Generalized Thouless Pumps in 1+1-dimensional Interacting Fermionic Systems

Abstract

The Thouless pump is a phenomenon in which $\mathrm{U}(1)$ charges are pumped from an edge of a fermionic system to another edge. The Thouless pump has been generalized in various dimensions and for various charges. In this paper, we investigate the generalized Thouless pumps of fermion parity in both trivial and non-trivial phases of $1+1$-dimensional interacting fermionic short range entangled (SRE) states. For this purpose, we use matrix product states (MPSs). MPSs describe many-body systems in $1+1$ dimensions, and can characterize SRE states algebraically. We prove fundamental theorems for fermionic MPSs (fMPSs) and use them to investigate the generalized Thouless pumps. We construct non-trivial pumps in both the trivial and non-trivial phases and we show the stability of the pumps against interactions. Furthermore, we define topological invariants for the generalized Thouless pumps in terms of fMPSs and establish consistency with existing results. These are invariants of the family of SRE states that are not captured by the higher dimensional Berry curvature. We also argue a relation between the topological invariants of the generalized Thouless pump and the twist of the $K$-theory in the Donovan-Karoubi formulation.

arXiv comment59 pages, 4 figures

References in corpus (38)

• Topological Insulators
• Topological insulators and superconductors
• Unpaired Majorana fermions in quantum wires
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Josephson Current and Noise at a Superconductor-Quantum Spin Hall Insulator-Superconductor Junction
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Topological Phases of One-Dimensional Fermions: An Entanglement Point of View
• Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory
• String order and symmetries in quantum spin lattices
• Classification of Interacting Topological Floquet Phases in One Dimension
• Classification of topological phases in periodically driven interacting systems
• Symmetry protected topological phases from decorated domain walls
• Phase Structure of 1d Interacting Floquet Systems I: Abelian SPTs
• Reflection positivity and invertible topological phases
• Exact ground states and topological order in interacting Kitaev/Majorana chains
• Quantum phase transitions in matrix product systems
• A quantum version of Wielandt's inequality
• Symmetry Protected Topological phases and Generalized Cohomology
• Fermionic Matrix Product States and One-Dimensional Topological Phases
• Floquet topological phases with symmetry in all dimensions
• Towards a complete classification of fermionic symmetry protected topological phases in 3D and a general group supercohomology theory
• Geometry of Matrix Product States: metric, parallel transport and curvature
• Rigorous RG algorithms and area laws for low energy eigenstates in 1D
• Berry Phase in Quantum Field Theory: Diabolical Points and Boundary Phenomena
• Higher-dimensional generalizations of the Berry curvature
• Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes
• Minimalist approach to the classification of symmetry protected topological phases
• Exact zero modes in twisted Kitaev chains
• Spin Topological Field Theory and Fermionic Matrix Product States
• Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries
• Fermionic quantum cellular automata and generalized matrix product unitaries
• The classification of symmetry protected topological phases of one-dimensional fermion systems
• On adiabatic cycles of quantum spin systems
• On ${\mathbb{Z}}_2$-indices for ground states of fermionic chains
• Gappability Index for Quantum Many-Body Systems

Cited by in corpus (6)

• Higher Berry Curvature from the Wave function II: Locally Parameterized States Beyond One Dimension
• Higher Berry Curvature from the Wave Function I: Schmidt Decomposition and Matrix Product States
• Charting the space of ground states with tensor networks
• Charge pumps, boundary modes, and the necessity of unnecessary criticality
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• Modulating Hamiltonian Approach to Quantum Many-Body Systems and Crystalline Topological Phases Protected by Generalized Magnetic Translations