Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · cond-mat/9812034

Scaling and infrared divergences in the replica field theory of the Ising spin glass

T. Temesvari·C. De Dominicis·I. Kondor
View original source ↗Read source document ↗Export metadata ↓

Abstract

Replica field theory for the Ising spin glass in zero magnetic field is studied around the upper critical dimension d=6. A scaling theory of the spin glass phase, based on Parisi's ultrametrically organised order parameter, is proposed. We argue that this infinite step replica symmetry broken (RSB) phase is nonperturbative in the sense that amplitudes of scaling forms cannot be expanded in term of the coupling constant w^2. Infrared divergent integrals inevitably appear when we try to compute amplitudes perturbatively, nevertheless the ε-expansion of critical exponents seems to be well-behaved. The origin of these problems can be traced back to the unusual behaviour of the free propagator having two mass scales, the smaller one being proportional to the perturbation parameter w^2 and providing a natural infrared cutoff. Keeping the free propagator unexpanded makes it possible to avoid producing infrared divergent integrals. The role of Ward-identities and the problem of the lower critical dimension are also discussed.

Explore related subjects

cond-mat.dis-nn

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

T. Temesvari, C. De Dominicis, I. Kondor. 1998-12-02. Scaling and infrared divergences in the replica field theory of the Ising spin glass. https://doi.org/10.1007/s100510051191

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Unifying Physical Backpropagation

Physical computing systems exploit device dynamics for computation, but their gradient-based optimization is challenging: backpropagation through a digital twin suffers from a model-reality gap. On-device gradient computation could resolve this issue, and a handful of theoretical and experimental studies have proposed ways to achieve it. Yet a unifying theory identifying when a physical system can compute the gradient of its own performance has been missing. Here we develop such a unification based on the adjoint method: we identify sufficient conditions under which the adjoint field required for formally exact gradients can be generated on the same hardware that performs the computation. Linear and nonlinear systems obey fundamentally different conditions: for linear systems, damping or gain is admissible provided reciprocity is preserved. For nonlinear trajectory systems, the sufficient conditions are reciprocity of the linearized system and the existence of a time-reversal mirror. Algorithmically, the nonlinear case requires infinitesimal nudging, whereas linear systems admit a finite-amplitude experiment. We recover (quantum) Equilibrium Propagation, Hamiltonian echo backpropagation, fully forward mode training and in situ gradient methods in integrated-photonic and free-space-optical systems. Finally, we show that reciprocity is a special case of more general intertwining conditions. For linear systems, these permit exact on-device gradients in a class of non-Hermitian, non-reciprocal systems. For nonlinear trajectories, they combine with generalized time-reversal mirrors to cover, e.g., PT-symmetric equations. The framework also includes time-dependent parameters and Onsager-reciprocal dynamics, providing a unified basis for formally exact physical learning.

cond-mat.dis-nn↗
arXiv2026

Rank-One Signal Recovery in Sparse Wishart Noise

We study the high-dimensional recovery of a signal vector $\mathbf{x}$ in the presence of sparse Wishart-like noise. We define an $N \times N$ matrix $A = J+(θ/N)\mathbf{xx}^{\top}$, where $\mathbf{xx}^{\top}$ is the rank-one deformation of the random noise matrix $J$. We consider a Wishart-like matrix $J={X}^{\top} X$, where $X$ is a sparse $M \times N$ random matrix with entries $X_{ij} = c_{ij}W_{ij}$, with $c_{ij}$ regulating the density of non-zero elements, and $W_{ij}$ the bond weights. Using the replica method, we compute analytically the top eigenpair statistics of $A$, and their dependence on the signal strength $θ$, the rectangularity ratio $α=\sqrt{M/N}$, and the average connectivity of the noise. The spectral observables are expressed in terms of a system of Recursive Distributional Equations, which are efficiently solved via a Population Dynamics algorithm. They allow us to compute the average largest eigenvalue $\langleλ_1\rangle_{A}$, the average top eigenvector component density, and the average overlap between the top eigenvector of $A$ and $\mathbf{x}$. We identify a critical threshold $θ_{\mathrm{crit}}$--depending on the average connectivity of the noise--that marks a BBP-like phase transition: below this value, $\langleλ_1\rangle_{A}$ is unaffected by the signal, and the overlap vanishes. Thus, the signal is not recoverable from the top eigenvector of $A$. For $θ>θ_{\mathrm{crit}}$, the signal-related outlier eigenvalue becomes $\langleλ_1\rangle_{A}$ and the overlap is nonzero, allowing for recovery of the signal. The results are in excellent agreement with numerical diagonalisation. We show that in the dense limit, the recovery threshold and eigen-statistics converge to the results predicted by the classical BBP transition for additive rank-one deformations of dense Wishart matrices.

cond-mat.dis-nn↗
arXiv2026

Topological Fingerprints of Commensurate Order in Twisted Moiré Lattices

Moiré patterns in twisted bilayer materials exhibit long-range periodic order that is highly sensitive to twist angle. Identifying commensurate angles from real-space atomic structures remains challenging, as spectral and geometric methods emphasize global periodicity and are sensitive to disorder and finite-size effects. Here, we introduce a data-driven topological framework that quantifies moiré periodicity by treating atomic configurations as point-cloud data and extracting multiscale signatures using persistent homology. At the core of our approach is a small-neighborhood separation filter that removes redundant local motifs in persistence diagrams while preserving key structural features, enabling sharp minima in Wasserstein distances between twisted and untwisted reference configurations that accurately recover commensurate angles. We benchmark this framework against geometric and spectral similarity measures and show that the resulting topological descriptors remain stable under positional disorder ranging from weak to strong perturbations in the bond length. We further transfer these descriptors using a constrained Gaussian process surrogate to sparsely sampled configurations of a twisted MoTe$_2$ dichalcogenide bilayer. These results establish topological descriptors with neighborhood separation as a robust framework for identifying commensurate order in moiré systems and tracking commensurate structural similarity under disorder, sparsity, and finite-size effects.

cond-mat.dis-nn↗