Reflection equations and exactly solvable lattice spin models

Reflection equations are used to obtain families of commuting double-row transfer matrices for interaction-round-a-face (IRF) models with fixed and free boundary conditions. We illustrate our methods for the Andrews-Baxter-Forrester (ABF) models which areL-state models associated with the quantum groupU _q (su(2)) at a root of unity. We construct elliptic solutions to the reflection equations for the ABF models by a procedure which uses fusion to build the solutions starting from a trivial solution.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.On leave from Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia.The authors thank Vladimir Rittenberg for his kind hospitality at Bonn. This work was supported by the Australian Research Council.     

Time-dependent exactly solvable models for quantum computing

A time-dependent periodic Hamiltonian admitting exact solutions is applied to construct a set of universal gates for a quantum computer. The time evolution matrices are obtained in an explicit form and used to construct logic gates for computation. A way of obtaining an entanglement operator is discussed, too. The method is based on transformation of soluble time-independent equations into time-dependent ones by employing a set of special time-dependent transformation operators. The text was submitted by the authors in English.     

Class of exactly solvable pairing models

We present three classes of exactly solvable models for fermion and boson systems, based on the pairing interaction. These models are solvable in any dimension. As an example we show the first results for fermions interacting with repulsive pairing forces in a two-dimensional square lattice. In spite of the repulsive pairing force the exact results show attractive pair correlations.     

Star-triangle relations in the exactly solvable statistical models

A regular method for analysis of lattice spin models with a nearest neighbour interaction is proposed. Star-triangle relations in the form of functional equations are used. Parametric families of transfer matrices commuting due to star-triangle relations are constructed. The eigenvalues of transfer matrices as functions of the spectral parameter are shown to obey two functional equations. The solution of these equations for the maximal eigenvalue yields the partition function of the model. The method is applied for evaluation of the partition function of the critical Potts models, the Ising model, the Ashkin-Teller model equivalent to the eight-vertex model.     

Lattice gases and exactly solvable models

We detail the construction of a family of lattice gas automata based on a model of 't Hooft, proceeding by use of symmetry principles to define first the kinematics of the model and then the dynamics. A spurious conserved quantity appears; we use it to effect a radical transformation of the model into one whose spacetime configurations are equivalent to the two-dimensional states of an exactly solvable statistical mechanics model, the symmetric eight-vertex model with parameters restricted to a disorder variety. We comment on the implications of this identification for the original lattice gas.     

Many-body forces in exactly solvable field-theoretical models

The forces acting among three, and more, baryons, are discussed, in the framework of the (exactly solvable) field theoretical models of Lee and of Wentzel.     

Entanglement and relaxation in exactly solvable models

A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ρ^ℐ(t) approaches an equilibrium state ρ_eq^ℐ, the latter depending only on macroscopic characteristics of the bath (e.g. temperature), but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present contribution we discuss the recently proven general theorems about thermalization and demonstrate how they work in exactly solvable models. In particular, we review a necessary condition for the system initial state independence (ISI) of ρ_eq^ℐ, which was proven in our previous work, and apply it for two exactly solvable models, the XX spin chain and a central spin model with a special interaction with the environment. In the latter case we are able to prove the absence of the system ISI. Also the Eigenstate Thermalization Hypothesis is discussed. It is pointed out that although it is supposed to be generically true in essentially not-integrable (chaotic) quantum systems, it is how-ever also valid in the integrable XX model.     

A class of exactly solvable Fokker Planck equations

A class of nonlinear,n-dimensional Fokker Planck equations with exact time dependent solution is presented. An equation of this class can be obtained from any function (q ₁, ,q _n ). Some examples are discussed. For a certain subclass, the associated Itô and Stratonovich stochastic differential equations coincide.     

Random matrix theories and exactly solvable models

A connection between random-matrix theories and exactly solvable models is discussed here. This is done in three parts: firstly, for theWigner—Dyson case; secondly, for the short-range Dyson case; and thirdly, for the pseudo-Hermitian one. The exactly solvable models are variants and extensions of Calogero—Sutherland—Moser systems.     

The new identity for the scattering matrix of exactly solvable models

We discovered a simple quadratic equation, which relates scattering phases of particles on Fermi surface. We consider one-dimensional Bose gas and XXZ Heisenberg quantum spin chain. Received: 4 December 1997 / Accepted: 17 March 1998     

Generalized algebraic transformations and exactly solvable classical-quantum models

The rigorous approach aimed at providing exact analytical results for hybrid classical-quantum models is elaborated on the grounds of generalized algebraic mapping transformations. This conceptually simple method allows one to obtain novel interesting exact results for the hybrid classical-quantum models, which may for instance describe interacting many-particle systems composed of the classical Ising spins and quantum Heisenberg spins, the localized Ising spins and delocalized electrons, or many other hybrid systems of a mixed classical-quantum nature.     

Spin-phonon coupling in intermediate valency: exactly solvable models

Two exactly solvable models are presented to describe the dynamic coupling of magnetic transitions within Kramer's degenerate multiplets and lattice vibrations. The magnetic susceptibilities are calculated. It is shown that the spin-boson coupling can reduce the single ion Curic constant at low temperature. For special symmetries the Curie constant may vanish, yielding a temperature independent susceptibility at lowT.     

Understanding finite size effects in quasi-long-range orders for exactly solvable chain models

In this paper, we investigate how much of the numerical artefacts introduced by finite system size and choice of boundary conditions can be removed by finite size scaling, for strongly correlated systems with quasi-long-range order. Starting from the exact ground-state wave functions of hardcore bosons and spinless fermions with infinite nearest-neighbor repulsion on finite periodic chains and finite open chains, we compute the two-point, density-density, and pair-pair correlation functions, and fit these to various asymptotic power laws. Comparing the finite-periodic-chain and finite-open-chain correlations with their infinite-chain counterparts, we find reasonable agreement among them for the power-law amplitudes and exponents, but poor agreement for the phase shifts. More importantly, for chain lengths on the order of 100, we find our finite-open-chain calculation overestimates some infinite-chain exponents (as did a recent density-matrix renormalization-group (DMRG) calculation on finite smooth chains), whereas our finite-periodic-chain calculation underestimates these exponents. We attribute this systematic difference to the different choice of boundary conditions. Eventually, both finite-chain exponents approach the infinite-chain limit: by a chain length of 1000 for periodic chains, and >2000 for open chains. There is, however, a misleading apparent finite size scaling convergence at shorter chain lengths, for both our finite-chain exponents, as well as the finite-smooth-chain exponents. Implications of this observation are discussed.     

Thermodynamic picture of the glassy state gained from exactly solvable models

A picture for thermodynamics of the glassy state was introduced recently by us [Phys. Rev. Lett. 79, 1317 (1997); 80, 5580 (1998)]. It starts by assuming that one extra parameter, the effective temperature, is needed to describe the glassy state. This approach connects responses of macroscopic observables to a field change with their temporal fluctuations, and with the fluctuation-dissipation relation, in a generalized, nonequilibrium way. Similar universal relations do not hold between energy fluctuations and the specific heat. In the present paper, the underlying arguments are discussed in greater length. The main part of the paper involves details of the exact dynamical solution of two simple models introduced recently: uncoupled harmonic oscillators subject to parallel Monte Carlo dynamics, and independent spherical spins in a random field with such dynamics. At low temperature, the relaxation time of both models diverges as an Arrhenius law, which causes glassy behavior in typical situations. In the glassy regime, we are able to verify the above-mentioned relations for the thermodynamics of the glassy state. In the course of the analysis, it is argued that stretched exponential behavior is not a fundamental property of the glassy state, though it may be useful for fitting in a limited parameter regime.     

A set of exactly solvable Ising models with half-odd-integer spin

We present a set of exactly solvable Ising models, with half-odd-integer spin-S on a square-type lattice including a quartic interaction term in the Hamiltonian. The particular properties of the mixed lattice, associated with mixed half-odd-integer spin-(S,1/2) and only nearest-neighbor interaction, allow us to map this system either onto a purely spin-1/2 lattice or onto a purely spin-S lattice. By imposing the condition that the mixed half-odd-integer spin-(S,1/2) lattice must have an exact solution, we found a set of exact solutions that satisfy the free fermion condition of the eight vertex model. The number of solutions for a general half-odd-integer spin-S is given by S+1/2. Therefore we conclude that this transformation is equivalent to a simple spin transformation which is independent of the coordination number.     

An exactly solvable model for the large polaron

We present an exactly diagonalizable model Hamiltonian for the large polaron derived by analyzing the variational ansatz by Haga-Larsen (HL) for the Fröhlich Hamiltonian. The lowest energy eigenvalue of the model Hamiltonian for fixed wave numbers reproduces the energy of the variational ansatz by Haga-Larsen and is, therefore, an upper bound with respect to the corresponding energy eigenvalue of the Fröhlich Hamiltonian. This is valid for any momentum which is proven by extending the Haga-Larsen approach. Furthermore, since all integrations can be performed analytically, the model Hamiltonian is easily tractable. The energy eigenvalue spectrum of the model Hamiltonian is studied below and above the phonon-emission threshold. The quality of the model Hamiltonian is determined by the variational ansatz of Haga and Larsen. Incorporating an improved energy-momentum relation, a generalized model Hamiltonian is derived possessing a larger validity range with respect to the coupling strength. Furthermore, a second exactly diagonalizable model Hamiltonian based on improved Wigner-Brillouin perturbation theory due to Warmenbol, Peeters, and Devreese (WPD) is presented. It is briefly demonstrated that one is able to construct all mentioned model Hamiltonians also in the 2D polaron problem. In contrast to the 3D case, where the HL-type model Hamiltonian possesses the higher quality for any momentum, in the 2D case, it works well only for small momenta. For large momenta, only the WPD-type model Hamiltonian describes the energy-momentum relation correctly. We demonstrate the usefulness of the model Hamiltonian concept by exactly calculating the one-electron Green’s function for all mentioned model Hamiltonians and comment why significant advantages of the model Hamilton concept for the treating of low-dimensional systems (planar semiconducting quantum-well structures) can be expected.