New perspectives on the Ising model

The Ising model, in presence of an external magnetic field, is isomorphic to a model of localized interacting particles satisfying the Fermi statistics. By using this isomorphism, we construct a general solution of the Ising model which holds for any dimensionality of the system. The Hamiltonian of the model is solved in terms of a complete finite set of eigenoperators and eigenvalues. The Green’s function and the correlation functions of the fermionic model are exactly known and are expressed in terms of a finite small number of parameters that have to be self-consistently determined. By using the equation of the motion method, we derive a set of equations which connect different spin correlation functions. The scheme that emerges is that it is possible to describe the Ising model from a unified point of view where all the properties are connected to a small number of local parameters, and where the critical behavior is controlled by the energy scales fixed by the eigenvalues of the Hamiltonian. By using algebra and symmetry considerations, we calculate the self-consistent parameters for the one-dimensional case. All the properties of the system are calculated and obviously agree with the exact results reported in the literature.     

Exact solution of the one-dimensional spin-\frac{\mathsf 3}{\mathsf 2} Ising model in magnetic field

In this paper, we study the Ising model with general spin S in presence of an external magnetic field by means of the equations of motion method and of the Green's function formalism. First, the model is shown to be isomorphic to a fermionic one constituted of 2S species of localized particles interacting via an intersite Coulomb interaction. Then, an exact solution is found, for any dimension, in terms of a finite, complete set of eigenoperators of the latter Hamiltonian and of the corresponding eigenenergies. This explicit knowledge makes possible writing exact expressions for the corresponding Green's function and correlation functions, which turn out to depend on a finite set of parameters to be self-consistently determined. Finally, we present an original procedure, based on algebraic constraints, to exactly fix these latter parameters in the case of dimension 1 and spin . For this latter case and, just for comparison, for the cases of dimension 1 and spin [F. Mancini, Eur. Phys. J. B 45, 497 (2005)] and spin 1 [F. Mancini, Eur. Phys. J. B 47, 527 (2005)], relevant properties such as magnetization 〈S 〉 and square magnetic moment 〈S² 〉, susceptibility and specific heat are reported as functions of temperature and external magnetic field both for ferromagnetic and antiferromagnetic couplings. It is worth noticing the use we made of composite operators describing occupation transitions among the 3 species of localized particles and the related study of single, double and triple occupancy per site.     

Universal correlations of one-dimensional interacting electrons in the gas phase

We consider dynamical correlation functions of short range interacting electrons in one dimension at finite temperature. Below a critical value of the chemical potential there is no Fermi surface anymore, and the system can no longer be described as a Luttinger liquid. Its low temperature thermodynamics is that of an ideal gas. We identify the impenetrable electron gas model as a universal model for the gas phase and present exact and explicit expressions for the asymptotics of correlation functions at small temperatures, in the presence of a magnetic field.     

Remarks on A 2 Toda theory

We study the Toda field theory with finite Lie algebras using an extension of the Goulian-Li technique. In this way, we show that, after integrating over the zero mode in the correlation functions of the exponential fields, the resulting correlation function resembles that of a free theory. Furthermore, it is shown that for some ratios of the charges of the exponential fields the four-point correlation functions which contain a degenerate field satisfy the Riemann ordinary differential equation. Using this fact and the crossing symmetry, we derive a set of functional equations for the structure constants of the A ₂ Toda field theory.     

Slave bosons in radial gauge: A bridge between path integral and Hamiltonian language

We establish a correspondence between the resummation of world lines and the diagonalization of the Hamiltonian for a strongly correlated electronic system. For this purpose, we analyze the functional integrals for the partition function and the correlation functions invoking a slave boson representation in the radial gauge. We show in the spinless case that the Green's function of the physical electron and the projected Green's function of the pseudofermion coincide. Correlation and Green's functions in the spinful case involve a complex entanglement of the world lines which, however, can be obtained through a strikingly simple extension of the spinless scheme. As a toy model we investigate the two-site cluster of the single impurity Anderson model which yields analytical results. All expectation values and dynamical correlation functions are obtained from the exact calculation of the relevant functional integrals. The hole density, the hole auto-correlation function and the Green's function are computed, and a comparison between spinless and spin 1/2 systems provides insight into the role of the radial slave boson field. In particular, the exact expectation value of the radial slave boson field is finite in both cases, and it is not related to a Bose condensate.     

Applications of Reflection Amplitudes in Toda-Type Theories

Reflection amplitudes are defined as two-point functions of certain class of conformal field theories where primary fields are given by vertex operators with real couplings. Among these, we consider (Super-) Liouville theory and simply and non-simply laced Toda theories. In this paper we show how to compute the scaling functions of effective central charge for the models perturbed by some primary fields which maintains integrability. This new derivation of the scaling functions are compared with the results from conventional TBA approach and confirms our approach along with other non-perturbative results such as exact expressions of the on-shell masses in terms of the parameters in the action, exact free energies. Another important application of the reflection amplitudes is a computation of one-point functions for the integrable models. Introducing functional relations between the one-point functions in terms of the reflection amplitudes, we obtain explicit expressions for simply-laced and non-simply-laced affine Toda theories. These nonperturbative results are confirmed numerically by comparing the free energies from the scaling functions with exact expressions we obtain from the one-point functions.     

Forced responses of solid axially polarized piezoelectric ceramic finite cylinders with internal losses

A method is presented to determine the forced responses of piezoelectric cylinders using weighted sums of only certain exact solutions to the equations of motion and the Gauss electrostatic conditions. One infinite set of solutions is chosen such that each field variable is expressed in terms of Bessel functions that form a complete set in the radial direction. Another infinite set of solutions is chosen such that each field variable is expressed in terms of trigonometric functions that form a complete set in the axial direction. Another solution is used to account for the electric field that can exist even when there is no vibration. The weights are determined by using the orthogonal properties of the functions and are used to satisfy specified, arbitrary, axisymmetric boundary conditions on all the surfaces. Special cases including simultaneous mechanical and electrical excitation of cylinders are presented. All numerical results are in excellent agreement with those obtained using the finite element software ATILA. For example, the five lowest frequencies at which the conductance and susceptance of a stress-free cylinder, of length 10 mm and radius 5 mm, reach a local maximum or minimum differ by less than 0.01% from those computed using ATILA.     

A dynamics-controlled truncation scheme for the hierarchy of density matrices in semiconductor optics

We discuss the interaction of coherent electromagnetic fields with the semiconductor band edge in a dynamic density matrix model. Due to the influence of the Coulomb-interaction then-point density matrices are coupled in an infinite hierarchy of equations of motion. We show how this hierarchy is related to an expansion of the density matrices in terms of powers of the exciting field. We make use of the above results to set up a closed set of equations of motion involving two-, four-and six-point correlation functions, from which all third order contributions to the polarization can be calculated exactly. Comparison of our treatment of the hierarchy with the widely used RPA decoupling on the two-point level, gives interesting insight into the validity of the RPA. In particular we find, that a RPA-like factorization for two of the relevant density-matrices yields a solution of their respective equations of motion to lowest order in the electric field.     

Quantum Harmonic Oscillator and Nonstationary Casimir Effect

We consider the relations between the theory of quantum nonstationary damped oscillator and nonstationary Casimir effect in view of the problem of photon creation from vacuum inside the cavity with periodical time-dependent conductivity of a thin semiconductor boundary layer, which simulates periodical displacements of the cavity boundaries. We develop a consistent model of quantum damped harmonic oscillator with arbitrary time-dependent frequency and damping coefficients within the framework of Heisenberg-Langevin equations with two noncommuting delta-correlated noise operators. For the minimum noise set of correlation functions, whose time dependence follows that of the damping coefficient, we obtain the exact solution, which is a generalization of the Husimi solution for undamped nonstationary oscillator. It yields the general formula for the photon-generation rate under the resonance condition in the presence of dissipation. We obtain a simple approximate formula for a time-dependent shift of the complex resonance frequency. It depends only on the total energy of a short laser pulse (which creates an effective time-dependent electron-hole “plasma mirror” on the semiconductor-slab surface) and the recombination time. We show that damping due to a finite conductivity of the material significantly diminishes the photon-generation rate in the selected field mode of the cavity. Nonetheless, we have found optimum values of the parameters (laser pulse power, recombination time, and cavity dimensions), for which the effect of photon generation from vacuum could be observed in the experimental set-up proposed in the University of Padua. We also provide with a list of publications from 2001 to 2005 devoted to the study on quantum-field interactions with moving boundaries (mirrors). Dedicated to Prof. Vladimir I. Man'ko on the occasion of his 65th birthday.     

A multidimensional pendulum in a nonconservative force field under the presence of linear damping

A nonconservative force field in the dynamics of a multidimensional solid is constructed according to the results from the dynamics of real solids occurring in the force field of the action of the medium. In this case, it becomes possible to generalize the equations of motion of a multidimensional solid in a similarly constructed field of forces and to obtain a complete list of, generally speaking, transcendental first integrals expressed through a finite combination of elementary functions. In the study, the integrability in elementary functions is shown for the simultaneous equations of motion of a dynamically symmetric fixed multidimensional solid under the action of a nonconservative pair of forces in the presence of the linear damping moment (the additional dependence of the force field on the tensor of angular velocity of the solid).     

Correlation effects,image charge effects and finite size in the macro-ion-electrolyte system: A field-theoretic approach

We consider a model of a macro-ion surrounded by small ions of an electrolyte solution. The finite size of ionic charge distributions of ions, and image charge effects are considered. From such a model it is possible to construct a statistical field theory with a single fluctuating field and derive physical interpretations for both the mean field and two-point correlation function. For point-like charges, at the level of a Gaussian (or saddle point) approximation, we recover the standard Poisson-Boltzmann equation. However, to include ionic correlation effects, as well as image charge effects of individual ions, we must go beyond this. From the field theory considered, it is possible to construct self-consistent approximations. We consider the simplest of these, namely the Hartree approximation. The Hartree equations take the form of two coupled equations. One is a modified Poisson-Boltzmann equation; the other describes both image charge effects on the individual ions, as well as correlations. Such equations are difficult to solve numerically, so we develop an (a WKB-like) approximation for obtaining approximate solutions. This, we apply to a uniformly charged rod in univalent electrolyte solution, for point like ions, as well as for extended spherically symmetric distributions of ionic charge on electrolyte ions. The solutions show how correlation effects and image charge effects modify the Poisson-Boltzmann result. Finite-size charge distributions of the ions reduce both the effects of correlations and image charge effects. For point charges, we test the WKB approximation by calculating a leading-order correction from the exact Hartree result, showing that the WKB-like approximation works reasonably well in describing the full solution to the Hartree equations. From these solutions, we also calculate an effective charge compensation parameter in an analytical formula for the interaction of two charged cylinders. Electronic supplementary material  Supplementary material in the form of a doc file available from the journal web page at and are accessible for authorised users.     

One-point functions in perturbed boundary conformal field theories

We consider the one-point functions of bulk and boundary fields in the scaling Lee–Yang model for various combinations of bulk and boundary perturbations. The one-point functions of the bulk fields are analysed using the truncated conformal space approach and the form-factor expansion. Good agreement is found between the results of the two methods, though we find that the expression for the general boundary state given by Ghoshal and Zamolodchikov has to be corrected slightly. For the boundary fields we use thermodynamic Bethe ansatz equations to find exact expressions for the strip and semi-infinite cylinder geometries. We also find a novel off-critical identity between the cylinder partition functions of models with differing boundary conditions, and use this to investigate the regions of boundary-induced instability exhibited by the model on a finite strip.     

Quantum transport equations for vibrating isotopically disordered crystals

A transport equation approach is developed for the linear response of the electron-phonon system in a vibrating isotopically disordered crystal to a frequency dependent electric field. The smallness of the mutual differences of the various masses is taken into account to derive a microscopic set of equations for the phonon and electron distribution functions. Applying an appropriate averaging technique, we obtain a set of equations from which the relevant macroscopic transport equation and hence physical quantities, such as the electrical conductivity, may be derived.     

The Generalized Thermo Jaynes–Cummings Model and?Its Energy Spectrum

By analyzing the characteristics of thermo field dynamics, we present a new generalized thermo Jaynes-Cummings (TJC) model at finite temperature including the thermal effects and expatiate on its physical explanation. By virtue of Lewis-Riesenfeld invariant method, we obtain its exact eigenenergy spectrum and the explicit expressions for the evolution operator of the wavefunctions. In addition, we evaluate various expectation values of physical quantities and find the pseudo-invariant eigen-operator for the generalized TJC Hamiltonian.     

The renormalized σ model

A renormalization procedure of the boson σ model based on the finite field equations of Brandt-Wilson is given. We first show that the current operators appearing in the field equations, which are finite local limit of sums of nonlocal field products and suitable subtraction terms, can be chosen to be the same form as the one given for the symmetric limit except for the symmetry breaking constant source term itself. The set of integral equations derived from the field equations is shown to be equivalent to the usual Bogoliubov-Parasiuk-Hepp renormalization theory, and gives us immediately all the renormalized Green's functions in each order of perturbation theory in clear and straightforward fashion. We then analyze the structures of the model in detail. In particular, Ward identities are shown to be satisfied to all orders of perturbation theory. The Goldstone theorem is a particular consequence of these identities.     

The method of finite differences for some operator field equations

We propose a new approach to the formulation of some operator field equations. In Section 1 we study the consistency of some particular difference equations and the convergence of the exact solutions. In Section 2 we apply these results to the quantum mechanical system in one dimension and obtain explicit and general expressions for the transfer operator and for the corresponding matrix elements.