Fate of Parisi–Sourlas supersymmetry in the random-field Ising model at $D=5$: exponentially small violations, or exact only above a critical dimension $D_{\mathrm{int}}\approx 5.1$?

The random-field Ising model (RFIM) is expected, from the Parisi–Sourlas supersymmetric formulation, to obey dimensional reduction: its critical behavior in $D$ dimensions should coincide with that of the pure Ising model in $D-2$ dimensions. Perturbation theory in $\epsilon=6-D$ confirms this to all orders, yet dimensional reduction fails at $D=3$ and $D=4$. Fytas, Martín-Mayor, Parisi, Picco, and Sourlas [Phys. Rev. Lett. 122, 240603 (2019)] performed the first direct test of the supersymmetric Ward identities at $D=5$ by zero-temperature ground-state simulations in a double limit of elongated geometries: all three tested dimensionless quantities matched the pure 3D Ising values to 0.2–2%, while at $D=4$ they fail by at least 10%. The paper discarded the scenario of strong nonperturbative destruction of supersymmetry but could not distinguish its two remaining scenarios: (2) supersymmetry violations are exponentially small, of order $\exp(-A/\epsilon)$, with the coefficient $A$ left as an unsolved analytical problem; or (3) the supersymmetric fixed point exists only for $D>D_{\mathrm{int}}\approx 5.1$, so that the $D=5$ agreement reflects weakly broken supersymmetry. Later literature has sharpened but not settled the question: the nonperturbative functional renormalization group of Balog, Tarjus, and Tissier gives $D_{\mathrm{int}}=5.11\pm0.09$ with supersymmetry broken at below the percent level at $D=5$, whereas the perturbative renormalization-group analysis of Kaviraj, Rychkov, and Trevisani finds SUSY-breaking perturbations relevant only below $d_c\approx4.2$–$4.7$, implying an exactly supersymmetric $D=5$ fixed point. Deciding between these scenarios — through the corrections-to-scaling exponent $\omega(D)$, the spectrum of nonsupersymmetric perturbations, the coefficient $A$, higher-precision or tuned simulations, and a quantitative mechanism for supersymmetry breaking — remains open.

Background

The random-field Ising model (RFIM) is an Ising ferromagnet coupled to a frozen random magnetic field: spins $S_x=\pm1$ on a hypercubic lattice in $D$ dimensions with Hamiltonian

$$H=-J\sum_{\langle xy\rangle}S_xS_y-\sum_x h_xS_x,$$

where the $h_x$ are independent, zero-mean random variables of variance $\sigma^2$ (quenched disorder, i.e. frozen and not in thermal equilibrium with the spins). Increasing the disorder strength $\sigma$ destroys the ferromagnetic phase at a critical point $\sigma_c$; the upper critical dimension (above which mean-field exponents hold) is $D_u=6$, so the natural expansion parameter is $\epsilon=6-D$.

In 1979, Parisi and Sourlas showed that the critical field theory of the RFIM possesses a supersymmetry — a symmetry mixing the ordinary (bosonic) spin field with ghost (anticommuting) fields representing response functions, which emerges in an equivalent stochastic description of the problem. A consequence is dimensional reduction: the critical behavior of the RFIM in $D$ dimensions should coincide with that of the pure (non-random) Ising model in $D-2$ dimensions. In particular, the anomalous dimensions $\eta$ and $\bar\eta$ governing the decay of the connected correlation function $\overline{\langle S_xS_y\rangle_c}$ and of the disconnected one $\overline{\langle S_x\rangle\langle S_y\rangle}$ (overline = disorder average) should be equal, and equal to the anomalous dimension of the pure Ising model in $D-2$ dimensions. Perturbation theory confirms dimensional reduction to all orders in $\epsilon$. For a related problem, branched polymers (tree-like statistical molecules), supersymmetry and dimensional reduction have even been mathematically proven. For the RFIM, however, dimensional reduction fails at low dimension: in $D=3$ the RFIM has a ferromagnetic phase while the $D=1$ pure Ising model has no transition at all, and large-scale simulations show clear violations at $D=3$ and $D=4$. The failure is attributed to nonperturbative effects invisible to the diagrammatic expansion, such as the interaction of replicas (copies of the system introduced to perform the disorder average, analytically continued to $n\to0$) studied by Brézin and De Dominicis, or instantons in replica space (nonperturbative saddle points) as studied by Parisi and Dotsenko.

Near the upper critical dimension, i.e. for $D=6-\epsilon$, Fytas, Martín-Mayor, Parisi, Picco, and Sourlas (2019) listed three possible scenarios: (1) nonperturbative effects destroy supersymmetry already at a finite order of the $\epsilon$ expansion, or even at $D=6$; (2) violations of supersymmetry are exponentially small in the distance from six dimensions, of order $\exp(-A/\epsilon)$, with the computation of the coefficient $A$ an unsolved analytical problem; (3) supersymmetry is exact only for $D>D_{\mathrm{int}}\approx5.1$, below which the supersymmetric fixed point (the scale-invariant solution governing the critical point in the renormalization-group sense) becomes unstable against nonsupersymmetric perturbations — a value obtained in the nonperturbative functional renormalization group (NP-FRG) of Tissier and Tarjus and of Tarjus, Balog, and Tissier, which treats the full functional dependence of the renormalized disorder cumulants rather than a finite set of couplings.

The 2019 test worked directly at zero temperature, where the RFIM critical fixed point lies, computing exact ground states by optimization methods for $10^5$ disorder samples per $(L,R)$ pair. The supersymmetric Ward identities (relations among correlation functions that follow from the symmetry) hold in finite volume only in a specific geometry and limit: an elongated hypercube $L^{D-2}\times(RL)^2$ with aspect ratio $R\ge1$, taking $R\to\infty$ before the thermodynamic limit $L\to\infty$. In that double limit, three dimensionless quantities of the $D$-dimensional RFIM must coincide with known values of the pure Ising model in $D-2$ dimensions: the ratio $\xi^{(\mathrm{dis})}/L$ of the second-moment correlation length (computed from the Fourier transform of the disconnected propagator at the smallest nonzero momentum) to the short linear size; the analogous ratio $\xi^{(\mathrm{con})}_{\sigma-\eta}/L$ for the connected propagator, fixed by a Ward identity to equal $\xi^{(\mathrm{dis})}/L$; and the dimensionless four-point coupling $U_4$. At $D=5$ (with $L\le14$, $R\le10$) all three agreed with the pure 3D Ising values at the 0.2–2% level, while at $D=4$ (with $L\le28$) they missed the pure 2D Ising values by at least 10%. Scenario 1 was thereby discarded. The corrections-to-scaling exponent $\omega$ — the exponent of the leading irrelevant operator, governing how finite-size observables approach their critical-limit values, with dimensional reduction predicting $\omega_{\mathrm{DR}}=\epsilon+O(\epsilon^2)$ while nonsupersymmetric irrelevant operators would give $\omega_{\mathrm{NS}}=2+O(\epsilon^2)$ — was proposed as the discriminator: the measured $\omega(D=5)\approx0.66$ seemed to favor Scenario 2, since Scenario 3 naively suggests $\omega(D=5)\sim D_{\mathrm{int}}-D\approx0.1$. The authors, however, explicitly left the choice between Scenarios 2 and 3 open.

After 2019 the question has been sharpened but not settled, by two competing frameworks. On one side, the NP-FRG analysis of Balog, Tarjus, and Tissier asserts that the supersymmetric/dimensional-reduction fixed point disappears below $d_{\mathrm{DR}}\approx5.1$ by annihilating with an unstable fixed point (with a square-root singularity of the leading stability eigenvalue), so that Scenario 3 holds, $\omega(D=5)\approx0.65$ is nevertheless large, and supersymmetry at $D=5$ is broken at below the 1% level — consistent with the 2019 data within their accuracy. On the other side, the perturbative renormalization-group program of Kaviraj, Rychkov, and Trevisani classifies all replica-singlet perturbations of the supersymmetric fixed point in a field basis with manifest proximity to it (the Cardy basis), where the lowest-dimension parts of perturbations ('leaders') are susy-writable, susy-null, or non-susy-writable; at two loops, SUSY-breaking perturbations become relevant only below $d_c\approx4.2$–$4.7$, so that the supersymmetric fixed point would exist (and be exact) at $D=5$, and even survive, though unstable, at $D=4$. The two frameworks disagree on the critical dimension, on the mechanism of supersymmetry breaking, and on whether the SUSY fixed point exists at all below $\approx5.1$, while both remain compatible with all existing $D=3,4,5$ numerical data.

Problem Statement

The problem, as left open by Fytas, Martín-Mayor, Parisi, Picco, and Sourlas [Phys. Rev. Lett. 122, 240603 (2019), DOI: 10.1103/PhysRevLett.122.240603, https://doi.org/10.1103/PhysRevLett.122.240603] and still undecided, is to determine the fate of Parisi–Sourlas supersymmetry in the random-field Ising model near five dimensions.

Decide between the two surviving scenarios: (2) the supersymmetric fixed point is exact at $D=5$ to all orders in perturbation theory, so that any violations of the supersymmetric Ward identities and of dimensional reduction are nonperturbatively small, of order $\exp(-A/\epsilon)$ with $\epsilon=6-D$, the coefficient $A$ being an unsolved analytical problem; or (3) supersymmetry and dimensional reduction hold only above a critical dimension $D_{\mathrm{int}}\approx5.1$, so that the observed $D=5$ agreement reflects weakly broken supersymmetry that should become resolvable at higher precision.

The open sub-questions are: (i) the corrections-to-scaling exponent $\omega(D)$ near $D=5$ and the scaling dimensions of the nonsupersymmetric (SUSY-breaking) perturbations — including whether $\omega(D=5)$ takes the dimensional-reduction value $\omega_{\mathrm{DR}}=\epsilon+O(\epsilon^2)$ or the value generated by a fixed-point collision near $D_{\mathrm{int}}$, and whether $\omega$ can discriminate the scenarios at all; (ii) the origin and quantitative description of the nonperturbative effects that break supersymmetry at low dimension — replica bound states or instantons in replica space, avalanche-induced cusp singularities in the renormalized disorder cumulants, and relevant non-susy-writable/susy-null operators — and their evolution with $D$; and (iii) the discriminating power of improved zero-temperature simulations of the $D=5$ (and $D=4$) RFIM at the critical disorder beyond $L\le14$, $R\le10$ — whether they would detect or bound the predicted sub-percent Ward-identity violation at $D=5$, or realize the proposed $D=4$ tuning of the SUSY-breaking couplings, distinguishing an existing-but-unstable supersymmetric fixed point from its disappearance below $D_{\mathrm{int}}$.

References

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