Exactly solvable one-dimensional inhomogeneous models

We present a simple way of constructing one-dimensional inhomogeneous models (random or quasiperiodic) which can be solved exactly. We treat the example of an Ising chain in a varying magnetic field, but our procedure can easily be extended to other one-dimensional inhomogeneous models. For all the models we can construct, the free energy and its derivatives with respect to temperature can be computed exactly at one particular temperature.     

Exactly solvable two-dimensional quantum spin models

A method is proposed for constructing an exact ground-state wave function of a two-dimensional model with spin 1/2. The basis of the method is to represent the wave function by a product of fourth-rank spinors associated with the nodes of a lattice and the metric spinors corresponding to bonds between nearest neighbor nodes. The function so constructed is an exact wave function of a 14-parameter model. The special case of this model depending on one parameter is analyzed in detail. The ground state is always a nondegenerate singlet, and the spin correlation functions decay exponentially with distance. The method can be generalized for models with spin 1/2 to other types of lattices. Zh. éksp. Teor. Fiz. 115, 249–267 (January 1999)     

Exactly solvable models for atom-molecule Hamiltonians

We present a family of exactly solvable generalizations of the Jaynes-Cummings model involving the interaction of an ensemble of SU(2) or SU(1,1) quasispins with a single boson field. They are obtained from the trigonometric Richardson-Gaudin models by replacing one of the SU(2) or SU(1,1) degrees of freedom by an ideal boson. The application to a system of bosonic atoms and molecules is reported.     

Exactly solvable models showing chaotic behavior

Several examples of the one-dimensional mapping which are exactly solvable and show chaotic behaviour are presented. The importance of the accuracy of the numerical calculation is stressed.     

Exactly solvable dicke models and radiation dragging

6. Conclusion We have investigated various exactly solvable Dicke-model cases describing the interaction of two- or three-level atoms with the electromagnetic field in an ideal cavity. In Secs. 2 and 3 are considered two-level atoms, one or two of which are excited at the initial instant of time; both symmetric and asymmetric excitations are considered. In addition, solutions are presented for the case when there are no excited atoms at the initial instant of time, but one or two photons are present. It is shown that in the case of symmetric initial conditions superradiance is present in a certain sense, whereas asymmetric initial conditions lead to the effect of "radiation dragging" from one or two atoms for a very large total number of atoms. In Sec. 4 it is shown that radiation dragging takes place also in the case of asymmetric initial excitation of an arbitrary number m of two-level atoms in the presence of a very large number n of atoms in the ground state. The radiation-dragging condition is the inequality m < < n. Furthermore, in Sec. 5 are considered symmetric and asymmetric excitations of one of N three-level atoms interacting with two or more field modes. The different configurations of the atomic levels (ladder and Λ cases) are considered). It turns out that in addition to radiation dragging due to asymmetric initial conditions, there appears for the ladder configuration an additional absence of radiation from the upper level of the atom in the case of symmetric initial conditions. This additional vanishing of the radiation occurs also in the equidistant ladder three-level case, when transitions between neighboring level pairs result from interaction with one field mode. The authors are deeply grateful to V. I. Man’ko for interest in the work, and to é. A. Akhundova and V. P. Karasev for helpful discussions. Translated from Trudy Fizicheskogo Instituta im. P. N. Lebedeva, Vol. 191, pp. 150–170, 1989.     

Exactly solvable models in atomic and molecular physics

We construct integrable generalized models in a systematic way exploring different representations of the gl(N)$\mathit{gl}(N)$ algebra. The models are then interpreted in the context of atomic and molecular physics, most of them related to different types of Bose–Einstein condensates. The spectrum of the models is given through the analytical Bethe ansatz method. We further extend these results to the case of the superalgebra gl(M|N)$\mathit{gl}(M|N)$, providing in this way models which also include fermions.     

Exactly solvable 1D frustrated quantum spin models

A class of the frustrated quantum s = ½ models with nearest and next nearest neighbor couplings is investigated. An exact wave function of the singlet ground state at the transition point from the ferromagnetic to the singlet state is presented. The recurrence technics of expectation value calculations is developed and the simple expressions for spin-correlation function at N → ∞ are obtained. A long range double-spiral ordering is demonstrated. We show that in one particular case the model reduces to the effective spin-1 model and the exact singlet ground state wave function is presented for this model. The behavior of the system in the vicinity of the transition point is investigated.     

Exactly solvable models of nonstationary turbulence in Bose condensate

In this Letter we study a turbulence decay mechanism in the superfluid liquid. We proceed with developement of master equation approach introduced Copeland, Kibble, Steer and Nemirovskii. We obtain the full rate of reconnection in presence of normal component. We also discuss different random-walk models of vortex filaments. We obtain the expression for the reconnection rate in the nonstationary vortex tangle for these models. The equation for the full number of vortex loops is derived. We also obtain the expression for the relaxation time.     

Exactly solvable reaction diffusion models on a Cayley tree

The most general reaction-diffusion model on a Cayley tree with nearest-neighbor interactions is introduced, which can be solved exactly through the empty-interval method. The stationary solutions of such models, as well as their dynamics, are discussed. Concerning the dynamics, the spectrum of the evolution Hamiltonian is found and shown to be discrete, hence there is a finite relaxation time in the evolution of the system towards its stationary state.     

Exactly solvable energy-dependent potentials

We introduce a method for constructing exactly-solvable Schrödinger equations with energy-dependent potentials. Our method is based on converting a general linear differential equation of second order into a Schrödinger equation with energy-dependent potential. Particular examples presented here include harmonic oscillator, Coulomb and Morse potentials with various types of energy dependence.     

Exactly solvable time-dependent models in quantum mechanics and their applications

Methods for obtaining exact and approximate solutions of the evolution of quantum-mechanical problems are discussed. The cyclic evolution of quantum systems described by time-periodic Hamiltonians is analyzed. A class of time-periodic Hamiltonians is constructed in the close analytical form. The corresponding cyclic solutions are calculated. Time-dependent Hamiltonians are generated whose expectation values calculated with cyclic solutions are time independent. It is shown that the expectation values of the spin projection calculated with the same cyclic solutions, as well as the probability density of finding a particle at a given space-time point, are also time independent. Therefore, the approach can be used to simulate quantum dynamic potential wells with the particle localization effect. Nonadiabatic geometric phases are expressed in terms of the cyclic solutions. Exactly solvable time-dependent problems are used to construct a universal set of gates for quantum computers. A method for obtaining entanglement operators is discussed.     

Exactly solvable models of conformal-invariant quantum field theory in D-dimensional space

The method for exact solution of a certain class of models of conformal quantum field theory in D-dimensional Euclidean space is proposed. The method allows one to derive closed differential equations for all the Green functions and also algebraic equations to scale dimensions of all field. A scalar field P of a scale dimension d_p = D − 2 is needed for nontrivial solutions to exist. At D ≠ 2 this field is converted to a constant that coincides with the central charge of two-dimensional theories. A new class of D = 2 models has been obtained, where the infinite-parametric symmetry is not manifest. The two-dimensional Wess-Zumino model is used to illustrate the method of solution.     

Exactly solvable sandpile with fractal avalanching

A simple one-dimensional sandpile model is constructed which possesses exact analytical solvability while displaying both scale-free behavior and fractal properties. The sandpile grows by avalanching on all scales, yet its shape and energy content are described by a simple, continuous (but nowhere differentiable) analytical formula. The avalanche energy distribution and the avalanche time series are both power laws with index -1 ("1/f spectra").