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.
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.
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.
Preparing an integrable system in a mixed state described by a thermal density matrix, we subject it to a sudden quench and explore the subsequent unitary dynamics. To address the question of whether the nonanalyticities, namely, the dynamical quantum phase transitions (DQPTs), persist when the initial state is mixed, we consider two versions of the generalized Loschmidt overlap amplitude (GLOA). Our study shows that the GLOA constructed using the Uhlmann approach does not show any signature of DQPTs at any nonzero initial temperature. On the other hand, a GLOA defined in the interferometric phase approach through the purifications of the time-evolved density matrix, indeed shows that nonanalyiticies in the corresponding ``dynamical free-energy density'' persist, thereby establishing the existence of mixed state dynamical quantum phase transitions (MSDQPTs). Our work provides a framework that perfectly reproduces both the nonanalyticities and also the emergent topological structure in the pure state limit. These claims are corroborated by analyzing the nonequilibrium dynamics of a transverse Ising chain initially prepared in a thermal state and subjected to a sudden quench of the transverse field.
We study the dynamical process of equilibration of topological properties in quantum many-body systems undergoing a parameter quench between two topologically inequivalent Hamiltonians. This scenario is motivated by recent experiments on ultracold-atomic gases, where a trivial initial state is prepared before the Hamiltonian is ramped into a topological insulator phase. While the many-body wave function must stay topologically trivial in the coherent postquench dynamics, here we show how the topological properties of the single-particle density matrix dynamically change and equilibrate in the presence of interactions. In this process, the single-particle density matrix goes through a characteristic level crossing as a function of time, which plays an analogous role to the gap closing of a Hamiltonian in an equilibrium topological quantum phase transition. As an exact case study exemplifying this mechanism, we numerically solve the quench dynamics of an interacting one-dimensional topological insulator.
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.
Understanding the properties of far-from-equilibrium quantum systems is becoming a major challenge of both fundamental and applied physics. For instance, the lack of thermalization in integrable and (many body) localized systems provides new insights in the understanding of the relaxation dynamics of quantum phases. On a more applicative side, the possibility of exploiting the properties of far-from-equilibrium states, for example in pump-probe experiments, opens unprecedented scenarios. The effort in providing a classification of far-from-equilibrium phases, in terms of local or topological order parameters, is hence intense. In this context, the concept of Dynamical Quantum Phase Transition (DQPT) has been introduced. A DQPT is (roughly) defined as a zero of the Loschmidt-Echo as a function of time and represents a natural non-equilibrium counterpart of a thermal phase transition. Here, we investigate the DQPTs occurring in the quantum xy chain subject to a quantum quench of finite duration. We show that the number of distinct DQPTs can vary as the duration of the quantum quench is varied. However, the parity of such number only depends on the pre-quench and post-quench Hamiltonians and is related to a topological invariant.
Recent experiments began to explore the topological properties of quench dynamics, i.e. the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. But it only works for topologically trivial initial state and is hard to generalize to multiband systems or other dimensions. Here we introduce the concept of loop unitary constructed from the unitary time-evolution operator, and show its homotopy invariant fully characterizes the dynamical topology. For two-band systems in two dimensions, we prove that the invariant is precisely equal to the change in the Chern number across the quench regardless of the initial state. We further show that the nontrivial dynamical topology manifests as hedgehog defects in the loop unitary, and also as winding and linking of its eigenvectors along a curve where dynamical quantum phase transition occurs. This opens up a systematic route to classify and characterize quantum quench dynamics.