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arXiv:1604.04698 · Physical Review Letters

Dynamical Quantum Phase Transitions: Role of Topological Nodes in Wave Function Overlaps

Zhoushen Huang, Alexander V. Balatsky

Published Aug 15, 2016• 104 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.117.086802

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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.
Dynamical quantum phase transitions (DQPTs) represent a counterpart in nonequilibrium quantum time evolution of thermal phase transitions at equilibrium, where real time becomes analogous to a control parameter such as temperature. In quenched quantum systems, recently the occurrence of DQPTs has been demonstrated, both with theory and experiment, to be intimately connected to changes of topological properties. Here, we contribute to broadening the systematic understanding of this relation between topology and DQPTs to multiorbital and disordered systems. Specifically, we provide a detailed ergodicity analysis to derive criteria for DQPTs in all spatial dimensions and construct basic counterexamples to the occurrence of DQPTs in multiband topological insulator models. As a numerical case study illustrating our results, we report on microscopic simulations of the quench dynamics in the Harper-Hofstadter model. Furthermore, going gradually from multiband to disordered systems, we approach random disorder by increasing the (super)unit cell within which random perturbations are switched on adiabatically. This leads to an intriguing order of limits problem which we address by extensive numerical calculations on quenched one-dimensional topological insulators and superconductors with disorder.
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.
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.
The notion of a dynamical quantum phase transition (DQPT) was recently introduced [Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)] as the nonanalytic behavior of the Loschmidt echo at critical times in the thermodynamic limit. In this work the quench dynamics in the ground state sector of the two-dimensional Kitaev honeycomb model is studied regarding the occurrence of DQPTs. For general two-dimensional systems of BCS type it is demonstrated how the zeros of the Loschmidt echo coalesce to areas in the thermodynamic limit, implying that DQPTs occur as discontinuities in the second derivative. In the Kitaev honeycomb model DQPTs appear after quenches across a phase boundary or within the massless phase. In the 1d limit of the Kitaev honeycomb model it becomes clear that the discontinuity in the higher derivative is intimately related to the higher dimensionality of the nondegenerate model. Moreover, there is a strong connection between the stationary value of the rate function of the Loschmidt echo after long times and the occurrence of DQPTs in this model.
We study the quench dynamics on cross-stitch flat band networks by a sudden change of the intercell hopping strength JJJ. For quench processes with JJJ changing as $J=0\ensuremath{\rightarrow}J\ensuremath{\ne}0$, we give the analytical expression to the Loschmidt echo which possesses a series of zero points at critical times t∗{t}^{*}t∗, indicating where the dynamical quantum phase transitions occur. We further study the converse quench process with $J\ensuremath{\ne}0\ensuremath{\rightarrow}J=0$, and find a nontrivial example that the prequench quantum state is not an eigenstate of the post-quench Hamiltonian, whereas the Loschmidt echo $\mathcal{L}(t)\ensuremath{\equiv}1$ during this process. For both situations, these results are also illustrated numerically. Finally, we give a brief discussion on the possible experimental observation of these predictions in the system of ultracold atoms in optical lattices.