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arXiv:1103.4812 · Physical Review Letters

Supersymmetry and Its Spontaneous Breaking in the Random Field Ising Model

Matthieu Tissier, Gilles Tarjus

Published Jul 18, 2011• 92 citations
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Abstract

We provide a resolution of one of the long-standing puzzles in the theory of disordered systems. By reformulating the functional renormalization group for the critical behavior of the random field Ising model in a superfield formalism, we are able to follow the associated supersymmetry and its spontaneous breaking along the functional renormalization group flow. Breaking is shown to occur below a critical dimension d(DR) ≃ 5.1 and leads to a breakdown of the "dimensional reduction" property. We compute the critical exponents as a function of dimension and give evidence that scaling is described by three independent exponents.

DOI
10.1103/physrevlett.107.041601

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We provide a resolution of one of the long-standing puzzles in the theory of disordered systems. By reformulating the functional renormalization group for the critical behavior of the random field Ising model in a superfield formalism, we are able to follow the associated supersymmetry and its spontaneous breaking along the functional renormalization group flow. Breaking is shown to occur below a critical dimension d(DR) ≃ 5.1 and leads to a breakdown of the "dimensional reduction" property. We compute the critical exponents as a function of dimension and give evidence that scaling is described by three independent exponents.
We apply the nonperturbative functional renormalization group (NP-FRG) in the superfield formalism that we have developed in the preceding paper to study long-standing issues concerning the critical behavior of the random field Ising model. Through the introduction of an appropriate regulator and a supersymmetry-compatible nonperturbative approximation, we are able to follow the supersymmetry, more specifically, the superrotational invariance first unveiled by Parisi and Sourlas [Phys. Rev. Lett. 43, 744 (1979)] and its spontaneous breaking along the RG flow. Breaking occurs below a critical dimension ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, and the supersymmetry-broken fixed point that controls the critical behavior then leads to a breakdown of the ``dimensional-reduction'' property. We solve the NP-FRG flow equations numerically and determine the critical exponents as a function of dimension down to $d\ensuremath{\lesssim}3$, with a good agreement in d=3d=3d=3 and 4 with the existing numerical estimates.
We investigate dimensional reduction, the property that the critical behavior of a system in the presence of quenched disorder in dimension ddd is the same as that of its pure counterpart in $d\ensuremath{-}2$, and its breakdown in the case of the random-field Ising model in which both the interactions and the correlations of the disorder are long ranged, i.e., power-law decaying. To some extent the power-law exponents play the role of spatial dimension in a short-range model, which allows us to probe the theoretically predicted existence of a nontrivial critical value separating a region where dimensional reduction holds from one where it is broken, while still considering the physical dimension d=3d=3d=3. By extending our recently developed approach based on a nonperturbative functional renormalization group combined with a supersymmetric formalism, we find that such a critical value indeed exists, provided one chooses a specific relation between the decay exponents of the interactions and of the disorder correlations. This transition from dimensional reduction to its breakdown should therefore be observable in simulations and numerical analyses, if not experimentally.
We discuss the breakdown of the Parisi-Sourlas supersymmetry (SUSY) and of the dimensional-reduction (DR) property in the random field Ising and O( N N ) models as a function of space dimension d d and/or number of components N N . The functional renormalization group (FRG) predicts that this takes place below a critical line d_\mathrm{DR}(N) d D R ( N ) . We revisit the perturbative FRG results for the RFO( N N )M in d=4+\epsilon d = 4 + ϵ and carry out a more comprehensive investigation of the nonperturbative FRG approximation for the RFIM. In light of this FRG description, we discuss the perturbative results in \epsilon=6-d ϵ = 6 − d recently derived for the RFIM by Kaviraj, Rychkov, and Trevisani. We stress in particular that the disappearance of the SUSY/DR fixed point below d_\mathrm{DR} d D R arises as a consequence of the nonlinearity of the FRG equations and cannot be found via the perturbative expansion in \epsilon=6-d ϵ = 6 − d (nor in 1/N 1 / N ). We also provide an error bar on the value of the critical dimension d_\mathrm{DR} d D R for the RFIM, which we find around 5.11±0.09 5.11 ± 0.09 , by studying several successive orders of the nonperturbative FRG approximation scheme.
We reformulate the nonperturbative functional renormalization group for the random field Ising model in a superfield formalism, extending the supersymmetric description of the critical behavior of the system first proposed by Parisi and Sourlas [Phys. Rev. Lett. 43, 744 (1979)]. We show that the two crucial ingredients for this extension are the introduction of a weighting factor, which accounts for ground-state dominance when multiple metastable states are present, and of multiple copies of the original system, which allows one to access the full functional dependence of the cumulants of the renormalized disorder and to describe rare events. We then derive exact renormalization group equations for the flow of the renormalized cumulants associated with the effective average action.
The random-field Ising model shows an extreme critical slowdown that has been described by activated dynamic scaling: The characteristic time of relaxation toward equilibrium diverges exponentially with the correlation length $ln\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{\psi}}/T$, with $\ensuremath{\psi}$ an a priori unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions ddd less than a critical value ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, also associated with dimensional-reduction breakdown, $\ensuremath{\psi}=\ensuremath{\theta}$ with $\ensuremath{\theta}$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For d>dDRd>{d}_{\text{DR}}d>dDR​, on the other hand, $\ensuremath{\psi}=\ensuremath{\theta}\ensuremath{-}2\ensuremath{\lambda}$, where $\ensuremath{\theta}=2$ and $\ensuremath{\lambda}>0$ an additional exponent. At the upper critical dimension $d=6,\ensuremath{\lambda}=1$, so that $\ensuremath{\psi}=0$, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches, and droplets. We also propose a way to check the two regimes by computer simulations of long-range one-dimensional systems.
The random-field Ising model is one of the few disordered systems where the perturbative renormalization group can be carried out to all orders of perturbation theory. This analysis predicts dimensional reduction, i.e., that the critical properties of the random-field Ising model in D dimensions are identical to those of the pure Ising ferromagnet in D-2 dimensions. It is well known that dimensional reduction is not true in three dimensions, thus invalidating the perturbative renormalization group prediction. Here, we report high-precision numerical simulations of the 5D random-field Ising model at zero temperature. We illustrate universality by comparing different probability distributions for the random fields. We compute all the relevant critical exponents (including the critical slowing down exponent for the ground-state finding algorithm), as well as several other renormalization-group invariants. The estimated values of the critical exponents of the 5D random-field Ising model are statistically compatible to those of the pure 3D Ising ferromagnet. These results support the restoration of dimensional reduction at D=5. We thus conclude that the failure of the perturbative renormalization group is a low-dimensional phenomenon. We close our contribution by comparing universal quantities for the random-field problem at dimensions 3≤D<6 to their values in the pure Ising model at D-2 dimensions, and we provide a clear verification of the Rushbrooke equality at all studied dimensions.