Correlation Length versus Gap in Frustration-Free Systems

Abstract

Hastings established exponential decay of correlations for ground states of gapped quantum many-body systems. A ground state of a (geometrically) local Hamiltonian with spectral gap ϵ has correlation length ξ upper bounded as ξ ¼ Oð1=ϵÞ. In general this bound cannot be improved. Here we study the scaling of the correlation length as a function of the spectral gap in frustration-free local Hamiltonians, and we prove a tight bound ξ ¼ Oð1= ffiffi ffi ϵ p Þ in this setting. This highlights a fundamental difference between frustration-free and frustrated systems near criticality. The result is obtained using an improved version of the combinatorial proof of correlation decay due to Aharonov, Arad, Vazirani, and Landau. DOI: 10.1103/PhysRevLett.116.097202 Exponential decay of correlations is a basic feature of the ground space in gapped quantum many-body systems. The setting is as follows. We consider a geometrically local Hamiltonian H which acts on particles of constant dimension s; i.e., the Hilbert space is ðC s Þ ⊗n , where n is the total number of particles. The particles are located at the sites of a finite lattice (of some arbitrary dimension). We write the Hamiltonian aswhere distinct terms H i ; H j are supported on distinct subsets of particles. Here the support of a term H i is the set of particles on which it acts nontrivially. We assume that H has constant range r with respect to the usual distance function d on the lattice, the shortest path metric. This means that the diameter of the support of each term H i is upper bounded by r (e.g., r ¼ 2 for nearest-neighbor interactions). Without loss of generality we assume that the smallest eigenvalue of each term H i is equal to zero, and that ∥H i ∥ ≤ 1.If H has a unique ground state jψi and spectral gap ϵ, connected correlation functions decay exponentially as a function of distance [1][2][3]. In particular, where dðA; BÞ denotes the distance between the supports of two (arbitrary) local observables A, B, and C is a positive constant which depends on r and the lattice. In the transverse field Ising chain the scaling ξ ¼ Θð1=ϵÞ is achieved [4,5], which shows that the upper bound on ξ in Eq. (2) cannot be improved. In gapped systems with (exactly) degenerate ground states, a modification of Eq. (1),holds with ξ ¼ Oð1=ϵÞ for any ground state jψi [6], where G is the projector onto the ground space. An overview of these results and the proof techniques used to obtain them is given in Ref. [7].Here we specialize to frustration-free geometrically local Hamiltonians. Frustration-freeness means that any ground state of H is also in the ground space of each term H i . Since we assume that H i has smallest eigenvalue zero, this means that any ground state jψi of H satisfies H i jψi ¼ 0 for all i. The ground energy of H is therefore zero and in general the ground space may be degenerate. The spectral gap ϵ of H is defined to be its smallest nonzero eigenvalue. Henceforth we assume (without loss of generality [8]) that each term H i in the Hamiltonian is a projector,...

Simple proof of the detectability lemma and spectral gap amplification

“…This tighter form of the DL has already been used in Ref. [22] to derive a quadratically improved upper bound on the correlation length of gapped ground states of frustration-free systems.…”

“…A direct application of both lemmas provides the result for a length scale O(γ −1 ); we quadratically improve the dependence on γ by employing a Chebyshev polynomial in a way analogous to recent work of Gosset and Huang [22].…”

“…Before proving the theorem, we note, following Ref. [22], that Theorem 1 is optimal in the sense that in general one cannot hope to amplify the gap of a frustration-free system to a constant by coarse graining into groups of r = O(γ −λ ) particles with λ < 1/2. Indeed, as was shown in Ref.…”

“…Our proof employs a recent "trick" by Gosset and Huang [22] to boost the effect of DL q (H ) by using a Chebyshev polynomial. This reduces the length scale of the required coarse graining from O(γ −1 ) to O(γ −1/2 ).…”

“…Such operators are referred to as approximate ground state projectors (AGSPs), and various constructions have been used to establish properties of gapped ground states such as exponential decay of correlations [14,22], area laws [21,26], and local reversibility [27]. Frustration-free systems can be given a very natural construction of AGSP, called the detectability lemma operator DL(H ).…”