Avalanche exponents and the breakdown of dimensional reduction in the Bethe-lattice loop expansion of the zero-temperature random-field Ising model

The loop expansion around the Bethe-lattice solution, generated by the M-layer construction, is a renormalization-group tool for strongly disordered systems whose criticality is controlled by a zero-temperature fixed point and is therefore poorly captured by the standard Landau-Ginzburg epsilon expansion. Applied to the random-field Ising model (RFIM) at zero temperature, its one-loop two-point calculation reproduces the usual $6-\epsilon$ expansion: the new cubic-vertex diagrams that encode avalanches are divergent but subdominant, so dimensional reduction (the Parisi-Sourlas prediction that RFIM exponents in $D$ dimensions equal pure-Ising exponents in $D-2$) survives at $O(\epsilon)$. What remains open is what the expansion predicts for avalanches: the source paper announces, but does not perform, the one-loop correction to $\chi_2$, the second-moment susceptibility of avalanches, from which the epsilon expansions of the avalanche exponent $\rho_{av}$ and avalanche fractal dimension $d_f$ would follow, testing the nonperturbative functional-renormalization-group predictions of Tarjus and Tissier. Two further questions remain as background: whether the two-point epsilon expansion itself is unaltered beyond $O(\epsilon)$ (the full two-loop question, tied to scenario 1 for the DR breakdown), and whether that breakdown arises from exponentially small corrections of order $\exp(-A/\epsilon)$ versus a critical dimension $D_{DR}\simeq 5.1$ at which the dimensional-reduction fixed point becomes unstable or disappears (scenarios 2 and 3, which the one-loop result cannot discriminate).

Background

The random-field Ising model (RFIM) places Ising spins $\sigma_i=\pm 1$ on a $D$-dimensional lattice with ferromagnetic nearest-neighbor couplings and a quenched (frozen, sample-dependent) random magnetic field $h_i$ at each site. It is the prototypical strongly disordered system: its critical behavior is governed by a zero-temperature renormalization-group (RG) fixed point (a scale-invariant theory that controls the long-distance behavior near the critical point), and its upper critical dimension - the dimension above which mean-field exponents hold - is $D=6$. A celebrated consequence of the field-theoretic description is dimensional reduction (DR): Parisi and Sourlas showed in 1979 that an emergent supersymmetry of the disorder-averaged theory implies that the critical exponents of the RFIM in $D$ dimensions equal those of the pure Ising model in $D-2$ dimensions. Dimensional reduction follows at all orders of the standard Landau-Ginzburg (LG) perturbation expansion in $\epsilon=6-D$, yet it is provably wrong in $D=2$ and $D=3$, where exponents differ from the DR prediction. How and where between $D=3$ and $D=6$ dimensional reduction breaks down has been debated for four decades.

One line of thought attributes the breakdown to avalanches: at $T=0$ the response to an infinitesimal change of the external field is a cascade of collectively flipping spins, with a power-law size distribution $P(s)\propto s^{-\rho_{av}}$ (mean-field value $\rho_{av}=3/2$ on the Bethe lattice) and a fractal dimension $d_f$ (the scaling of the typical avalanche spatial extent). Tarjus, Baczyk and Tissier argued that DR breaks down when critical avalanches are 'big enough', more precisely when $d_f=D-d_\phi$, with $d_\phi$ the scaling dimension of the magnetization near the relevant $T=0$ fixed point. Standard LG field theory cannot describe avalanches: for instance, it predicts the wrong scaling ($\tau^{-2}$ instead of $\tau^{-3}$, with $\tau$ the distance from criticality) for the fluctuations of the susceptibility. The nonperturbative functional renormalization group (NP-FRG) of Tarjus and Tissier, which keeps the full field-dependence of the disorder cumulants and can be formulated directly at $T=0$, predicts that DR breaks down below a critical dimension $D_{DR}\simeq 5.1$.

The source paper approaches the puzzle from a new direction. The Bethe lattice (an infinite loop-free tree) admits an exact mean-field solution of the RFIM at $T=0$ via the cavity method, which iterates 'cavity messages' - the probability distributions of effective local fields reaching a site from each neighbor - on a large representative population of messages, a scheme known as population dynamics. The M-layer construction of Altieri et al. builds, for any finite-dimensional lattice, a family of random lattices that interpolates between the Bethe lattice and the physical lattice, yielding a loop expansion in the small parameter $1/M$ around the Bethe solution. Expanding around the Bethe solution rather than the fully-connected one is essential because the Bethe solution already contains the rare-region fluctuations that dominate at $T=0$. For the RFIM at $T=0$ this expansion generates an effective theory containing, besides the standard quartic vertices, cubic $\phi^3$ vertices that encode avalanches and have no counterpart in LG theory.

Angelini, Lucibello, Parisi, Ricci-Tersenghi and Rizzo (PNAS 2020) performed the one-loop computation of the two-point correlation functions in this expansion, semi-analytically and numerically. The cubic-vertex diagrams do give new contributions that diverge for $D<6$, but these are subdominant with respect to the standard supersymmetric quartic diagrams: at one loop the usual $\epsilon$-expansion results and dimensional reduction are recovered. The paper distinguishes three scenarios for the DR breakdown: (1) the $\epsilon$-expansion is wrong even as a Taylor series near $D=6$ - refuted at $O(\epsilon)$ by their result; (2) the $\epsilon$-expansion is correct but receives exponentially small nonperturbative corrections of order $\exp(-A/\epsilon)$, invisible at any finite perturbative order (a suggestion of Parisi and Dotsenko); (3) there is a critical dimension $D_{DR}\simeq 5.1$ below which the DR fixed point becomes unstable, as advocated by Tarjus and Tissier on the basis of the NP-FRG. The one-loop result cannot discriminate scenarios 2 and 3, and the paper announces - but does not perform - the computation of the one-loop correction to $\chi_2$ (the susceptibility associated with the second moment of the avalanche size distribution, a three-point object whose scaling determines $d_f$), which would yield the $\epsilon$-expansion for $\rho_{av}$ and $d_f$.

Problem Statement

Within the Bethe-lattice (M-layer) loop expansion of the zero-temperature RFIM - the expansion in the small parameter $1/M$ generated by the M-layer construction, which interpolates between the Bethe lattice and the physical lattice - perform the computation announced by Angelini et al. (PNAS 2020) but left open: the one-loop correction to $\chi_2$, the susceptibility associated with the second moment of the avalanche size distribution, in $D=6-\epsilon$. $\chi_2$ is a three-point object defined through sample-resolved fluctuations of the integrated $T=0$ response, i.e., the disorder average of the product of connected two-point functions within the same sample, $\chi_2=\sum_{x,y}\overline{\langle\sigma_0\sigma_x\rangle_c\,\langle\sigma_0\sigma_y\rangle_c}$ (the overline denotes the disorder average; this is the second cumulant of the susceptibility, not the square of the disorder-averaged connected susceptibility), whose scaling encodes the avalanche fractal dimension. Derive from this result the $\epsilon$-expansions of the avalanche exponent $\rho_{av}$ - the exponent of the power-law avalanche size distribution $P(s)\propto s^{-\rho_{av}}$, whose mean-field (Bethe-lattice) value is $3/2$ - and of the avalanche fractal dimension $d_f$ (the scaling of the typical avalanche spatial extent), and check the results against the nonperturbative FRG predictions of Tarjus and Tissier, in particular the avalanche criterion $d_f=D-d_\phi$ (with $d_\phi$ the scaling dimension of the magnetization near the relevant $T=0$ fixed point) for the breakdown of dimensional reduction.

As background: the source paper's one-loop two-point result establishes that dimensional reduction is valid at least at $O(\epsilon)$, refuting at that order scenario 1 for the DR breakdown (that the $\epsilon$-expansion is wrong even as a Taylor series near $D=6$); it cannot discriminate between the two other scenarios - exponentially small nonperturbative corrections of order $\exp(-A/\epsilon)$ (scenario 2, Parisi-Dotsenko) versus a critical dimension $D_{DR}\simeq 5.1$ at which the dimensional-reduction fixed point becomes unstable or disappears (scenario 3, Tarjus-Tissier). A complete two-loop two-point calculation, deciding whether the cubic-vertex sector remains subdominant beyond one loop, i.e., whether the standard $\epsilon$-expansion of the two-point critical behavior is altered beyond $O(\epsilon)$, is a natural continuation of the same program but is not announced as a goal by the source paper.

The objective is to determine what the Bethe-lattice loop expansion predicts for avalanche criticality beyond leading order: whether it merely reproduces the Landau-Ginzburg $\epsilon$-expansion in the cubic (avalanche) sector, or corrects it in a way that bears on the competing pictures of the dimensional-reduction breakdown.

References

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