Rational indices for quantum ground state sectors

We study the stability with respect to a broad class of perturbations of gapped ground-state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi–Hastings–Michalakis (BHM) strategy that under a condition of local topological quantum order (LTQO), the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work, we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of an indistinguishability radius, which we introduce. Using the uniform finite-volume results, we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state.

Section: Introduction 1stability Of the Ground-state Gapmentioning

confidence: 99%

Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States

Nachtergaele

Sims

Young

2021

“…As already pointed out, in order to obtain volume independent bounds, we use a local parallel transport instead of the traditional one of Kato. We further bypass the geometric argument, in particular the Chern-Simons formula and the need for averaging, by using the many-body index of [10].…”

Section: Charge Transport and The Adiabatic Theoremmentioning

confidence: 99%

“…for all L and . See the appendix of [10] for an extended discussion. In the remaining text we omit the index L, but keep track of .…”

Section: Equality Of Charge Transportsmentioning

confidence: 99%

“…Quantization of the Hall conductance in the bulk is a well understood phenomenon under a spectral gap assumption, both in the absence and presence of interactions between the electrons, see [1,2,3,4,5] and [6,7,8,9,10].…”

Section: Introductionmentioning

confidence: 99%

“…without any error in either or L −1 . Hence, for any fixed , all assumptions of the index theorem in[10] hold, yielding an integer-valued index associated with P and W n, (s).The map s → W n, (s) being differentiable, it is a fortiori continuous. It follows that s → Ind P (W n, (s)) is constant up to O (L −∞ ), see [9, Proposition 2.2].…”

mentioning

confidence: 99%

See 2 more Smart Citations

Exactness of Linear Response in the Quantum Hall Effect

Bachmann

Roeck

Fraas

et al. 2021

Self Cite

In general, linear response theory expresses the relation between a driving and a physical system's response only to first order in perturbation theory. In the context of charge transport, this is the linear relation between current and electromotive force expressed in Ohm's law. We show here that, in the case of the quantum Hall effect, all higher order corrections vanish. We prove this in a fully interacting setting and without flux averaging.

A Bulk Spectral Gap in the Presence of Edge States for a Truncated Pseudopotential

Warzel

Young

2022

We study the low-energy properties of a truncated Haldane pseudopotential with maximal half filling, which describes a strongly correlated system of spinless bosons in a cylinder geometry. For this Hamiltonian with either open or periodic boundary conditions, we prove a spectral gap above the highly degenerate ground-state space which is uniform in the volume and particle number. Our proofs rely on identifying invariant subspaces to which we apply gap-estimate methods previously developed only for quantum spin Hamiltonians. In the case of open boundary conditions, the lower bound on the spectral gap accurately reflects the presence of edge states, which do not persist into the bulk. Customizing the gap technique to the invariant subspace, we avoid the edge states and establish a more precise estimate on the bulk gap in the case of periodic boundary conditions.