Reply to Comment on "Nature of Spin Glass State"
This is a reply to cond-mat/0010033 by Newman and Stein, which is a comment on on our paper Phys. Rev. Lett. 85, 3017 (2000), cond-mat/0002134.
cond-mat.dis-nn↗
arXiv subjects
A. P. Young
Publications and source records attributed to A. P. Young.
Computer Science in Physics
This talk describes how techniques developed by Computer Scientists have helped our understanding of certain problems in statistical physics which involve randomness and ``frustration''. Examples will be given from two problems that have been widely studied: the ``spin glass'' and the ``random field model''.
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Reply to Comment on "Triviality of the Ground State Structure in Ising Spin Glasses"
We reply to the comment of Marinari and Parisi [cond-mat/0002457 v2] on our paper [Phys. Rev. Lett. 83, 5126 (1999) and cond-mat/9906323]. We show that the data in the comment are affected by strong finite-size corrections. Therefore the original conclusion of our paper still stands.
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Nature of the Spin Glass State
The nature of the spin glass state is investigated by studying changes to the ground state when a weak perturbation is applied to the bulk of the system. We consider short range models in three and four dimensions and the infinite range Sherrington-Kirkpatrick (SK) and Viana-Bray models. Our results for the SK and Viana-Bray models agree with the replica symmetry breaking picture. The data for the short range models fit naturally a picture in which there are large scale excitations which cost a finite energy but whose surface has a fractal dimension, $d_s$, less than the space dimension $d$. We also discuss a possible crossover to other behavior at larger length scales than the sizes studied.
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Finite Temperature Ordering in the Three-Dimensional Gauge Glass
We present results of Monte Carlo simulations of the gauge glass model in three dimensions using exchange Monte Carlo. We show for the first time clear evidence of the vortex glass ordered phase at finite temperature. Using finite size scaling we obtain estimates for the correlation length exponent, nu = 1.39 +/- 0.20, the correlation function exponent, eta = -0.47 +/- 0.07, and the dynamic exponent z = 4.2 +/- 0.6. Using our values for z and nu we calculate the resistivity exponent to be s = 4.5 +/- 1.1. Finally, we provide a plausible lower bound on the the zero-temperature stiffness exponent, theta >= 0.18.
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Triviality of the Ground State Structure in Ising Spin Glasses
We investigate the ground state structure of the three-dimensional Ising spin glass in zero field by determining how the ground state changes in a fixed finite block far from the boundaries when the boundary conditions are changed. We find that the probability of a change in the block ground state configuration tends to zero as the system size tends to infinity. This indicates a trivial ground state structure, as predicted by the droplet theory. Similar results are also obtained in two dimensions.
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Ground State Structure of Spin Glasses
We investigate the ground state structure of Ising spin glasses in zero magnetic field by determining how the ground state changes in a fixed finite block far from the boundaries when the boundary conditions are changed. We find, both in two and three dimensions, that the probability of a change in the block ground state configuration tends to zero as the system size tends to infinity. This indicates a trivial ground state structure, as predicted by the droplet theory.
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Monte Carlo Study of the Critical Behavior of Random Bond Potts Models
We present results of Monte Carlo simulations of random bond Potts models in two dimensions, for different numbers of Potts states, q. We introduce a simple scheme which yields continuous self-dual distributions of the interactions. As expected, we find multifractal behavior of the correlation functions at the critical point and obtain estimates of the exponent eta_n for several moments, n, of the correlation functions, including typical (n -> 0), average (n=1) and others. In addition, for q=8, we find that there is only a single correlation length exponent describing the correlation length away from criticality. This is numerically very close to the pure Ising value of unity.
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Trivial Ground State Structure in the Two-Dimensional Ising Spin Glass
We study how the ground state of the two-dimensional Ising spin glass with Gaussian interactions in zero magnetic field changes on altering the boundary conditions. The probability that relative spin orientations change in a region far from the boundary goes to zero with the (linear) size of the system L like L^{-lambda}, where lambda = -0.70 +/- 0.08. We argue that lambda is equal to d-d_f where d (=2) is the dimension of the system and d_f is the fractal dimension of a domain wall induced by changes in the boundary conditions. Our value for d_f is consistent with earlier estimates. These results show that, at zero temperature, there is only a single pure state (plus the state with all spins flipped) in agreement with the predictions of the droplet model.
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Critical Behavior and Griffiths-McCoy Singularities in the Two-Dimensional Random Quantum Ising Ferromagnet
We study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one-dimension. At the critical point, the dynamical exponent is infinite and the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point there are Griffiths-McCoy singularities, characterized by a single, continuously varying exponent, z', which diverges at the critical point, as in one-dimension. Consequently, the zero temperature susceptibility diverges for a RANGE of parameters about the transition.
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Dynamical Critical Properties of the Random Transverse-Field Ising Spin Chain
We study the dynamical properties of the random transverse-field Ising chain at criticality using a mapping to free fermions, with which we can obtain numerically exact results for system sizes, L, as large as 256. The probability distribution of the local imaginary time correlation function S(tau) is investigated and found to be simply a function of alpha = -log S(tau) / log(tau). This scaling behavior implies that the typical correlation function decays algebraically where the exponent is determined from the distribution of alpha. The precise value for the exponent depends on exactly how the ``typical'' correlation function is defined. The form of P(alpha) for small alpha gives a contribution to the average correlation function, which varies as a power of the logarithm of time, which was obtained recently in Europhys. Lett. 39, 135 (1997). These results represent a type of ``multiscaling'' different from the well-known ``multifractal'' behavior.
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Distributions of gaps and end-to-end correlations in random transverse-field Ising spin chains
A previously introduced real space renormalization-group treatment of the random transverse-field Ising spin chain is extended to provide detailed information on the distribution of the energy gap and the end-to-end correlation function for long chains with free boundary conditions. Numerical data, using the mapping of the problem to free fermions, are found to be in good agreement with the analytic finite size scaling predictions.
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Critical Behavior of the Two-Dimensional Random Quantum Ising Ferromagnet
We study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one-dimension: the dynamical exponent is infinite and, at the critical point, the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point, there may be different exponents for the divergence of the average and typical correlation lengths, again as in one-dimension, but the evidence for this is less strong.
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Dynamical universality classes of the superconducting phase transition
We present a finite temperature Monte Carlo study of the XY-model in the vortex representation, and study its dynamical critical behavior in two limits. The first neglects magnetic field fluctuations, corresponding to the absence of screening, which should be a good approximation in high $T_c$ superconductors ($κ\to \infty$) except extremely close to the critical point. Here, from finite size scaling of the linear resistivity we find the dynamical critical exponent of the vortex motion to be $z\approx 1.5$. The second limit includes magnetic field fluctuations in the strong screening limit ($κ\to 0$) corresponding to the true asymptotic inverted XY critical regime, where we find the unexpectedly large value $z\approx 2.7$. We compare these results, obtained from dissipative dynamics in the vortex representation, with the universality class of the corresponding model in the phase representation with propagating (spin wave) modes. We also discuss the effect of disorder and the relevance of our results for experiments.
cond-mat.supr-con↗
Finite Temperature and Dynamical Properties of the Random Transverse-Field Ising Spin Chain
We study numerically the paramagnetic phase of the spin-1/2 random transverse-field Ising chain, using a mapping to non-interacting fermions. We extend our earlier work, Phys. Rev. 53, 8486 (1996), to finite temperatures and to dynamical properties. Our results are consistent with the idea that there are ``Griffiths-McCoy'' singularities in the paramagnetic phase described by a continuously varying exponent $z(δ)$, where $δ$ measures the deviation from criticality. There are some discrepancies between the values of $z(δ)$ obtained from different quantities, but this may be due to corrections to scaling. The average on-site time dependent correlation function decays with a power law in the paramagnetic phase, namely $τ^{-1/z(δ)}$, where $τ$ is imaginary time. However, the typical value decays with a stretched exponential behavior, $\exp(-cτ^{1/μ})$, where $μ$ may be related to $z(δ)$. We also obtain results for the full probability distribution of time dependent correlation functions at different points in the paramagnetic phase.
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Low temperature relaxational dynamics of the Ising chain in a transverse field
We present asymptotically exact results for the real time order parameter correlations of a class of d=1 Ising models in a transverse field at low temperatures (T) on both sides of the quantum critical point. The correlations are a product of a T-independent factor determined by quantum effects, and a T-dependent relaxation function which comes from a classical theory. We confirm our predictions by a no-free-parameter comparison with numerical studies on the nearest neighbor spin-1/2 model.
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Quantum Spin Glasses
Ising spin glasses in a transverse field exhibit a zero temperature quantum phase transition, which is driven by quantum rather than thermal fluctuations. They constitute a universality class that is significantly different from the classical, thermal phase transitions. Most interestingly close to the transition in finite dimensions a quantum Griffiths phase leads to drastic consequences for various physical quantities: for instance diverging magnetic susceptibilities are observable over a whole range of transverse field values in the disordered phase.
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Monte Carlo Study of a Three-Dimensional Vortex Glass Model with Screening
We investigate the critical behavior of the gauge glass model for the vortex glass transition in three-dimensional superconductors, including screening of the interaction between vortices. A Monte Carlo study of the linear resistivity and a scaling analysis of current--voltage characteristics indicates that screening destroys the finite--temperature transition found earlier when screening was neglected. The correlation length exponent at the resulting zero temperature transition is found to be $ν=1.05 \pm 0.1$.
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