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Domain-Wall Renormalization-Group Study of the Edwards-Anderson Ising Spin Glass in a Magnetic Field

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Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)

Abstract

In spite of extensive studies for more than two decades, a basic problem on the field-temperature phase diagram of the short-range Ising spin glass is still controversial. The mean-field picture insists the existence of the spin glass phase in magnetic field. This means that a transition from paramagnetic phase to spin glass phase occurs at a finite temperature if field is smaller than some critical value $H_c$. On the other hand, the droplet theory, which is a phenomenological theory for short-range spin glasses, predicts that the spin-glass order is unstable even against an infinitesimal field. It is a long-standing question whether the spin glass phase in field exists in real Ising spin glasses or not. In this work, we study the Edwards-Anderson (EA) short-range Ising spin glass in field $H$ by a numerical domain-wall renormalization-group method 1. This method enables us to measure effective couplings $J_eff$ and effective fields $H_eff$ of length scale $L$ within the block spin picture. Because $J_eff$ and the free-energy difference $F$ caused by changing the boundary condition from periodic to anti-periodic are related by $J_eff=-F/2$ in zero field, we consider that $J_eff$ represents the strength of the spin-glass order. Since $J_eff$ is either positive or negative, we calculate the standard deviation of sample-to-sample fluctuations of the effective couplings, $\sigma_J(L,H)$. The figure shows result of the three-dimensional $J$ EA Ising spin glass. In the inset, $\sigma_J$ is plotted as a function of $L$. In the main frames, we test the scaling equation \sigma_J(L,H) /\ell(H)^þeta =\sigma_JL/\ell(H), equation predicted by the droplet theory. In this equation, $\ell(H)=H^1d/2-þeta$ is the overlap length, $þeta$ is the stiffness exponent, $d$ is the dimension, and $\tilde\sigma_J$ is scaling function. $d$ is fixed to $3$ and $þeta$ is estimated by fitting. Although the value of $þeta$ is a bit higher than previous estimations (around 0.25), the scaling works nicely. We also find that the scaling function $\sigma_J(X)$ drops to zero for large $X$. This means that the spin-glass order is destroyed by field beyond the crossover length. Since the crossover length obeys a power law of $H$ which diverges as $H 0$ but remains finite for any non-zero $H$, the scaling implies that the spin-glass phase is absent even in an infinitesimal field.2mm 1) M.Sasaki, K. Hukushima, H. Yoshino and H. Takayama, cond-mat/0702302.

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