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arXiv:1610.02674 · Physical review. B./Physical review. B

Interconnections between equilibrium topology and dynamical quantum phase transitions in a linearly ramped Haldane model

Utso Bhattacharya, Amit Dutta

Published May 30, 2017• 52 citations
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Abstract

We study the behavior of Fisher zeros (FZs) and dynamical quantum phase transitions (DQPTs) for a linearly ramped Haldane model occurring in the subsequent temporal evolution of the same, and we probe the intimate connection with the equilibrium topology of the model. Here, we investigate the temporal evolution of the final state of the Haldane Hamiltonian (evolving with the time-independent final Hamiltonian) reached following a linear ramping of the staggered (Semenoff) mass term from an initial to a final value, first selecting a specific protocol, so chosen that the system is ramped from one nontopological phase to the other through a topological phase. We establish the existence of three possible behaviors of areas of FZs corresponding to a given sector: (i) no-DQPT, (ii) one-DQPT (intermediate), and (iii) two-DQPTs (reentrant), depending on the inverse quenching rate $\ensuremath{\tau}$. Our study also reveals that the appearance of the areas of FZs is an artefact of the nonzero (quasi-momentum-dependent) Haldane mass (${M}_{H}$), whose absence leads to an emergent one-dimensional behavior indicated by the shrinking of the area's FZs to lines and the nonanalyticity in the dynamical ``free energy'' itself. Moreover, the characteristic rates of crossover between the three behaviors of FZs are determined by the time-reversal-invariant quasimomentum points of the Brillouin zone where ${M}_{H}$ vanishes. Thus, we observe that through the presence or absence of ${M}_{H}$, there exists an intimate relation to the topological properties of the equilibrium model even when the ramp drives the system far away from equilibrium.

DOI
10.1103/physrevb.95.184307

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We study the behavior of Fisher zeros (FZs) and dynamical quantum phase transitions (DQPTs) for a linearly ramped Haldane model occurring in the subsequent temporal evolution of the same, and we probe the intimate connection with the equilibrium topology of the model. Here, we investigate the temporal evolution of the final state of the Haldane Hamiltonian (evolving with the time-independent final Hamiltonian) reached following a linear ramping of the staggered (Semenoff) mass term from an initial to a final value, first selecting a specific protocol, so chosen that the system is ramped from one nontopological phase to the other through a topological phase. We establish the existence of three possible behaviors of areas of FZs corresponding to a given sector: (i) no-DQPT, (ii) one-DQPT (intermediate), and (iii) two-DQPTs (reentrant), depending on the inverse quenching rate \ensuremathτ\ensuremath{\tau}\ensuremathτ. Our study also reveals that the appearance of the areas of FZs is an artefact of the nonzero (quasi-momentum-dependent) Haldane mass (MH{M}_{H}MH​), whose absence leads to an emergent one-dimensional behavior indicated by the shrinking of the area's FZs to lines and the nonanalyticity in the dynamical ``free energy'' itself. Moreover, the characteristic rates of crossover between the three behaviors of FZs are determined by the time-reversal-invariant quasimomentum points of the Brillouin zone where MH{M}_{H}MH​ vanishes. Thus, we observe that through the presence or absence of MH{M}_{H}MH​, there exists an intimate relation to the topological properties of the equilibrium model even when the ramp drives the system far away from equilibrium.
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.
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 demonstrate the existence of a dynamical quantum phase transition (DQPT) in a dissipative collective-spin model that exhibits the boundary time crystal (BTC) phase. We initialize the system in the ground state of the Hamiltonian in either the BTC or the non-BTC phase, and drive it across the BTC transition. The driving is done by an abrupt quench or by a finite-time linear ramp of a Hamiltonian control parameter under Markovian Lindblad dynamics. We diagnose DQPTs through zeros of the fidelity-based Loschmidt echo between the initial state and the evolving mixed state, which induce nonanalytic cusp-like features in the associated rate function. For quenches into the BTC phase, the Loschmidt echo exhibits repeated zeros due to the emergent time-periodic steady state, whereas for quenches into the non-BTC phase, the overlap vanishes and remains zero once the dynamics relaxes to a stationary state. We further show that the DQPT persists under the ramp protocol followed by unitary evolution with the final Hamiltonian. Finally, we analyze the finite-size scaling of the first critical time and find convergence to a constant in the thermodynamic limit, with distinct power-law approaches for the quench and the ramp protocols.
The short time evolution of a topological system after a sudden quench is considered, calculating the Loschmidt rate and studying in detail the distribution of Fisher zeros. Singular behavior in the Loschmidt rate has been predicted and found in several systems and, in particular, in one dimension (1D) one finds sharp features in the time evolution of the Loschmidt rate. These are associated with critical times that may be seen as the imaginary part of the Fisher zeros, under the condition that their real part vanishes. In the complex plane one finds lines of zeros that cross the imaginary axis at certain points. In a 2D system, in general, the Fisher zeros occupy areas instead of lines, as shown before, with the consequences that a continuous set of critical times results along the imaginary axis and that one does not find a well defined singular behavior in the Loschmidt rate. Here we show that several 2D topological systems also display very narrow regions (effectively lines) of zeros, in addition to finite extent areas of zeros. As a consequence, well defined sharp critical times are found, under appropriate conditions. An example is a 2Dp2\text{D}\phantom{\rule{4pt}{0ex}}p2Dp-wave superconductor in the presence of a magnetic field, but other systems share the same property, such as a Haldane insulator, a system with an anomalous Hall effect, and topological insulators. Details on the momentum distribution of the real and imaginary parts of the Fisher zeros are discussed, clarifying the existence of the lines of zeros and taking as an example the ppp-wave superconductor in magnetic field. The time dependence of the Loschmidt rate phase and of the Pancharatnam geometrical phase is considered, for different times after the quench and as a function of momentum. Pairs of vortices and antivortices in the phase associated with the geometric phase and the Loschmidt rate phase, that have been proposed for some 2D systems, are also shown to appear in the regime where we find lines of zeros.
Abstract Dynamical quantum phase transitions (DQPTs) can occur following quenches in quantum systems when the rate function, a dynamical analogue of the free energy, becomes non-analytic at critical times. Here we exhaustively investigate in an exemplary model how the dynamically evolving state responds to a second quench. We demonstrate that for quenches where the initial and final Hamiltonian belong to different phases always result in DQPTs, irrespective of the intermediate quench and dynamics or the time of the second quench. However, if the initial and final Hamiltonian belong to the same equilibrium phase then the intermediate Hamiltonian must belong to a different phase. In this case, the second quench time in relation to the critical times of the first quench becomes crucial to the existence of DQPTs.
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.