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arXiv:1105.4955 · Europhysics Letters (EPL)

Real-space Renormalization Group analysis of a non-mean-field spin-glass

Michele Castellana

Published Jul 29, 2011• 29 citations
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Abstract

A real-space Renormalization Group approach is presented for a non-mean-field spin-glass. This approach has been conceived in the effort to develop an alternative method to the Renormalization Group approaches based on the replica method. Indeed, non-perturbative effects in the latter are quite generally out of control, in such a way that such approaches are non-predictive. On the contrary, we show that the real-space method presented here yields a precise prediction of the critical behavior and exponents of the model.

DOI
10.1209/0295-5075/95/47014

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A real-space Renormalization Group approach is presented for a non-mean-field spin-glass. This approach has been conceived in the effort to develop an alternative method to the Renormalization Group approaches based on the replica method. Indeed, non-perturbative effects in the latter are quite generally out of control, in such a way that such approaches are non-predictive. On the contrary, we show that the real-space method presented here yields a precise prediction of the critical behavior and exponents of the model.
Abstract We focus on two real-space renormalization-group (RG) methods recently proposed for a hierarchical model of a spin glass: a sample-by-sample method, in which the RG transformation is performed separately on each disorder sample, and an ensemble RG (ERG) method (Angelini M C, Parisi G and Ricci-Tersenghi F 2013 Ensemble renormalization group for disordered systems Phys. Rev . B 87 134201) in which the transformation is based on an average over samples. Above the upper critical dimension, the sample-by-sample method yields the correct mean-field value for the critical exponent related to the divergence of the correlation length, while it does not predict the correct qualitative behavior of below the upper critical dimension. On the other hand, the ERG procedure has been claimed to predict the correct behavior of both above and below the upper critical dimension. Here, we straighten out the reasons for the discrepancy between the two methods above, by demonstrating that the ERG method predicts a marginally stable critical fixed point, thus implying a prediction for the critical exponent given by . This prediction disagrees, on a qualitative and quantitative level, both with the mean-field value of above the critical dimension, and with numerical estimates of below the upper critical dimension. Therefore, our results show that finding a real-space RG method for spin glasses which yields the correct prediction for universal quantities below the upper critical dimension is still an open problem, for which our analysis may provide some general guidance for future studies.
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.
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.
By using real-space renormalization group (RG) methods, we show that spin glasses in a field display a new kind of transition in high dimensions. The corresponding critical properties and the spin-glass phase are governed by two nonperturbative zero-temperature fixed points of the RG flow. We compute the critical exponents and discuss the RG flow and its relevance for three-dimensional systems. The new spin-glass phase we discovered has unusual properties, which are intermediate between the ones conjectured by droplet and full replica symmetry-breaking theories. These results provide a new perspective on the long-standing debate about the behavior of spin glasses in a field.
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.