Critical fluctuations of the link overlap versus the specific heat in the zero-field Edwards-Anderson spin glass: higher-order confirmation of the exponent ordering and numerical detection of the anomalous logarithmic correction

Altieri, Parisi, and Rizzo (Phys. Rev. B 93, 024422, 2016) showed, at first order in the $\epsilon=6-D$ expansion of the replicated cubic field theory of the zero-field Edwards-Anderson (EA) Ising spin glass, that the fluctuations of the link overlap and the specific heat are governed by different critical exponents ($\alpha_{\rm link}=-1-\epsilon$ versus $\alpha=-1-2\epsilon$), even though the average energy and the average link overlap share their critical behavior. They further found that the link-overlap fluctuations contain a subdominant power-law contribution with an anomalous logarithmic prefactor (amplitude $\Delta\eta_2$), carried by the difference channel of the degenerate squared-longitudinal and squared-anomalous sectors, which is absent from the specific heat and masked by the dominant squared-replicon power law. The open problem, acknowledged in the paper's conclusions, is to establish this one-loop picture in physically relevant dimensions: (i) compute the squared-sector anomalous dimensions and the logarithmic amplitude beyond one loop, deciding whether the exponent ordering and the log correction survive; and (ii) devise and implement a numerical/data-analysis procedure that detects the logarithmic correction and tests the ordering of the fluctuation exponents in three dimensions. No published work is known to compute these exponents at higher order or to attempt the numerical isolation of the logarithmic correction; the problem remains open.

Background

The Edwards-Anderson (EA) model places Ising spins on a $D$-dimensional lattice with quenched random nearest-neighbor couplings, and its zero-field spin-glass transition is studied largely by numerical simulation. A central family of observables is built from the overlap $q_x^{ab}=\sigma_x^a\sigma_x^b$ between two replicas $a,b$ (independent spin configurations sharing the same disorder); its spatial average is the spin-glass order parameter. The link overlap, the product of overlaps on two neighboring sites, $Q_{x,\mu}^{ab}=q_x^{ab}q_{x+\hat e_\mu}^{ab}$, is a standard observable in simulations (Krzakala and Martin 2000; Katzgraber and Young 2002). For Gaussian couplings one can show by integration by parts that the average energy and the average link overlap have identical critical behavior; the question investigated by Altieri, Parisi, and Rizzo (2016) is whether their fluctuations also do.

Below the upper critical dimension $D_u=6$, the continuum theory of the zero-field EA model is a replicated cubic field theory for the order-parameter field $q_{ab}(x)$ (a real symmetric $n\times n$ matrix field, with the quenched disorder average taken in the limit $n\to0$) with a single cubic invariant, the setting of the classic $\epsilon=6-D$ expansion of Harris, Lubensky, and Chen (1976). Quadratic observables such as the energy density and the link-overlap field correspond to composite operators $q_{ab}q_{cd}$, whose renormalization determines critical exponents beyond those of the order parameter. Replica symmetry decomposes the space of such operators into orthogonal sectors; within the 'squared' subspace $\{q_{ab}^2\}$ the diagonal propagators are named squared-longitudinal, squared-anomalous, and squared-replicon, in analogy with the replicon/longitudinal/anomalous eigenvalues of the replica-symmetric mass matrix of spin-glass theory (the replicon being the softest mode of the mean-field low-temperature phase).

Altieri, Parisi, and Rizzo proved two structural facts that hold at all orders: the energy-energy correlation function, whose integral gives the specific heat, coincides exactly with the squared-longitudinal propagator $g_L^{(2)}(x)$; and the link-overlap fluctuations $G_1^{(2)}(x)$ also contain the squared-replicon contribution $g_R^{(2)}(x)$. At one loop they found $\eta_L^{(2)}=-5\epsilon/3$ and $\eta_R^{(2)}=-2\epsilon/3+O(n)$, so $\eta_R^{(2)}>\eta_L^{(2)}$: the squared-replicon dominates, and the link-overlap susceptibility exponent $\alpha_{\rm link}=-1-\epsilon$ exceeds the specific-heat exponent $\alpha=-1-2\epsilon$ (with $\nu=1/2+5\epsilon/12$ and $\alpha_{\rm link}=\alpha+2\nu(\eta_R^{(2)}-\eta_L^{(2)})$). The fluctuations of the link overlap are therefore stronger than those of the energy near six dimensions, although the two averages share their critical behavior. In the quenched limit the squared-longitudinal and squared-anomalous eigenvalues are degenerate; their difference quotient $\Delta g(x)=\lim_{n\to0}[g_A^{(2)}(x)-g_L^{(2)}(x)]/n$ thus acquires an anomalous logarithmic prefactor, $\Delta g(x)\propto\Delta\eta_2\ln x/x^{2D-4-2\eta_L^{(2)}}$ with $\Delta\eta_2=\lim_{n\to0}(\eta_A^{(2)}-\eta_L^{(2)})/n$, an instance of Cardy's general mechanism (1999) by which degenerate eigenvalues in $n\to0$ replicated theories generate logarithmic corrections. This log term is absent from the specific heat and is masked in the raw overlap correlations by the dominant squared-replicon power law; the paper gives explicit invertible linear combinations of the multi-replica correlation functions $G_1^{(2)},G_2^{(2)},G_3^{(2)}$ that isolate $g_R^{(2)}$, $g_L^{(2)}$, and $\Delta g$ (its eqs. (30) and (35)-(37)).

$D=3$ lies far from the upper critical dimension, so the one-loop $\epsilon$-series is qualitative there. The third-order $\epsilon$-expansion for the basic exponents $\nu$ and $\eta$, corrected by Yeo, Moore, and Aspelmeier (2005), whose large-order analysis found alternating leading coefficients but irregular subdominant ones, is of uncertain resummability in $D=3$. The authors note that the ordering of the link-overlap and energy fluctuations 'seems to remain true in $D=3$ according to preliminary numerical observations' of the Janus collaboration, a consortium running large-scale dedicated-hardware simulations of the three-dimensional EA model (a 2013 private communication, unpublished; the published Janus critical-parameter study gives $T_c=1.1019(29)$, $\nu=2.562(42)$, $\eta=-0.3900(36)$, and $\alpha=-5.69(13)$, but no link-overlap fluctuation exponent). The conclusions of the paper pose the open problem explicitly: the anomalous logarithmic prefactor sits on a subdominant term, and 'one should devise an appropriate procedure to observe it in a numerical simulation', while the one-loop picture awaits confirmation in realistic dimensions.

Problem Statement

Establish, beyond the one-loop order at which it was derived, the critical behavior of the link-overlap fluctuations of the zero-field Edwards-Anderson Ising spin glass relative to the specific heat, in physically relevant dimensions, in particular $D=3$. The single scientific objective is to confirm or refute the one-loop picture of Altieri, Parisi, and Rizzo, namely (a) the ordering $\eta_R^{(2)}>\eta_L^{(2)}$, equivalently $\alpha_{\rm link}>\alpha$: the link-overlap fluctuations are stronger than the specific heat even though the two averages share their critical behavior, and (b) the presence of an anomalous logarithmic prefactor, with amplitude $\Delta\eta_2=\lim_{n\to0}(\eta_A^{(2)}-\eta_L^{(2)})/n$, on the subdominant squared-longitudinal-sector difference channel $\Delta g(x)$ of the link-overlap fluctuations. Two complementary and jointly supporting lines of work address this objective:

(1) Field-theoretic determination: compute the squared-subspace anomalous dimensions $\eta_L^{(2)}$, $\eta_R^{(2)}$ and the logarithmic amplitude $\Delta\eta_2$ beyond one loop in the replicated cubic field theory, deciding whether the exponent ordering and the logarithmic correction survive beyond leading order.

(2) Numerical detection: devise and implement a numerical/data-analysis procedure that isolates the subdominant log-corrected channel from the dominant squared-replicon power law in simulations of the three-dimensional EA model, and use it, together with finite-size-scaling analyses of the link-overlap susceptibility and the specific heat, to test the predicted exponent ordering and the presence, sign, and magnitude of the logarithmic term in $D=3$.

References

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