This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

/pdf/many-body-physics-with-ultracold-gases-552s10zfdn.pdf

Many-Body Physics with Ultracold Gases

Topological quantum computation has emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations that are necessary for quantum computation are carried out by braiding quasiparticles and then measuring the multiquasiparticle states. The fault tolerance of a topological quantum computer arises from the nonlocal encoding of the quasiparticle states, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the $\ensuremath{\nu}=5∕2$ state, although several other prospective candidates have been proposed in systems as disparate as ultracold atoms in optical lattices and thin-film superconductors. In this review article, current research in this field is described, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. Both the mathematical underpinnings of topological quantum computation and the physics of the subject are addressed, using the $\ensuremath{\nu}=5∕2$ fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

Non-Abelian Anyons and Topological Quantum Computation

Hybrid Monte Carlo

Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as {it Non-Abelian anyons}, meaning that they obey {it non-Abelian braiding statistics}. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations which are necessary for quantum computation are carried out by braiding quasiparticles, and then measuring the multi-quasiparticle states. The fault-tolerance of a topological quantum computer arises from the non-local encoding of the states of the quasiparticles, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the nu=5/2 state, although several other prospective candidates have been proposed in systems as disparate as ultra-cold atoms in optical lattices and thin film superconductors. In this review article, we describe current research in this field, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. We address both the mathematical underpinnings of topological quantum computation and the physics of the subject using the nu=5/2 fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

This is a tutorial review on the Potts model aimed at bringing out in an organized fashion the essential and important properties of the standard Potts model. Emphasis is placed on exact and rigorous results, but other aspects of the problem are also described to achieve a unified perspective. Topics reviewed include the mean-field theory, duality relations, series expansions, critical properties, experimental realizations, and the relationship of the Potts model with other lattice-statistical problems.

The Potts model

This article is an interdisciplinary review of lattice gauge theory and spin systems. It discusses the fundamentals, both physics and formalism, of these related subjects. Spin systems are models of magnetism and phase transitions. Lattice gauge theories are cutoff formulations of gauge theories of strongly interacting particles. Statistical mechanics and field theory are closely related subjects, and the connections between them are developed here by using the transfer matrix. Phase diagrams and critical points of continuous transitions are stressed as the keys to understanding the character and continuum limits of lattice theories. Concepts such as duality, kink condensation, and the existence of a local, relativistic field theory at a critical point of a lattice theory are illustrated in a thorough discussion of the two-dimensional Ising model. Theories with exact local (gauge) symmetries are introduced following Wegner's Ising lattice gauge theory. Its gauge-invariant "loop" correlation function is discussed in detail. Three---dimensional Ising gauge theory is studied thoroughly. The renormalization group of the two dimensional planar model is presented as an illustration of a phase transition driven by the condensation of topological excitations. Parallels are drawn to Abelian lattice gauge theory in four dimensions. Non-Abelian gauge theories are introduced and the possibility of quark confinement is discussed. Asymptotic freedom of O(n) Heisenberg spin systems in two dimensions is verified for $n\ensuremath{\ge}$ and is explained in simple terms. The direction of present-day research is briefly reviewed.

/pdf/an-introduction-to-lattice-gauge-theory-and-spin-systems-2j635nh47y.pdf

An introduction to lattice gauge theory and spin systems

The discovery of narrow resonances at 3.1 and 3.7 GeV and their interpretation as charmed quark-antiquark bound states suggest additional narrow states between 3.0 and 4.3 GeV. A model which incorporates quark confinement is used to determine the quantum numbers and estimate masses and decay widths of these states. Their existence should be revealed by γ-ray transitions among them.

The Spectrum of Charmonium

The author reviews in a pedagogical fashion some of the recent developments in lattice quantum chromodynamics. This review emphasizes explicit examples and illustrations rather than general proofs and analyses. It begins with a discussion of the heavy-quark potential in continuum quantum chromodynamics. Asymptotic freedom and renormalization-group improved perturbation theory are discussed. A simple dielectric model of confinement is considered as an intuitive guide to the vacuum of non-Abelian gauge theories. Next, the Euclidean form of lattice gauge theory is introduced, and an assortment of calculational methods are reviewed. These include high-temperature expansions, duality, Monte Carlo computer simulations, and weak coupling expansions. A $\ensuremath{\Lambda}$- parameter calculation for asymptotically free-spin models is presented. The Hamiltonian formulation of lattice gauge theory is presented and is illustrated in the context of flux tube dynamics. Roughening transitions, Casimir forces, and the restoration of rotational symmetry are discussed. Mechanisms of confinement in lattice theories are illustrated in the two-dimensional electrodynamics of the planar model and the U(1) gauge theory in four dimensions. Generalized actions for SU(2) gauge theories and the relevance of monopoles and strings to crossover phenomena are considered. A brief discussion of the continuity of fields and topological charge in asymptotically free lattice models is presented. The final major topic of this review concerns lattice fermions. The species doubling problem and its relation to chiral symmetry are illustrated. Staggered Euclidean fermion methods are discussed in detail, with an emphasis on species counting, remnants of chiral symmetry, Block spin variables, and the axial anomaly. Numerical methods for including fermions in computer simulations are considered. Jacobi and Gauss-Siedel inversion methods to obtain the fermion propagator in a background gauge field are reviewed. Langevin, pseudofermion Monte Carlo methods, and random walk methods are discussed, as are proposals for including the fermion determinant in computer simulations. Finally, spontaneous chiral symmetry breaking in lattice models is discussed. The review ends with brief remarks about present research directions.

The lattice gauge theory approach to quantum chromodynamics

Using a formulation of quantum electrodynamics in the infinite-momentum frame, we develop a theory to describe the scattering of energetic electrons or photons off an external field. A physical picture emerges which proves to be a realization of Feymnan's "parton" ideas. In this picture the incoming electron is composed of bare constituents (the quanta of the Schrodinger fields) which, at high laboratory energies, interact slowly with one another. Each bare constitutent is scattered from the external field in a simple way and then the constituents again interact among themselves to form the final state. This formalism is applied to elastic electron and photon scattering, bremsstrahlung and pair production, and deep-inelastic electro-production of lepton pairs, and the results of Cheng and Wu and others are recovered in a simple way. In these applications, perturbation theory is used to construct the wave functions of the constituents in the initial and final states.

https://inspirehep.net/files/d0368a411484b6083e223f06ddcac0ff

John B. Kogut

Papers

An introduction to lattice gauge theory and spin systems

The Spectrum of Charmonium

The lattice gauge theory approach to quantum chromodynamics

Quantum Electrodynamics at Infinite Momentum: Scattering from an External Field

QCD-like theories at finite baryon density