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arXiv:1812.03417 · Physical review. B./Physical review. B

Dynamical quantum phase transitions in Weyl semimetals

Aritra Lahiri, Soumya Bera

Published May 23, 2019• 19 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.99.174311

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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.
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.
Dynamical quantum phase transitions occur in dynamically evolving quantum systems when nonanalyticities occur at critical times in the return rate, a dynamical analog of the free energy. This extension of the concept of phase transitions can be brought into contact with another, namely, that of topological phase transitions in which the phase transition is marked by a change in a topological invariant. Following a quantum quench dynamical quantum phase transitions can happen in topological matter, a fact which has already been explored in one-dimensional topological insulators and in two-dimensional Chern insulators. Additionally, in one-dimensional systems a dynamical bulk boundary correspondence has been seen, related to the periodic appearance of zero modes of the Loschmidt echo itself. Here we extend both of these concepts to two-dimensional higher-order topological matter in which the topologically protected boundary modes are corner modes. We consider a minimal model which encompasses all possible forms of higher-order topology in two-dimensional topological band structures. We find that dynamical quantum phase transitions can still occur, and can occur for quenches which cross both bulk and boundary gap closings. Furthermore, a dynamical bulk boundary correspondence is also found, which takes a different form to that in one dimension.
The nonequilibrium dynamics of two-dimensional Su-Schrieffer-Heeger model, in the presence of staggered chemical potential, is investigated using the notion of dynamical quantum phase transition. We contribute to expanding the systematic understanding of the interrelation between the equilibrium quantum phase transition and the dynamical quantum phase transition (DQPT). Specifically, we find that dynamical quantum phase transition relies on the existence of massless propagating quasiparticles as signaled by their impact on the Loschmidt overlap. These massless excitations are a subset of all gapless modes, which leads to quantum phase transitions. The underlying two-dimensional model reveals gapless modes, which do not couple to the dynamical quantum phase transitions, while relevant massless quasiparticles present periodic nonanalytic signatures on the Loschmidt amplitude. The topological nature of DQPT is verified by the quantized integer values of the topological order parameter, which gets even values. Moreover, we have shown that the dynamical topological order parameter truly captures the topological phase transition on the zero Berry curvature line, where the Chern number is zero and the two-dimensional Zak phase is not the proper indicator.
We study the nonequilibrium time evolution of a variety of one-dimensional (1D) and two-dimensional (2D) systems (including SSH model, Kitaev-chain, Haldane model, p+ipp+ipp+ip superconductor, etc.) following a sudden quench. We prove analytically that topology-changing quenches are always followed by nonanalytical temporal behavior of return rates (logarithm of the Loschmidt echo), referred to as dynamical phase transitions (DPTs) in the literature. Similarly to edge states in topological insulators, DPTs can be classified as being topologically protected or not. In 1D systems the number of topologically protected nonequilibrium time scales are determined by the difference between the initial and final winding numbers, while in 2D systems no such relation exists for the Chern numbers. The singularities of dynamical free energy in the 2D case are qualitatively different from those of the 1D case; the cusps appear only in the first time derivative.
We study the slow quenching dynamics (characterized by an inverse rate ${\ensuremath{\tau}}^{\ensuremath{-}1}$) of a one-dimensional transverse Ising chain with nearest neighbor ferromagentic interactions across the quantum critical point (QCP) and analyze the Loschmidt overlap measured using the subsequent temporal evolution of the final wave function (reached at the end of the quenching) with the final time-independent Hamiltonian. Studying the Fisher zeros of the corresponding generalized ``partition function,'' we probe nonanalyticities manifested in the rate function of the return probability known as dynamical phase transitions (DPTs). In contrast to the sudden quenching case, we show that DPTs survive in the subsequent temporal evolution following the quenching across two critical points of the model for a sufficiently slow rate; furthermore, an interesting ``lobe'' structure of Fisher zeros emerge. We have also made a connection to topological aspects studying the dynamical topological order parameter $[{\ensuremath{\nu}}_{D}(t)]$ as a function of time (t)(t)(t) measured from the instant when the quenching is complete. Remarkably, the time evolution of ${\ensuremath{\nu}}_{D}(t)$ exhibits drastically different behavior following quenches across a single QCP and two QCPs. In the former case, ${\ensuremath{\nu}}_{D}(t)$ increases stepwise by unity at every DPT (i.e., $\mathrm{\ensuremath{\Delta}}{\ensuremath{\nu}}_{D}=1$). In the latter case, on the other hand, ${\ensuremath{\nu}}_{D}(t)$ essentially oscillates between 0 and 1 (i.e., successive DPTs occur with $\mathrm{\ensuremath{\Delta}}{\ensuremath{\nu}}_{D}=1$ and $\mathrm{\ensuremath{\Delta}}{\ensuremath{\nu}}_{D}=\ensuremath{-}1$, respectively), except for instants where it shows a sudden jump by a factor of unity when two successive DPTs carry a topological charge of the same sign.
We introduce and study the dynamical probes of band-structure topology in the postquench time evolution of quantum many-body systems initialized in mixed states. Our construction generalizes the notion of dynamical quantum phase transitions (DQPTs), a real-time counterpart of conventional equilibrium phase transitions in quantum dynamics, to finite temperatures and generalized Gibbs ensembles. The nonanalytical signatures hallmarking these mixed-state DQPTs are found to be characterized by observable phase singularities manifesting in the dynamical formation of vortex-antivortex pairs in the interferometric phase of the density matrix. Studying quenches in Chern insulators, we find that changes in the topological properties of the Hamiltonian can be identified in this scenario, without ever preparing a topologically nontrivial or low-temperature initial state. Our observations are of immediate relevance for current experiments aimed at realizing topological phases in ultracold atomic gases.