the order parameter with the "time distance" from the critical point. As a result, scaling of quench-generated excitations with quench rates can exhibit behavior that is locally (i.e., in the neighborhood of any given quench rate) well approximated by the power law, but with exponents that depend on that rate, and that are quite different from the naive prediction based on the critical exponents relevant for asymptotically long quench times. Kosterlitz-Thouless scaling (that governs e.g. Mott insulator to superfluid transition in the Bose-Hubbard model in one dimension) is investigated as an example of this phenomenon.

DOI
10.1038/srep05950

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Quench in 1D Bose-Hubbard Model: Topological Defects and Excitations from Kosterlitz-Thouless Phase Transition Dynamics

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We propose a theory to explain the experimental observed deviation from the Kibble-Zurek mechanism (KZM) scaling in rapidly quenched critical phase transition dynamics. There is a critical quench rate $τ_{Q}^{c1}$ above it the KZM scaling begins to appear. Smaller than $τ_Q^{c1}$, the defect density $n$ is a constant independent of the quench rate but depends on the final temperature $T_f$ as $n \propto L^d ε_{T_f} ^{d ν}$, the freeze out time $\hat{t}$ admits the scaling law $\hat{t} \propto ε_{T_f}^{-νz}$ where $d$ is the spatial dimension, $ε_{T_f}= (1-T_f/T_c)$ is the dimensionless reduced temperature, $L$ is the sample size, $ν$ and $z$ are spatial and dynamical critical exponents. Quench from $T_c$, the critical rate is determined by the final temperature $T_f$ as $τ_Q^{c1} \propto ε_{T_f}^{-(1+z ν)} $. All the scaling laws are verified in a rapidly quenched superconducting ring via the AdS/CFT correspondence.

arXiv:2110.07969•Oct 15, 2021•3 citations
Kibble-Zurek mechanism (KZM) uses critical scaling to predict density of topological defects and other excitations created in second order phase transitions. We point out that simply inserting asymptotic critical exponents deduced from the immediate vicinity of the critical point to obtain predictions can lead to results that are inconsistent with a more careful KZM analysis based on causality - on the comparison of the relaxation time of the order parameter with the "time distance" from the critical point. As a result, scaling of quench-generated excitations with quench rates can exhibit behavior that is locally (i.e., in the neighborhood of any given quench rate) well approximated by the power law, but with exponents that depend on that rate, and that are quite different from the naive prediction based on the critical exponents relevant for asymptotically long quench times. Kosterlitz-Thouless scaling (that governs e.g. Mott insulator to superfluid transition in the Bose-Hubbard model in one dimension) is investigated as an example of this phenomenon.
Kibble-Zurek mechanism (KZM) uses critical scaling to predict density of topological defects and other excitations created in second order phase transitions. We point out that simply inserting asymptotic critical exponents deduced from the immediate vicinity of the critical point to obtain KZM predictions can lead to results that are inconsistent with a more careful KZM-like analysis based on causality -- on the comparison of the relaxation time of the order parameter with the time "distance" from the critical point. As a result, scaling of quench-generated excitations with quench rates can exhibit behavior that is locally (i.e., in the neighborhood of any given quench rate) well approximated by the power law, but with exponents that depend on that rate, and that are quite different from the naive prediction based on the critical exponents relevant for asymptotically long quench times. Kosterlitz-Thouless scaling (that governs e.g. Mott insulator to superfluid transition in the Bose-Hubbard model in one dimension) is investigated as an example of this phenomenon.
The ground state of the one-dimensional Bose-Hubbard model at unit filling undergoes the Mott-superfluid quantum phase transition. It belongs to the Kosterlitz-Thouless universality class with an exponential divergence of the correlation length in place of the usual power law. We present numerical simulations of a linear quench both from the Mott insulator to superfluid and back. The results satisfy the scaling hypothesis that follows from the Kibble-Zurek mechanism (KZM). In the superfluid-to-Mott quenches there is no significant excitation in the superfluid phase despite its gaplessness. Since all critical superfluid ground states are qualitatively similar, the excitation begins to build up only after crossing the critical point when the ground state begins to change fundamentally. The last process falls into the KZM framework.
The ground state of the one-dimensional Bose-Hubbard model at unit filling undergoes the Mott-superfluid quantum phase transition. It belongs to the Kosterlitz-Thouless universality class with an exponential divergence of the correlation length in place of the usual power law. We present numerical simulations of a linear quench both from the Mott insulator to superfluid and back. The results satisfy the scaling hypothesis that follows from the Kibble-Zurek mechanism (KZM). In the superfluid-to-Mott quenches there is no significant excitation in the superfluid phase despite its gaplessness. Since all critical superfluid ground states are qualitatively similar, the excitation begins to build up only after crossing the critical point when the ground state begins to change fundamentally. The last process falls into the KZM framework.
The crossing of a continuous phase transition gives rise to the formation of topological defects described by the Kibble-Zurek mechanism (KZM) in the limit of slow quenches. The KZM predicts a universal power-law scaling of the defect density as a function of the quench time. We focus on the deviations from KZM experimentally observed in rapid quenches and establish their universality. While KZM scaling holds below a critical quench rate, for faster quenches the defect density and the freeze-out time become independent of the quench rate and exhibit a universal power-law scaling with the final value of the control parameter. These predictions are verified in several paradigmatic scenarios in both the classical and quantum domains.
Abstract We study the dynamics of second-order phase transitions that may have taken place in the early universe and analyze the Kibble-Zurek (KZ) scaling as well as the formation of topological defects emerging from a quench in the one-dimensional Bose-Hubbard model. We describe the universal dynamics of the KZ mechanism (KZM) and the topological defect formation by simulating the Bose–Hubbard model at zero temperature. We study the effects of thermalization on the ground state by analyzing the transition between Mott and superfluid phases. The out-of-equilibrium dynamics induced by a quench of the hopping parameter is presented. We find a KZ scaling behaviour at zero temperature and analyze the KZ dynamics.
We propose a theory to explain the experimental observed deviation from the Kibble-Zurek mechanism (KZM) scaling in rapidly quenched critical phase transition dynamics. There is a critical quench rate τQc1τ_{Q}^{c1}τQc1​ above it the KZM scaling begins to appear. Smaller than τQc1τ_Q^{c1}τQc1​, the defect density nnn is a constant independent of the quench rate but depends on the final temperature TfT_fTf​ as n∝LdεTfdνn \propto L^d ε_{T_f} ^{d ν}n∝LdεTf​dν​, the freeze out time t^\hat{t}t^ admits the scaling law t^∝εTf−νz\hat{t} \propto ε_{T_f}^{-νz}t^∝εTf​−νz​ where ddd is the spatial dimension, εTf=(1−Tf/Tc)ε_{T_f}= (1-T_f/T_c)εTf​​=(1−Tf​/Tc​) is the dimensionless reduced temperature, LLL is the sample size, ννν and zzz are spatial and dynamical critical exponents. Quench from TcT_cTc​, the critical rate is determined by the final temperature TfT_fTf​ as $τ_Q^{c1} \propto ε_{T_f}^{-(1+z ν)} $. All the scaling laws are verified in a rapidly quenched superconducting ring via the AdS/CFT correspondence.