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arXiv:2601.07926

Low energy excitations in a long prism geometry: testing the lower critical dimension of the Ising spin glass

Massimo Bernaschi, L. A. Fernández, I. González-Adalid Pemartín, Víctor Martín-Mayor, Giorgio Parisi A, Federico Ricci‐Tersenghi

Published Jan 12, 2026• 0 citations
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Abstract

We propose a general method for studying systems that display low-energy excitations in their low-temperature phase. We argue that in a rectangular right-prism geometry (the 3-dimensional generalization of a strip), with one longitudinal size much larger than the transverse size, correlations decay exponentially (at all temperatures) along the longitudinal direction. Still, the scaling of the correlation length with the transverse size carries crucial information from which the lower critical dimension can be inferred. The method is applied in the particularly demanding context of Ising spin glasses at zero magnetic field. The lower critical dimension and the multifractal spectrum for the correlation functions are computed from large-scale numerical simulations. Several technical novelties (such as the unexpectedly crucial performance of Houdayer's cluster method or the convenience of using open---rather than periodic---boundary conditions) allow us to study three-dimensional prisms with transverse dimensions up to $L=24$ and effectively infinite longitudinal dimensions, down to low temperatures. The value that we find for the exponent controlling the behavior of the correlation length agrees with predictions from the Replica Symmetry Breaking (RSB) theory. We argue that our novel setting holds promise in clarifying which of the two competing theories, RSB or the Droplet Model, more accurately describes three-dimensional spin glasses.

DOI
10.48550/arxiv.2601.07926

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We propose a general method for studying systems that display low-energy excitations in their low-temperature phase. We argue that in a rectangular right-prism geometry (the 3-dimensional generalization of a strip), with one longitudinal size much larger than the transverse size, correlations decay exponentially (at all temperatures) along the longitudinal direction. Still, the scaling of the correlation length with the transverse size carries crucial information from which the lower critical dimension can be inferred. The method is applied in the particularly demanding context of Ising spin glasses at zero magnetic field. The lower critical dimension and the multifractal spectrum for the correlation functions are computed from large-scale numerical simulations. Several technical novelties (such as the unexpectedly crucial performance of Houdayer's cluster method or the convenience of using open---rather than periodic---boundary conditions) allow us to study three-dimensional prisms with transverse dimensions up to L=24L=24L=24 and effectively infinite longitudinal dimensions, down to low temperatures. The value that we find for the exponent controlling the behavior of the correlation length agrees with predictions from the Replica Symmetry Breaking (RSB) theory. We argue that our novel setting holds promise in clarifying which of the two competing theories, RSB or the Droplet Model, more accurately describes three-dimensional spin glasses.
We study large-scale, low-energy excitations in the Ising spin glass with Gaussian interactions in two dimensions at zero temperature, using an optimization algorithm to determine exact ground states. Periodic boundary conditions are applied. Our results for the fractal dimension of the surface, ds,{d}_{s},ds​, and stiffness exponent, ${\ensuremath{\theta}}^{\ensuremath{'}},$ for ``droplet'' excitations, are in reasonable agreement with estimates from ``domain wall'' calculations, and so support the predictions of the ``droplet theory.'' Restricting our analysis to small lattices, we do not find an effective value of ${\ensuremath{\theta}}^{\ensuremath{'}}$ close to $\ensuremath{-}0.47$ as has been recently proposed. The effects of averaging over droplets of different sizes are studied and are also found to be too small to give ${\ensuremath{\theta}}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{-}0.47$ for smaller sizes. Larger corrections to finite-size scaling would be needed in three and four dimensions in order for the numerical data used to support the ``TNT'' scenario to be compatible with the droplet theory prediction that the stiffness exponent is the same for droplets and domain walls.
We investigate spin-spin correlation functions in the low-temperature phase of spin-glasses. Using the replica field theory formalism, we examine in detail their infrared (long-distance) behavior. In particular we identify a longitudinal mode that behaves as massive at intermediate-length scales (near-infrared region). These issues are then addressed by numerical simulation; the analysis of our data is compatible with the prediction that the longitudinal mode appears as massive, i.e., that it undergoes an exponential decay, for distances smaller than an appropriate correlation length.
Results are presented for the geometry of low-energy excitations in the one-dimensional Ising spin chain with power-law interactions, in which the model parameters are chosen to yield a finite spin-glass transition temperature. Both finite-temperature and ground-state studies are carried out. For the range of sizes studied the data cannot be fitted to any of the standard spin-glass scenarios without including corrections to scaling. Incorporating such corrections we find that the fractal dimension of the surface of the excitations, is either equal to the space dimension, consistent with replica symmetry breaking, or very slightly less than it. The latter case is consistent with the droplet and ``trivial-nontrivial'' pictures.
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We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at Tc{T}_{c}Tc​=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero Tc{T}_{c}Tc​ but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that Tc{T}_{c}Tc​ is nonzero with a long-range-ordered phase below Tc{T}_{c}Tc​. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.
Spin glasses are the paradigm of complex systems. These materials present really slow dynamics. However, the nature of the spin glass phase in finite dimensional systems is still controversial. Different theories describing the low temperature phase have been proposed: droplet, replica symmetry breaking and chaotic pairs. We present analytical studies of critical properties of spin glasses, in particular, critical exponents at and below the phase transition, existence of a phase transition in a magnetic field, computation of the lower critical dimension (in presence/absence of a magnetic field). We also introduce some rigorous results based on the concept of metastate. Finally, we report some numerical results regarding the construction of the Aizenman-Wehr metastate, scaling of the correlation functions in the spin glass phase and existence of a phase transition in a field, confronting these results with the predictions of different theories.