arXiv subjects

Manaka Okuyama

Publications and source records attributed to Manaka Okuyama.

At least 19 recordsLinked to original sources

arXiv2026

Absence of Spin-Glass Order on Migdal--Kadanoff Hierarchical Lattices near Three Dimensions

We derive a rigorous sufficient condition for the absence of spin-glass order in the Ising spin glass with symmetric binary couplings on Migdal--Kadanoff (MK) hierarchical lattices with even branching number. The key observation is that a single exact renormalization-group step creates zero effective bonds with positive probability, thereby reducing the problem to bond percolation on the corresponding hierarchical lattice. When the induced dilution exceeds the percolation threshold, both spin-glass order and stiffness are absent at all temperatures, including zero temperature. As a consequence, our criterion gives a rigorous proof that the Ising spin glass on the square lattice does not exhibit a spin-glass phase within the MK approximation. More unexpectedly, by choosing sufficiently large scale factors and branching numbers, we construct MK hierarchical lattices whose fractal dimensions are arbitrarily close to three from below but still exhibit no spin-glass order. This sharply contrasts with numerical estimates obtained for MK hierarchical lattices with relatively small scale factors and branching numbers, which placed the lower critical dimension near $2.52$.

cond-mat.dis-nn↗

arXiv2026

Existence of a Phase Transition in the One-Dimensional Ising Spin Glass Model with Long-Range Interactions on the Nishimori Line

Dyson [Commun. Math. Phys. 12, 91 (1969)] rigorously proved the existence of a phase transition in the one-dimensional Ising model with long-range interactions of the form $r^{-α}$ for $1 < α< 2$. In the present study, we extend this result to the Ising spin glass model with Gaussian disorder on the Nishimori line. Following Dyson's method, we first prove the existence of long-range order at finite low temperatures in the Dyson hierarchical Ising spin glass model on the Nishimori line, with power-law-like interactions $J(r) \sim r^{-α}$ for $1 < α< 3/2$. The key ingredients of the proof are the interpolation method developed in the rigorous analysis of mean-field spin glass models, the Gibbs--Bogoliubov inequality on the Nishimori line, and the Tsirelson--Ibragimov--Sudakov inequality (Gaussian concentration inequality). We then use the Griffiths inequality on the Nishimori line to rigorously establish the existence of a phase transition in the one-dimensional Ising spin glass model with long-range interactions on the Nishimori line for $1 < α< 3/2$. For $3/2 \le α\le 2 $, the existence of a phase transition remains an open problem.

math-ph↗

arXiv2026

Griffiths inequalities and Gibbs-Bogoliubov inequality for general gauge glasses with Gaussian disorder on Nishimori line

We consider a class of gauge glass models with Gaussian disorder on the Nishimori line, including the Ising spin glass, the $XY$ gauge glass, the $Z_q$ gauge glass, and the gauge-invariant Potts model. We prove that the first and second Griffiths inequalities hold for these models on arbitrary lattice structures. As a consequence, both the pressure and the correlation functions are monotonically increasing with respect to the inverse temperature along the Nishimori line. Furthermore, we establish an analogue of the Gibbs--Bogoliubov inequality for this class of models. This result implies that, on the Nishimori line, the approximate quenched free energy obtained via the replica method with a replica-symmetric mean-field approximation is always greater than the true quenched free energy. Our results provide a broad generalization of previous results established for the Ising spin glass with Gaussian disorder on the Nishimori line.

math-ph↗

arXiv2025

Temperature chaos as a logical consequence of the reentrant transition in spin glasses

Temperature chaos is a striking phenomenon in spin glasses, where even slight changes in temperature lead to a complete reconfiguration of the spin state. Another intriguing effect is the reentrant transition, in which lowering the temperature drives the system from a ferromagnetic phase into a less ordered spin-glass or paramagnetic phase. In the present paper, we reveal an unexpected connection between these seemingly unrelated phenomena in the finite-dimensional Edwards-Anderson model of spin glasses by introducing a generalized formulation that incorporates correlations among disorder variables. Assuming the existence of a spin glass phase at finite temperature, we establish that temperature chaos arises as a logical consequence of reentrance in the Edwards-Anderson model. Our findings uncover a previously hidden mathematical structure relating reentrance and temperature chaos, offering a new perspective on the physics of spin glasses beyond the mean-field theory.

cond-mat.dis-nn↗

arXiv2025

Free energy equivalence between mean-field models and nonsparsely diluted mean-field models

We studied nonsparsely diluted mean-field models that differ from sparsely diluted mean-field models, such as the Viana--Bray model. When the existence probability of each edge follows a Bernoulli distribution, we rigorously prove that the free energy of nonsparsely diluted mean-field models with appropriate parameterization coincides exactly with that of the corresponding mean-field models in ferromagnetic and spin-glass models composed of any discrete spin $S$ in the thermodynamic limit. Our results is a broad generalization of the result of a previous study [Bovier and Gayrard, J. Stat. Phys. 72, 643 (1993)], where the densely diluted mean-field ferromagnetic Ising model (diluted Curie--Weiss model) with appropriate parameterization was analyzed rigorously, and it was proven that its free energy was exactly equivalent to that of the corresponding mean-field model (Curie--Weiss model).

cond-mat.dis-nn↗

arXiv2025

Existence of long-range order in random-field Ising model on Dyson hierarchical lattice

We study the random-field Ising model on a Dyson hierarchical lattice, where the interactions decay in a power-law-like form, $J(r)\sim r^{-α}$, with respect to the distance. Without a random field, the Ising model on the Dyson hierarchical lattice has a long-range order at finite low temperatures when $1<α<2$. In this study, for $1<α<3/2$, we rigorously prove that there is a long-range order in the random-field Ising model on the Dyson hierarchical lattice at finite low temperatures, including zero temperature, when the strength of the random field is sufficiently small but nonzero. Our proof is based on Dyson's method for the case without a random field, and the concentration inequalities in probability theory enable us to evaluate the effect of a random field.

math-ph↗

arXiv2024

Replica bound for Ising spin glass models in one dimension

The interpolation method is a powerful tool for rigorous analysis of mean-field spin glass models, both with and without dilution. In this study, we show that the interpolation method can be applied to Ising spin glass models in one dimension, such as a one-dimensional chain and a two-leg ladder. In one dimension, the replica symmetric (RS) cavity method is naturally expected to be rigorous for Ising spin glass models. Using the interpolation method, we rigorously prove that the RS cavity method provides lower bounds on the quenched free energies of Ising spin glass models in one dimension at any finite temperature in the thermodynamic limit.

cond-mat.dis-nn↗

arXiv2024

Toward mean-field bound for critical temperature on Nishimori line

The critical inverse temperature of the mean-field approximation establishes a lower bound of the true critical inverse temperature in a broad class of ferromagnetic spin models. This is referred to as the mean-field bound for the critical temperature. In this study, we explored the possibility of a corresponding mean-field bound for the critical temperature in Ising spin glass models with Gaussian randomness on the Nishimori line. On this line, the critical inverse temperature of the mean-field approximation is given by $β_{MF}^{NL}=\sqrt{1/z}$, where $z$ is the coordination number. Using the Griffiths inequalities on the Nishimori line, we proved that there is zero spontaneous magnetization in the high-temperature region $β< β_{MF}^{NL}/2$. In other words, the true critical inverse temperature $β_c^{NL}$ on the Nishimori line is always bounded by $β_c^{NL} \ge β_{MF}^{NL}/2$. Unfortunately, we have not succeeded in obtaining the corresponding mean-field bound $β_c^{NL} \ge β_{MF}^{NL}$ on the Nishimori line.

cond-mat.dis-nn↗

arXiv2023

Exact solution of free entropy for matrix-valued geometric Brownian motion with non-commutative matrices via the replica method

Geometric Brownian motion (GBM) is a standard model in stochastic differential equations. In this study, we consider a matrix-valued GBM with non-commutative matrices. Introduction of non-commutative matrices into the matrix-valued GBM makes it difficult to obtain an exact solution because the existence of noise terms prevents diagonalization. However, we show that the replica method enables us to overcome this difficulty. We map the trace of the time evolution operator of the matrix-valued GBM with non-commutative matrices into the partition function of the isotropic Lipkin-Meshkov-Glick model used in quantum spin systems. Then, solving the eigenvalue problem of the isotropic Lipkin-Meshkov-Glick model, we obtain an analytical expression of the free entropy. Numerical simulation is consistent with our analytical result. Thus, our expression is the exact solution of the free entropy for the matrix-valued GBM with non-commutative matrices.

cond-mat.stat-mech↗

arXiv2023

Gibbs-Bogoliubov inequality on Nishimori line

The Gibbs-Bogoliubov inequality states that the free energy of a system is always lower than that calculated by a trial function. In this study, we show that a counterpart of the Gibbs-Bogoliubov inequality holds on the Nishimori line for Ising spin-glass models with Gaussian randomness. Our inequality states that the quenched free energy of a system is always lower than that calculated using a quenched trial function. The key component of the proof is the convexity of the pressure function $\mathbb{E}\left[\log Z_{} \right]$ with respect to the parameters along the Nishimori line, which differs from the conventional convexity with respect to the inverse temperature. When our inequality was applied to mean-field models, such as the Sherrington-Kirkpatrick model and $p$-spin model, the bound coincided with the replica-symmetric solution indicating that the equality holds.

cond-mat.stat-mech↗

arXiv2023

Threshold theorem in quantum annealing with deterministic analog control errors

We investigate the effect of deterministic analog control errors in the time-dependent Hamiltonian on isolated quantum dynamics. Deterministic analog control errors are formulated as time-dependent operators in the Schrodinger equation. We give an upper bound on the distance between two states in time evolution with and without deterministic analog control errors. As a result, we prove that, if the strength of deterministic analog control errors is less than the inverse of computational time, the final state in quantum dynamics without deterministic analog control errors can be obtained through a constant-order number of measurements in quantum dynamics with deterministic analog control errors.

quant-ph↗

arXiv2023

Mean-field theory is exact for Ising spin glass models with Kac potential in non-additive limit on Nishimori line

Recently, Mori [Phys. Rev. E 84, 031128 (2011)] has conjectured that the free energy of Ising spin glass models with the Kac potential in the non-additive limit, such as the power-law potential in the non-additive regime, is exactly equal to that of the Sherrington-Kirkpatrick model in the thermodynamic limit. In this study, we prove that his conjecture is true on the Nishimori line at any temperature in any dimension. One of the key ingredients of the proof is the use of the Gibbs-Bogoliubov inequality on the Nishimori line. We also consider the case in which the probability distribution of the interaction is symmetric, where his conjecture is true at any temperature in one dimension but is an open problem in the low-temperature regime in two or more dimensions.

cond-mat.dis-nn↗

arXiv2022

Threshold theorem in isolated quantum dynamics with stochastic control errors

We investigate the effect of stochastic control errors in the time-dependent Hamiltonian on isolated quantum dynamics. The control errors are formulated as time-dependent stochastic noise in the Schrodinger equation. For a class of stochastic control errors, we establish a threshold theorem that provides a sufficient condition to obtain the target state, which should be determined in noiseless isolated quantum dynamics, as a relation between the number of measurements and noise strength. The theorem guarantees that if the sum of the noise strengths is less than the inverse of computational time, the target state can be obtained through a constant-order number of measurements. If the opposite is true, the number of measurements to guarantee obtaining the target state increases exponentially with computational time. Our threshold theorem can be applied to any isolated quantum dynamics such as quantum annealing and adiabatic quantum computation.

quant-ph↗

arXiv2020

Upper bound on the second derivative of the quenched pressure in spin-glass models: weak Griffiths second inequality

The Griffiths first and second inequalities have played an important role in the analysis of ferromagnetic models. In spin-glass models, although the counterpart of the Griffiths first inequality has been obtained, the counterpart of the Griffiths second inequality has not been established. In this study, we generalize the method in the previous work [J. Phys. Soc. Jpn. 76, 074711 (2007)] to the case with multi variables for both symmetric and non-symmetric distributions of the interactions, and derive some correlation inequalities for spin-glass models. Furthermore, by combining the acquired equalities in symmetric distributions, we show that there is a non-trivial positive upper bound on the second derivative of the quenched pressure with respect to the strength of the randomness, which is a weak result of the counterpart of the Griffiths second inequality in spin-glass models for general symmetric distributions.

cond-mat.dis-nn↗

arXiv2020

Some inequalities for correlation functions of Ising models with quenched randomness

Correlation inequalities have played an essential role in the analysis of ferromagnetic models but have not been established in spin glass models. In this study, we obtain some correlation inequalities for the Ising models with quenched randomness, where the distribution of the interactions is symmetric. The acquired inequalities can be regarded as an extension of the previous results, which were limited to the local energy for a spin set, to the local energy for a pair of spin sets. Besides, we also obtain some correlation inequalities for asymmetric distribution.

cond-mat.dis-nn↗

arXiv2020

Inequality for local energy of Ising models with quenched randomness and its application

In this study, we extend the lower bound on the average of the local energy of the Ising model with quenched randomness [J. Phys. Soc. Jpn. 76, 074711 (2007)] obtained for a symmetric distribution to an asymmetric one. Compared with the case of symmetric distribution, our bound has a non-trivial term. By applying the acquired bound to a Gaussian distribution, we obtain the lower bounds on the expectation of the square of the correlation function. Thus, we demonstrate that in the Ising model in a Gaussian random field, the spin-glass order parameter generally has a finite value at any temperature, regardless of the forms of the other interactions.

cond-mat.dis-nn↗

arXiv2018

An exact solution of the partition function for mean-field quantum spin systems without the static approximation

Suzuki-Trotter decomposition is a well-known technique used to calculate the partition function of quantum spin systems, in which the imaginary-time dependence of the partition function occurs inevitably. Since it is very difficult to explicitly treat the imaginary-time dependence of the partition function, we usually neglect the imaginary-time dynamical effect, which is called the static approximation. Although the static approximation is the first approach, it is not even clear when the static approximation is justified for mean-field quantum spin systems, that is, mean-field quantum spin systems have not been solved exactly so far. In this study, we solve exactly the partition function for a particular class of mean-field quantum spin systems including randomness without the static approximation. The partition function can be regarded as a result of time evolution in the imaginary-time Schrödinger equation, and solving the exact solution of the partition function is equivalent to solving the optimal control problem in the imaginary-time Schrödinger equation. As the result, the solution of the optimal control problem coincides exactly with the static approximate solution of the partition function and, therefore, the static approximation is exact for the particular class of mean-field quantum spin systems including randomness in general. Furthermore, we prove that the analysis of the previous study in quantum annealing is exact where the non-stoquastic interaction and the inhomogeneous transverse field accelerate the computational time exponentially for mean-field quantum spin systems.

cond-mat.stat-mech↗

arXiv2018

A useful fundamental speed limit for the imaginary-time Schrodinger equation

The quantum speed limit (QSL), or the energy-time uncertainty relation, gives a fundamental speed limit for quantum dynamics. Recently, Kieu [arXiv:1702.00603] derived a new class of QSL which is not only formal but also suitable for actually evaluating the speed limit. Inspired by his work, we obtain a similar speed limit for the imaginary-time Schrödinger equation. Using this new bound, we show that the optimal computational time of the Grover problem in imaginary-time quantum annealing is bounded from below by $\log N$, which is consistent with a result of previous study.

cond-mat.stat-mech↗