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Journal of Physics A Mathematical and General

Spherical random-field systems with long-range interactions: general results and application to the Coulomb glass

Thomas Vojta

Published Jun 21, 1993• 23 citations
Publisher

Abstract

A classical spherical random-field Hamiltonian with long-range (power-law) interactions is investigated by means of the replica theory. Both ferromagnetic and anti-ferromagnetic interactions are considered. The use of continuous variables instead of Ising variables in the spherical version of the model allows one to calculate the free energy exactly. The existence of an equilibrium phase transition is investigated based on the replica-symmetric solution. The results are applied to the Coulomb-glass model of interacting localized electrons in a disordered solid. This model is shown not to have an equilibrium phase transition for spatial dimensions D<or=4. For D>4 the model has a phase transition to an ordered phase; however, it does not have a phase transition to a 'glassy' phase.

DOI
10.1088/0305-4470/26/12/025

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A classical spherical random-field Hamiltonian with long-range (power-law) interactions is investigated by means of the replica theory. Both ferromagnetic and anti-ferromagnetic interactions are considered. The use of continuous variables instead of Ising variables in the spherical version of the model allows one to calculate the free energy exactly. The existence of an equilibrium phase transition is investigated based on the replica-symmetric solution. The results are applied to the Coulomb-glass model of interacting localized electrons in a disordered solid. This model is shown not to have an equilibrium phase transition for spatial dimensions D<or=4. For D>4 the model has a phase transition to an ordered phase; however, it does not have a phase transition to a 'glassy' phase.
We discuss the spin glass states of the deterministic spin model with long range anti-ferromagnetic interactions. To apply the replica method, the partial statistical summation with fixed overlap parameters is introduced. By this formulation, we find that the replica theory for this model reduces to that of the anti-Hebbian model in the long range limit, suggesting the existence of dynamical phase transition and glassy states. Results of simulated annealing are also presented to confirm these results.
We study analytically the equilibrium properties of the spherical hierarchical model in the presence of random fields. The expression for the critical line separating a paramagnetic from a ferromagnetic phase is derived. The critical exponents characterising this phase transition are computed analytically and compared with those of the corresponding D -dimensional short-range model, leading to conclude that the usual mapping between one dimensional long-range models and D -dimensional short-range models holds exactly for this system, in contrast to models with Ising spins. Moreover, the critical exponents of the pure model and those of the random field model satisfy a relationship that mimics the dimensional reduction rule. The absence of a spin-glass phase is strongly supported by the local stability analysis of the replica symmetric saddle-point as well as by an independent computation of the free-energy using a renormalization-like approach. This latter result enlarges the class of random field models for which the spin-glass phase has been recently ruled out.
In the present paper we analyze the critical properties of a quantum spherical spin-glass model with short-range random interactions. Since the model allows for rigorous detailed calculations, we can show how the effective partition function calculated with the help of the replica method for the spin-glass fluctuating fields ${Q}_{\ensuremath{\alpha}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$ separates into a mean-field contribution for ${Q}_{\ensuremath{\alpha}\ensuremath{\alpha}}(0;\ensuremath{\omega};\ensuremath{-}\ensuremath{\omega})$ and a strictly short-range partition function for the fields ${Q}_{\ensuremath{\alpha}\ensuremath{\ne}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$. Here $\ensuremath{\alpha},\ensuremath{\gamma}=1,\dots{},n$ are replica indices. The mean-field part WMF{W}_{\text{MF}}WMF​ coincides with previous results. The short-range part WSR{W}_{\text{SR}}WSR​ describes a phase transition in a Q3{Q}^{3}Q3-field theory, where the fluctuating fields depend on a space variable $\stackrel{P\vec}{r}$ and times ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$. This we analyze using the renormalization group with dimensional regularization and minimal subtraction of dimensional poles. By generalizing standard field theory methods to our particular situation, we can identify the critical dimensionality as dc=5{d}_{c}=5dc​=5 at very low temperatures due to the dimensionality shift Dc=dc+1=6{D}_{c}={d}_{c}+1=6Dc​=dc​+1=6. We then perform an ${ϵ}^{\ensuremath{'}}$ expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.
We analyze the critical properties of a quantum spherical spin-glass model with random inverse power-law interactions ${r}^{\ensuremath{-}(d+\ensuremath{\sigma})}$. It was shown in a previous publication that the effective partition function calculated with help of the replica method for the spin-glass fluctuating fields ${Q}_{\ensuremath{\alpha}\ensuremath{\gamma}}(\stackrel{P\vec}{k},{\ensuremath{\omega}}_{1},{\ensuremath{\omega}}_{2})$ separates into a mean-field contribution for the ${Q}_{\ensuremath{\alpha}\ensuremath{\alpha}}(0,\ensuremath{\omega},\ensuremath{-}\ensuremath{\omega})$ and a strictly long-range partition function for the fields ${Q}_{\ensuremath{\alpha}\ensuremath{\ne}\ensuremath{\gamma}}(\stackrel{P\vec}{k},{\ensuremath{\omega}}_{1},{\ensuremath{\omega}}_{2})$. Here $\ensuremath{\alpha},\ensuremath{\gamma}=1,\dots{},n$ are replica indices. The long-range part WLR{W}_{\text{LR}}WLR​ describes a phase transition in a Q3{Q}^{3}Q3-field theory that we analyze using the renormalization group with dimensional regularization and minimal subtraction of dimensional poles. By generalizing standard field theory methods to our particular situation, we can identify the upper critical dimensionality as ${d}_{c}=\frac{5\ensuremath{\sigma}}{2}$ at very low temperatures due to the dimensionality shift ${D}_{c}={d}_{c}+\frac{\ensuremath{\sigma}}{2}=3\ensuremath{\sigma}$. We then perform an ${ϵ}^{\ensuremath{'}}={d}_{c}\ensuremath{-}d$ expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.
We study the random-field Ising model with long-range interactions and show the exactness of the mean-field theory under certain mild conditions. This is a generalization of the result of Mori for the non-random and spin-glass cases. To treat random fields, we evoke the self-averaging property of a function of random fields, without recourse to the replica method. The result is that the mean-field theory gives the exact expression of the canonical free energy for systems with power-decaying interactions if the power is smaller than or equal to the spatial dimension.