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arXiv:2102.10986 · Physical review. B./Physical review. B

Flat-band full localization and symmetry-protected topological phase on bilayer lattice systems

Ikuo Ichinose, Takahiro Orito, Yoshihito Kuno

Published May 27, 2021• 4 citations
arXivHTMLPDF

Abstract

In this work, we present bilayer flat-band Hamiltonians, in which all bulk states are localized and specified by extensive local integrals of motion (LIOMs). The present systems are bilayer extension of the Creutz ladder, which is studied previously. In order to construct models, we employ building blocks, cube operators, which are linear combinations of fermions defined in each cube of the bilayer lattice. There are eight cubic operators, and the Hamiltonians are composed of the number operators of them, the LIOMs. A suitable arrangement of locations of the cube operators is needed to have exact projective Hamiltonians. The projective Hamiltonians belong to a topological classification class, the BDI class. With the open boundary condition, the constructed Hamiltonians have gapless edge modes, which commute with each other as well as the Hamiltonian. This result comes from a symmetry analogous to the one-dimensional chiral symmetry of the BDI class. These results indicate that the projective Hamiltonians describe a kind of symmetry-protected topological phase matter. Careful investigation of topological indexes, such as the Berry phase and string operator, is given. We also show that, by using the gapless edge modes, a generalized Sachdev-Ye-Kitaev model is constructed.

DOI
10.1103/physrevb.103.184113

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In this work, we present bilayer flat-band Hamiltonians, in which all bulk states are localized and specified by extensive local integrals of motion (LIOMs). The present systems are bilayer extension of the Creutz ladder, which is studied previously. In order to construct models, we employ building blocks, cube operators, which are linear combinations of fermions defined in each cube of the bilayer lattice. There are eight cubic operators, and the Hamiltonians are composed of the number operators of them, the LIOMs. A suitable arrangement of locations of the cube operators is needed to have exact projective Hamiltonians. The projective Hamiltonians belong to a topological classification class, the BDI class. With the open boundary condition, the constructed Hamiltonians have gapless edge modes, which commute with each other as well as the Hamiltonian. This result comes from a symmetry analogous to the one-dimensional chiral symmetry of the BDI class. These results indicate that the projective Hamiltonians describe a kind of symmetry-protected topological phase matter. Careful investigation of topological indexes, such as the Berry phase and string operator, is given. We also show that, by using the gapless edge modes, a generalized Sachdev-Ye-Kitaev model is constructed.
In this work, we construct an exact projector Hamiltonian with interactions, which is given by a sum of mutually commuting operators called stabilizers. The model is based on the recently studied Creutz ladder of fermions, in which flat-band structure and strong localization are realized. These stabilizers are local integrals of motion from which many-body localization (MBL) is realized. All energy eigenstates are explicitly obtained even in the presence of local disorders. All states are MBL states, that is, this system is a full many-body localized (FMBL) system. We show that this system has a topological order and stable gapless edge modes exist under the open boundary condition. By the numerical study, we investigate stability of the FMBL and topological order.
Symmetry-protected topological (SPT) phases in insulators and superconductors are known for their robust edge modes, linked to bulk invariants through the bulk-boundary correspondence. While this principle traditionally applies to gapped phases, recent advances have extended it to gapless systems, where topological edge-states persist even in the absence of a bulk gap. We extend this framework to periodically driven chains with chiral symmetry, revealing the existence of topological edge zero and $\ensuremath{\pi}$ modes despite the lack of bulk gaps in the quasienergy spectrum. By examining the half-period decomposition of chiral evolutions, we construct topological invariants that circumvent the need to define the Floquet Hamiltonian, making them more suitable for generalization to the gapless regime. We provide explicit examples, including generalizations of the Kitaev chain and related spin models, where localized $\ensuremath{\pi}$ modes emerge even when the bulk is gapless at the same quasienergy as the edge modes. We numerically study the effect of interactions, which give a finite lifetime to the edge modes in the thermodynamic limit with the decay rate consistent with Fermi's golden rule.
Using the decorated domain wall procedure, we construct finite-depth local unitaries that realize fermionic symmetry-protected topological (SPT) phases. This results in explicit full commuting projector Hamiltonians, where ``full'' implies the fact that the ground state as well as all excited states of these Hamiltonians realize the nontrivial SPT phase. We begin by constructing explicit examples of 1+1D phases protected by the symmetry group $G={\mathbb{Z}}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{F}$, which also has a free fermion realization in class BDI, and by the symmetry group $G={\mathbb{Z}}_{4}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}^{F}$, which does not. We then turn to 2+1D, and construct the square roots of the Levin-Gu bosonic SPT phase, protected by ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{F}$ symmetry, in a concrete model of fermions and spins on the triangular lattice. Edge states and the anomalous symmetry action on them are explicitly derived. Although this phase has a free fermion representation as two copies of p+ipp+ipp+ip superconductors combined with their $p\ensuremath{-}ip$ counterparts with a different symmetry charge, the full set of commuting projectors is only realized in the strongly interacting version, which also implies that it admits a many-body localized realization.
Intrinsically gapless symmetry-protected topological phases (igSPT) are gapless systems with SPT edge states with properties that could not arise in a gapped system with the same symmetry and dimensionality. igSPT states arise from gapless systems in which an anomaly in the low-energy (infrared) symmetry group emerges from an extended anomaly-free microscopic (ultraviolet) symmetry. We construct a general framework for constructing lattice models for igSPT phases with emergent anomalies classified by group cohomology, and establish a direct connection between the emergent anomaly, group-extension, and topological edge states by gauging the extending symmetry. In many examples, the edge-state protection has a physically transparent mechanism: the extending UV symmetry operations pump lower-dimensional SPTs onto the igSPT edge, tuning the edge to a (multi)critical point between different SPTs protected by the IR symmetry. In two- and three-dimensional systems, an additional possibility is that the emergent anomaly can be satisfied by an anomalous symmetry-enriched topological order, which we call a quotient-symmetry-enriched topological order (QSET) that is sharply distinguished from the nonanomalous UV SETs by an edge phase transition. We construct exactly solvable lattice models with QSET order.
We show how 1+11+11+1-dimensional fermionic symmetry-protected topological states (SPTs, i.e., nontrivial short-range entangled gapped phases of quantum matter whose boundary exhibits 't Hooft anomaly and whose bulk cannot be deformed into a trivial tensor product state under finite-depth local unitary transformations only in the presence of global symmetries), indeed can be unwound to a trivial state by enlarging the Hilbert space via adding extra degrees of freedom and suitably extending the global symmetries. The extended projective global symmetry on the boundary can become supersymmetric in a specific sense, i.e., it contains group elements that do not commute with the fermion number parity ${(\ensuremath{-}1)}^{F}$, while the antiunitary time-reversal symmetry becomes fractionalized. This also means we can uplift and remove certain exotic fermionic anomalies (e.g., ``parity'' anomaly in time-reversal or reflection symmetry) via appropriate supersymmetry extensions in terms of group extensions. We work out explicit examples for multilayers of 1+1d1+1\mathrm{d}1+1d Majorana fermion chains, then comment on models with Sachdev-Ye-Kitaev (SYK) interactions, intrinsic fermionic gapless SPTs protected by supersymmetry, and generalizations to higher space-time dimensions via a cobordism theory.
Symmetry-protected topological (SPT) phases are gapped phases of matter that cannot be deformed to a trivial phase without breaking the symmetry or closing the bulk gap.Here we introduce a notion of a topological obstruction that is not captured by bulk energy gap closings in periodic boundary conditions.More specifically, given a symmetric boundary termination we say two bulk Hamiltonians belong to distinct boundary obstructed topological phases (BOTPs) if they can be deformed to each other on a system with periodic boundaries, but cannot be deformed to each other in the open system without closing the gap at at least one high-symmetry surface.BOTPs are not topological phases of matter in the standard sense since they are adiabatically deformable to each other on a torus, but, similar to SPTs, they are associated with boundary signatures in open systems such as surface states or fractional corner charges.In contrast to SPTs, these boundary signatures are not anomalous and can be removed by symmetrically adding lower-dimensional SPTs on the boundary, but they are stable as long as the spectral gap at high-symmetry edges/surfaces remains open.We show that the double-mirror quadrupole model of [W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Science 357, 61 (2017)] is a prototypical example of such phases, and present a detailed analysis of several aspects of boundary obstructions in this model.In addition, we introduce several three-dimensional models having boundary obstructions, which are characterized either by surface states or fractional corner charges.Furthermore, we provide a complete characterization of boundary obstructed phases in terms of symmetry representations.Namely, two distinct BOTP phases correspond to equivalent band representations in the periodic system which become inequivalent upon restricting the symmetry group to that of the open system.This is used to shown that for a given open boundary, there is only one class of BOTPs which corresponds to a local representation of the symmetry of the open system and thus can be designated as the trivial phase.All other BOTP classes do not correspond to local representation of the open system and as a result necessarily exhibit a filling anomaly or gapless surface states.