Lieb-Robinson bounds, the spectral flow, and
                    stability of the spectral gap for lattice fermion
                    systems

Nachtergaele, Bruno; Sims, Robert; Young, Amanda

doi:10.1090/conm/717/14443

Cited by 37 publications

(49 citation statements)

References 34 publications

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“…Secondly, the Heisenberg dynamics of an observable A ∈ A X , generated by a local Hamiltonian, satisfies a Lieb-Robinson bound [29]. These two properties, which are crucial for our results, are the main reasons why A is chosen to be the even algebra in the fermionic case, see [13,35].…”

Section: Spin Systemsmentioning

confidence: 90%

A Many-Body Index for Quantum Charge Transport

Bachmann

Bols

Roeck

et al. 2019

Commun. Math. Phys.

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We propose an index for pairs of a unitary map and a clustering state on manybody quantum systems. We require the map to conserve an integer-valued charge and to leave the state, e.g. a gapped ground state, invariant. This index is integer-valued and stable under perturbations. In general, the index measures the charge transport across a fiducial line. We show that it reduces to (i) an index of projections in the case of non-interacting fermions, (ii) the charge density for translational invariant systems, and (iii) the quantum Hall conductance in the two-dimensional setting without any additional symmetry. Example (ii) recovers the Lieb-Schultz-Mattis theorem, and (iii) provides a new and short proof of quantization of Hall conductance in interacting many-body systems.

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Section: Spin Systemsmentioning

confidence: 90%

A Many-Body Index for Quantum Charge Transport

Bachmann

Bols

Roeck

et al. 2019

Commun. Math. Phys.

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“…(This implies that interaction terms of disjoint support commute.) For background on these models, see, e.g., [31].…”

Section: Possible Extensionsmentioning

confidence: 99%

Spectral gaps of frustration-free spin systems with boundary

Lemm

Mozgunov

2019

Journal of Mathematical Physics

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In quantum many-body systems, the existence of a spectral gap above the ground state has far-reaching consequences. In this paper, we discuss "finite-size" criteria for having a spectral gap in frustration-free spin systems and their applications.We extend a criterion that was originally developed for periodic systems by Knabe and Gosset-Mozgunov to systems with a boundary. Our finite-size criterion says that if the spectral gaps at linear system size n exceed an explicit threshold of order n −3/2 , then the whole system is gapped. The criterion takes into account both "bulk gaps" and "edge gaps" of the finite system in a precise way. The n −3/2 scaling is robust: it holds in 1D and 2D systems, on arbitrary lattices and with arbitrary finite-range interactions. One application of our results is to give a rigorous foundation to the folklore that 2D frustration-free models cannot host chiral edge modes (whose finite-size spectral gap would scale like n −1 ). arXiv:1801.08915v2 [quant-ph] 3 May 2019It is well-known that the existence of a spectral gap may depend on the imposed boundary conditions, and this fact is at the core of our work.In general, the question whether a quantum spin system is gapped or gapless is difficult: In 1D, the Haldane conjecture [15,16] ("antiferromagnetic, integer-spin Heisenberg chains are gapped") remains open after 30 years of investigation. In 2D, the "gapped versus gapless" dichotomy is in fact undecidable in general [10], even among the class of translation-invariant, nearest-neighbor Hamiltonians.This paper studies the spectral gaps of a comparatively simple class of models: 1D and 2D frustration-free (FF) quantum spin systems with a non-trivial boundary (i.e., open boundary conditions). (A famous FF spin system is the AKLT chain [1]. One general reason why FF systems arise is that any quantum state which is only locally correlated can be realized as the ground state of an appropriate FF "parent Hamiltonian" [12,32,35].)Specifically, we are interested in finite-size criteria for having a spectral gap in such systems. Let us explain what we mean by this.Let γ m denote the spectral gap of the Hamiltonian of interest, when it acts on systems of linear size m. Letγ n be the "local gap", i.e., the spectral gap of a subsystem of linear size up to n. (We will be more precise later.) The finite-size criterion is a bound of the form γ m ≥ c n (γ n − t n ), (1.1) for all m sufficiently large compared to n (say m ≥ 2n).Here c n > 0 is an unimportant constant, but the value of t n (in particular its ndependence) is critical. Indeed, if for some fixed n 0 , we know from somewhere that γ n 0 > t n 0 , then (1.1) gives a uniform lower bound on the spectral gap γ m for all sufficiently large m. Accordingly, we call t n the "local gap threshold".The general idea to prove a finite-size criterion like (1.1) is that the Hamiltonian on systems of linear size m can be constructed out of smaller Hamiltonians acting on subsystems of linear size up to n, and these can be controlled in terms ofγ n . We emph...

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“…, n. Equations (19) and (20) can be solved for A µ , since L µ , Q µ ,Ṽ µ−1 , and H 1,µ−1 depend only on A ν for ν < µ. First, note that in the block decomposition with respect to P * (the full spectral projection), the P ⊥ * (· · · )P ⊥ * -blocks on both sides of (19) resp. (20) vanish identically, independently of the choice of A µ .…”

Section: Inserting the Expansions Into (18) Leaves Us Withmentioning

confidence: 99%

“…(20) vanish identically, independently of the choice of A µ . Second, the off-diagonal blocks of (19) resp. (20) determine A µ uniquely.…”

Section: Inserting the Expansions Into (18) Leaves Us Withmentioning

confidence: 99%

“…To implement these ideas mathematically, we heavily use and partly extend a technical machinery that has experienced important new developments during recent years. This includes Lieb-Robinson bounds for interacting fermions [19,6], the quasi-local inverse of the Liouvillian introduced in the context of the so called spectral-flow or quasi-adiabatic evolution [12,3], as well as ideas and technical lemmas from [2] and [18]. The mathematical justification of linear response theory in similar situations for non-interacting systems of fermions was studied e.g.…”

Section: Introductionmentioning

confidence: 99%

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Non-equilibrium Almost-Stationary States and Linear Response for Gapped Quantum Systems

Teufel

2019

Commun. Math. Phys.

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We prove the validity of linear response theory at zero temperature for perturbations of gapped Hamiltonians describing interacting fermions on a lattice. As an essential innovation, our result requires the spectral gap assumption only for the unperturbed Hamiltonian and applies to a large class of perturbations that close the spectral gap. Moreover, we prove formulas also for higher order response coefficients.Our justification of linear response theory is based on a novel extension of the adiabatic theorem to situations where a time-dependent perturbation closes the gap. According to the standard version of the adiabatic theorem, when the perturbation is switched on adiabatically and as long as the gap does not close, the initial ground state evolves into the ground state of the perturbed operator. The new adiabatic theorem states that for perturbations that are either slowly varying potentials or small quasi-local operators, once the perturbation closes the gap, the adiabatic evolution follows non-equilibrium almost-stationary states (NEASS) that we construct explicitly.

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Lieb-Robinson bounds, the spectral flow, and stability of the spectral gap for lattice fermion systems

Cited by 37 publications

References 34 publications

A Many-Body Index for Quantum Charge Transport

A Many-Body Index for Quantum Charge Transport

Spectral gaps of frustration-free spin systems with boundary

Non-equilibrium Almost-Stationary States and Linear Response for Gapped Quantum Systems

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