%0 Book Section %1 statphys23_0304 %A Sasaki, M. %A Hukushima, K. %A Yoshino, H. %A Takayama, H. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K almeida-thouless domain-wall edwards-anderson glass group line method model renormalization spin statphys23 topic-7 %T Domain-Wall Renormalization-Group Study of the Edwards-Anderson Ising Spin Glass in a Magnetic Field %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=304 %X 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.

@incollection{statphys23_0304, 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 $H_{\rm c}$. 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 $J_{\rm eff}$ and effective fields $H_{\rm eff}$ of length scale $L$ within the block spin picture. Because $J_{\rm eff}$ and the free-energy difference $\delta F$ caused by changing the boundary condition from periodic to anti-periodic are related by $J_{\rm eff}=-\delta F/2$ in zero field, we consider that $J_{\rm eff}$ represents the strength of the spin-glass order. Since $J_{\rm eff}$ is either positive or negative, we calculate the standard deviation of sample-to-sample fluctuations of the effective couplings, $\sigma_{\rm J}(L,H)$. The figure shows result of the three-dimensional $\pm J$ EA Ising spin glass. In the inset, $\sigma_{\rm J}$ is plotted as a function of $L$. In the main frames, we test the scaling \begin{equation} \sigma_{\rm J}(L,H) /\ell(H)^{\theta} =\tilde{\sigma}_{\rm J}[L/\ell(H)], \end{equation} predicted by the droplet theory. In this equation, $\ell(H)=H^{\frac{1}{d/2-\theta}}$ is the overlap length, $\theta$ is the stiffness exponent, $d$ is the dimension, and $\tilde\sigma_{\rm J}$ is scaling function. $d$ is fixed to $3$ and $\theta$ is estimated by fitting. Although the value of $\theta$ is a bit higher than previous estimations (around 0.25), the scaling works nicely. We also find that the scaling function $\tilde{\sigma}_{\rm J}(X)$ 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 $H \rightarrow 0$ but remains finite for any non-zero $H$, the scaling implies that the spin-glass phase is absent even in an infinitesimal field.\vspace{2mm} 1) M.Sasaki, K. Hukushima, H. Yoshino and H. Takayama, cond-mat/0702302.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Sasaki, M. and Hukushima, K. and Yoshino, H. and Takayama, H.}, biburl = {https://www.bibsonomy.org/bibtex/261e5c76025e2fff02f105552e1f38851/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {5a5ec1dcba6532b222411b853a9b9928}, intrahash = {61e5c76025e2fff02f105552e1f38851}, keywords = {almeida-thouless domain-wall edwards-anderson glass group line method model renormalization spin statphys23 topic-7}, month = {9-13 July}, timestamp = {2007-06-20T10:16:16.000+0200}, title = {Domain-Wall Renormalization-Group Study of the Edwards-Anderson Ising Spin Glass in a Magnetic Field}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=304}, year = 2007 }