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Science

One model to rule them all?

Jelena Stajic

Published Mar 10, 2016• 0 citations
Publisher

Abstract

Spin Models The idea of finding an all-encompassing model to describe the world around us has appealed to generations of scientists. Although we are very far from this ultimate goal, more modest steps can be taken if we focus on particular problems. De las Cuevas and Cubitt used concepts borrowed from computer science to show that all classical spin models (introduced initially to study magnetism) can be solved by tackling a slightly more complex, universal model (see the Perspective by Wehner). Thanks to these findings, it may be possible to physically simulate systems with complex interactions, using the well-understood two-dimensional Ising model with fields. Science , this issue p. [1180][1]; see also p. [1156][2] [1]: /lookup/doi/10.1126/science.aab3326 [2]: /lookup/doi/10.1126/science.aaf0748

DOI
10.1126/science.351.6278.1162-c

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Spin Models
The idea of finding an all-encompassing model to describe the world around us has appealed to generations of scientists. Although we are very far from this ultimate goal, more modest steps can be taken if we focus on particular problems. De las Cuevas and Cubitt used concepts borrowed from computer science to show that all classical spin models (introduced initially to study magnetism) can be solved by tackling a slightly more complex, universal model (see the Perspective by Wehner). Thanks to these findings, it may be possible to physically simulate systems with complex interactions, using the well-understood two-dimensional Ising model with fields.
Science , this issue p. 1180; see also p. 1156
Though the idea to use numerical techniques, in order to solve complex three-dimensional problems, has become quite old, computational techniques have gained immense importance in past few decades because of the advent of new generation fast and efficient computers and development of algorithms as parallel computing.Many mathematical problems have no exact solutions.Depending on the complexity of the equations, one needs to use approximate methods.But there are problems, which are beyond our limits, and need support of computers.Ernst Ising published his PhD dissertation in the form of a scientific report in 1925.He used a string of magnetic moments; spin up (+1/2) and spin down (À1/2), and applied periodic boundary conditions to prove that magnetic phase transition does not exist in one dimensions.Lars Onsager, latter, exactly solved the phase transition problem in two dimensions in 1944.It is going to be a century-old problem now.A variety of potential applications of Ising model are possible now a days; classified as Ising universality class models.It has now become possible to solve phase transition problems in complex three-dimensional geometries.Though the area of spinotronics still needs more engagements of computational techniques, its limited use have provided good insights at molecular scale in recent past.This chapter gives a brief introduction to Ising model and its applications, highlighting the developments in the field of magnetism relevant to the area of solid state physics.
Spin models are used in many studies of complex systems because they exhibit rich macroscopic behavior despite their microscopic simplicity. Here, we prove that all the physics of every classical spin model is reproduced in the low-energy sector of certain "universal models," with at most polynomial overhead. This holds for classical models with discrete or continuous degrees of freedom. We prove necessary and sufficient conditions for a spin model to be universal and show that one of the simplest and most widely studied spin models, the two-dimensional Ising model with fields, is universal. Our results may facilitate physical simulations of Hamiltonians with complex interactions.
We study a set of exactly soluble spin models in one and two dimensions for any spin SSS. Its ground state, the excitation spectrum, quantum phase transition points, as well as dimensional crossover are determined.
The theory of spin models intersects with condensed matter physics, complex systems, graph theory, combinatorial optimization, computational complexity and neural networks. Many ensuing applications rely on transforming complicated spin models to simpler ones. What is the structure of such transformations? Here, we provide a framework to study and construct emulations between spin models. A spin model is a set of spin systems, and emulations are efficiently computable families of simulations with arbitrary energy cut-off, where a source spin system simulates a target system if, below the cut-off, the target Hamiltonian is encoded in the source Hamiltonian. We prove that emulations preserve important properties, as they induce reductions between computational problems such as computing ground states, approximating partition functions and approximate sampling from Boltzmann distributions. Emulations are modular (they can be added, scaled and composed), and allow for universality, i.e., under emulations, certain spin models have maximal reach. We prove that a spin model is universal if and only if it is scalable, closed and functional complete. Because the characterization is constructive, it provides a step-by-step guide to construct emulations. We prove that the 2d Ising model with fields is universal, for which we also provide two new crossing gadgets. Finally, we show that simulations can be computed by linear programs. While some ideas of this work are contained in G. De las Cuevas and T. S. Cubitt, Science 351, 1180 (2016), we provide new definitions and theorems. This framework provides a toolbox for applications involving emulations of spin models.
We study a set of exactly soluble spin models in one and two dimensions for any spin SSS. Its ground state, the excitation spectrum, quantum phase transition points, as well as dimensional crossover are determined.
Quantum Walks Generations of physics students have been taught to think of one-dimensional random walks in terms of a drunken sailor taking random steps to the right or to the left. But that doesn't compare with the complexity of a quantum walker, who can propagate down multiple paths at the same time. Preiss et al. detected particles in single sites of an optical lattice to study the dynamics of two interacting atoms of 87Rb performing a quantum walk (see the Perspective by Widera). Depending on the initial conditions and the interaction strength between the atoms, the atoms either ignored each other, stuck to each other, or tried to get as far away from each other as possible. Science , this issue p. [1229][1]; see also p. [1200][2] [1]: /lookup/doi/10.1126/science.1260364 [2]: /lookup/doi/10.1126/science.aaa6885