The AKLT spin chain is the prototypical example of a frustration-free quantum spin system with a spectral gap above its ground state. Affleck, Kennedy, Lieb, and Tasaki also conjectured that the two-dimensional version of their model on the hexagonal lattice exhibits a spectral gap. In this paper, we introduce a family of variants of the two-dimensional AKLT model depending on a positive integer n, which is defined by decorating the edges of the hexagonal lattice with one-dimensional AKLT spin chains of length n. We prove that these decorated models are gapped for all n ≥ 3.
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...
Nematic elastomers and glasses are solids that display spontaneous distortion under external stimuli. Recent advances in the synthesis of sheets with controlled heterogeneities have enabled their actuation into nontrivial shapes with unprecedented energy density. Thus, these have emerged as powerful candidates for soft actuators. To further this potential, we introduce the key metric constraint which governs shape-changing actuation in these sheets. We then highlight the richness of shapes amenable to this constraint through two broad classes of examples which we term nonisometric origami and lifted surfaces. Finally, we comment on the derivation of the metric constraint, which arises from energy minimization in the interplay of stretching, bending, and heterogeneity in these sheets.
The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations. Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map. The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, 2014] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, 2003]. Our paper constitutes partial progress towards a remainder term which features just the Petz recovery map (not a rotated Petz map), a conjecture which would have many consequences in quantum information theory. A well known result states that the monotonicity of relative entropy with respect to quantum operations is equivalent to each of the following inequalities: strong subadditivity of entropy, concavity of conditional entropy, joint convexity of relative entropy, and monotonicity of relative entropy with respect to partial trace. We show that this equivalence holds true for refinements of all these inequalities in terms of the Petz recovery map. So either all of these refinements are true or all are false.
We consider a divergence-form elliptic difference operator on the lattice Z d , with a coefficient matrix that is an i.i.d. perturbation of the identity matrix. Recently, Bourgain introduced novel techniques from harmonic analysis to prove the convergence of the Feshbach-Schur perturbation series related to the averaged Green's function of this model. Our main contribution is a refinement of Bourgain's approach which improves the key decay rate from −2d + ǫ to −3d + ǫ. (The optimal decay rate is conjectured to be −3d.) As an application, we derive estimates on higher derivatives of the averaged Green's function which go beyond the second derivatives considered by Delmotte-Deuschel and related works. Preliminaries2.1 Invertibility of L on some function spaces
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