Inverse-square Interaction Research Articles

Q-state Potts model with power-law decaying interactions: Along the tricritical line

By relying on a recently proposed multicanonical algorithm adapted to long-ranged models, we shed new light on the critical behavior of the long-ranged q-state Potts model. We refine the controversial phase diagram by an order of magnitude, over a large range of q values, by applying finite-size scaling arguments to spinodal curves. We further offer convincing evidence that the phase transition on the line of inverse-square interactions is not of the first order, by virtue of a very unusual, previously unnoticed, finite-size effect. Finally, we obtain estimates of critical couplings near the mean-field region, which clearly reinforce Tsallis conjecture.

On the real spectra of the Calogero model with complex coupling

We study the eigenvalue problem of the rational Calogero model with the coupling of the inverse-square interaction as a complex number. We show that although this model is manifestly non-invariant under the combined parity and time-reversal symmetry PT , the eigenstates corresponding to the zero value of the generalized angular momentum have real energies.

Generalization of the Darboux transformation and generalized harmonic oscillators

The Darbroux transformation is generalized for time-dependent Hamiltonian systems which include a term linear in momentum and a time-dependent mass. The formalism for the $N$-fold application of the transformation is also established, and these formalisms are applied for a general quadratic system (a generalized harmonic oscillator) and a quadratic system with an inverse-square interaction up to N=2. Among the new features found, it is shown, for the general quadratic system, that the shape of potential difference between the original system and the transformed system could oscillate according to a classical solution, which is related to the existence of coherent states in the system.

Exact ground-state number fluctuations of trapped ideal and interacting fermions

We consider a small and fixed number of fermions in an isolated one-dimensional trap (microcanonical ensemble). The ground state of the system is defined at T=0, with the lowest single-particle levels occupied. The number of particles in this ground state fluctuates as a function of excitation energy. By breaking up the energy spectrum into particle and hole sectors, and mapping the problem onto the classic number partitioning theory, we formulate a new method to calculate the exact particle number fluctuation more efficiently than the direct combinatorics method. The exact ground state number fluctuation for particles interacting via an inverse-square pair-wise interaction is also calculated.

Spin Chains from Super-Models

We construct and study a class of N particle supersymmetric Hamiltonians with nearest and next-nearest neighbor inverse-square interaction in one dimension. We show that inhomogeneous XY models in an external non-uniform magnetic field can be obtained from these super-Hamiltonians in a particular limit decoupling the fermionic degrees of freedom from the kinematic ones. We further consider a suitable deformation of these super-models such that inhomogeneous XXZ Hamiltonians in an external non-uniform magnetic field are obtained in the same limit. We show that this deformed Hamiltonian with rational potential is, (i) mapped to a set of free super-oscillators through a similarity transformation and (ii) supersymmetric in terms of a new, non-standard realization of the supercharge. We construct many exact eigenstates of this Hamiltonian and discuss about the applicability of this technique to other models.

Reflection coefficients of light ions for the inverse-square interaction potential

The linear Boltzmann transport equation for diffusion and slowing down of low-energy light ions in solids is Laplace-transformed in relative path-length and solved by applying the DP0 technique. The ion–target atom interaction potential is assumed to have a form of the inverse-square law and furthermore, the collision integral of the transport equation is replaced by the P 3 approximation in angular variable. The approximative Laplace-transformed solution for the reflection function is found and inverted leading to the distribution of backscattered particles in the relative path-length. Analytic expressions for the particle and energy reflection coefficients were derived and our results are compared with a large number of computer simulation data.

Exact coherent states in one-dimensional quantum many-body systems with inverse-square interactions

For the models of $N$-body identical harmonic oscillators interacting through potentials of homogeneous degree -2, the unitary operator that transforms a system of time-dependent parameters into that of unit spring constant and unit mass of different timescale is found. If the interactions can be written in terms of the differences between positions of two particles, it is also shown that the Schrodinger equation is invariant under a unitary transformation. These unitary relations can be used not only in finding coherent states from the given stationary states in a system, but also in finding exact wave functions of the Hamiltonian systems of time-dependent parameters from those of time-independent Hamiltonian systems. Both operators are invariant under the exchange of any pair of particles. The transformations are explicitly applied for some of the Calogero-Sutherland models to find exact coherent states.

Dynamical Properties of the 1/r2-Type Supersymmetrict-JModel in a Magnetic Field: Manifestation of Spin-Charge Separation

Quasi-particle picture in a magnetic field is pursued for dynamical spin and charge correlation functions of the one-dimensional supersymmetric t-J model with inverse-square interaction. With use of exact diagonalization and the asymptotic Bethe-ansatz equations for finite systems, excitation contents of relevant excited states are identified which are valid in the thermodynamic limit. The excitation contents are composed of spinons, antispinons, holons and antiholons obeying fractional statistics. Both longitudinal and transverse components of the dynamical spin structure factor are independent of the electron density in the region where only quasi-particles with spin degrees of freedom (spinons and antispinons) contribute. The dynamical charge structure factor does not depend on the spin-polarization density in the region where only quasi-particles with charge (holons and antiholons) are excited. These features indicate the strong spin-charge separation in dynamics, reflecting the high symmetry of the model.

HIDDEN NONLINEAR SUPERSYMMETRIES IN PURE PARABOSONIC SYSTEMS

The existence of intimate relations between generalized statistics and supersymmetry is established by the observation of hidden supersymmetric structure in pure parabosonic systems. This structure is characterized generally by a nonlinear superalgebra. The nonlinear supersymmetry of parabosonic systems may be realized, in turn, by modifying appropriately the usual supersymmetric quantum mechanics. The relation of nonlinear parabosonic supersymmetry to the Calogero-like models with exchange interaction and to the spin chain models with inverse-square interaction is pointed out.

Alternative technique for complex spectra analysis

The choice of a suitable random matrix model of a complex system is very sensitive to the nature of its complexity. The statistical spectral analysis of various complex systems requires, therefore, a thorough probing of a wide range of random matrix ensembles which is not an easy task. It is highly desirable, if possible, to identify a common mathematical structure among all the ensembles and analyze it to gain information about the ensemble properties. Our successful search in this direction leads to the Calogero Hamiltonian, a one-dimensional quantum Hamiltonian with inverse-square interaction, as the common base. This is because both the eigenvalues of the ensembles and a general state of the Calogero Hamiltonian evolve in an analogous way for arbitrary initial conditions. The varying nature of the complexity is reflected in different forms of the evolution parameter in each case. A complete investigation of the Calogero Hamiltonian can then help us in the spectral analysis of complex systems.

Dynamics of the t– J model in a magnetic field

Quasi-particle picture is pursued for dynamical charge and spin correlation functions of the one-dimensional supersymmetric t– J model with inverse-square interaction in a magnetic field. With use of exact diagonalization and the asymptotic Bethe-ansatz equations for finite systems, excitation contents of intermediate states are identified in the thermodynamic limit. The dynamical charge structure factor is independent of the spin-polarization density in the region where only quasi-particles with charge (holons and antiholons) are excited. The spin dynamics do not depend on the electron density in the region where only quasi-particles with spin degrees of freedom (spinons and antispinons) contribute. These features indicate the strong spin–charge separation in dynamics, reflecting the high symmetry of the model.

Rodrigues formulas for the non-symmetric multivariable polynomials associated with the BCN-type root system

The non-symmetric Macdonald–Koornwinder polynomials are joint eigenfunctions of the commuting Cherednik operators which are constructed from the representation theory for the affine Hecke algebra corresponding to the BC N -type root system. We present the Rodrigues formula for the non-symmetric Macdonald–Koornwinder polynomials. The raising operators are derived from the realizations of the corresponding double affine Hecke algebra. In the quasi-classical limit, the above theory reduces to that of the BC N -type Sutherland model which describes many particles with inverse-square long-range interactions on a circle with one impurity. We also present the Rodrigues formula for the non-symmetric Jacobi polynomials of type BC N which are eigenstates of the BC N -type Sutherland model.

Yangians and transition operators

Yangian has emerged as a underlying algebraic structure for quantum spin chains with inverse-square exchange interactions. Recently, it has also been found that the Yangian can be regarded as an extended dipole operator or transition operator connecting different highest weight states. In this report, we describe how Yangian can be used to construct the shift operators associated with the spin chain.

Exact Charge Dynamics in the Supersymmetrict-JModel with Inverse-Square Interaction

Dynamical charge structure factor $N(Q,\omega)$ with $Q$ smaller than the Fermi wave number is derived analytically for the one-dimensional supersymmetric t-J model with $1/r^2$ interaction. Strong spin-charge separation in dynamics is found. $N(Q,\omega)$ with given electron density does not depend on the spin polarization for arbitrary size of the system. In the thermodynamic limit, only two holons and one antiholon contribute to $N(Q,\omega)$. These results together with generalized ones for the SU($K$,1) supersymmetric t-J model are derived via solution of the Sutherland model in the strong coupling limit.

Dynamical Properties of the One-Dimensional Supersymmetrict-JModel: A View from Elementary Excitations

Dynamical properties, such as dynamical spin and charge structure factors and single-particle spectral functions, are studied for the one-dimensional supersymmetric t-J model with inverse-square interaction. Exact diagonalization and the recursion method are used for finite systems up to 16 sites. A simple rule is proposed to understand the supermultiplet structure in the excitation spectrum. Numerical calculations show that the dynamical spin structure factor is independent of the electron density in the region where only two spinons contribute. In the electron removal spectrum from half-filling, close correspondence is found between the dominant contribution from three-spinon plus one-holon states in the t-J model and the two-spinon excitations in the Haldane-Shastry model. Comparison is made with dynamics of the nearest-neighbor supersymmetric t-J model.

A quantum many-body problem in two dimensions: ground state

We obtain the exact ground state for the Calogero-Sutherland problem in arbitrary dimensions. In the special case of two dimensions, we show that the problem is connected to the random matrix problem for complex matrices, provided the strength of the inverse-square interaction g = 2. In the thermodynamic limit, we obtain the ground state energy and the pair-correlation function and show that in this case there is no long-range order.

Orthogonality of the Hi-Jack Polynomials Associated with the Calogero Model

The Calogero model is a one-dimensional quantum integrable system with inverse-square long-range interactions confined in an external harmonic well. It shares the same algebraic structure with the Sutherland model, which is also a one-dimensional quantum integrable system with inverse-sine-square interactions. Inspired by the Rodrigues formula for the Jack polynomials, which form the orthogonal basis of the Sutherland model, recently found by Lapointe and Vinet, we construct the Rodrigues formula for the Hi-Jack (hidden-Jack) polynomials that form the orthogonal basis of the Calogero model.

QUASIHOLE WAVE FUNCTIONS FOR THE CALOGERO MODEL

The one-quasihole wave functions and their norms are derived for the system of particles on the line with inverse-square interactions and harmonic confining potential.

Exactly solvable model of interacting electrons confined by the Morse potential

The integrals of motion are constructed for the Sutherland hyperbolic quantum systems of particles with internal degrees of freedom (SU(n) spins) interacting with an external field with the Morse potential of an arbitrary strength τ2. These systems are confined if some constraint is imposed on τ, the strength λ of the pairwise interaction and the number of particles. The ground state is described by the wave function of the Jastrow form. The known results for the systems with inverse-square interaction within harmonic well are reproduced in the limit τ → ∞.

Exact dynamical correlation functions of Calogero-Sutherland model and one-dimensional fractional statistics.

One-dimensional model of non-relativistic particles with inverse-square interaction potential known as Calogero-Sutherland Model (CSM) is shown to possess fractional statistics. Using the theory of Jack symmetric polynomial the exact dynamical density-density correlation function and the one-particle Green's function (hole propagator) at any rational interaction coupling constant $\lambda = p/q$ are obtained and used to show clear evidences of the fractional statistics. Motifs representing the eigenstates of the model are also constructed and used to reveal the fractional {\it exclusion} statistics (in the sense of Haldane's ``Generalized Pauli Exclusion Principle''). This model is also endowed with a natural {\it exchange } statistics (1D analog of 2D braiding statistics) compatible with the {\it exclusion} statistics. (Submitted to PRL on April 18, 1994)