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arXiv:1906.10058

Comment on "Evidence for Supersymmetry in the Random-Field Ising Model at D=5''

Ivan Balog, Gilles Tarjus, Matthieu Tissier

Published Jun 24, 2019• 2 citations
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Abstract

In a recent letter, Fytas et al. [Phys. Rev. Lett. 122, 240603 (2019)] study the critical point of the equilibrium random-field Ising model (RFIM) in $D=5$ by means of state-of-art zero-temperature lattice simulations. We show that their numerical results concerning the virtual absence of violation of supersymmetry and the characterization of the main corrections to scaling are in full agreement with our predictions by means of a nonperturbative functional renormalization group and with the existence of a critical dimension $D_{DR}\approx 5.1$ above which supersymmetry and dimensional reduction are valid at the fixed point and below which they break down.

DOI
10.48550/arxiv.1906.10058

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In a recent letter, Fytas et al. [Phys. Rev. Lett. 122, 240603 (2019)] study the critical point of the equilibrium random-field Ising model (RFIM) in D=5D=5D=5 by means of state-of-art zero-temperature lattice simulations. We show that their numerical results concerning the virtual absence of violation of supersymmetry and the characterization of the main corrections to scaling are in full agreement with our predictions by means of a nonperturbative functional renormalization group and with the existence of a critical dimension DDR≈5.1D_{DR}\approx 5.1DDR​≈5.1 above which supersymmetry and dimensional reduction are valid at the fixed point and below which they break down.
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 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 provide a nontrivial test of supersymmetry in the random-field Ising model at five spatial dimensions, by means of extensive zero-temperature numerical simulations. Indeed, supersymmetry relates correlation functions in a D-dimensional disordered system with some other correlation functions in a D-2 clean system. We first show how to check these relationships in a finite-size scaling calculation and then perform a high-accuracy test. While the supersymmetric predictions are satisfied even to our high accuracy at D=5, they fail to describe our results at D=4.
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 show that, contrary to previous suggestions based on computer simulations or erroneous theoretical treatments, the critical points of the random-field Ising model out of equilibrium, when quasistatically changing the applied source at zero temperature, and in equilibrium are not in the same universality class below some critical dimension ${d}_{DR}\ensuremath{\approx}5.1$. We demonstrate this by implementing a nonperturbative functional renormalization group for the associated dynamical field theory. Above dDR{d}_{DR}dDR​, the avalanches, which characterize the evolution of the system at zero temperature, become irrelevant at large distance, and hysteresis and equilibrium critical points are then controlled by the same fixed point. We explain how to use computer simulation and finite-size scaling to check the correspondence between in and out of equilibrium criticality in a far less ambiguous way than done so far.
Abstract Perturbation theory for the random-field Ising model (RFIM) has the infamous attribute that it predicts at all orders a dimensional-reduction property for the critical behavior that turns out to be wrong in low dimensions. Guided by our previous work based on the nonperturbative functional renormalization group (NP-FRG), we show that one can still make some use of the perturbation theory for a finite range of dimensions below the upper critical dimension, d = 6. The new twist is to account for the influence of large-scale zero-temperature events known as avalanches. These avalanches induce nonanalyticities in the field dependence of the correlation functions and renormalized vertices, and we compute in a loop expansion the eigenvalue associated with the corresponding anomalous operator. The outcome confirms the NP-FRG prediction that the dimensional-reduction fixed point correctly describes the dominant critical scaling of the RFIM above some dimension close to 5 but not below.