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Physical Review B

Quantum spherical spin glass with inverse power-law disorder

Pedro Castro Menezes, Alba Theumann

Published Mar 11, 2009• 2 citations
Publisher

Abstract

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 ${W}_{\text{LR}}$ describes a phase transition in a ${Q}^{3}$-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.

DOI
10.1103/physrevb.79.094412

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We analyze the critical properties of a quantum spherical spin-glass model with random inverse power-law interactions r\ensuremath−(d+\ensuremathσ){r}^{\ensuremath{-}(d+\ensuremath{\sigma})}r\ensuremath−(d+\ensuremathσ). 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\ensuremathα\ensuremathα(0,\ensuremathω,\ensuremath−\ensuremathω){Q}_{\ensuremath{\alpha}\ensuremath{\alpha}}(0,\ensuremath{\omega},\ensuremath{-}\ensuremath{\omega})Q\ensuremathα\ensuremathα​(0,\ensuremathω,\ensuremath−\ensuremathω) 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 \ensuremathα,\ensuremathγ=1,…,n\ensuremath{\alpha},\ensuremath{\gamma}=1,\dots{},n\ensuremathα,\ensuremathγ=1,…,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 dc=5\ensuremathσ2{d}_{c}=\frac{5\ensuremath{\sigma}}{2}dc​=25\ensuremathσ​ at very low temperatures due to the dimensionality shift Dc=dc+\ensuremathσ2=3\ensuremathσ{D}_{c}={d}_{c}+\frac{\ensuremath{\sigma}}{2}=3\ensuremath{\sigma}Dc​=dc​+2\ensuremathσ​=3\ensuremathσ. We then perform an ϵ\ensuremath′=dc\ensuremath−d{ϵ}^{\ensuremath{'}}={d}_{c}\ensuremath{-}dϵ\ensuremath′=dc​\ensuremath−d expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.
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 investigate the balanced M=4M=4M=4, p=4p=4p=4 spin-glass model for a one-dimensional long-range proxy for the finite dimensional short-range ppp-spin glass model to examine the nature of the glass transition beyond mean-field theory. We perform large-scale Monte Carlo equilibrated simulations for both fully connected and power-law diluted versions of the model. The critical temperatures extracted from the finite-size scaling (FSS) analysis of spin-glass susceptibility are in good agreement with theoretical predictions for σ=0,0.25σ= 0, 0.25σ=0,0.25, and 0.55. For these values of the long-range exponent σσσ (which is the power of the decrease of the interactions between the spins with their separation), one might have expected that mean-field theory would provide a good description of the system. However, the spin-overlap distribution and the value of the λλλ-parameter do not provide numerical evidence for a one-step replica symmetry breaking (1RSB) phase transition. Instead, our results indicate a direct transition from the paramagnetic state to a full replica symmetry broken phase, with a renormalized value of λ≡ω2/ω1<1λ\equiv ω_2/ω_1 < 1λ≡ω2​/ω1​<1 suggesting a continuous FRSB transition, despite this ratio being equal to 2 at mean-field level. A value of λ>1λ> 1λ>1 is required for the discontinuous 1RSB transition. We argue that strong finite-size effects and closely spaced transition temperatures remove the expected 1RSB transition for the system sizes which we can study. For values of the exponent σ=0.85σ= 0.85σ=0.85, which roughly corresponds to a three dimensional system, we find that the renormalized value of λλλ is again less than 1, with no signs of either the 1RSB transition or the continuous FRSB transition, suggesting that the Kauzmann temperature TKT_KTK​ in three dimensions might be zero and the complete absence of phase transitions in structural glasses.
We consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.
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.
A mean-field spherical model with random couplings between pairs, quartets, and possibly higher multiplets of spins is considered. It has the same critical behavior as the Sherrington-Kirkpatrick model. It thus exhibits replica symmetry breaking. The order parameter function is solved exactly in the whole low temperature phase. The zero-field-cooled susceptibility remains finite at low TTT. Next a quantum version of the system is considered. Whereas the magnetic properties are not altered qualitatively, the thermodynamics is now regular at small temperatures.
The large scale behavior of the simplest non-mean-field spin-glass system is analyzed, and the critical exponent related to the divergence of the correlation length is computed at two loops within the ε-expansion technique with two independent methods. The techniques presented show how the underlying ideas of the renormalization group apply also in this disordered model, in such a way that an ε-expansion can be consistently set up. By pushing such calculation to high orders in ε, a consistent non-mean-field theory for such disordered system could be established, giving a substantial contribution the development of a predictive theory for real spin glasses.