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Hisamitsu Mukaida

Publications and source records attributed to Hisamitsu Mukaida.

18 recordsLinked to original sources
arXiv2024

Griffiths-type theorems for short-range spin glass models

We establish relations between different characterizations of order in spin glass models. We first prove that the broadening of the replica overlap distribution indicated by a nonzero standard deviation of the replica overlap $R^{1,2}$ implies the non-differentiability of the two-replica free energy with respect to the replica coupling parameter $λ$. In $\mathbb Z_2$ invariant models such as the standard Edwards-Anderson model, the non-differentiability is equivalent to the spin glass order characterized by a nonzero Edwards-Anderson order parameter. This generalization of Griffiths' theorem is proved for any short-range spin glass models with classical bounded spins. We also prove that the non-differentiability of the two-replica free energy mentioned above implies replica symmetry breaking in the literal sense, i.e., a spontaneous breakdown of the permutation symmetry in the model with three replicas. This is a general result that applies to a large class of random spin models, including long-range models such as the Sherrington-Kirkpatrick model and the random energy model.

math-ph↗
arXiv2015

Non-differentiability of the effective potential and the replica symmetry breaking in the random energy model

The effective potential for the two-replica system of the random energy model is exactly derived. It is an analytic function of the magnetizations of two replicas, $φ^1$ and $φ^2$ in the high-temperature phase. In the low-temperature phase, where the replica symmetry breaking takes place, the effective potential becomes non-analytic when $φ^1=φ^2$. The non-analyticity is considered as a consequence of the condensation of the Boltzmann measure, which is a typical property of a glass phase.

cond-mat.dis-nn↗
arXiv2014

The generating function for the connected correlator in the random energy model and its effective potential

Starting with two copies of the random energy model coupled with independent magnetic fields, the generating function for the connected correlator of the magnetization is exactly derived. Without use of the replica trick, it is shown that the Hessian of the generating function is symmetric under exchanging the two copies when the system is finite, but the symmetry is spontaneously broken in the low-temperature phase. It can be regarded as a rigorous realization of the replica symmetry breaking. The corresponding effective potential, which has two independent variables conjugate to the magnetic fields, is also calculated. It is singular when the two variables coincide. The singularity is consistent with that observed in the effective potentials of short-ranged disordered systems in the context of the functional renormalization group.

cond-mat.dis-nn↗
arXiv2012

Essential on-Brane Equations for the Braneworld Gravity under the Schwarzschild Ansatz

It is argued that the braneworld gravity under the Schwarzschild ansatz should obey three essential equations on the brane, and they are solved exactly. We express the general solution of the fundamental equations of the braneworld in power series of the brane normal coordinate. The power series are derived by solving three independent components of the bulk Einstein equation, and the coefficients are recursively determined in terms of five functions on the brane. The other components of the bulk Einstein equation are automatically satisfied, as far as the five on-brane functions obey three essential equations. They are the radial-extra and the extra-extra components of the bulk Einstein equation on the brane, and the equation of motion of the brane. Therefore the solution includes two arbitrary functions on the brane. We show that the essential equations are exactly solved by choosing an appropriate set of the two arbitrary functions. The arbitrariness may affect the predictive powers on the Newtonian and the post-Newtonian evidences.

gr-qc↗
arXiv2010

General Solutions of Braneworlds under the Schwarzschild Ansatz

General solutions for braneworld dynamics coupled with the bulk Einstein equation are derived under the Schwarzschild ansatz. They relate the brane metric to the exterior configurations, and establish fine tuning conditions for the braneworlds to reproduce successes of the Einstein gravity.

hep-th↗
arXiv2009

Exactness of the replica method in perturbation

The replica method for a quenched disordered system is considered in a perturbative field theory. Since correction in a finite-order perturbation is given in a polynomial of the replica number $n$, the zero-replica limit $n \to 0$ is regarded as extracting the constant term from the polynomial, which mathematically makes sense. The meaning of the extraction is clarified comparing with a direct calculation.

cond-mat.dis-nn↗
arXiv2006

Stability of fixed points in the (4+ε)-dimensional random field O(N) spin model for sufficiently large N

We study the stability of fixed points in the two-loop renormalization group for the random field O($N$) spin model in $4+ε$ dimensions. We solve the fixed-point equation in the 1/N expansion and $ε$ expansion. In the large-N limit, we study the stability of all fixed points. We solve the eigenvalue equation for the infinitesimal deviation from the fixed points under physical conditions on the random anisotropy function. We find that the fixed point corresponding to dimensional reduction is singly unstable and others are unstable or unphysical. Therefore, one has no choice other than dimensional reduction in the large-N limit. The two-loop $β$ function enables us to find a compact area in the $(d, N)$ plane where the dimensional reduction breaks down. We calculate higher-order corrections in the 1/N and $ε$ expansions to the fixed point. Solving the corrected eigenvalue equation nonperturbatively, we find that this fixed point is singly unstable also for sufficiently large $N$ and the critical exponents show a dimensional reduction.

cond-mat.dis-nn↗
arXiv2005

Effect of second-rank random anisotropy on critical phenomena of random field O(N) spin model in the large N limit

We study the critical behavior of a random field O($N$) spin model with a second-rank random anisotropy term in spatial dimensions $4<d<6$, by means of the replica method and the 1/N expansion. We obtain a replica-symmetric solution of the saddle-point equation, and we find the phase transition obeying dimensional reduction. We study the stability of the replica-symmetric saddle point against the fluctuation induced by the second-rank random anisotropy. We show that the eigenvalue of the Hessian at the replica-symmetric saddle point is strictly positive. Therefore, this saddle point is stable and the dimensional reduction holds in the 1/N expansion. To check the consistency with the functional renormalization group method, we obtain all fixed points of the renormalization group in the large $N$ limit and discuss their stability. We find that the analytic fixed point yielding the dimensional reduction is practically singly unstable in a coupling constant space of the given model with large $N$. Thus, we conclude that the dimensional reduction holds for sufficiently large $N$.

cond-mat.dis-nn↗
arXiv2003

Renormalization group for the probability distribution of magnetic impurities in a random-field $ϕ^4$ model

Extending the usual Ginzburg-Landau theory for the random-field Ising model, the possibility of dimensional reduction is reconsidered. A renormalization group for the probability distribution of magnetic impurities is applied. New parameters corresponding to the extra $ϕ^4$ coupling constants in the replica Hamiltonian are introduced. Although they do not affect the critical phenomena near the upper critical dimension, they can when dimensions are lowered.

cond-mat.stat-mech↗
arXiv2000

Stability of a fixed point in the replica action for the random field Ising model

We reconsider stability of the non-trivial fixed point in $6-ε$ dimensional effective action for the random field Ising model derived by Brézin and De Dominicis. After expansion parameters of physical observables are clarified, we find that the non-trivial fixed point in $6-ε$ dimensions is stable, contrary to the argument by Brézin and De Dominicis. We also computed the exponents $ν$ and $η$ by the $ε$ expansion. The results are consistent with the argument of the dimensional reduction at least in the leading order.

cond-mat.stat-mech↗
arXiv1999

Renormalization group for renormalization-group equations toward the universality classification of infinite-order phase transitions

We derive a new renormalization group to calculate a non-trivial critical exponent of the divergent correlation length which gives a universality classification of essential singularities in infinite-order phase transitions. This method resolves the problem of a vanishing scaling matrix. The exponent is obtained from the maximal eigenvalue of a scaling matrix in this renormalization group, as in the case of ordinary second-order phase transitions. We exhibit several nontrivial universality classes in infinite-order transitions different from the well-known Berezinski\uı-Kosterlitz-Thouless transition.

cond-mat.stat-mech↗
arXiv1998

Replica Method for Wide Correlators in Gaussian Orthogonal, Unitary And Symplectic Random Matrix Ensembles

We calculate connected correlators in Gaussian orthogonal, unitary and symplectic random matrix ensembles by the replica method in the 1/N-expansion. We obtain averaged one-point Green's functions up to the next-to-leading order O(1/N) and wide two-level correlators up to the first nontrivial order O(1/N^2) and wide three-level correlators up to the first nontrivial order $O(1/N^4)$ by carefully treating fluctuations in saddle-point evaluation.

cond-mat↗
arXiv1996

Convex effective potential of $O(N)$-symmetric $phi^4$ theory for large $N$

We obtain effective potential of $O(N)$-symmetric $ϕ^4$ theory for large $N$ starting with a finite lattice system and taking the thermodynamic limit with great care. In the thermodynamic limit, it is globally real-valued and convex in both the symmetric and the broken phases. In particular, it has a flat bottom in the broken phase. Taking the continuum limit, we discuss renormalization effects to the flat bottom and exhibit the effective potential of the continuum theory in three and four dimensions.On the other hand the effective potential is nonconvex in a finite lattice system. Our numerical study shows that the barrier height of the effective potential flattens as a linear size of the system becomes large. It decreases obeying power law and the exponent is about $-2$. The result is clearly understood from dominance of configurations with slowly-rotating field in one direction.

hep-th↗