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arXiv:cond-mat/0207460 · Physical review. B, Condensed matter

Eigenvalue analysis of the density matrix of four-dimensional spin glasses supports replica symmetry breaking

Lorenzo Correale, Enzo Marinari, Víctor Martín-Mayor

Published Nov 4, 2002• 2 citations
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Abstract

We present a general and powerful numerical method useful to study the density matrix of spin models. We apply the method to finite-dimensional spin glasses, and analyze in detail the four-dimensional Edwards-Anderson model with Gaussian quenched random couplings. Our results clearly support the existence of replica symmetry breaking in the thermodynamical limit.

DOI
10.1103/physrevb.66.174406

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We present a general and powerful numerical method useful to study the density matrix of spin models. We apply the method to finite-dimensional spin glasses, and analyze in detail the four-dimensional Edwards-Anderson model with Gaussian quenched random couplings. Our results clearly support the existence of replica symmetry breaking in the thermodynamical limit.
The nature of spin-glass phase of the four-dimensional Edwards-Anderson Ising model is numerically studied by eigenmode analysis of the susceptibility matrix up to the lattice size 10^4. Unlike the preceding results on smaller lattices, our result suggests that there exist multiple extensive eigenvalues of the matrix, which does not contradict replica-symmetry-breaking scenarios. The sensitivity of the eigenmodes with respect to a temperature change is examined using finite-size-scaling analysis and an evidence of anomalous sensitivity is found. A computational advantage of dual formulation of the eigenmode analysis in the study of large lattices is also discussed.
We characterize numerically the properties of the phase transition of the three-dimensional Ising spin glass with Gaussian couplings and of the low-temperature phase. We compute critical exponents on large lattices. We study in detail the overlap probability distribution and the equilibrium overlap-overlap correlation functions. We find a clear agreement with off-equilibrium results from previous work. These results strongly support the existence of a continuous spontaneous replica symmetry breaking in three-dimensional spin glasses.
We apply the real space renormalization group (RG) to the replica Hamiltonian of the two dimensional Ising Edwards–Anderson model to discuss the existence of the spin glass (SG) phase. We derive the RG equations under the replica symmetric (RS) ansatz and the resulting flow diagram indicates the existence of the SG phase. The critical exponent for the SG transition has a plausible value while that of the multicritical point (MCP) is a complex number. We consider that this failure at the MCP is due to not taking the RS breaking into account. Indeed we find that the RS breaking parameter is relevant both at the SG critical point and the MCP, indicating the nontriviality of the SG phase of this model.
By using numerical simulations we show that the four-dimensional Edwards Anderson spin glass in magnetic field undergoes a mean-field-like phase transition. We use a dynamical approach: we simulate large lattices (of volume V ) and work out the behaviour of the system in a limit where both t and V go to infinity, but where the limit is taken first. By showing that the dynamic overlap q converges to a value smaller than the static one we exhibit replica symmetry breaking. The critical exponents are compatible with those obtained by mean-field computations.