Charge pumps, pivot Hamiltonians and symmetry-protected topological phases

Abstract

Generalised charge pumps are topological obstructions to trivialising loops in the space of symmetric gapped Hamiltonians. We show that given mild conditions on such pumps, the associated loop has high-symmetry points which must be in distinct symmetry-protected topological (SPT) phases. To further elucidate the connection between pumps and SPTs, we focus on closed paths, `pivot loops', defined by two Hamiltonians, where the first is unitarily evolved by the second `pivot' Hamiltonian. While such pivot loops have been studied as entanglers for SPTs, here we explore their connection to pumps. We construct families of pivot loops which pump charge for various symmetry groups, often leading to SPT phases -- including dipole SPTs. Intriguingly, we find examples where non-trivial pumps do not lead to genuine SPTs but still entangle representation-SPTs (RSPTs). We use the anomaly associated to the non-trivial pump to explain the a priori `unnecessary' criticality between these RSPTs. We also find that particularly nice pivot families form circles in Hamiltonian space, which we show is equivalent to the Hamiltonians satisfying the Dolan-Grady relation -- known from the study of integrable models. This additional structure allows us to derive more powerful constraints on the phase diagram. Natural examples of such circular loops arise from pivoting with the Onsager-integrable chiral clock models, containing the aforementioned RSPT example. In fact, we show that these Onsager pivots underlie general group cohomology-based pumps in one spatial dimension. Finally, we recast the above in the language of equivariant families of Hamiltonians and relate the invariants of the pump to the candidate SPTs. We also highlight how certain SPTs arise in cases where the equivariant family is labelled by spaces that are not manifolds.

arXiv commentv2 close to published version

References in corpus (52)

• Persistent entanglement in arrays of interacting particles
• Generalized Global Symmetries
• Entanglement spectrum of a topological phase in one dimension
• Symmetry protected topological orders and the group cohomology of their symmetry group
• An Area Law for One Dimensional Quantum Systems
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Classifying quantum phases using Matrix Product States and PEPS
• Symmetry Protected Topological phases of Quantum Matter
• Braiding statistics approach to symmetry-protected topological phases
• The effects of interactions on the topological classification of free fermion systems
• Topological Phases of One-Dimensional Fermions: An Entanglement Point of View
• Theta, Time Reversal, and Temperature
• Detection of Symmetry Protected Topological Phases in 1D
• Classification of topological phases in periodically driven interacting systems
• Symmetry protected topological phases from decorated domain walls
• Anomalies in the Space of Coupling Constants and Their Dynamical Applications II
• One-Dimensional Symmetry Protected Topological Phases and their Transitions
• Anomalies in the Space of Coupling Constants and Their Dynamical Applications I
• Phase Structure of 1d Interacting Floquet Systems I: Abelian SPTs
• Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge
• Gapless topological phases and symmetry-enriched quantum criticality
• Statistical mechanics of the Cluster-Ising model
• Quantum phase transition between cluster and antiferromagnetic states
• Symmetry Protected Topological phases and Generalized Cohomology
• Floquet topological phases with symmetry in all dimensions
• Dynamically enriched topological orders in driven two-dimensional systems
• Berry Phase in Quantum Field Theory: Diabolical Points and Boundary Phenomena
• Twisted Injectivity in PEPS and the Classification of Quantum Phases
• Global inconsistency, 't Hooft anomaly, and level crossing in quantum mechanics
• Pivot Hamiltonians as generators of symmetry and entanglement
• Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism
• The phase transitions between $Z_n\times Z_n$ bosonic topological phases in 1+1 D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases
• Rokhsar-Kivelson Models of Bosonic Symmetry-Protected Topological States
• Matrix Product Representation of Locality Preserving Unitaries
• Onsager symmetries in $U(1)$-invariant clock models
• Interaction Induced Topological Charge Pump
• Building models of topological quantum criticality from pivot Hamiltonians
• The "not-A", RSPT and Potts phases in an $S_3$-invariant chain
• A classification of phases of bosonic quantum lattice systems in one dimension
• On adiabatic cycles of quantum spin systems
• Bulk-boundary correspondence and singularity-filling in long-range free-fermion chains
• A UV perspective on mixed anomalies at critical points between bosonic symmetry-protected phases
• Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps
• Generalized Thouless Pumps in 1+1-dimensional Interacting Fermionic Systems
• Modified Szego-Widom Asymptotics for Block Toeplitz Matrices with Zero Modes
• Charting the space of ground states with tensor networks
• Pivoting through the chiral-clock family
• A Long Exact Sequence in Symmetry Breaking: order parameter constraints, defect anomaly-matching, and higher Berry phases
• Counting operator analysis of the discrete spectrum of some model Hamiltonians
• A multiplet analysis of spectra in the presence of broken symmetries

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