Time Reversal Polarization and a Z_2 Adiabatic Spin Pump

Abstract

We introduce and analyze a class of one dimensional insulating Hamiltonians which, when adiabatically varied in an appropriate closed cycle, define a "$Z_2$ pump". For an isolated system a single closed cycle of the pump changes the expectation value of the spin at each end even when spin orbit interactions violate the conservation of spin. A second cycle, however returns the system to its original state. When coupled to leads, we show that the $Z_2$ pump functions as a spin pump in a sense which we define, and transmits a finite, though non quantized spin in each cycle. We show that the $Z_2$ pump is characterized by a $Z_2$ topological invariant that is analogous to the Chern invariant that characterizes a topological charge pump. The $Z_2$ pump is closely related to the quantum spin Hall effect, which is characterized by a related $Z_2$ invariant. This work presents an alternative formulation which clarifies both the physical and mathematical meaning of that invariant. A crucial role is played by time reversal symmetry, and we introduce the concept of the time reversal polarization, which characterizes time reversal invariant Hamiltonians and signals the presence or absence of Kramers degenerate end states. For non interacting electrons we derive a formula for the time reversal polarization which is analogous to the Berry's phase formulation of the charge polarization. For interacting electrons, we show that abelian bosonization provides a simple formulation of the time reversal polarization. We discuss implications for the quantum spin Hall effect, and argue in particular that the $Z_2$ classification of the quantum spin Hall effect is valid in the presence of electron electron interactions.

14 pages, 6 figures

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• $8\pi$-periodic dissipationless ac Josephson effect on a quantum spin-Hall edge via a Quantum magnetic impurity
• Spin-orbit coupling in the kagome lattice with flux and time-reversal symmetry
• Competing Adiabatic Thouless Pumps in Enlarged Parameter Spaces
• Topological band theory of a generalized eigenvalue problem with Hermitian matrices: Symmetry-protected exceptional rings with emergent symmetry
• Atoms to topological electronic materials: A bedtime story for beginners
• Correlation-Assisted Quantized Charge Pumping
• Revealing the spatial nature of sublattice symmetry
• A symmetry-protected exceptional ring in a photonic crystal with negative index media
• Notes on photonic topological insulators and scattering-protected edge states - a brief introduction
• Topological charge pumping in spin-dependent superlattices with glide symmetry
• Entropy Signatures of Topological Phase Transitions
• Green's Function Approach to Interacting Higher-order Topological Insulators
• A semiclassical treatment of spinor topological effects in driven, inhomogeneous insulators under external electromagnetic fields
• Edge states and topological properties of electrons on the bismuth on silicon surface with giant spin-orbit coupling
• Vanishing of the quantum spin Hall phase in a semi-Dirac Kane Mele model
• A diagnosis of explicit symmetry breaking in the tight-binding constructions for symmetry-protected topological systems
• Model Hamiltonian for Altermagnetic Topological Insulators
• Topological viewpoint of two-dimensional group III-V and IV-IV compounds in the presence of electric field and spin-orbit coupling by density functional theory and tight-binding model
• Quantum Spin Hall Effect and Su-Schrieffer-Heeger Model Implementation in Novel C3N-based Dumbbell Morphologies
• Cubic Dirac and quadruple Weyl points in screw-symmetric materials
• Holstein polaron in a pseudospin-$1$ quantum spin Hall system: first and second order topological phase transitions
• Surface Step Defect in 3D Topological Insulators: Electric Manipulation of Spin and Quantum Spin Hall Effect
• A NOT operation on Majorana qubits with mobilizable solitons in an extended Su-Schrieffer-Heeger model
• Quantization of non-Abelian Berry phase for time reversal invariant systems
• A Z${}_2$ index of Dirac operator with time reversal symmetry
• Topological indicators for systems with open boundaries: Application to the Kitaev wire
• Entanglement of a 3D generalization of the Kitaev model on the diamond lattice
• Topological Spin and Valley Pumping in Silicene
• ${\mathbb{Z}}_2$ topological invariants for mixed states of fermions in time-reversal invariant band structures
• Anomalous pumping in the non-Hermitian Rice-Mele model
• Alloying-induced topological transition in 2D transition-metal dichalcogenide semiconductors
• Tuning topology in thin films of topological insulators by strain gradients
• Hidden-symmetry-protected quantum pseudo-spin Hall effect in optical lattices
• Band alignment effect in the topological photonic alloy
• Pfaffian invariant identifies magnetic obstructed atomic insulators
• Fractional quantization of charge and spin in topological quantum pumps
• The topological invariants of rotationally symmetric crystals
• Control of Edge Currents at a Ferromagnet - Triplet Superconductor Interface by Multiple Helical Majorana Modes
• Simulating Higher-Order Topological Insulators in Density Wave Insulators
• Pseudo-time-reversal-symmetry-protected topological Bogoliubov excitations of Bose-Einstein condensates in optical lattices
• Bulk-boundary-transport correspondence of the second-order topological insulators
• BerryEasy: A GPU enabled python package for diagnosis of nth-order and spin-resolved topology in the presence of fields and effects
• Dimerization, Trimerization and Quantum pumping
• Electric-field-induced Z2 topological phase transition in strained single bilayer Bi(111)
• Fractionalized Metals and Superconductors in Three Dimensions
• Chiral Hall effect in the kink states in topological insulators with magnetic domain walls
• Entanglement Chern number for three-dimensional topological insulators: Characterization by Weyl points of entanglement Hamiltonians
• 2d Fu-Kane-Mele invariant as Wess-Zumino action of the sewing matrix
• Catalyst-free MBE growth of PbSnTe nanowires with tunable aspect ratio
• Green's Function Method for Line Defects and Gapless Modes in Topological Insulators : Beyond Semiclassical Approach
• Computing the ${\mathbb{Z}}_2$ Invariant in Two-Dimensional Strongly-Correlated Systems
• Hierarchical Topological States in Thermal Diffusive Networks
• Topological insulator on honeycomb lattices and ribbons without inversion symmetry
• Pressure-Driven Topological Phase Transition in the Yb Chalcogenides YbO and YbS
• Flux sensitivity of quantum spin Hall rings
• Interface currents and magnetization in singlet-triplet superconducting heterostructures: Role of chiral and helical domains
• A complementary screening for quantum spin Hall insulators in 2D exfoliable materials
• Substrate-limited helical edge states
• Realizing robust large-gap quantum spin Hall state in 2D HgTe monolayer on insulating substrate
• Anomalous bulk-edge correspondence of nonlinear Rice-Mele model
• Topological Critical Materials of Ternary Compounds
• The Stiefel--Whitney theory of topological insulators
• Topological optical Raman superlattices
• Topological phase transition driven by magnetic field in one-dimensional topological superconductor rings
• Synthesizing Weyl semimetals in weak topological insulator and topological crystalline insulator multilayers
• Interaction Correction to the Magneto-Electric Polarizability of $Z_2$ Topological Insulators
• Topological electronic properties of silicon
• Universal quantum noise in adiabatic pumping
• Topological property of a $t_{2g}^5$ system with a honeycomb lattice structure
• Symmetry, topology, and geometry: The many faces of the topological magnetoelectric effect
• Dual topological insulator with mirror symmetry protected helical edge states
• Topological photon pumping in quantum optical systems
• Electron transport in a ferromagnetic/normal/ferromagnetic tunnel junction based on the surface of a topological insulator
• Stability of Topological Insulators with Non-Abelian Edge Excitations
• Quantum spin Hall phase in multilayer graphene
• Interactions and the Theta Term in One-Dimensional Gapped Systems
• Geometric pumping and dephasing at topological phase transition
• Tailoring higher-order topological phases via orbital hybridization
• $Z_2$ Dirac points with topologically protected multihelicoid surface states
• Additive rule of real and reciprocal space topologies at disclinations
• Fractional transconductance via non-adiabatic topological Cooper pair pumping
• Resistively detected NMR as a probe of the topological nature of conducting edge/surface states
• Spin Weyl Topological Insulators
• Subgap states at ferromagnetic and spiral-ordered magnetic chains in two-dimensional superconductors. II. Topological classification
• Tailoring topological states in silicene using different halogen-passivated Si(111) substrates
• Quantum geometric bounds in spinful systems with trivial band topology
• 4$\pi$ and 8$\pi$ dual Josephson effects induced by symmetry defects
• Discriminant indicator with generalized rotational symmetry
• Quantum spin Hall insulators and quantum valley Hall insulators of BiX/SbX (X = H, F, Cl, and Br) monolayers with a record bulk band gap
• $K$-theory classification of Wannier localizability and detachable topological boundary states
• Non-Hermitian ${\mathbb{Z}}_4$ skin effect protected by glide symmetry
• The effects of dissipation on topological mechanical systems
• Optical properties, plasmons, and orbital Skyrme textures in twisted TMDs
• General corner charge formulas in various tetrahedral and cubic space groups
• Layer dependent topological phases and transitions in TaRhTe${}_4$: From monolayer and bilayer to bulk
• Tilted Dirac Fermions
• Homotopy invariant in time-reversal and twofold rotation symmetric systems
• Synthesizing the Quantum Spin Hall Phase for Ultracold Atoms in Bichromatic Chiral Optical Ladders
• Two-dimensional flat band on the (011) surface of UTe${}_2$: Implication for STM measurements with a superconducting tip
• Theory of fractional corner charges in cylindrical crystal shapes
• Critical phase boundary and finite-size fluctuations in Su-Schrieffer-Heeger model with random inter-cell couplings
• Topological Aspects of Antiferromagnets
• Transmission of charge and spin in a topological-insulator-based magnetic structure
• Topological charge and spin pumping in a semiconductor nanowire
• Variational Tensor Wavefunctions for the Interacting Quantum Spin Hall Phase
• Majorana modes in helical altermagnet without net magnetism and spin-orbit coupling
• Spontaneous loop-spin current with topological characters in the Hubbard model
• Many-Body Quadrupolar Sum Rule for Higher-Order Topological Insulator
• A spin pump characterized by entanglement Chern numbers
• Differential geometric invariants for time-reversal symmetric Bloch-bundles II: The low dimensional "Quaternionic" case
• Gauge Dynamics and Topological Insulators
• Stability of the X-Y-phase of the two-dimensional C4 point group insulator
• Band structure picture for topology in strongly correlated systems with the ghost Gutzwiller ansatz
• A Coherent Physics Picture of Topological Insulators at Single-Particle Level
• Generalized Josephson effect with arbitrary periodicity in quantum magnets
• Topological transition in monolayer blue phosphorene with transiton-metal adatom under stain
• A non-perturbative expression for the transmission through a leaky chiral edge mode
• Hybrid Higher-Order Skin Topological Modes in the Two-Dimensional Su-Schrieffer-Heeger Model with Nonreciprocal Hoppings
• Topological Insulators and the Kane-Mele Invariant: Obstruction and Localisation Theory
• Real and imaginary edge states in stacked Floquet honeycomb lattices
• The magnetoelectric effect due to a semispherical capacitor surrounded by a spherical topologically insulating shell
• Nonlinear orbital response across topological phase transition in centrosymmetric materials
• Laughlin charge pumping from interplay of chiral Dirac and chiral Majorana modes
• Local expression of fractional corner charges in obstructed atomic insulators and relationship with the fractional disclination charges
• Topological constraints on the electronic band structure of hexagonal lattice in a magnetic field
• Regularized universal topological markers for Dirac systems
• Fragile dislocation modes in obstructed atomic topological phases
• Current Modulator based on Topological Insulator with Sliding Magnetic Superlattice
• Hidden wallpaper fermion and third-order topological insulator
• Network model for magnetic higher-order topological phases
• Entanglement entropy as a probe of topological phase transitions
• Hidden chiral symmetry protected $\mathbb{Z}\oplus \mathbb{Z}$ topological insulators in a ladder dimer model
• Topological properties of domain walls in antiferromagnetic topological insulators
• Index theory and noncommutative geometry of topological insulator
• The antiferromagnetic Chern insulator phase in the Kane-Mele-Hubbard model
• Surface states of gapped electron systems and semi-metals
• Quantized single-particle Thouless pump induced by topology transfer from a Chern insulator at finite temperature
• Systematic study for two-dimensional $Z_2$ topological phase transitions at high-symmetry points in all layer groups
• The enabling electronic motif for topological insulation in ABO3 perovskites and its structural stability
• Topological Classification of Gapped/Gap-preserving Rational Space-Time Crystal Systems
• Momentum non-conservation in a scalar quantum field theory with a planar $\theta$ interface
• Fractional spin Josephson effect in topological spin superconductors
• The Construction of Dirac Operators on Orientifolds
• Immutable quantized transport in Floquet chains
• Combined spontaneous symmetry-breaking and symmetry-protected topological order from cluster charge interaction
• Coarse geometric approach to topological phases: Invariants from real-space representations
• Band Topology or Geometry?
• Glide-resolved photoemission spectroscopy: Measuring topological invariants in nonsymmorphic space groups
• Spin transport in a normal meta-altermagnetic superconducting nanowire junction
• Simple Models for All Topological Phases
• Topological energy pumping in a quasi-periodically driven four-level system