Lie–Schwinger Block-Diagonalization and Gapped Quantum Chains with Unbounded Interactions

Vecchio, Simone Del; Fröhlich, Jürg; Pizzo, Alessandro; Rossi, Stefano

doi:10.1007/s00220-020-03878-y

Cited by 8 publications

(11 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(4. 37) In what follows, we will frequently make reference to the following decay class: Definition 4.5. Let η, ξ, and θ be positive numbers.…”

Section: Lemma 43 Letmentioning

confidence: 99%

“…A generalization we do not pursue in the this paper is the inclusion of models with unbounded on-site Hamiltonians of the type considered in [41,105] and unbounded interactions as in [37]. Fröhlich and Pizzo introduced a method that handles a class of unbounded one-dimensional lattice Hamiltonians with ease as long as the unperturbed ground state is unique and given by a product state.…”

Section: Introduction 1stability Of the Ground-state Gapmentioning

confidence: 99%

See 1 more Smart Citation

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

Nachtergaele

Sims

Young

2021

Ann. Henri Poincaré

View full text Add to dashboard Cite

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.

show abstract

“…(4. 37) In what follows, we will frequently make reference to the following decay class: Definition 4.5. Let η, ξ, and θ be positive numbers.…”

Section: Lemma 43 Letmentioning

confidence: 99%

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

Ann. Henri Poincaré

View full text Add to dashboard Cite

show abstract

“…To prove our main result, Theorem 3.3, we employ the scheme of the local Lie-Schwinger block-diagonalization algorithm, developed in [FP], [DFPR1], [DFPR2], [DFPR3], which is here adapted to suit the present situation. We here briefly recap the strategy of the algorithm.…”

Section: Transformed Hamiltoniansmentioning

confidence: 99%

“…The construction of S = S (t) is inspired by a novel technique introduced in [FP] for quantum chains and in [DFPR3] for systems in arbitrary spatial dimensions larger than 1. In [DFPR1] the scheme of [FP] was extended to one dimensional bosons systems with relatively bounded interactions analogous to those discussed in the present paper.…”

Section: Introductionmentioning

confidence: 99%

“…The extension to unbounded operators for quantum chains (e.g. the massive φ 4 model on a one-dimensional lattice) has been discussed in [DFPR1], where we specify the general structure of the degenerate ground-state subspace that allows us to implement our block-diagonalization scheme. Under the same assumption on the ground-state subspace, our method works in any dimensions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Block-diagonalization of infinite-volume lattice Hamiltonians with unbounded interactions

Vecchio¹,

Fröhlich²,

Pizzo³

2021

Preprint

View full text Add to dashboard Cite

In this paper we extend the local iterative Lie-Schwinger block-diagonalization methodintroduced in [DFPR3] for quantum lattice systems with bounded interactions in arbitrary dimension-to systems with unbounded interactions, i.e., systems of bosons. We study Hamiltonians that can be written as the sum of a gapped operator consisting of a sum of on-site terms and a perturbation given by relatively bounded (but unbounded) interaction potentials of short range multiplied by a real coupling constant t. For sufficiently small values of |t| independent of the size of the lattice, we prove that the spectral gap above the ground-state energy of such Hamiltonians remains strictly positive. As in [DFPR3], we iteratively construct a sequence of local block-diagonalization steps based on unitary conjugations of the original Hamiltonian and inspired by the Lie-Schwinger procedure. To control the ranges and supports of the effective potentials generated in the course of our block-diagonalization steps, we use methods introduced in [DFPR3] for Hamiltonians with bounded interactions potentials. However, due to the unboundedness of the interaction potentails, weighted operator norms must be introduced, and some of the steps of the inductive proof by which we control the weighted norms of the effective potentials require special care to cope with matrix elements of unbounded operators. We stress that no "large-field problems" appear in our construction. In this respect our operator methods turn out to be an efficient tool to separate the low-energy spectral region of the Hamiltonian from other spectral regions, where the unbounded nature of the interaction potentials would become manifest.

show abstract

Stability of the Spectral Gap and Ground State Indistinguishability for a Decorated AKLT Model

Lucia,

Moon,

Young

2023

Ann. Henri Poincaré

View full text Add to dashboard Cite

show abstract

Lie–Schwinger Block-Diagonalization and Gapped Quantum Chains with Unbounded Interactions

Cited by 8 publications

References 17 publications

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

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

Block-diagonalization of infinite-volume lattice Hamiltonians with unbounded interactions

Stability of the Spectral Gap and Ground State Indistinguishability for a Decorated AKLT Model

Contact Info

Product

Resources

About