Numerical ground state calculations are used to study four models with quenched disorder in finite samples with free boundary conditions. Extrapolation to the infinite volume limit indicates that the configurations in ``windows'' of fixed size converge to a unique configuration, up to global symmetries. The scaling of this convergence is consistent with calculations based on the fractal dimension of domain walls. These results provide strong evidence for the ``two-state'' picture of the low temperature behavior of these models. Convergence in three-dimensional systems can require relatively large windows.
Numerical ground state calculations are used to study four models with quenched disorder in finite samples with free boundary conditions. Extrapolation to the infinite volume limit indicates that the configurations in windows'' of fixed size converge to a unique configuration, up to global symmetries. The scaling of this convergence is consistent with calculations based on the fractal dimension of domain walls. These results provide strong evidence for the two-state'' picture of the low temperature behavior of these models. Convergence in three-dimensional systems can require relatively large windows.
We have performed numerical simulation of a three-dimensional elastic medium, with scalar displacements, subject to quenched disorder. In the absence of topological defects this system is equivalent to a (3+1)-dimensional interface subject to a periodic pinning potential. We have applied an efficient combinatorial optimization algorithm to generate exact ground states for this interface representation. Our results indicate that this Bragg glass is characterized by power law divergences in the structure factor $S(k)\ensuremath{\sim}{\mathrm{Ak}}^{\ensuremath{-}3}.$ We have found numerically consistent values of the coefficient A for two lattice discretizations of the medium, supporting universality for A in the isotropic systems considered here. We also examine the response of the ground state to the change in boundary conditions that corresponds to introducing a single dislocation loop encircling the system. The rearrangement of the ground state caused by this change is equivalent to the domain wall of elastic deformations which span the dislocation loop. Our results indicate that these domain walls are highly convoluted, with a fractal dimension df=2.60(5). We also discuss the implications of the domain wall energetics for the stability of the Bragg glass phase. Elastic excitations similar to these domain walls arise when the pinning potential is slightly perturbed. As in other disordered systems, perturbations of relative strength \ensuremath{\delta} introduce a new length scale ${L}^{*}\ensuremath{\sim}{\ensuremath{\delta}}^{\ensuremath{-}1/\ensuremath{\zeta}}$ beyond which the perturbed ground state becomes uncorrelated with the reference (unperturbed) ground state. We have performed a scaling analysis of the response of the ground state to the perturbations and obtain $\ensuremath{\zeta}=0.385(40).$ This value is consistent with the scaling relation $\ensuremath{\zeta}{=d}_{f}/2\ensuremath{-}\ensuremath{\theta},$ where \ensuremath{\theta} characterizes the scaling of the energy fluctuations of low energy excitations.
The four dimensional Gaussian random field Ising magnet is investigated numerically at zero temperature, using samples up to size 644, to test scaling theories and to investigate the nature of domain walls and the thermodynamic limit. As the magnetization exponent β is more easily distinguishable from zero in four dimensions than in three dimensions, these results provide a useful test of conventional scaling theories. Results are presented for the critical behavior of the heat capacity, magnetization, and stiffness. The fractal dimensions of the domain walls at criticality are estimated. A notable difference from three dimensions is the structure of the spin domains: frozen spins of both signs percolate at a disorder magnitude less than the value at the ferromagnetic to paramagnetic transition. Hence, in the vicinity of the transition, there are two percolating clusters of opposite spins that are fixed under any boundary conditions. This structure changes the interpretation of the domain walls for the four dimensional case. The scaling of the effect of boundary conditions on the interior spin configuration is found to be consistent with the domain wall dimension. There is no evidence of a glassy phase: there appears to be a single transition from two ferromagnetic states to a single paramagnetic state, as in three dimensions. The slowing down of the ground state algorithm is also used to study this model and the links between combinatorial optimization and critical behavior.
Disordered systems arise in many physical contexts. Not all matter is uni- form, and impurities or heterogeneities can be modeled by fixed random disor- der. Numerous complex networks also possess fixed disorder, leading to appli- cations in transportation systems [1], telecommunications [2], social networks [3, 4], and epidemic modeling [5], to name a few. Due to their random nature and power law critical behavior, disordered systems are difficult to study analytically. Numerical simulation can help overcome this hurdle by allowing for the rapid computation of system states. In order to get precise statistics and extrapolate to the thermodynamic limit, large systems must be studied over many realizations. Thus, innovative al- gorithm development is essential in order reduce memory or running time requirements of simulations. This thesis presents a review of disordered systems, as well as a thorough study of two particular systems through numerical simulation, algorithm de- velopment and optimization, and careful statistical analysis of scaling proper- ties. Chapter 1 provides a thorough overview of disordered systems, the his- tory of their study in the physics community, and the development of tech- niques used to study them. Topics of quenched disorder, phase transitions, the renormalization group, criticality, and scale invariance are discussed. Several prominent models of disordered systems are also explained. Lastly, analysis techniques used in studying disordered systems are covered. In Chapter 2, minimal spanning trees on critical percolation clusters are studied, motivated in part by an analytic perturbation expansion by Jackson and Read [6] that I check against numerical calculations. This system has a direct mapping to the ground state of the strongly disordered spin glass [7]. We compute the path length fractal dimension of these trees in dimensions d = {2, 3, 4, 5} and find our results to be compatible with the analytic results suggested by Jackson and Read. In Chapter 3, the random bond Ising ferromagnet is studied, which is es- pecially useful since it serves as a prototype for more complicated disordered systems such as the random field Ising model and spin glasses. We investigate the effect that changing boundary spins has on the locations of domain walls in the interior of the random ferromagnet system. We provide an analytic proof that ground state domain walls in the two dimensional system are de- composable, and we map these domain walls to a shortest paths problem. By implementing a multiple-source shortest paths algorithm developed by Philip Klein [8], we are able to efficiently probe domain wall locations for all possible configurations of boundary spins. We consider lattices with uncorrelated dis- order, as well as disorder that is spatially correlated according to a power law. We present numerical results for the scaling exponent governing the probabil- ity that a domain wall can be induced that passes through a particular location in the system’s interior, and we compare these results to previous results on the directed polymer problem.
The phase transition in the q -state Potts model with homogeneous ferromagnetic couplings is strongly first order for large q, while it is rounded in the presence of quenched disorder. Here we study this phenomenon on different two-dimensional lattices by using the fact that the partition function of the model is dominated by a single diagram of the high-temperature expansion, which is calculated by an efficient combinatorial optimization algorithm. For a given finite sample with discrete randomness the free energy is a piecewise linear function of the temperature, which is rounded after averaging, however, the discontinuity of the internal energy at the transition point (i.e., the latent heat) stays finite even in the thermodynamic limit. For a continuous disorder, instead, the latent heat vanishes. At the phase transition point the dominant diagram percolates and the total magnetic moment is related to the size of the percolating cluster. Its fractal dimension is found d(f) = ( 5 + square root of 5)/4 and it is independent of the type of the lattice and the form of disorder. We argue that the critical behavior is exclusively determined by disorder and the corresponding fixed point is the isotropic version of the so-called infinite randomness fixed point, which is realized in random quantum spin chains. From this mapping we conjecture the values of the critical exponents as beta=2- d(f), beta(s) =1/2, and nu=1.
The XY model with quenched random disorder is studied by a zero-temperature domain wall renormalization group method in two dimensions (2D) and three dimensions (3D). Instead of the usual phase representation we use the charge (vortex) representation to compute the domain wall, or defect, energy. For the gauge glass corresponding to the maximum disorder we reconfirm earlier predictions that there is no ordered phase in 2D but an ordered phase can exist in 3D at low temperature. However, our simulations yield spin stiffness exponents ${\ensuremath{\theta}}_{s}\ensuremath{\approx}\ensuremath{-}0.36$ in 2D and ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.31$ in 3D, which are considerably larger than previous estimates and strongly suggest that the lower critical dimension is less than three. For the $\ifmmode\pm\else\textpm\fi{}J \mathrm{XY}$ spin glass in 3D, we obtain a spin stiffness exponent ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.10$ which supports the existence of spin glass order at finite temperature in contrast with previous estimates which obtain ${\ensuremath{\theta}}_{s}<0.$ Our method also allows us to study renormalization group flows of both the coupling constant and the disorder strength with a length scale L. Our results are consistent with recent analytic and numerical studies suggesting the absence of a reentrant transition in 2D at low temperature. Some possible consequences and connections with real vortex systems are discussed.
We study large-scale, low-energy excitations in the Ising spin glass with Gaussian interactions in two dimensions at zero temperature, using an optimization algorithm to determine exact ground states. Periodic boundary conditions are applied. Our results for the fractal dimension of the surface, ds, and stiffness exponent, ${\ensuremath{\theta}}^{\ensuremath{'}},$ for ``droplet'' excitations, are in reasonable agreement with estimates from ``domain wall'' calculations, and so support the predictions of the ``droplet theory.'' Restricting our analysis to small lattices, we do not find an effective value of ${\ensuremath{\theta}}^{\ensuremath{'}}$ close to $\ensuremath{-}0.47$ as has been recently proposed. The effects of averaging over droplets of different sizes are studied and are also found to be too small to give ${\ensuremath{\theta}}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{-}0.47$ for smaller sizes. Larger corrections to finite-size scaling would be needed in three and four dimensions in order for the numerical data used to support the ``TNT'' scenario to be compatible with the droplet theory prediction that the stiffness exponent is the same for droplets and domain walls.