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arXiv:2002.10296 · Physical review. B./Physical review. B

Geometrical quench and dynamical quantum phase transition in the α−T3 lattice

Balázs Gulácsi, Markus Heyl, Balázs Dóra

Published May 20, 2020• 11 citations
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Abstract

We investigate quantum quenches and the Loschmidt echo in the two-dimensional, three-band $\ensuremath{\alpha}\ensuremath{-}{T}_{3}$ model, a close descendant of the dice lattice. By adding a chemical potential to the central site, the integral of the Berry curvature of the bands in different valleys is continously tunable by the ratio of the hopping integrals between the sublattices. By investigating one and two filled bands, we find that dynamical quantum phase transition (DQPT), i.e., nonanalytical temporal behavior in the rate function of the return amplitude, occurs for a certain range of parameters, independent of the band filling. By focusing on the effective low-energy description of the model, we find that DQPTs happen not only in the time derivative of the rate function, which is a common feature in two-dimensional models, but also in the rate function itself. This feature is not related to the change of topological properties of the system during the quench but rather follows from population inversion for all momenta. This is accompanied by the appearance of dynamical vortices in the time-momentum space of the Pancharatnam geometric phase. The positions of the vortices form an infinite vortex ladder, i.e., a macroscopic phase structure, which allows us to identify the dynamical phases that are separated by the DQPT.

DOI
10.1103/physrevb.101.205135

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We investigate quantum quenches and the Loschmidt echo in the two-dimensional, three-band \ensuremathα\ensuremath−T3\ensuremath{\alpha}\ensuremath{-}{T}_{3}\ensuremathα\ensuremath−T3​ model, a close descendant of the dice lattice. By adding a chemical potential to the central site, the integral of the Berry curvature of the bands in different valleys is continously tunable by the ratio of the hopping integrals between the sublattices. By investigating one and two filled bands, we find that dynamical quantum phase transition (DQPT), i.e., nonanalytical temporal behavior in the rate function of the return amplitude, occurs for a certain range of parameters, independent of the band filling. By focusing on the effective low-energy description of the model, we find that DQPTs happen not only in the time derivative of the rate function, which is a common feature in two-dimensional models, but also in the rate function itself. This feature is not related to the change of topological properties of the system during the quench but rather follows from population inversion for all momenta. This is accompanied by the appearance of dynamical vortices in the time-momentum space of the Pancharatnam geometric phase. The positions of the vortices form an infinite vortex ladder, i.e., a macroscopic phase structure, which allows us to identify the dynamical phases that are separated by the DQPT.
We show that dynamical quantum phase transitions (DQPTs) in the quench dynamics of two-dimensional topological systems can be characterized by a dynamical topological invariant defined along an appropriately chosen closed contour in momentum space. Such a dynamical topological invariant reflects the vorticity of dynamical vortices responsible for the DQPTs, and thus serves as a dynamical topological order parameter in two dimensions. We demonstrate that when the contour crosses topologically protected fixed points in the quench dynamics, an intimate connection can be established between the dynamical topological order parameter in two dimensions and those in one dimension. We further define a reduced rate function of the Loschmidt echo on the contour, which features nonanalyticities at critical times and is sufficient to characterize DQPTs in two dimensions. We illustrate our results using the Haldane honeycomb model and the quantum anomalous Hall model as concrete examples, both of which have been experimentally realized using cold atoms.
Dynamical quantum phase transitions (DQPTs) manifested in the nonanalyticities in the temporal evolution of a closed quantum system generated by the time-independent final Hamiltonian, following a quench (or ramping) of a parameter of the Hamiltonian, is an emerging frontier of nonequilibrium quantum dynamics. We, here, introduce the notion of a dynamical topological order parameter (DTOP) that characterizes these DQPTs occurring in quenched (or ramped) two-dimensional closed quantum systems; this is quite a nontrivial generalization of the notion of DTOP introduced in Budich and Heyl [Phys. Rev. B 93, 085416 (2016)] for one-dimensional situations. This DTOP is obtained from the ``gauge-invariant'' Pancharatnam phase extracted from the Loschmidt overlap, i.e., the modulus of the overlap between the initially prepared state and its time-evolved counterpart reached following a temporal evolution generated by the time-independent final Hamiltonian. This generic proposal is illustrated considering DQPTs occurring in the subsequent temporal evolution following a sudden quench of the staggered mass of the topological Haldane model on a hexagonal lattice where it stays fixed to zero or unity and makes a discontinuous jump between these two values at critical times at which DQPTs occur. What is remarkable is that while the topology of the equilibrium model is characterized by the Chern number, the emergent topology associated with the DQPTs is characterized by a generalized winding number.
Over the past decade, dynamical quantum phase transitions (DQPTs) have emerged as a paradigm shift in understanding nonequilibrium quantum many-body systems. However, the challenge lies in identifying order parameters that effectively characterize the associated dynamic phases. In this study we investigate the behavior of vortex singularities in the phase of the Green's function for a broad class of fermion lattice models in three dimensions after an instantaneous quench in both interacting and noninteracting systems. We find that the full set of vortices form one-dimensional dynamical objects, which we call vortex loops. We propose that the number of such vortex loops can be interpreted as a quantized order parameter that distinguishes between different nonequilibrium phases. Our results establish an explicit link between variations in the order parameter and DQPTs in the noninteracting scenario. Moreover, we show that the vortex loops are robust in the weakly interacting case, even though there is no direct relation between the Loschmidt amplitude and the Green's function. Finally, we observe that vortex loops can form complex dynamical patterns in momentum space. Our findings provide valuable insights for developing definitions of dynamical order parameters in nonequilibrium systems.
We establish a unified framework for dynamical quantum phase transitions (DQPTs) in non-Hermitian systems that encompasses both biorthogonal and self-norm nonbiorthogonal formulations for pure and mixed states under quantum quench protocols. Our framework provides explicit expressions for the Loschmidt amplitude, Loschmidt echo, and rate function, revealing a universal geometric signature of DQPTs in the two-band model: orthogonality of two related vectors in two-dimensional real space. Strikingly, we demonstrate that nonbiorthogonal quenches from non-Hermitian to Hermitian Hamiltonians under chiral symmetry exhibit emergent topological characteristics of DQPTs, unveiling their fundamental features beyond conventional Hermitian regimes. This work establishes fundamental geometric and topological principles governing quantum criticality in open systems, with implications for quantum sensing and many-body physics in dissipative environments.
A sudden quantum quench of a Bloch band from one topological phase toward another has been shown to exhibit an intimate connection with the notion of a dynamical quantum phase transition (DQPT), where the returning probability of the quenched state to the initial state-i.e., the Loschmidt echo-vanishes at critical times {t^{*}}. Analytical results to date are limited to two-band models, leaving the exact relation between topology and DQPT unclear. In this Letter, we show that, for a general multiband system, a robust DQPT relies on the existence of nodes (i.e., zeros) in the wave function overlap between the initial band and the postquench energy eigenstates. These nodes are topologically protected if the two participating wave functions have distinctive topological indices. We demonstrate these ideas in detail for both one and two spatial dimensions using a three-band generalized Hofstadter model. We also discuss possible experimental observations.
The quench dynamics in type-I inversion-symmetric Weyl semimetals (WSM) is explored in this work which, due to the form of the Hamiltonian, may be readily extended to two-dimensional Chern insulators. We analyze the role of equilibrium topological properties characterized by the Chern number of the prequench ground state in dictating the nonequilibrium dynamics of the system, specifically, the emergence of dynamical quantum phase transitions (DQPT). By investigating the ground-state fidelity, it is found that a change in the signed Chern number constitutes a sufficient but not necessary condition for the occurrence of DQPTs. Depending on the ratio of the transverse and longitudinal hopping parameters, DQPTs may also be observed for quenches lying entirely within the initial Chern phase. Additionally, we analyze the zeros of the boundary partition function discovering that while the zeros generally form two-dimensional structures resulting in one-dimensional critical times, infinitesimal quenches may lead to one-dimensional zeros with zero-dimensional critical times, provided the quench distance scales appropriately with the system size. This is strikingly manifested in the nature of nonanalyticities of the dynamical free energy, revealing a logarithmic singularity. Moreover, we show that these zero-dimensional critical times may be detected only by a direct observation of the dynamical free energy. Finally, following recent advances in observing the dynamical zeros of the Loschmidt overlap through azimuthal Bloch phase vortices, we rigorously investigate the same in WSMs, demonstrating that only one-dimensional critical times arising from two-dimensional manifolds of zeros of the boundary partition function lead to dynamical vortices.