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arXiv:1901.08473 · Physical Review Letters

Evidence for Supersymmetry in the Random-Field Ising Model at D = 5

Nikolaos G. Fytas, Víctor Martín-Mayor, Giorgio Parisi A, Marco Picco, Nicolas Sourlas

Published Jun 20, 2019• 53 citations
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Abstract

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.

DOI
10.1103/physrevlett.122.240603

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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.
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 investigate the critical properties of the d=3d=3d=3 random-field Ising model with a Gaussian field distribution at zero temperature. By implementing suitable graph-theoretical algorithms, we perform a large-scale numerical simulation of the model for a vast range of values of the disorder strength hhh and system sizes $\mathcal{V}=L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$, with $L\ensuremath{\le}156$. Using the sample-to-sample fluctuations of various quantities and proper finite-size scaling techniques we estimate with high accuracy the critical disorder strength hc{h}_{\mathrm{c}}hc​ and the correlation length exponent $\ensuremath{\nu}$. Additional simulations in the area of the estimated critical-field strength and relevant scaling analysis of the bond energy suggest bounds for the specific heat critical exponent $\ensuremath{\alpha}$ and the violation of the hyperscaling exponent $\ensuremath{\theta}$. Finally, a data collapse analysis of the order parameter and disconnected susceptibility provides accurate estimates for the critical exponent ratios $\ensuremath{\beta}/\ensuremath{\nu}$ and $\overline{\ensuremath{\gamma}}/\ensuremath{\nu}$, respectively.
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.
By performing a high-statistics simulation of the D=4 random-field Ising model at zero temperature for different shapes of the random-field distribution, we show that the model is ruled by a single universality class. We compute to a high accuracy the complete set of critical exponents for this class, including the correction-to-scaling exponent. Our results indicate that in four dimensions (i) dimensional reduction as predicted by the perturbative renormalization group does not hold and (ii) three independent critical exponents are needed to describe the transition.
Finite-size scaling above the upper critical dimension is a long-standing puzzle in the field of statistical physics. Even for pure systems various scaling theories have been suggested, partially corroborated by numerical simulations. In the present manuscript we address this problem in the even more complicated case of disordered systems. In particular, we investigate the scaling behavior of the random-field Ising model at dimension D=7, i.e., above its upper critical dimension D_{u}=6, by employing extensive ground-state numerical simulations. Our results confirm the hypothesis that at dimensions D>D_{u}, linear length scale L should be replaced in finite-size scaling expressions by the effective scale L_{eff}=L^{D/D_{u}}. Via a fitted version of the quotients method that takes this modification, but also subleading scaling corrections into account, we compute the critical point of the transition for Gaussian random fields and provide estimates for the full set of critical exponents. Thus, our analysis indicates that this modified version of finite-size scaling is successful also in the context of the random-field problem.