%0 Book Section %1 statphys23_0322 %A Hartmann, A.K. %A Amoruso, C. %A Hastings, M.B. %A Houdayer, J. %A Moore, M.A. %A Young, A.P. %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 algorithms droplets glasses ground optimization sle spin states statphys23 topic-9 %T Using optimization algorithms to understand the low-temperature behavior of two-dimensional Ising spin glasses %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=322 %X For many statistical-mechanics models, by using model-tailored mappings to graph-theoretical problems and applying sophisticated problem-dependent optimization algorithms 1, one can obtain results which allow to understand the behavior numerically much better in comparison to applying standard approaches like Monte Carlo simulations. Here, as an example for such an approach, we study the two-dimensional Ising spin glass, which is a prototypical model for complex systems with quenched disorder. In particular we are interested in the behavior near the phase transition point $T_c=0$, where many properties of this model are not well understood. We review how, by using a mapping to the minimum-weight perfect matching problem and by applying fast matching algorithms from computer science, one can study the model at and close to $T=0$ in perfect equilibrium for large system sizes, easily up to $N=512^2$ spins. By using this approach, recently much progress has been obtained for the system with Gaussian disorder by studying domain-wall excitations 2 (see figure), droplet excitations 3, energy barriers 4 and the SLE properties of domain walls 5. All results show that the behavior of the Gaussian model is well described by the famous droplet ansatz, and characterized by one single exponent $þeta\approx-0.29$, which is related to the correlation-length exponent via $\nu=1/þeta$, as confirmed recently by Monte Carlo simulations 6. For the model with bimodal $J$ distribution, the situation is less clear, in particular it is currently heavily discussed whether the correlation length diverges algebraically or exponentially when approaching $T_c$. Many contradicting results from finite-temperature approaches exist. Here, again the droplet approach 3 is applied, taking advantage of working exactly at $T=0$. For large systems, the droplet energy shows a power-law behavior $EL^þeta$ with $þeta=-0.234(6)$, corresponding to an algebraic correlation-length divergence with $\nu=-1/=4.1(1)$. This behavior is qualitatively similar to the Gaussian system, but the value of $þeta$ is different, hence non-universal. Furthermore, the disorder-averaged spin-spin correlation exponent $\eta$ defined via $S_i S_i+l \rangle^2_J l^-\eta$ is determined here via the probability to have a non-zero-energy droplet, and $\eta=0.223(7)>0$ is found.\\ 1) AKH and H. Rieger, Optimization Algorithms in Physics, (Wiley-VCH, Berlin 2001)\\ 2) AKH and A.P. Young, Phys. Rev. B 64, 180404 (2001)\\ 3) AKH and M.A. Moore, Phys. Rev. Lett. 90, 127201 (2003)\\ 4) C. Amoruso, AKH, and M.A. Moore, Phys. Rev. B 73, 184405 (2006)\\ 5) C. Amoruso, AKH, M.B. Hastings and M.A. Moore, Phys. Rev. Lett. 97, 267202 (2006)\\ 6) J. Houdayer and AKH, Phys. Rev. B 70, 014418 (2004)

@incollection{statphys23_0322, abstract = {For many statistical-mechanics models, by using model-tailored mappings to graph-theoretical problems and applying sophisticated problem-dependent optimization algorithms [1], one can obtain results which allow to understand the behavior numerically much better in comparison to applying standard approaches like Monte Carlo simulations. Here, as an example for such an approach, we study the two-dimensional Ising spin glass, which is a prototypical model for complex systems with quenched disorder. In particular we are interested in the behavior near the phase transition point $T_c=0$, where many properties of this model are not well understood. We review how, by using a mapping to the {\em minimum-weight perfect matching} problem and by applying fast matching algorithms from computer science, one can study the model at and close to $T=0$ in perfect equilibrium for large system sizes, easily up to $N=512^2$ spins. By using this approach, recently much progress has been obtained for the system with Gaussian disorder by studying domain-wall excitations [2] (see figure), droplet excitations [3], energy barriers [4] and the SLE properties of domain walls [5]. All results show that the behavior of the Gaussian model is well described by the famous droplet ansatz, and characterized by one single exponent $\theta\approx-0.29$, which is related to the correlation-length exponent via $\nu=1/\theta$, as confirmed recently by Monte Carlo simulations [6]. For the model with bimodal $\pm J$ distribution, the situation is less clear, in particular it is currently heavily discussed whether the correlation length diverges algebraically or exponentially when approaching $T_c$. Many contradicting results from finite-temperature approaches exist. Here, again the droplet approach [3] is applied, taking advantage of working exactly at $T=0$. For large systems, the droplet energy shows a power-law behavior $E\sim L^{\theta}$ with $\theta=-0.234(6)$, corresponding to an algebraic correlation-length divergence with $\nu=-1/\theta =4.1(1)$. This behavior is qualitatively similar to the Gaussian system, but the value of $\theta$ is different, hence non-universal. Furthermore, the disorder-averaged spin-spin correlation exponent $\eta$ defined via $[\langle S_i S_{i+l} \rangle^2]_J \sim l^{-\eta}$ is determined here via the probability to have a non-zero-energy droplet, and $\eta=0.223(7)>0$ is found.\\ 1) AKH and H. Rieger, Optimization Algorithms in Physics, (Wiley-VCH, Berlin 2001)\\ 2) AKH and A.P. Young, Phys. Rev. B {\bf 64}, 180404 (2001)\\ 3) AKH and M.A. Moore, Phys. Rev. Lett. {\bf 90}, 127201 (2003)\\ 4) C. Amoruso, AKH, and M.A. Moore, Phys. Rev. B {\bf 73}, 184405 (2006)\\ 5) C. Amoruso, AKH, M.B. Hastings and M.A. Moore, Phys. Rev. Lett. {\bf 97}, 267202 (2006)\\ 6) J. Houdayer and AKH, Phys. Rev. B {\bf 70}, 014418 (2004)}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Hartmann, A.K. and Amoruso, C. and Hastings, M.B. and Houdayer, J. and Moore, M.A. and Young, A.P.}, biburl = {https://www.bibsonomy.org/bibtex/2ff5bfdb0693f1144b170addfc19c9036/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {04db68b0317cfb3d629178e18134a84d}, intrahash = {ff5bfdb0693f1144b170addfc19c9036}, keywords = {algorithms droplets glasses ground optimization sle spin states statphys23 topic-9}, month = {9-13 July}, timestamp = {2007-06-20T10:16:17.000+0200}, title = {Using optimization algorithms to understand the low-temperature behavior of two-dimensional Ising spin glasses}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=322}, year = 2007 }