# Browsing by Subject "Ising model"

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Item Equilibrium and non-equilibrium molecular absorption: A study of the Ising Model and the infinite parking limit problem(2012-12-07) Beckman, Erin; John Stanton; Lorenzo SadunShow more Randomness is present in so many everyday systems that we often forget its importance in both mathematical and chemical situations. Chemical reactions depend on random interactions and collisions, the movement of particles is often randomly determined, and randomness plays a role in the way in which diffusing particles interact with a solid surface. Looking specifically into this last situation, we know that there are many ways in which diffusing particles can interact with solids. Particles can diff use through either water or air, and at low concentrations, this process is well-modeled by random processes. When looking at this situation, there are two distinct types of molecular absorption to consider: equilibrium or nonequilibrium absorption. That is, when the diffusing particle comes into contact with the surface, it can either stick exactly where it lands or move around a little to come to a more stable equilibrium arrangement. These processes might look very similar on the macroscopic level, but on the microscopic level, they are studied using very different mathematical techniques. Sometimes the process is modeled by a simpler problem in order to use more rigorous mathematics to address things which are very complicated. Other times, it is more beneficial to solve a problem computationally with programming. These different approaches each have benefits and are both used in comprehensive studies of natural processes.To get a feel for these two different ways of studying this problem, we will examine two different models: an equilibrium model and a non-equilibrium model. One of the most common models used to study equilibrium systems is the Ising model. We start by thinking of the Ising model as a infinite system with boundary conditions, and then we consider the limit as the size of the grid goes to infinity. The value of the spin at each lattice site depends on two different things-the values of the neighboring lattice sites and any applied external magnetic fields. The contributions of each of these pieces depends on the system and is controlled by constants. It was originally proposed by Ising to model the spontaneous magnetization in ferromagnetic substances. The restriction on the spin states and the ordered pattern of the grid points allow this model to be analyzed mathematically, while still giving interesting information about the behavior of many different chemical systems. As the number of points in the grid goes to infinity, we use this model to look at phase transitions, switches between one defined state and another.Show more Item Ferromagnetic properties of partially filled two-dimensional Ising lattices(2003) Faraggi, Eshel; Reichl, L. E.Show more Item Flow equation approach to periodically driven quantum systems(Physical Review X, 2019) Vogl, Michael; Laurell, Pontus; Barr, Aaron D.; Fiete, Gregory A.Show more We present a theoretical method to generate a highly accurate time-independent Hamiltonian governing the finite-time behavior of a time-periodic system. The method exploits infinitesimal unitary transformation steps, from which renormalization-group–like flow equations are derived to produce the effective Hamiltonian. Our tractable method has a range of validity reaching into frequency—and drive strength—regimes that are usually inaccessible via high-frequency ω expansions in the parameter h=ω, where h is the upper limit for the strength of local interactions. We demonstrate exact properties of our approach on a simple toy model and test an approximate version of it on both interacting and noninteracting many-body Hamiltonians, where it offers an improvement over the more well-known Magnus expansion and other high-frequency expansions. For the interacting models, we compare our approximate results to those found via exact diagonalization. While the approximation generally performs better globally than other high-frequency approximations, the improvement is especially pronounced in the regime of lower frequencies and strong external driving. This regime is of special interest because of its proximity to the resonant regime where the effect of a periodic drive is the most dramatic. Our results open a new route towards identifying novel nonequilibrium regimes and behaviors in driven quantum many-particle systems.Show more Item An information theoretic approach to structured high-dimensional problems(2013-12) Das, Abhik Kumar; Vishwanath, SriramShow more A majority of the data transmitted and processed today has an inherent structured high-dimensional nature, either because of the process of encoding using high-dimensional codebooks for providing a systematic structure, or dependency of the data on a large number of agents or variables. As a result, many problem setups associated with transmission and processing of data have a structured high-dimensional aspect to them. This dissertation takes a look at two such problems, namely, communication over networks using network coding, and learning the structure of graphical representations like Markov networks using observed data, from an information-theoretic perspective. Such an approach yields intuition about good coding architectures as well as the limitations imposed by the high-dimensional framework. Th e dissertation studies the problem of network coding for networks having multiple transmission sessions, i.e., multiple users communicating with each other at the same time. The connection between such networks and the information-theoretic interference channel is examined, and the concept of interference alignment, derived from interference channel literature, is coupled with linear network coding to develop novel coding schemes off ering good guarantees on achievable throughput. In particular, two setups are analyzed – the first where each user requires data from only one user (multiple unicasts), and the second where each user requires data from potentially multiple users (multiple multicasts). It is demonstrated that one can achieve a rate equalling a signi ficant fraction of the maximal rate for each transmission session, provided certain constraints on the network topology are satisfi ed. Th e dissertation also analyzes the problem of learning the structure of Markov networks from observed samples – the learning problem is interpreted as a channel coding problem and its achievability and converse aspects are examined. A rate-distortion theoretic approach is taken for the converse aspect, and information-theoretic lower bounds on the number of samples, required for any algorithm to learn the Markov graph up to a pre-speci fied edit distance, are derived for ensembles of discrete and Gaussian Markov networks based on degree-bounded graphs. The problem of accurately learning the structure of discrete Markov networks, based on power-law graphs generated from the con figuration model, is also studied. The eff ect of power-law exponent value on the hardness of the learning problem is deduced from the converse aspect – it is shown that discrete Markov networks on power-law graphs with smaller exponent values require more number of samples to ensure accurate recovery of their underlying graphs for any learning algorithm. For the achievability aspect, an effi cient learning algorithm is designed for accurately reconstructing the structure of Ising model based on power-law graphs from the con figuration model; it is demonstrated that optimal number of samples su ffices for recovering the exact graph under certain constraints on the Ising model potential values.Show more Item Latent slice sampling(2022-05-06) Li, Yanxin; Walker, Stephen G., 1945-; Linero, Antonio; Viswanathan, Bindu; Guyot, Layla; Zhou, Mingyuan; Williamson, SineadShow more The thesis develops a new and generic Markov chain Monte Carlo sampling methodology, naming latent slice sampling, that originates from slice sampling and is capable of efficient sampling. More specifically, three angles are studied to cover different types of random variables: (i). We develop a latent slice sampler for discrete variables by designing a transition probability function that can perform direct sampling without knowing the exact form of target distributions. (ii). We manage to derive a latent slice sampler for continuous variables which has the potential to be a more efficient alternative to the Metropolis-Hasting algorithm, obviates the need for a proposal distribution, and has no accept/reject component. (iii). We further propose a novel algorithm based on latent slice sampling methodology which copes well with multi-modal problem, which can approach well-studied problems from a different angle and provide new perspectives. All the methods bring clear gains, which demonstrate the benefits of applying latent slice sampling to improve Markov chain simulation.Show more Item Time-dependent properties of solids : non-perturbative Floquet Hamiltonians and beyond(2020-06-22) Vogl, Michael; Fiete, Gregory A.; MacDonald, Allan H; Niu, Qian; Sadun, Lorenzo AShow more In the different chapters of this dissertation we investigate multiple non-perturbative approaches that allow the study of solids subjected to time dependent fields. In the second chapter, we develop a very general non-perturbative and flexible approach that allows to study periodically driven quantum systems via an effective time-independent theory. It makes use of renormalization-group-like flow equations to find highly accurate effective Hamiltonians[Phys. Rev. X 9, 021037 (2019)]. The range of validity of the approach is checked numerically for various 1D spin systems and confirmed to be beyond the reaches of ordinary perturbative methods. It also compares favourably to another non-perturbative approach. In the third chapter we examine the wealth of different approximations to the time evolution operator that are known from the literature. Inspired by Hamilton-Jacobi theory we find a useful reformulation of the equation for the time evolution operator [Phys. Rev. A 100, 012132 (2019)]. This allows us to find the various known approximations of the time evolution operator in a unified fashion and new ones. Interestingly the RG-like flow equations from the second chapter are found to be one specific limit of the more general approach developed here. A periodically driven Ising model is used to verify the range of validity of the different approximations. This allows us to put them into a hierarchy. Up until here the methods that were explored are most valid in the high to mid frequency regime. In the fourth chapter we show how for the case of a weakly driven system a simple approach can be used to treat the low frequency regime [Phys. Rev. B 101, 024303 (2020)]. To achieve this we derive a quasi-energy dependent Hamiltonians and very crucially require self-consistency for the quasi-energies. For the simple two band example of single layer graphene it is found to be highly accurate approach that also allows for accurate descriptions of even sensitive quantities like Chern numbers For the last fifth chapter of the thesis we switch gears and consider a specific system of much recent interest - twisted bilayer graphene. We investigate how light that is altered by the boundary conditions of a metallic waveguide can change the strength of interlayer couplings [arXiv:2001.04416 (2020)]. We investigate how this change affects the twist angles at which flat bands appear and find very simple analytic results that are in good agreement with a numerical treatment. In the sixth chapter of the thesis, we investigate the effects circularly polarized light has on twisted bilayer graphene[ arXiv:2002.05124 (2020)]. We derive two effective Hamiltonians valid in different driving regimes. One is valid for weak drives but low frequencies and the other for relatively strong drives. The Hamiltonian for the strongly driven regime is compared to exact numeric results and found to perform better than other common methods. We analyse the newly generated terms in both Hamiltonians for their symmetry.Show more Item Unsupervised learning for large-scale data(2019-09-20) Wu, Shanshan, Ph. D.; Sanghavi, Sujay Rajendra, 1979-; Dimakis, Alexandros G.; Caramanis, Constantine; Klivans, Adam R; Ward, Rachel AShow more Unsupervised learning involves inferring the inherent structures or patterns from unlabeled data. Since there is no label information, the fundamental challenge of unsupervised learning is that the objective function is not explicitly defined. The ubiquity of large-scale datasets adds another layer of complexity to the overall learning problem. When the data size or dimension is large, even algorithms with quadratic runtime may be prohibitive. This thesis presents four large-scale unsupervised learning problems. We start with two density estimation problems: given samples from a one-layer ReLU generative model or a discrete pairwise graphical model, the goal is to recover the parameters of the generative model. We then move to representation learning of high-dimensional sparse data coming from one-hot encoded categorical features. We assume that there are additional but a-priori unknown structures in their support. The goal is to learn a lossless low-dimensional embedding for the given data. Our last problem is to compute low-rank approximations of a matrix product given the individual matrices. We are interested in the setting where the matrices are too large and can only be stored in the disk. For every problem presented in this thesis, we (i) design novel and efficient algorithms to capture the inherent structure from data in an unsupervised manner; (ii) establish theoretical guarantees and compare the empirical performance with the state-of-the-art methods; and (iii) provide source code to support our experimental findingsShow more