The recent proof by F. Guerra that the Parisi ansatz provides a lower bound on the free energy of the SK spin-glass model could have been taken as offering some support to the validity of the purported solution. In this work we present a broader variational principle, in which the lower bound, as well as the actual value, are expressed through an optimization procedure for which ultrametic/hierarchal structures form only a subset of the variational class. The validity of Parisi's ansatz for the SK model is still in question. The new variational principle may be of help in critical review of the issue.PACS numbers: 75.50.Lk IntroductionThe statistical mechanics of spin-glass models is characterized by the existence of a diverse collection of competing states, very slow relaxation of the quenched dynamics, and a rather involved picture of the equilibrium state.A great deal of insight on the subject has been produced through the study of the Sherrington Kirkpatrick (SK) model [1]. After some initial attempts, a solution was proposed by G. Parisi which has the requisite stability and many other attractive features [2]. Its development has yielded a plethora of applications of the method, in which a key structural assumption is a particular form of the replica symmetry-breaking (i.e., the assumption of "ultrametricity", or the hierarchal structure, of the overlaps among the observed spin configurations) [3].Yet to this day it was not established that this very appealing proposal does indeed provide the equilibrium structure of the SK model. A recent breakthrough is the proof by F. Guerra [4] that the free energy provided by Parisi's purported solution is a rigorous lower bound for the SK free energy.More completely, the result of Guerra is that for any value of the order parameter, which within the assumed ansatz is a function, the Parisi functional provides a rigorous lower bound. Thus, this relation is also valid for the maximizer which yields the Parisi solution.In this work we present a variational principle for the free energy of the SK model which makes no use of a Parisi-type order parameter, and which yields the result of Guerra as a particular implication. More explicitly, the new principle allows more varied bounds on the free energy, for which there is no need to assume a hierarchal organization of the Gibbs state (e.g., as expressed in the assumed ultrametricity of the overlaps [3]). Guerra's results follow when the variational principle is tested against the Derrida-Ruelle hierarchal probability cascade models (GREM) [5].This leads us to a question which is not new: is the ultrametricity an inherent structue of the SK mean-field model, or is it only a simplifiying assumption. The new

We give a Lieb-Robinson bound for the group velocity of a large class of discrete quantum systems which can be used to prove that a non-vanishing spectral gap implies exponential clustering in the ground state of such systems.Comment: v2: corrected proof of Theorem 2. v3: slightly better bound in Theorem 2; updated proo

Abstract. Gapped ground states of quantum spin systems have been referred to in the physics literature as being 'in the same phase' if there exists a family of Hamiltonians H(s), with finite range interactions depending continuously on s ∈ [0, 1], such that for each s, H(s) has a non-vanishing gap above its ground state and with the two initial states being the ground states of H(0) and H(1), respectively. In this work, we give precise conditions under which any two gapped ground states of a given quantum spin system that 'belong to the same phase' are automorphically equivalent and show that this equivalence can be implemented as a flow generated by an s-dependent interaction which decays faster than any power law (in fact, almost exponentially). The flow is constructed using Hastings' 'quasi-adiabatic evolution' technique, of which we give a proof extended to infinitedimensional Hilbert spaces. In addition, we derive a general result about the locality properties of the effect of perturbations of the dynamics for quantum systems with a quasi-local structure and prove that the flow, which we call the spectral flow, connecting the gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a co-cycle of automorphisms of the algebra of quasi-local observables of the infinite spin system. This proves that the ground state phase structure is preserved along the curve of models H(s), 0 ≤ s ≤ 1.

Abstract. We provide a simple proof of the Lieb-Robinson bound and use it to prove the existence of the dynamics for interactions with polynomial decay. We then use our results to demonstrate that there is an upper bound on the rate at which correlations between observables with separated support can accumulate as a consequence of the dynamics.

We prove Lieb-Robinson bounds for systems defined on infinite dimensional Hilbert spaces and described by unbounded Hamiltonians. In particular, we consider harmonic and certain anharmonic lattice systems.