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arXiv:2212.05839 · Physical review. B./Physical review. B

Finite-temperature dynamical quantum phase transition in a non-Hermitian system

Debashish Mondal, Tanay Nag

Published May 24, 2023• 37 citations
arXivHTMLPDF

Abstract

We investigate the interplay between non-Hermiticity and finite temperature in the context of a mixed-state dynamical quantum phase transition (MSDQPT). We consider a $p$-wave superconductor model, encompassing complex hopping and non-Hermiticity, that can lead to gapless phases in addition to gapped phases, to examine the MSDQPT and winding number via the intraphase quench. We find that the MSDQPT is always present irrespective of the gap structure of the underlying phase; however, the profile of Fisher zeros changes between the above phases. Such occurrences of MSDQPT are in contrast to the zero-temperature case in which a DQPT does not take place for the gapped phase. Surprisingly, the half-integer jumps in winding number at zero temperature are washed away for finite temperature in the gapless phase. We study the evolution of the minimum time required by the system to experience MSDQPT with the inverse temperature such that gapped and gapless phases can be differentiated. Our study indicates that the minimum time shows monotonic (nonmonotonic) behavior for the gapped (gapless) phase.

DOI
10.1103/physrevb.107.184311

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We investigate the interplay between non-Hermiticity and finite temperature in the context of a mixed-state dynamical quantum phase transition (MSDQPT). We consider a ppp-wave superconductor model, encompassing complex hopping and non-Hermiticity, that can lead to gapless phases in addition to gapped phases, to examine the MSDQPT and winding number via the intraphase quench. We find that the MSDQPT is always present irrespective of the gap structure of the underlying phase; however, the profile of Fisher zeros changes between the above phases. Such occurrences of MSDQPT are in contrast to the zero-temperature case in which a DQPT does not take place for the gapped phase. Surprisingly, the half-integer jumps in winding number at zero temperature are washed away for finite temperature in the gapless phase. We study the evolution of the minimum time required by the system to experience MSDQPT with the inverse temperature such that gapped and gapless phases can be differentiated. Our study indicates that the minimum time shows monotonic (nonmonotonic) behavior for the gapped (gapless) phase.
The dynamical quantum phase transitions (DQPTs) and the associated winding numbers have been extensively studied in the context Hermitian system. We consider the non-Hermitian analog of ppp-wave superconductor, supporting Hermitian gapless phase with complex hopping, in presence of on-site or superconducting loss term. This allows us to investigate the effect of non-Hermitian gapless phases on the DQPTs in addition to the Hermitian gapless phases. Our findings indicate that contour analysis of the underlying Hamiltonian, enclosing the origin and/or exceptional points, can predict the occurrences of DQPTs except the quench within the gapless phases. For the Hermitian case with initial and final Hamiltonians both being Hermitian, we find nonmonotonic integer jump for the winding number as the hallmark signature of the gapless phase there. For the hybrid case with initial and final Hamiltonians being Hermitian and non-Hermitian respectively, winding number exhibits integer spike in addition to the nonmonotonic integer jumps. For the non-Hermitian case with initial and final Hamiltonians both being non-Hermitian, the winding number show half-integer jumps for lossy superconductivity that does not have any Hermitian analog. On the other hand, the integer jumps in winding number is observed for lossy chemical potential. We understand our findings by connecting them with the profile of Fisher zeros and number of exceptional points and/or origin.
The dynamical quantum phase transitions (DQPTs) in quantum spin chains with gapless phases after a sudden quench are studied. We mainly consider the general systems with asymmetrical quasiparticle excitation spectra and obtain the general expression of the Loschmidt echo as well as the general conditions for the occurrence of DQPTs. As two examples, we study the DQPTs in the \emph{XY} chains with Dzyaloshinskii-Moriya interaction and \emph{XZY-YZX} type of three-site interaction. It's found that the DQPTs may not occur in the quench across the quantum phase transitions regardless of whether the quench is from the gapless phase to gapped phase or from the gapped phase to gapless phase. This is different from the DQPTs in the case of quench from the gapped phase to gapped phase, in which the DQPTs will always appear. Besides, we also analyze the different reasons for the absence of DQPTs in the quench from the gapless phase and the gapped phase.
We investigate dynamical quantum phase transitions (DQPTs) in both pure and mixed states within the extended Su-Schrieffer-Heeger (SSH) model framework, focusing on the SSH-3 and SSH-4 models, which differ in symmetry properties. The SSH-3 model, characterized by a chiral-like point symmetry rather than true chiral symmetry, supports robust localized edge states tied to its topological nature. Our results show that for pure states, DQPTs occur after quenches crossing the topological transition, even when the energy band gap remains open. For mixed states, DQPT behavior aligns with pure states at low temperatures but undergoes significant changes at higher temperatures, including the emergence of multiple critical times. In contrast, the SSH-4 model, which possesses chiral symmetry, features four distinct energy-spectrum configurations. We find that pure-state DQPTs arise only when the quench starts from the initial state, chosen from a band that becomes gapless with another band at a certain critical point, and crosses the critical point. At finite temperature, mixed-state DQPTs persist at low temperatures only if the corresponding pure-state quench induces DQPTs, but they disappear at elevated temperatures. These findings elucidate the interplay between symmetry, band structures, and temperature in governing DQPTs within generalized SSH models.
By deriving a general framework and analyzing concrete examples, we demonstrate a class of dynamical quantum phase transitions (DQPTs) in one-dimensional two-band systems going through double-quench processes. When this type of DQPT occurs, the Loschmidt amplitude vanishes and the rate function remains singular after the second quench, meaning the final state continually has no overlap with the initial state. This type of DQPT is named metamorphic DQPT to differentiate it from ordinary DQPTs that only exhibit zero Loschmidt amplitude and singular rate function at discrete time points. The metamorphic DQPTs occur at zero as well as finite temperatures. Our examples of the Su-Schrieffer-Heeger (SSH) model and Kitaev chain illustrate the conditions and behavior of the metamorphic DQPT. Since ordinary DQPTs have been experimentally realized in many systems, similar setups with double quenches will demonstrate the metamorphic DQPT. Our findings thus provide additional controls of dynamical evolution of quantum systems.
Dynamical quantum phase transitions (DQPT) provide a fundamental framework for characterizing nonequilibrium critical phenomena, and have recently been extended to non-Hermitian systems. Half-integer jumps of the dynamical topological order parameter can arise in both Hermitian and non-Hermitian systems, yet their origin in non-Hermitian settings remains unclear. Here, we investigate DQPTs in a two-band non-Hermitian model and show that these fractional jumps originate from the discontinuity in the complex energy spectrum. We demonstrate that this behavior is intrinsically linked to the steady-state phase diagram of the quench Hamiltonian, and also classify the occurrence of DQPT. We further extend our analysis to the multiband Hamiltonian and the two-dimensional model, providing a unified understanding of fractional dynamical topological signatures.
Preparing an integrable system in a mixed state described by a thermal density matrix, we subject it to a sudden quench and explore the subsequent unitary dynamics. To address the question of whether the nonanalyticities, namely, the dynamical quantum phase transitions (DQPTs), persist when the initial state is mixed, we consider two versions of the generalized Loschmidt overlap amplitude (GLOA). Our study shows that the GLOA constructed using the Uhlmann approach does not show any signature of DQPTs at any nonzero initial temperature. On the other hand, a GLOA defined in the interferometric phase approach through the purifications of the time-evolved density matrix, indeed shows that nonanalyiticies in the corresponding ``dynamical free-energy density'' persist, thereby establishing the existence of mixed state dynamical quantum phase transitions (MSDQPTs). Our work provides a framework that perfectly reproduces both the nonanalyticities and also the emergent topological structure in the pure state limit. These claims are corroborated by analyzing the nonequilibrium dynamics of a transverse Ising chain initially prepared in a thermal state and subjected to a sudden quench of the transverse field.