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arXiv:2411.11147 · SciPost Physics

On the breakdown of dimensional reduction and supersymmetry in random-field models

Gilles Tarjus, Matthieu Tissier, Ivan Balog

Published Jul 1, 2025• 2 citations

Abstract

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.

DOI: 10.21468/scipostphys.19.1.001

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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 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=3

and 4 with the existing numerical estimates.

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.

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.

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 study the critical behavior and phase diagram of the

d

-dimensional random field

O(N)

model by means of the nonperturbative functional renormalization group approach presented in the preceding paper. We show that the dimensional-reduction predictions, which are obtained from conventional perturbation theory, break down below a critical dimension

{d}_{\text{DR}}(N)

and we provide a description of criticality, ferromagnetic ordering, and quasi-long range order in the whole

(N,d)

plane. Below

{d}_{\text{DR}}(N)

, our formalism gives access to both the typical behavior of the system, which is controlled by zero-temperature fixed points at which the effective action has a nonanalytic field dependence, and to the physics of rare low-energy excitations (``droplets''), which is described at nonzero temperature by the rounding of the nonanalyticity in a thermal boundary layer.

We develop a nonperturbative functional renormalization group approach for the random-field O(N) model that allows us to investigate the ordering transition in any dimension and for any value of N including the Ising case. We show that the failure of dimensional reduction and standard perturbation theory is due to the nonanalytic nature of the zero-temperature fixed point controlling the critical behavior, nonanalyticity, which is associated with the existence of many metastable states. We find that this nonanalyticity leads to critical exponents differing from the dimensional reduction prediction only below a critical dimension dc(N)<6, with dc(N=1)>3.