Classical statistical mechanics of a lattice model of superradiance

Barry M. McCoy

David V. Chudnovsky, Gregory V. Chudnovsky

Using the method of Backlund transformations the investigators studied in detail classical and quantum two dimensional completely integrable systems, and lattice models equivalent to and approximating these continuous systems were constructed and examined. Among the continuous systems thoroughly examined in this way are the following important models of quantum field theory and statistical mechanics: the matrix (quantum and classical) nonlinear Schrodinger equation, the sine-Gordon equation, the matrix (non-Abelian) Toda lattice, the generalized XXX-and XXZ-quantum models and models of statistical mechanics, and the quantum generalized XYZ-lattice system (whose classical counterpart is the Landau-Lifshitz equation of superconductivity and whose statistical mechanics reductions include the eight-vertex model of Baxter.

Vincenzo Gino Benza, E. Montaldi, Mikael Ciftan

L. I. Men’shikov

Norman Margolus

In statistical mechanics, it is well known that finite-state classical lattice models can be recast as quantum models, with distinct classical configurations identified with orthogonal basis states. This mapping makes classical statistical mechanics on a lattice a special case of quantum statistical mechanics, and classical combinatorial entropy a special case of quantum entropy. In a similar manner, finite-state classical dynamics can be recast as finite-energy quantum dynamics. This mapping translates continuous quantities, concepts and machinery of quantum mechanics into a simplified finite-state context in which they have a purely classical and combinatorial interpretation. For example, in this mapping quantum average energy becomes the classical update rate. Interpolation theory and communication theory help explain the truce achieved here between perfect classical determinism and quantum uncertainty, and between discrete and continuous dynamics.

Gemma De las Cuevas

We review some connections between quantum information and statistical mechanics. We focus on three sets of results for classical spin models. First, we show that the partition function of all classical spin models (including models in different dimensions, different types of many-body interactions, different symmetries, etc) can be mapped to the partition function of a single model. Second, we give efficient quantum algorithms to estimate the partition function of various classical spin models, such as the Ising or the Potts model. The proofs of these two results are based on a mapping from partition functions to quantum states and to quantum circuits, respectively. Finally, we show how classical spin models can be used to describe certain fluctuating lattices appearing in models of discrete quantum gravity.