Nontrivial topological invariant of bulk electronic wavefunctions in two-dimensional quantum crystals leaves its footprints on the edge, dislocation, and corner modes. Here we investigate non-unitary time dynamics of these topological modes in square lattice-based open quantum systems in which the time-dependent Hamiltonian smoothly interpolates between topologically distinct insulators across band gap closing quantum critical points. The temporal dynamics of these modes is described by a Lindblad equation in which the instantaneous Hamiltonian plays the role of the Lindblad operator, thereby allowing the environment to couple with the system through the energy channels (weak measurement protocol). We show that in the presence of such a real time ramp, the survival probability of these modes decreases (increases) in short (long) time scale where the dephasing (quantum Zeno) effect dominates with the increasing amplitude of the system-to-environment coupling, for both slow and fast ramps from a topological to a normal insulating state. For a reverse course of the time evolution, the revival or condensation probability of nucleating such topological modes, otherwise absent in the initial system, increases for stronger system-to-environment coupling. This phenomenon can be attributed to the strong decoherence of the initial mixed state among all the energy eigenstates of the final Hamiltonian which also includes the topological modes, causing their enhanced condensation probability. Our findings can be germane to real open topological materials with time-tunable band gap, and should be applicable to open topological crystals of arbitrary dimension and belonging to any symmetry class.
Nontrivial topological invariant of bulk electronic wavefunctions in two-dimensional quantum crystals leaves its footprints on the edge, dislocation, and corner modes. Here we investigate non-unitary time dynamics of these topological modes in square lattice-based open quantum systems in which the time-dependent Hamiltonian smoothly interpolates between topologically distinct insulators across band gap closing quantum critical points. The temporal dynamics of these modes is described by a Lindblad equation in which the instantaneous Hamiltonian plays the role of the Lindblad operator, thereby allowing the environment to couple with the system through the energy channels (weak measurement protocol). We show that in the presence of such a real time ramp, the survival probability of these modes decreases (increases) in short (long) time scale where the dephasing (quantum Zeno) effect dominates with the increasing amplitude of the system-to-environment coupling, for both slow and fast ramps from a topological to a normal insulating state. For a reverse course of the time evolution, the revival or condensation probability of nucleating such topological modes, otherwise absent in the initial system, increases for stronger system-to-environment coupling. This phenomenon can be attributed to the strong decoherence of the initial mixed state among all the energy eigenstates of the final Hamiltonian which also includes the topological modes, causing their enhanced condensation probability. Our findings can be germane to real open topological materials with time-tunable band gap, and should be applicable to open topological crystals of arbitrary dimension and belonging to any symmetry class.
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.
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. However, it only works for a 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.
Topological phase transitions represent a new paradigm beyond conventional Landau-Ginzburg symmetry breaking. Even in the absence of symmetry breaking, the symmetries of a Hamiltonian play an important role in stabilizing topological signatures. In this thesis, we will study the interplay between symmetry and topology in many-body systems, with three distinct physical setups in mind. First, we investigate a two-dimensional system of ultracold bosons which is condensed in the p-band of a triangular lattice. Certain smooth interaction profiles result in non-Abelian symmetry generators of the order parameter manifold, leading to anomalous physical consequences. These include a lack of Berezinskii-Kosterlitz-Thouless transition due to unstable vortex configurations, additional Goldstone modes, and a marginally divergent normal fluid density which implies lack of superfluidity in the thermodynamic limit. Our results suggest possible connections to solid state experiments on 4He films. We then show that disorder which keeps a Hamiltonian in the same Altland-Zirnbauer symmetry class is capable of inducing a topological transition. This is achieved by considering the normalizability of Majorana edge modes in a topological superconductor, and corroborated with entanglement metrics. While degeneracies of the many-body spectrum are robust with respect to weak perturbations, strong disorder can both promote and destroy a topological index. Lastly, we study topological edge modes and degeneracies in atomic, molecular, and optical systems which evolve non-unitarily in time or space due to non-equilibrium effects, including: condensate instabilities after a quench, decoherence channels from an external environment, and dissipation into a medium. Mathematically, the non-unitary evolution is generated by effective Hamiltonians which break Hermiticity. These systems can possess uniquely non-Hermitian symmetries which are absent from equilibrium counterparts. We present a symmetry-based classification, and investigate several systems which can be solved exactly.
We investigate dissipation-driven topological phase transitions in one-dimensional quantum open systems governed by the Lindblad equation with linear dissipation operators, which ensure the density matrix retains its Gaussian form throughout the dynamics. By employing the modular Hamiltonian framework, we rigorously demonstrate that the Z2 topological invariant characterizing steady states in one-dimensional class D systems is exclusively dependent on the dissipation operators, rather than the system Hamiltonian. Through a sudden quench protocol where the system evolves from the steady state of one Lindbladian to another, we reveal that topological transitions can occur at analytically predictable critical times, even when the initial and final steady states share identical topological indices. These transitions are shown, both analytically and numerically, to depend solely on dissipation parameters. Entanglement spectrum analysis demonstrates bulk-edge correspondence in non-equilibrium density matrices via coexisting single-particle gap closures (periodic boundaries) and topologically protected zero modes (open boundaries), directly underpinning the detection of dissipation-induced topology in quantum simulators.
Nontrivial geometry of electronic Bloch states gives rise to topological insulators which are robust against sufficiently weak randomness inevitably present in any quantum material. However, increasing disorder triggers a quantum phase transition into a featureless normal insulator. As the underlying quantum critical point is approached from the topological side, small scattered droplets of normal insulators start to develop in the system and their coherent nucleation causes ultimate condensation into a trivial insulator. Unless disorder is too strong, the normal insulator accommodates disjoint tiny topological puddles. Furthermore, in the close vicinity of such a transition the emergent islands of topological and trivial insulators display spatial fractal structures, a feature that is revealed only by local topological markers. Here, we showcase this (possibly) generic phenomenon that should be apposite to dirty topological crystals of any symmetry class in any dimension from the Bott index and local Chern marker for a square-lattice-based disordered Chern insulator model.
Topological phases of matter are understood to be characterized by particular configurations of entanglement encoded within the ground states of many-body quantum systems. In this thesis, we discuss how novel phenomena associated with topological phases can arise in systems that are driven out of equilibrium or coupled to their surroundings, which are not described by ground state physics. Firstly, we consider systems undergoing unitary time evolution due to some external driving. We show how the entanglement features of the time-dependent wavefunction can be topologically characterized, and demonstrate that these topological properties are reflected in the dynamics of various quantities usually associated with topological phases in equilibrium. We introduce a new non-equilibrium topological classification scheme which can be used to predict whether or not the topological features of a given system will be preserved under time evolution. In brief, this classification captures the fact that certain symmetries (namely antiunitary symmetries) are inevitably broken once the system is driven out of equilibrium; thus any topological phenomena protected by such symmetries cannot be expected to persist in non-equilibrium scenarios. Secondly, we investigate how topologically non-trivial systems are affected by dissipative effects, i.e.~coupling to an external environment. We demonstrate that system-environment interactions facilitate processes in the system that effectively break any antiunitary symmetries, regardless of the symmetries of the microscopic Hamiltonian. Accordingly, those phases that were shown to be unstable against time-dependent driving are also fragile against coupling to an environment. To illustrate the consequences of this fragility, we consider the effects of dissipation on the coherence properties of topological bound states, as well as the conductance properties of chiral and helical topological edge modes. We find that the decoherence rate of the former and the deviation from quantized conductivity of the latter are (not) exponentially suppressed in the inverse temperature when the phase in question is protected by unitary (antiunitary) symmetries. These results regarding open systems can be connected to the same non-equilibrium classification developed in the context of isolated systems undergoing unitary dynamics. Our findings highlight a distinction between topological phases that are either robust or fragile against non-equilibrium effects and/or system-environment coupling. The ramifications for the use of such systems in quantum technologies are discussed.