Topological Invariants for Quantum Quench Dynamics from Unitary Evolution

Abstract

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.

6+8 pages, 3+3 figures

References in corpus (10)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• Topological characterization of periodically-driven quantum systems
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Topological classification of dynamical phase transitions
• Topological invariants of Floquet systems: General formulation, special properties, and Floquet topological defects
• Topological surface states in three-dimensional magnetic insulators
• Topological index for periodically driven time-reversal invariant 2D systems
• Observation of the Hopf Links and Hopf Fibration in a 2D topological Raman Lattice
• Quench dynamics of Hopf insulators

Cited by in corpus (45)

• Knots and Non-Hermitian Bloch Bands
• Topological Euler class as a dynamical observable in optical lattices
• Floquet dynamical phase transition and entanglement spectrum
• Floquet dynamical quantum phase transition in the extended XY model: nonadiabatic to adiabatic topological transition
• Realization of Qi-Wu-Zhang model in spin-orbit-coupled ultracold fermions
• Unified theory to characterize Floquet topological phases by quench dynamics
• Simulation of Higher-Order Topological Phases and Related Topological Phase Transitions in a Superconducting Qubit
• Edge State, Band Topology, and Time Boundary Effect in the Fine-Grained Categorization of Chern Insulators
• Floquet Non-Abelian Topological Insulator and Multifold Bulk-Edge Correspondence
• Dynamical quantum phase transitions in a mesoscopic superconducting system
• Quantum dynamical characterization and simulation of topological phases with high-order band inversion surfaces
• Topological holographic quench dynamics in a synthetic dimension
• Signatures of topology in quantum quench dynamics and their interrelation
• Universal topological quench dynamics: Altland-Zirnbauer tenfold classes
• Dissipative preparation of many-body Floquet Chern insulators
• Mirror-symmetry-protected dynamical quantum phase transitions in topological crystalline insulators
• Emergent topology under slow non-adiabatic quantum dynamics
• Lattice gauge theory and dynamical quantum phase transitions using noisy intermediate scale quantum devices
• Nested-sphere description of the N-level Chern number and the generalized Bloch hypersphere
• Detecting Bulk Topology of Quadrupolar Phase from Quench Dynamics
• Experimental quantum simulation of non-Hermitian dynamical topological states using stochastic Schrödinger equation
• Dynamically characterizing topological phases by high-order topological charges
• Unconventional Floquet topological phases from quantum engineering of band inversion surfaces
• Wave packet dynamics and edge transport in anomalous Floquet topological phases
• Direct measurement of topological invariants through temporal adiabatic evolution of bulk states in the synthetic Brillouin zone
• Quench dynamics of Hopf insulators
• Disorder-induced topology in quench dynamics
• Unitary preparation of many body Chern insulators: Adiabatic bulk boundary correspondence
• Quantum quenches in a pseudo-Hermitian Chern insulator
• Stability of dynamical topology against dynamical noise in quantum quenches
• Generic theory of characterizing topological phases under quantum slow dynamics
• Universal Landau-Zener regimes in dynamical topological phase transitions
• Topological spin-orbit-coupled fermions beyond rotating wave approximation
• Dynamical detection of mean-field topological phases in an interacting Chern insulator
• Characterizing Floquet topological phases by quench dynamics: A multiple-subsystem approach
• Dynamical characterization of $Z_2$ Floquet topological phases via quantum quenches
• Dynamical characterization of topological phases beyond the minimal models
• Dynamical signatures of symmetry protected topology following symmetry breaking
• Quench-Probe Setup as Analyzer of Fractionalized Entanglement Spreading
• Loop unitary and phase band topological invariant in generic multi-band Chern insulators
• Tuning the topology of $p$-wave superconductivity in an analytically solvable two-band model
• From orthogonal link to phase vortex in generalized dynamical Hopf insulators
• Full investigation of nonadiabatic dynamical characterization in arbitrary quenching process
• Non-Markovianity Benefits Quantum Dynamics Simulation
• Phase transitions from Heating to non-heating in SU(1, 1) quantum dynamics: applications to Bose-Einstein condensates and periodically driven coupled oscillators