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arXiv:2303.03781 · Physical review. B./Physical review. B

Adiabaticity enhancement in the classical transverse field Ising chain, and its effective non-Hermitian description

Balázs Dóra, Roderich Moessner

Published Sep 7, 2023• 0 citations
arXivHTMLPDF

Abstract

We analyze the near-adiabatic dynamics in a ramp through the critical point (CP) of the classical transverse field Ising chain. This is motivated, conceptually, by the fact that this CP---unlike its quantum counterpart---experiences no thermal or quantum fluctuations, and technically by the tractability of its effective model. For a ``half ramp'' from a ferromagnet to the CP, the longitudinal and transverse magnetizations scale as ${\ensuremath{\tau}}^{\ensuremath{-}1/3}$ and ${\ensuremath{\tau}}^{\ensuremath{-}2/3}$, respectively, with $1/\ensuremath{\tau}$ the ramp rate, in accord with Kibble-Zurek theory. For ferro- to paramagnetic ramps across the CP, however, they stay closer, ${\ensuremath{\tau}}^{\ensuremath{-}1/2}$ and ${\ensuremath{\tau}}^{\ensuremath{-}1}$, to adiabaticity. This adiabaticity enhancement compared to the half ramp is understood by casting the dynamics in the paramagnet in the form of a non-Hermitian Dirac Hamiltonian, with the CP playing the role of an exceptional point, opening an additional decay channel.

DOI
10.1103/physrevb.108.l121105

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We analyze the near-adiabatic dynamics in a ramp through the critical point (CP) of the classical transverse field Ising chain. This is motivated, conceptually, by the fact that this CP---unlike its quantum counterpart---experiences no thermal or quantum fluctuations, and technically by the tractability of its effective model. For a ``half ramp'' from a ferromagnet to the CP, the longitudinal and transverse magnetizations scale as \ensuremathτ\ensuremath−1/3{\ensuremath{\tau}}^{\ensuremath{-}1/3}\ensuremathτ\ensuremath−1/3 and \ensuremathτ\ensuremath−2/3{\ensuremath{\tau}}^{\ensuremath{-}2/3}\ensuremathτ\ensuremath−2/3, respectively, with 1/\ensuremathτ1/\ensuremath{\tau}1/\ensuremathτ the ramp rate, in accord with Kibble-Zurek theory. For ferro- to paramagnetic ramps across the CP, however, they stay closer, \ensuremathτ\ensuremath−1/2{\ensuremath{\tau}}^{\ensuremath{-}1/2}\ensuremathτ\ensuremath−1/2 and \ensuremathτ\ensuremath−1{\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremathτ\ensuremath−1, to adiabaticity. This adiabaticity enhancement compared to the half ramp is understood by casting the dynamics in the paramagnet in the form of a non-Hermitian Dirac Hamiltonian, with the CP playing the role of an exceptional point, opening an additional decay channel.
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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.
Abstract We drive the one-dimensional quantum Ising chain in the transverse field from the paramagnetic phase to the critical point and study its free evolution there. We analyze excitation of such a system at the critical point and dynamics of its transverse magnetization and Loschmidt echo during free evolution. We discuss how the system size and quench-induced scaling relations from the Kibble–Zurek theory of non-equilibrium phase transitions are encoded in quasi-periodic time evolution of the transverse magnetization and Loschmidt echo.
We analyse the crossing of a quantum critical point based on exact results for the transverse XY model. In dependence of the change rate of the driving field, the evolution of the ground state is studied while the transverse magnetic field is tuned through the critical point with a linear ramping. The excitation probability is obtained exactly and is compared to previous studies and to the Landau–Zener formula, a long time solution for non-adiabatic transitions in two-level systems. The exact time dependence of the excitations density in the system allows us to identify the adiabatic and diabatic regions during the sweep and to study the mesoscopic fluctuations of the excitations. The effect of white-noise is investigated, where the critical point transmutes into a non-Hermitian 'degenerate region'. Besides an overall increase of the excitations during and at the end of the sweep, the most destructive effect of the noise is the decay of the state purity that is enhanced by the passage through the degenerate region.
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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.