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Our aim here is to define the limit t→ ∞ of the two-time autocorrelation function so as to obtain the Edwards-Anderson order parameter qEA in the limit τ→∞ (see the subsequent Sec. ). In the original definition of qEA [see Refs. and], one ought to take the average over many values of the waiting time t, thus effectively recovering time-translation invariance and only a dependence on τ survives. From a formal point of view, this is equivalent to allowing | P (t)〉 to reach its equilibrium distribution, Š |0. This in turn is equivalent to taking the limit t→〉 whilst holding the system size fixed, and then sending N→∞ before taking the limit τ→∞.
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Note that if the classical energy E C is local in the microscopic degrees of freedom labeling the configurations { C }, and if the dynamics are local (i.e., only a finite portion of the system can be rearranged by the same instantaneous thermal process), one can then guarantee that the associated quantum Hamiltonian is local in the basis { | C 〉 } (Ref.). Conversely, when the classical energy is nonlocal, one can readily find examples of nonlocal quantum Hamiltonians where the closing of a gap does not lead to a signature in the fidelity susceptibility. For instance, in the spherical p = 2 spin glass model the gap closes at the Thouless-Almeida-Palmer temperature T TAP, whereas one can prove that there are no singularities in the free energy of the classical system (and therefore no singularities in the heat capacity) down to a finitely lower dynamical transition temperature T D (Ref.).
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It is interesting to notice an important difference between this constructive scheme of a gaugeable glass and other disordered glassy systems. So long as the degeneracy can be understood in terms of essentially decoupled subsystems undergoing an ordering transition, it is bound to scale at most as exp ( Lθ ), with θ
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It is interesting to notice an important difference between this constructive scheme of a gaugeable glass and other disordered glassy systems. So long as the degeneracy can be understood in terms of essentially decoupled subsystems undergoing an ordering transition, it is bound to scale at most as exp (L θ), with θ < d - 1, where d is the dimensionality of the system (if the interactions are short ranged). Indeed, an extensive degeneracy ∼ exp (L d) would require the existence of an ordering transition in subsystems composed of a finite number of degrees of freedom, which is not possible. The significance of this difference is underlined by the fact that several phenomenological approaches to understand dynamical glass transitions [for instance in terms of configurational entropy (Ref.)] rely on the extensivity of the glass degeneracy.
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We note in passing that if one promotes the J zx i /J=±1 and J yz i /J=±1 to dynamical variables as opposed to quenched random variables, with the contraint J zx i J yz i+ x J zx i+ y J yz i = J4 yet enforced, the discussion above remains largely unchanged. However, in this case one has notably an example of a thermodynamic phase transition into a massively degenerate liquidlike phase (i.e., with no local order) in a nondisordered system.
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We note in passing that if one promotes the J z x i / J = ± 1 and J y z i / J = ± 1 to dynamical variables as opposed to quenched random variables, with the contraint J z x i J y z i + x J z x i + y J y z i = J 4 yet enforced, the discussion above remains largely unchanged. However, in this case one has notably an example of a thermodynamic phase transition into a massively degenerate liquidlike phase (i.e., with no local order) in a nondisordered system.
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It is worth pointing out that, in this correspondence, a d -dimensional classical system endowed with a given Markov dynamics (i.e., with a specific "time dimension") maps onto a d -dimensional quantum system, encoding all the information from d spatial dimensions plus 1 (time)
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It is worth pointing out that, in this correspondence, a d -dimensional classical system endowed with a given Markov dynamics (i.e., with a specific "time dimension") maps onto a d -dimensional quantum system, encoding all the information from d spatial dimensions plus 1 (time).
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For the sake of clarity, what we mean here by "local perturbation" is an energy term proportional to the sum of operators with finite support, which differ only by translations. For instance, ∑ i σ i or ∑ i σ i σ i + x σ i + y σ i + x + y. According to this definition, a random field does not qualify as a local perturbation.
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