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C. Itoi

Publications and source records attributed to C. Itoi.

At least 19 recordsLinked to original sources
arXiv2026

Self-averaging of replica overlaps in the random field Edwards-Anderson model

The self-averaging of the replica overlap is proven in the Edwards-Anderson (EA) model under random field almost everywhere in the coupling constant space in any dimension. The EA order parameter is represented in terms of the derivative of the free energy density with respect to the random field strength, regardless of boundary conditions. Tasaki's correlation inequality for finite-dimensional spin glass models shows that the expectation of the squared replica overlap is bounded by the squared EA order parameter. These simple evaluations enable us to prove that the variance of the replica overlap vanishes in the infinite-volume limit. The self-averaging of the replica bond overlap is proven also in the EA model with Gaussian exchange interaction without random field. Short-range spin glass models have been shown to behave differently from mean-field spin glass models with RSB phase.

math-ph↗
arXiv2025

Interpolations for a quantum Parisi formula in transverse field mean-field spin glass models

A quantum Parisi formula for the transverse field Sherrington-Kirkpatrick (SK) model is proven with an elementary mathematical method. First, a self-overlap corrected quantum model of the transverse field SK model is represented in terms of the Hamiltonian with annealed random interactions. The interpolation given by Guerra and Toninelli is extended to the self-overlap corrected quantum model. It is proven that the infinite-volume limit of the free energy density exists in the operator formalism. Next, another interpolation developed by Guerra and Talagrand is applied to obtain a finite step replica-symmetry breaking (RSB) bound on the free energy density in the transverse field SK model. The interpolation enables us to show that the deviation of the RSB solution from the exact solution vanishes in the self-overlap corrected quantum model in a functional representation of the quantum spin operators. Finally, the corrected terms are removed by the Hopf-Lax formula for a nonlinear partial differential equation to show the quantum Parisi formula for the original transverse field SK model. The formula is extended to that for the transverse field mean-field $p$-spin glass model.

math-ph↗
arXiv2024

Existence of de Almeida-Thouless-type instability in the transverse field Sherrington-Kirkpatrick model

The interpolation method for mean field spin glass models developed by Guerra and Talagrand is extended to a quantum mean field spin glass model. This extension enables us to obtain both replica-symmetric (RS) and one step replica-symmetry breaking (1RSB) solutions of the free energy density in the transverse field Sherrington-Kirkpatrick model. It is shown that the RS solution is exact in the paramagnetic phase. We provide a sufficient condition on coupling constants where the 1RSB solution gives better bound than the RS one. This condition reduced to physical quantities in disordered single spin systems allows a simple computer-assisted proof for the existence of the de Almeida-Thouless-type instability.

math-ph↗
arXiv2023

Gauge theory for quantum XYZ spin glasses

Nishimori's gauge theory is extended to the quantum XYZ $p$-spin glass model in finite dimensions. This enables us to obtain useful correlation equalities, which show also that Duhamel correlation functions at an arbitrary temperature are bounded by those in the corresponding classical model on the Nishimori line. These bounds give that the spontaneous magnetization vanishes in any low temperature even if the model enters the $\mathbb Z_2$-symmetry broken spin glass phase. This theory explains well-known fact from experiments and numerical calculations that the magnetic susceptibility does not diverge in the spin glass transition. The new gauge theory together with the known phase diagram of the Edwards-Anderson model can specify the spin glass region in the coupling constant space of the quantum Heisenberg XYZ spin glass model.

math-ph↗
arXiv2023

Gauge theory for mixed $p$-spin glasses

Physical quantities in the mixed $p$-spin glasses are evaluated with Nishimori's gauge theory and several variance inequalities. The $\mathbb Z_2$-symmetry breaking and the replica-symmetry breaking are studied in finite and infinite dimensions. Obtained bounds on the expectation of the square of the magnetization and spontaneous magnetization enable us to clarify properties of paramagnetic and spin glass phases. It is proven that variances of ferromagnetic and spin glass order parameters vanish on the Nishimori line in the infinite volume limit. These results imply the self-averaging of these order parameters on the Nishimori line. The self-averaging of the spin glass order parameter rigorously justifies already argued absence of replica-symmetry breaking on the Nishimori line.

math-ph↗
arXiv2022

Universality of replica-symmetry breaking in the transverse field Sherrington-Kirkpatrick model

The existence theorem for replica-symmetry breaking (RSB) in the transverse field Sherrington-Kirkpatrick (SK) model is extended to the model with a general random exchange interactions. The relation between the expectation value of the exchange interaction energy and the Duhamel correlation function of spin operators can be obtained by an approximate integration by parts for general random interactions. In addition to the Falk-Bruch inequality, these explicit evaluations enable us to prove that the variance of overlap between two replica spin operators does not vanish under sufficiently weak transverse field in sufficiently low temperature. The absence of the ferromagnetic long range order is also shown to distinguish RSB from the $\mathbb Z_2$-symmetry breaking.

math-ph↗
arXiv2021

Expansion for quantum perturbations in random spin systems

Energy eigenstates in the random transverse field Edwards-Anderson (EA) model and the random bond quantum Heisenberg XYZ model in a $d$-dimensional finite cubic lattice are obtained for sufficiently weak interactions. The Datta-Kennedy-Kirkwood-Thomas convergent perturbative expansion using the contraction mapping theorem is developed for quantum spin systems with site- and bond-dependent interactions. This expansion enables us to obtain energy eigenstates in the random transverse field free spin model perturbed by sufficiently weak longitudinal exchange interactions. This expansion is useful also for the EA model perturbed by sufficiently weak transverse fields and bond-dependent XY exchange interactions. In these models, their perturbations split the two fold degenerate energy eigenvalues because of the ${\mathbb Z}_2$ symmetry in the unperturbed EA model. It is shown that the energy gap between split energy eigenvalues is exponentially small in the system size. We provide a sufficient condition on the perturbation for absence of level crossing between arbitrary energy eigenstates.

math-ph↗
arXiv2021

Uniqueness of ground state in the Edwards-Anderson spin glass model

It is proven rigorously that the ground state in the Edwards-Anderson spin glass model is unique in any dimension for almost all continuous random exchange interactions under a condition that a single spin breaks the global ${\mathbb Z}_2$ symmetry. This theorem implies that replica symmetry breaking does not occur at zero temperature. The site- and bond-overlap are concentrated at their maximal values. It is argued that behaviors of short range spin glass models are much different from those of mean field spin glass models near zero temperature. Errata have been attached at the final page.

cond-mat.dis-nn↗
arXiv2021

Absence of replica symmetry breaking in the Edwards-Anderson model near zero temperature

It is proven that the ground state is unique in the Edwards-Anderson model for almost all continuous random exchange interactions, and any excited state with the overlap less than its maximal value has large energy in dimensions higher than two with probability one. Since the spin overlap is shown to be concentrated at its maximal value in the ground state, replica symmetry breaking does not occur in the Edwards-Anderson model near zero temperature.

math-ph↗
arXiv2020

Absence of replica symmetry breaking in the transverse and longitudinal random field Ising model

It is proved that replica symmetry is not broken in the transverse and longitudinal random field Ising model. In this model, the variance of spin overlap of any component vanishes in any dimension almost everywhere in the coupling constant space in the infinite volume limit. The weak Fortuin-Kasteleyn-Ginibre property in this model and the Ghirlanda-Guerra identities in artificial models in a path integral representation based on the Lie-Trotter-Suzuki formula enable us to extend Chatterjee's proof for the random field Ising model to the quantum model.

math-ph↗
arXiv2020

The second law of thermodynamics from concavity of energy eigenvalues

Quantum dynamics controlled by a time-dependent coupling constant are studied. It is proven that an energy eigenstate expectation value of work done by the system in a quench process cannot exceed the work in the corresponding quasi-static process, if and only if the energy eigenvalue is a concave function of the coupling constant. We propose this concavity of energy eigenvalues as a new universal criterion for quantum dynamical systems to satisfy the second law of thermodynamics. We argue simple universal conditions on quantum systems for the concavity, and show that every energy eigenvalue is indeed concave in some specific quantum systems. These results agree with the maximal work principle for adiabatic quench and quasi-static processes as an expression of the second law of thermodynamics. Our result gives a simple example of an integrable system satisfying an analogue to the strong eigenstate thermalization hypothesis (ETH) with respect to the principle of maximum work.

cond-mat.stat-mech↗
arXiv2020

The Schwartz-Soffer and more inequalities for random fields

A new series of correlation inequalities for random field spin systems is proven rigorously. First one corresponds to the well-known Schwartz-Soffer inequality. These are expected to rule out incorrect results calculated in effective theories and numerical studies. The large $N$ expansion with the replica method for random field systems as an example is checked by these inequalities. It is shown that several critical exponents of multiple-point correlation functions at critical point satisfy obtained inequalities.

math-ph↗
arXiv2019

The second law of thermodynamics from concave energy in classical mechanics

A recently proposed quantum mechanical criterion `concavity of energy' for the second law of thermodynamics is studied also for classical particle systems confined in a bounded region by a potential with a time-dependent coupling constant. It is shown that the time average of work done by particles in a quench process cannot exceed that in the corresponding quasi-static process, if the energy is a concave function of the coupling constant. It is proven that the energy is indeed concave for a general confining potential with certain properties. This result implies that the system satisfies the principle of maximum work in the adiabatic environment as an expression of the second law of thermodynamics.

cond-mat.stat-mech↗
arXiv2019

Absence of replica symmetry breaking in disordered FKG-Ising models under uniform field

We prove that the variance of spin overlap vanishes in disordered Ising models satisfying the Fortuin-Kasteleyn-Ginibre (FKG) inequality under a uniform field, such as generally distributed random field Ising model, site- and bond-diluted Ising models with the Bernoulli distribution. Chatterjee's proof for the Gaussian random field Ising model is generalized to other independent identically distributed quenched disorder under a uniform field.

math-ph↗
arXiv2019

Self-averaging of perturbation Hamiltonian density in perturbed spin systems

It is shown that the variance of a perturbation Hamiltonian density vanishes in the infinite-volume limit of the perturbed spin systems with quenched disorder. This is proven in a simpler way and under less assumptions than before. A corollary of this theorem indicates the impossibility of non-spontaneous replica symmetry-breaking in disordered spin systems. The commutativity between the infinite-volume limit and the switched-off limit of a replica symmetry-breaking perturbation implies that the variance of the spin overlap vanishes in the replica symmetric Gibbs state.

math-ph↗
arXiv2019

Zero-variance of perturbation Hamiltonian density in perturbed spin systems

We study effects of perturbation Hamiltonian to quantum spin systems which can include quenched disorder. Model-independent inequalities are derived, using an additional artificial disordered perturbation. These inequalities enable us to prove that the variance of the perturbation Hamiltonian density vanishes in the infinite volume limit even if the artificial perturbation is switched off. This theorem is applied to spontaneous symmetry breaking phenomena in a disordered classical spin model, a quantum spin model without disorder and a disordered quantum spin model.

math-ph↗
arXiv2017

Universal nature of replica symmetry breaking in quantum systems with Gaussian disorder

We study quantum spin systems with quenched Gaussian disorder. We prove that the variance of all physical quantities in a certain class vanishes in the infinite volume limit. We study also replica symmetry breaking phenomena, where the variance of an overlap operator in the other class does not vanish in the replica symmetric Gibbs state. On the other hand, it vanishes in a spontaneous replica symmetry breaking Gibbs state defined by applying an infinitesimal replica symmetry breaking field. We prove also that the finite variance of the overlap operator in the replica symmetric Gibbs state implies the existence of a spontaneous replica symmetry breaking.

math-ph↗