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arXiv:1512.00088 · Journal of Mathematical Physics

Local gap threshold for frustration-free spin systems

David Gosset, Evgeny Mozgunov

Published Sep 1, 2016• 54 citations
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Abstract

We improve Knabe’s spectral gap bound for frustration-free translation-invariant local Hamiltonians in 1D. The bound is based on a relationship between global and local gaps. The global gap is the spectral gap of a size-m chain with periodic boundary conditions, while the local gap is that of a subchain of size n < m with open boundary conditions. Knabe proved that if the local gap is larger than the threshold value 1/(n − 1) for some n > 2, then the global gap is lower bounded by a positive constant in the thermodynamic limit m → ∞. Here we improve the threshold to 6n(n+1), which is better (smaller) for all n > 3 and which is asymptotically optimal. As a corollary we establish a surprising fact about 1D translation-invariant frustration-free systems that are gapless in the thermodynamic limit: for any such system the spectral gap of a size-n chain with open boundary conditions is upper bounded as O(n−2). This contrasts with gapless frustrated systems where the gap can be Θ(n−1). It also limits the extent to which the area law is violated in these frustration-free systems, since it implies that the half-chain entanglement entropy is O(1/ϵ) as a function of spectral gap ϵ. We extend our results to frustration-free systems on a 2D square lattice.

DOI
10.1063/1.4962337

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We improve Knabe’s spectral gap bound for frustration-free translation-invariant local Hamiltonians in 1D. The bound is based on a relationship between global and local gaps. The global gap is the spectral gap of a size-m chain with periodic boundary conditions, while the local gap is that of a subchain of size n < m with open boundary conditions. Knabe proved that if the local gap is larger than the threshold value 1/(n − 1) for some n > 2, then the global gap is lower bounded by a positive constant in the thermodynamic limit m → ∞. Here we improve the threshold to 6n(n+1), which is better (smaller) for all n > 3 and which is asymptotically optimal. As a corollary we establish a surprising fact about 1D translation-invariant frustration-free systems that are gapless in the thermodynamic limit: for any such system the spectral gap of a size-n chain with open boundary conditions is upper bounded as O(n−2). This contrasts with gapless frustrated systems where the gap can be Θ(n−1). It also limits the extent to which the area law is violated in these frustration-free systems, since it implies that the half-chain entanglement entropy is O(1/ϵ) as a function of spectral gap ϵ. We extend our results to frustration-free systems on a 2D square lattice.
In 1988, Knabe found a "finite-size criterion" to determine whether a frustration-free quantum spin chain with periodic boundary conditions is uniformly gapped in the thermodynamic limit. The criterion provides a threshold for the spectral gap at a finite system size such that, if the threshold is exceeded for a fixed system size, then the chain with periodic boundary conditions is uniformly gapped. We extend Knabe's result to frustration-free spin chains equipped with open boundary conditions. We now obtain two finite-size criteria: The first one is identical to Knabe's criterion and we interpret it as a bulk criterion. The second one controls the spectral gaps at smaller system sizes and can be interpreted as a new edge criterion. Heuristically, it excludes the presence of thermodynamically gapless excitations living near the edge.
Knabe's theorem lower bounds the spectral gap of a one-dimensional frustration-free local Hamiltonian in terms of the local spectral gaps of finite regions. It also provides a local spectral gap threshold for Hamiltonians that are gapless in the thermodynamic limit, showing that the local spectral gap must scale inverse linearly with the length of the region for such systems. Recent works have further improved upon this threshold, tightening it in the one-dimensional case and extending it to higher dimensions. Here, we show a local spectral gap threshold for frustration-free Hamiltonians on a finite-dimensional lattice that is optimal up to a constant factor that depends on the dimension of the lattice. Our proof is based on the detectability lemma framework and uses the notion of a coarse-grained Hamiltonian (introduced in [Anshu et al., Phys. Rev. B 93, 205142]) as a link connecting the (global) spectral gap and the local spectral gap.
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).
We consider a family of translation-invariant quantum spin chains with nearest-neighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let ψψψ be an arbitrary two-qubit state. We consider a chain of nnn qubits with open boundary conditions and Hamiltonian Hn(ψ)H_n(ψ)Hn​(ψ) which is defined as the sum of rank-1 projectors onto ψψψ applied to consecutive pairs of qubits. We show that the spectral gap of Hn(ψ)H_n(ψ)Hn​(ψ) is upper bounded by 1/(n−1)1/(n-1)1/(n−1) if the eigenvalues of a certain two-by-two matrix simply related to ψψψ have equal non-zero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of nnn (depending only on ψψψ). A key ingredient in the proof is a new operator inequality for the ground space projector which expresses a monotonicity under the partial trace. This monotonicity property appears to be very general and might be interesting in its own right. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustration-free translation-invariant spin-1/2 chains with nearest-neighbor interactions.
Abstract Providing system-size independent lower bounds on the spectral gap of local Hamiltonian is in general a hard problem. For the case of finite-range, frustration free Hamiltonians on a spin lattice of arbitrary dimension, we show that a property of the ground state space is sufficient to obtain such a bound. We furthermore show that such a condition is necessary and equivalent to a constant spectral gap. Thanks to this equivalence, we can prove that for gapless models in any dimension, the spectral gap on regions of diameter n is at most o ( log ⁡ ( n ) 2 + ε n ) for any positive ϵ.
Estimating spectral gaps of quantum many-body Hamiltonians is a highly challenging computational task, even under assumptions of locality and translation-invariance. Yet, the quest for rigorous gap certificates is motivated by their broad applicability, ranging from many-body physics to quantum computing and classical sampling techniques. Here we present a general method for obtaining lower bounds on the spectral gap of frustration-free quantum Hamiltonians in the thermodynamic limit. We formulate the gap certification problem as a hierarchy of optimization problems (semidefinite programs) in which the certificate – a proof of a lower bound on the gap – is improved with increasing levels. Our approach encompasses existing finite-size methods, such as Knabe's bound and its subsequent improvements, as those appear as particular possible solutions in our optimization, which is thus guaranteed to either match or surpass them. We demonstrate the power of the method on one-dimensional spin-chain models where we observe an improvement by several orders of magnitude over existing finite size criteria in both the accuracy of the lower bound on the gap, as well as the range of parameters in which a gap is detected.