arXiv subjects

Tommaso Rizzo

Publications and source records attributed to Tommaso Rizzo.

At least 19 recordsLinked to original sources

arXiv2026

Non-Monotonic Dynamical Correlations Across The Glass Crossover

The dramatic slowing down of structural relaxation in supercooled liquids is accompanied by the emergence of dynamic heterogeneity. A monotonically increasing dynamical correlation length, measured at the $\alpha$-timescale, is one of the remarkable features of this phenomenon. Here we show that this picture is incomplete: the dynamical correlation length measured in the $\beta$-relaxation regime exhibits a striking non-monotonic temperature dependence, reaching a maximum near the mode-coupling crossover temperature $T_c$ and decreasing upon further cooling, even as local dynamical fluctuations continue to intensify. This behavior suggests a crossover from spatially extended, maximally cooperative motion near $T_c$ to increasingly compact and localized relaxation events below it. We demonstrate that this evolution is quantitatively captured by stochastic beta-relaxation theory, an extension of mode-coupling theory beyond mean-field that explicitly predicts an avoided dynamical transition in finite dimensions. Our results provide the first direct spatial evidence in favor of the avoided-transition picture of the mode-coupling crossover, and establish the peak of the $\beta$-regime correlation length as a robust indicator of the mode-coupling crossover.

cond-mat.soft↗

arXiv2026

Long-range spin glass in a field at zero temperature

We compute the critical exponents of the zero-temperature spin glass transition in a field on a one-dimensional long-range model, a proxy for higher-dimensional systems. Our approach is based on a novel loop expansion within the Bethe $M$-layer formalism, whose adaptation to this specific case is detailed here. The resulting estimates provide crucial benchmarks for numerical simulations that can access larger system sizes in one dimension, thus offering a key test of the theory of spin glasses in a field.

cond-mat.dis-nn↗

arXiv2025

Solution of the Critical Dynamics of the Mean-Field Kob-Andersen Model

We analytically solve the critical dynamics of the Kob-Andersen kinetically constrained model of supercooled liquids on the Bethe lattice, employing a combinatorial argument based on the cavity method. For arbitrary values of graph connectivity z and facilitation parameter m, we demonstrate that the critical behavior of the order parameter is governed by equations of motion equivalent to those found in Mode-Coupling Theory. The resulting predictions for the dynamical exponents are validated through direct comparisons with numerical simulations that include both continuous and discontinuous transition scenarios.

cond-mat.dis-nn↗

arXiv2025

Rare Trajectories in a Prototypical Mean-field Disordered Model: Insights into Landscape and Instantons

For disordered systems within the random first-order transition (RFOT) universality class, such as structural glasses and certain spin glasses, the role played by activated relaxation processes is rich to the point of perplexity. Over the last decades, various efforts have attempted to formalize and systematize such processes in terms of instantons similar to the nucleation droplets of first-order phase transitions. In particular, Kirkpatrick, Thirumalai, and Wolynes proposed in the late '80s an influential nucleation theory of relaxation in structural glasses. Already within this picture, however, the resulting structures are far from the compact objects expected from the classical droplet description. In addition, an altogether different type of single-particle hopping-like instantons has recently been isolated in molecular simulations. Landscape studies of mean-field spin glass models have further revealed that simple saddle crossing does not capture relaxation in these systems. We present here a landscape-agnostic study of rare dynamical events, which delineates the richness of instantons in these systems. Our work not only captures the structure of metastable states, but also identifies the point of irreversibility, beyond which activated relaxation processes become a fait accompli. An interpretation of the associated landscape features is articulated, thus charting a path toward a complete understanding of RFOT instantons.

cond-mat.dis-nn↗

arXiv2025

Critical exponents of the spin glass transition in a field at zero temperature

We analyze the spin glass transition in a field in finite dimension $D$ below the upper critical dimension directly at zero temperature using a recently introduced perturbative loop expansion around the Bethe lattice solution. The expansion is generated by the so-called $M$-layer construction, and it has $1/M$ as the associated small parameter. Computing analytically and numerically these non-standard diagrams at first order in the $1/M$ expansion, we construct an $\epsilon$-expansion around the upper critical dimension $D_\text{uc}=8$, with $\epsilon=D_\text{uc}-D$. Following standard field theoretical methods, we can write a $\beta$ function, finding a new zero-temperature fixed-point associated with the spin glass transition in a field in dimensions $D<8$. We are also able to compute, at first order in the $\epsilon$-expansion, the three independent critical exponents characterizing the transition, plus the correction-to-scaling exponent.

cond-mat.dis-nn↗

arXiv2024

Temperature-Resilient Analog Neuromorphic Chip in Single-Polysilicon CMOS Technology

In analog neuromorphic chips, designers can embed computing primitives in the intrinsic physical properties of devices and circuits, heavily reducing device count and energy consumption, and enabling high parallelism, because all devices are computing simultaneously. Neural network parameters can be stored in local analog non-volatile memories (NVMs), saving the energy required to move data between memory and logic. However, the main drawback of analog sub-threshold electronic circuits is their dramatic temperature sensitivity. In this paper, we demonstrate that a temperature compensation mechanism can be devised to solve this problem. We have designed and fabricated a chip implementing a two-layer analog neural network trained to classify low-resolution images of handwritten digits with a low-cost single-poly complementary metal-oxide-semiconductor (CMOS) process, using unconventional analog NVMs for weight storage. We demonstrate a temperature-resilient analog neuromorphic chip for image recognition operating between 10$^{\circ}$C and 60$^{\circ}$C without loss of classification accuracy, within 2\% of the corresponding software-based neural network in the whole temperature range.

eess.IV↗

arXiv2024

Bethe $M$-layer construction on the Ising model

In statistical physics, one of the standard methods to study second order phase transitions is the renormalization group that usually leads to an expansion around the corresponding fully connected solution. Unfortunately, often in disordered models, some important finite dimensional second-order phase transitions are qualitatively different or absent in the corresponding fully connected model: in such cases the standard expansion fails. Recently, a new method, the $M$-layer one, has been introduced that performs an expansion around a different soluble mean field model: the Bethe lattice one. This new method has been already used to compute the upper critical dimension $D_U$ of different disordered systems such as the Random Field Ising model or the Spin glass model with field. If then one wants to go beyond and construct an expansion around $D_U$ to understand how critical quantities get renormalized, the actual computation of all the numerical factors is needed. This next step has still not been performed, being technically more involved. In this paper we perform this computation for the ferromagnetic Ising model without quenched disorder, in finite dimensions: we show that, at one-loop order inside the $M$-layer approach, we recover the continuum quartic field theory and we are able to identify the coupling constant $g$ and the other parameters of the theory, as a function of macroscopic and microscopic details of the model such as the lattice spacing, the physical lattice dimension and the temperature. This is a fundamental step that will help in applying in the future the same techniques to more complicated systems, for which the standard field theoretical approach is impracticable.

cond-mat.stat-mech↗

arXiv2024

Thermodynamics of the Fredrickson-Andersen Model on the Bethe Lattice

The statics of the Fredrickson-Andersen model (FAM) of the liquid-glass transition is solved on the Bethe lattice (BL). The kinetic constraints of the FAM imply on the BL an ergodicity-breaking transition to a (glassy) phase where a fraction of spins of the system is permanently blocked, and the remaining "free" spins become non-trivially correlated. We compute several observables of the ergodicity-broken phase, such as the self-overlap, the configurational entropy and the spin-glass susceptibility, and we compare the analytical predictions with numerical experiments. The cavity equations that we obtain allow to define algorithms for fast equilibration and accelerated dynamics. We find that at variance with spin-glass models, the correlations inside a state do not exhibit a critical behavior.

cond-mat.dis-nn↗

arXiv2024

Bethe $M$-layer construction for the percolation problem

The major difference between percolation and other phase transition models is the absence of an Hamiltonian and of a partition function. For this reason it is not straightforward to identify the corresponding field theory to be used as starting point of Renormalization Group computations. Indeed, it could be identified with the field theory of $n+1$ states Potts model in the limit of $n \to 0$ only by means of the mapping discovered by Kasteleyn and Fortuin for bond percolation. In this paper we show that it is possible to recover the epsilon expansion for critical exponents in finite dimension directly using the $M$-layer expansion, without the need to perform any analytical continuation. Moreover, we also show explicitly that the critical exponents for site and bond percolation are the same. This computation provides a reference for applications of the $M$-layer method to systems where the underlying field theory is unknown or disputed.

cond-mat.stat-mech↗

arXiv2024

The localized phase of the Anderson model on the Bethe lattice

In this paper, we investigate the Anderson model on the Bethe lattice, focusing on the localized regime. Employing the cavity approach, we derive compact expressions for the inverse participation ratios (IPRs) that are equivalent to those obtained using the supersymmetric formalism and naturally facilitate a highly efficient computational scheme. This method yields numerical results with unprecedented accuracy, even very close to the localization threshold. Our approach allows for high-precision validation of all theoretical predictions from the analytical solution, including the finite jump of the IPRs at the transition. Additionally, we reveal a singular behavior of the IPRs near the critical point that has not been previously reported in the literature. This singular behavior is further confirmed by the numerical solution of the non-linear $σ$ model on the Bethe lattice, which provides an effective description of Anderson localization.

cond-mat.dis-nn↗

arXiv2024

Instantons in $ϕ^4$ Theories: Transseries, Virial Theorems and Numerical Aspects

We discuss numerical aspects of instantons in two- and three-dimensional $ϕ^4$ theories with an internal $O(N)$ symmetry group, the so-called $N$-vector model. Combining asymptotic transseries expansions for large argument with convergence acceleration techniques, we obtain high-precision values for certain integrals of the instanton that naturally occur in loop corrections around instanton configurations. Knowledge of these numerical properties are necessary in order to evaluate corrections to the large-order factorial growth of perturbation theory in $ϕ^4$ theories. The results contribute to the understanding of the mathematical structures underlying the instanton configurations.

hep-th↗

arXiv2024

Logarithmic critical slowing down in complex systems: from statics to dynamics

We consider second-order phase transitions in which the order parameter is a replicated overlap matrix. We focus on a tricritical point that occurs in a variety of mean-field models and that, more generically, describes higher order liquid-liquid or liquid-glass transitions. We show that the static replicated theory implies slowing down with a logarithmic decay in time. The dynamical equations turn out to be those predicted by schematic Mode Coupling Theory for supercooled viscous liquids at a $A_3$ singularity, where the parameter exponent is $λ=1$. We obtain a quantitative expression for the parameter $μ$ of the logarithmic decay in terms of cumulants of the overlap, which are physically observable in experiments or numerical simulations.

cond-mat.dis-nn↗

arXiv2023

Loop expansion around the Bethe approximation through the $M$-layer construction

For every physical model defined on a generic graph or factor graph, the Bethe $M$-layer construction allows building a different model for which the Bethe approximation is exact in the large $M$ limit and it coincides with the original model for $M=1$. The $1/M$ perturbative series is then expressed by a diagrammatic loop expansion in terms of so-called fat-diagrams. Our motivation is to study some important second-order phase transitions that do exist on the Bethe lattice but are either qualitatively different or absent in the corresponding fully connected case. In this case the standard approach based on a perturbative expansion around the naive mean field theory (essentially a fully connected model) fails. On physical grounds, we expect that when the construction is applied to a lattice in finite dimension there is a small region of the external parameters close to the Bethe critical point where strong deviations from mean-field behavior will be observed. In this region, the $1/M$ expansion for the corrections diverges and it can be the starting point for determining the correct non-mean-field critical exponents using renormalization group arguments. In the end, we will show that the critical series for the generic observable can be expressed as a sum of Feynman diagrams with the same numerical prefactors of field theories. However, the contribution of a given diagram is not evaluated associating Gaussian propagators to its lines as in field theories: one has to consider the graph as a portion of the original lattice, replacing the internal lines with appropriate one-dimensional chains, and attaching to the internal points the appropriate number of infinite-size Bethe trees to restore the correct local connectivity of the original model.

cond-mat.dis-nn↗

arXiv2023

Corrections to the Bethe lattice solution of Anderson localization

We study numerically Anderson localization on lattices that are tree-like except for the presence of one loop of varying length $L$. The resulting expressions allow us to compute corrections to the Bethe lattice solution on i) Random-Regular-Graph (RRG) of finite size $N$ and ii) euclidean lattices in finite dimension. In the first case we show that the $1/N$ corrections to to the average values of observables such as the typical density of states and the inverse participation ratio have prefactors that diverge exponentially approaching the critical point, which explains the puzzling observation that the numerical simulations on finite RRGs deviate spectacularly from the expected asymptotic behavior. In the second case our results, combined with the $M$-layer expansion, predict that corrections destroy the exotic critical behavior of the Bethe lattice solution in any finite dimension, strengthening the suggestion that the upper critical dimension of Anderson localization is infinity. This approach opens the way to the computation of non-mean-field critical exponents by resumming the series of diverging diagrams through the same recipes of the field-theoretical perturbative expansion.

cond-mat.dis-nn↗

arXiv2022

Theory of Kinetically-Constrained-Models Dynamics

The mean-field theory of Kinetically-Constrained-Models is developed by considering the Fredrickson-Andersen model on the Bethe lattice. Using certain properties of the dynamics observed in actual numerical experiments we derive asymptotic dynamical equations equal to those of Mode-Coupling-Theory. Analytical predictions obtained for the dynamical exponents are successfully compared with numerical simulations in a wide range of models, including the case of generic values of the connectivity and the facilitation, random pinning and fluctuating facilitation. The theory is thus validated for both continuous and discontinuous transitions and also in the case of higher order critical points characterized by logarithmic decays.

cond-mat.dis-nn↗

arXiv2022

The Ising spin glass on random graphs at zero temperature: not all spins are glassy in the glassy phase

We investigate the replica symmetry broken (RSB) phase of spin glass (SG) models in a random field defined on Bethe lattices at zero temperature. From the properties of the RSB solution we deduce a closed equation for the extreme values of the cavity fields. This equation turns out not to depend on the parameters defining the RSB, and it predicts that the spontaneous RSB does not take place homogeneously on the whole system. Indeed, there exist spins having the same effective local field in all local ground states, exactly as in the replica symmetric (RS) phase, while the spontaneous RSB manifests only on the remaining spins, whose fraction vanishes at criticality. The characterization in terms of spins having fixed or fluctuating local fields can be extended also to the random field Ising model (RFIM), in which case the fluctuating spins are the only responsible for the spontaneous magnetization in the ferromagnetic phase. Close to criticality we are able to connect the statistics of the local fields acting on the spins in the RSB phase with the correlation functions measured in the paramagnetic phase. Identifying the two types of spins on given instances of SG and RFIM, we show that they participate very differently to avalanches produced by flipping a single spin. From the scaling of the number of spins inducing RSB effects close to the critical point and using the $M$-layer expansion we estimate the upper critical dimension $D_U \geq 8$ for SG.

cond-mat.dis-nn↗

arXiv2022

Stochastic equations and dynamics beyond mean-field theory

The dynamical transition occurring in spin-glass models with one step of Replica-Symmetry-Breaking is a mean-field artifact that disappears in finite systems and/or in finite dimensions. The critical fluctuations that smooth the transition are described in the $β$ regime by dynamical stochastic equations. The quantitative parameters of the dynamical stochastic equations have been computed analytically on the 3-spin Bethe lattice Spin-Glass by means of the (static) cavity method and the equations have been solved numerically. The resulting parameter-free dynamical predictions are shown here to be in excellent agreement with numerical simulation data for the correlation and its fluctuations.

cond-mat.dis-nn↗

arXiv2022

Unexpected upper critical dimension for spin glass models in a field predicted by the loop expansion around the Bethe solution at zero temperature

The spin-glass transition in a field in finite dimension is analyzed directly at zero temperature using a perturbative loop expansion around the Bethe lattice solution. The loop expansion is generated by the $M$-layer construction whose first diagrams are evaluated numerically and analytically. The generalized Ginzburg criterion reveals that the upper critical dimension below which mean-field theory fails is $D_U \le 8$, at variance with the classical result $D_U = 6$ yielded by finite-temperature replica field theory. Our expansion around the Bethe lattice has two crucial differences with respect to the classical one. The finite connectivity $z$ of the lattice is directly included from the beginning in the Bethe lattice, while in the classical computation the finite connectivity is obtained through an expansion in $1/z$. Moreover, if one is interested in the zero temperature ($T = 0$) transition, one can directly expand around the $T = 0$ Bethe transition. The expansion directly at $T = 0$ is not possible in the classical framework because the fully connected spin glass does not have a transition at $T = 0$, being in the broken phase for any value of the external field.

cond-mat.dis-nn↗