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arXiv:2205.12859 · Physical review. B./Physical review. B

Anomaly in the dynamical quantum phase transition in a non-Hermitian system with extended gapless phases

Debashish Mondal, Tanay Nag

Published Aug 15, 2022• 38 citations
arXivHTMLPDF

Abstract

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 $p$-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.

DOI
10.1103/physrevb.106.054308

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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.
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.
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.
In closed quantum systems, dynamical phase transitions are identified by the nonanalytic behavior of the return probability as a function of time. In this work, we study the nonunitary dynamics following quenches across exceptional points in a non-Hermitian lattice realizable by optical resonators. Dynamical quantum phase transitions with topological signatures are found when an isolated exceptional point is crossed during the quench. A winding number defined by a real, noncyclic geometric phase is introduced, whose value features quantized jumps at critical times of these phase transitions and remains constant elsewhere, playing the role of a topological order parameter. This work provides a simple framework to study dynamical and topological quantum phase transitions in non-Hermitian systems.
Abstract Dynamical quantum phase transitions (DQPTs), which serve as a theoretical framework for understanding far-from-equilibrium physics in quantum many-body systems, have recently been observed experimentally. Their topological properties are typically characterized by the winding number, which acts as an order parameter. While DQPTs exhibiting both integer and half-integer jumps in the winding number have been reported, the underlying mechanisms behind these distinct topological behaviors, as well as the potential existence of other topological classes, remain open questions. To address this, we investigate DQPTs in the one-dimensional (1D) XY model under a quench protocol. We show that the observed topological diversity originates from the nature of the critical modes, which we classify into two categories: boundary modes and interior modes. Specifically, critical interior modes always lead to DQPTs with an integer winding number, while critical boundary modes always result in DQPTs characterized by a half-integer winding number. By analyzing the number and classification of critical modes, we provide a classification of the topological properties of DQPTs in the 1D XY model. According to their distinct topological features, we categorize DQPTs into six types, three of which have not been previously identified in the literature. We discuss in detail the conditions associated with each type and present the corresponding dynamical phase diagrams. Our framework is not restricted to the XY model; it is applicable to other two-band models in 1D systems, including the SSH model, Kitaev chain, Rice–Mele model, and Creutz model.
Dynamical quantum phase transitions (DQPTs) of periodic chains are determined by real Bloch momenta. We identify non-Bloch DQPTs, which occur only in open chains and are determined by complex momenta instead. They arise in a Hermitian chain coupled to a simple substrate, whose memory is treated exactly, and originate from the point-gap winding number WWW of the projected propagator: for W≠0W\neq0W=0, open boundaries change both the dynamical free energy and the critical times, even in the thermodynamic limit. We find that both follow from an analytic continuation to complex momenta, and show that WWW can be measured by particle counting. This reveals a dynamical bulk-boundary correspondence, in which non-Hermitian topology emerges from Hermitian dynamics with memory.
Dynamical quantum phase transitions (DQPTs), characterized by non-analytic behavior in rate function and abrupt changes in dynamic topological order parameters (DTOPs) over time, have garnered enormous attention in recent decades. However, in non-Hermitian systems, the special biorthogonality of the bases makes the definition of DQPTs complex. In this work, we delve into the comprehensive investigation of self-normal DQPTs (originally used in Hermitian systems) to compare them with their biorthogonal counterpart, within the context of non-Hermitian quantum walks (QWs). We present a detailed analysis of the behaviors of Loschmidt rate functions and DTOPs under these two distinct theoretical approaches. While both self-normal and biorthogonal methods can be used to detect DQPTs in quench dynamics between different topological phases, we theoretically present their differences in the definition of critical momenta and critical times by analyzing the Fisher zeros and fixed points. Finally, we present an experiment that observes both types of DQPTs using one-dimensional discrete-time QWs with single photons.