Spin glasses have been the subject of intense study and considerable controversy for decades, and the low-temperature phase of short-range spin glasses is still poorly understood. Our main goal is to improve our understanding in this area and find an answer to the following question: Are there only a single pair or a countable infinity of pure states in the low temperature phase of the EA spin glass? To that aim we first start by introducing spin glasses and provide a brief history of their research, then proceed to describe our method of simulation, the parallel tempering Monte Carlo algorithm. Next, we present the results of a large-scale numerical study of the equilibrium three-dimensional Edwards-Anderson Ising spin glass with Gaussian disorder. In order to understand how the parallel tempering algorithm works, we measure various static, as well as dynamical quantities, such as the autocorrelation times and round-trip times for the parallel tempering Monte Carlo method. We examine the correlation between static and dynamic observables for _ 5000 disorder realizations and up to 1000 spins down to temperatures at 20% of the critical temperature, and our results show that autocorrelation times are directly correlated with the roughness of the free-energy landscape. In the following chapters, the three- and four-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied again via large-scale Monte Carlo simulations at low temperatures, deep within the spinglass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. We arrive to the following conclusion: The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states. Finally we present results for several new observables, along with a few preliminary studies and suggestions for future research.Spin glasses have been the subject of intense study and considerable controversy for decades, and the low-temperature phase of short-range spin glasses is still poorly understood. Our main goal is to improve our understanding in this area and find an answer to the following question: Are there only a single pair or a countable infinity of pure states in the low temperature phase of the EA spin glass? To that aim we first start by introducing spin glasses and provide a brief history of their research, then proceed to describe our method of simulation, the parallel tempering Monte Carlo algorithm. Next, we present the results of a large-scale numerical study of the equilibrium three-dimensional Edwards-Anderson Ising spin glass with Gaussian disorder. In order to understand how the parallel tempering algorithm works, we measure various static, as well as dynamical quantities, such as the autocorrelation times and round-trip times for the parallel tempering Monte Carlo method. We examine the correlation between static and dynamic observables for _ 5000 disorder realizations and up to 1000 spins down to temperatures at 20% of the critical temperature, and our results show that autocorrelation times are directly correlated with the roughness of the free-energy landscape. In the following chapters, the three- and four-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied again via large-scale Monte Carlo simulations at low temperatures, deep within the spinglass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. We arrive to the following conclusion: The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states. Finally we present results for several new observables, along with a few preliminary studies and suggestions for future research.

Apr 23, 20190 citations

Spin Glasses at Low Temperatures

Helmut G. Katzgraber, Matteo Palassini, A. P. Young

We report the results of Monte Carlo simulations on several spin glass models at low temperatures. By using the parallel tempering (Exchange Monte Carlo) technique we are able to equilibrate down to low temperatures, for moderate sizes, and hence the data should not be affected by critical fluctuations. Our results for short range models are consistent with a picture proposed earlier that there are large scale excitations which cost only a finite energy in the thermodynamic limit, and these excitations have a surface whose fractal dimension is less than the space dimension. For the infinite range Viana-Bray model, our results obtained for a similar number of spins are consistent with standard replica symmetry breaking.We report the results of Monte Carlo simulations on several spin glass models at low temperatures. By using the parallel tempering (Exchange Monte Carlo) technique we are able to equilibrate down to low temperatures, for moderate sizes, and hence the data should not be affected by critical fluctuations. Our results for short range models are consistent with a picture proposed earlier that there are large scale excitations which cost only a finite energy in the thermodynamic limit, and these excitations have a surface whose fractal dimension is less than the space dimension. For the infinite range Viana-Bray model, our results obtained for a similar number of spins are consistent with standard replica symmetry breaking.

arXiv:cond-mat/0007113Jul 6, 20002 citations

Spin stiffness of vector spin glasses

Frank Beyer, Martin Weigel

Computer Physics CommunicationsDec 8, 20102 citations

Low-temperature behavior of two-dimensional Gaussian Ising spin glasses

J. Houdayer, Alexander K. Hartmann

We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $\ensuremath{\beta}=1∕T=50$. Equilibration is ensured by using a cluster algorithm including Monte Carlo moves consisting of flipping fundamental excitations. We study the thermodynamic behavior using the Binder cumulant, the spin-glass susceptibility, the distribution of overlaps, the overlap with the ground state, and the specific heat. We confirm that ${T}_{c}=0$ . All results are compatible with an algebraic divergence of the correlation length with an exponent $\ensuremath{\nu}$. We find $\ensuremath{-}1∕\ensuremath{\nu}=\ensuremath{-}0.295(40)$, which is compatible with the value for the domain-wall and droplet exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $\ensuremath{\beta}=1∕T=50$. Equilibration is ensured by using a cluster algorithm including Monte Carlo moves consisting of flipping fundamental excitations. We study the thermodynamic behavior using the Binder cumulant, the spin-glass susceptibility, the distribution of overlaps, the overlap with the ground state, and the specific heat. We confirm that ${T}_{c}=0$. All results are compatible with an algebraic divergence of the correlation length with an exponent $\ensuremath{\nu}$. We find $\ensuremath{-}1∕\ensuremath{\nu}=\ensuremath{-}0.295(40)$, which is compatible with the value for the domain-wall and droplet exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.

arXiv:cond-mat/0402036Physical Review BJul 1, 2004133 citations

Numerical studies of spin glasses

R. N. Bhatt, A. P. Young

Lecture notes in physicsJan 1, 19873 citations

{T}_{c}=0

. All results are compatible with an algebraic divergence of the correlation length with an exponent $\ensuremath{\nu}$. We find $\ensuremath{-}1∕\ensuremath{\nu}=\ensuremath{-}0.295(40)$, which is compatible with the value for the domain-wall and droplet exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.