Bethe Wave Functions Research Articles

Line-shape predictions via Bethe ansatz for the one-dimensional spin-12Heisenberg antiferromagnet in a magnetic field

The spin fluctuations parallel to the external magnetic field in the ground state of the one-dimensional (1D) $s=\frac{1}{2}$ Heisenberg antiferromagnet are dominated by a two-parameter set of collective excitations. In a cyclic chain of N sites and magnetization $0<{M}_{z}<N/2,$ the ground state, which contains ${2M}_{z}$ spinons, is reconfigured as the physical vacuum for a different species of quasiparticles, identifiable in the framework of the coordinate Bethe ansatz by characteristic configurations of Bethe quantum numbers. The dynamically dominant excitations are found to be scattering states of two such quasiparticles. For $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty},$ these collective excitations form a continuum in $(q,\ensuremath{\omega})$ space with an incommensurate soft mode. Their matrix elements in the dynamic spin structure factor ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ are calculated directly from the Bethe wave functions for finite N. The resulting line- shape predictions for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ complement the exact results previously derived via algebraic analysis for the exact two-spinon part of ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ in the zero-field limit. They are relevant for neutron-scattering experiments on quasi-1D antiferromagnetic compounds in a strong magnetic field.

Gaudin magnet and off-mass-shell Bethe wave functions

The general Bethe equation (or off-shell Bethe ansatz equation) is proved for the Gaudin magnet for a broad class of simple complex Lie algebras.

Properties of eigenstates of the six-vertex model with twisted and open boundary conditions

We use the method proposed by Izergin and Korepin to discuss the Bethe ansatz equations of the six-vertex model with twisted and open boundary conditions. Let two parameters of the Bethe wave functions be equal, then an additional Bethe ansatz equation arises. For the open boundary condition case, we also have discussed the dual pseudovacuum state and have written out the simplest scalar products of the Bethe wave function.

Spectrum of the supersymmetric t-J model

Abstract The t-J model is soluble at the supersymmetric point J = −2 t in one dimension using the Bethe ansatz (BA). The BA equations for finite chains are solved for the quasimomenta k j , which are needed to build the Bethe wave function. Bound states are found explicitly when pairs of the quasimomenta are complex. In the thermodynamic limit, they occur in string form. In this limit, elementary excitations above the ground state can be written in terms of so-called ‘BA holes’. By considering certain excitations of the model, the holon and spinon dispersion curves are obtained for finite systems. Comparisons with exact diagonalisation are made to demonstrate the validity of the data.

Integrable time-discretisation of the Ruijsenaars-Schneider model

An exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the Calogero-Moser system, that was introduced for the first time by Ruijsenaars and Schneider. For the discrete-time model the equations of motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2 Heisenberg magnet. We present a Lax pair, the symplectic structure and prove the involutivity of the invariants. Exact solutions are investigated in the rational and hyperbolic (trigonometric) limits of the system that is given in terms of elliptic functions. These solutions are connected with discrete soliton equations. The results obtained allow us to consider the Bethe Ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding Bethe wavefunction.

Comultiplication inABCD algebra and scalar products of bethe wave functions

The representation of scalar products of Bethe wave functions in terms of dual fields, proved by A. G. Izergin and V. E. Korepin in 1987, plays an important role in the theory of completely integrable models. The proof in [A. G. Izergin, Dokl. Akad. Nauk SSSR,297, No. 2, 331 (1987)] and [V. E. Korepin, Commun. Math. Phys.,113, 177–190 (1978)] is based on the explicit expression for the “senior” coefficient, which was guessed in the Izergin paper and then proved to satisfy some recurrent relations, which determine it unambiguously. In this paper we present an alternative proof based on direct computation. It uses the operation of comultiplication in the ABCD-algebra.

Comultiplication in ABCD algebra and scalar products of Bethe wave functions

The representation of scalar products of Bethe wave functions in terms of the Dual Fields, proven by A.G.Izergin and V.E.Korepin in 1987, plays an important role in the theory of completely integrable models. The proof in \cite{Izergin87} and \cite{Korepin87} is based on the explicit expression for the "senior" coefficient which was guessed in \cite{Izergin87} and then proven to satisfy some recurrent relations, which determine it unambiguously. In this paper we present an alternative proof based on the direct computation.

Exact multimagnon states in one-dimensional ferromagnetic spin chains with a short-range interaction

The authors propose the procedure of constructing the exact wavefunctions of M magnons on an infinite one-dimensional lattice in the one-parameter model with an exponentially decreasing interaction between spins. For M<or=4 the problem is solved explicitly. The Bethe wavefunctions in the Heisenberg XXX chain appear in the limit of nearest-neighbour interaction.

On the Bethe ansatz for random directed polymers

We show that the inclusion of the (gapless) center-of-mass motion together with a functional integral representation of the Bethe wave function allows one to predict exactly the critical exponents for random directed polymers in (1+1) dimensions. The corresponding amplitudes are computed; they compare satisfactorily with existing numerical data. Within a replica-symmetric theory, we find that the Green function of the polymer has the form recently proposed by Parisi.

Norms of Bethe wave functions for the nonlinear Schrödinger model of spin-1/2 particles

A formula for scalar products of Bethe wave functions in the nonlinear Schrödinger model of spin- (1)/(2) particles is proposed. It is shown, in addition, that one can replace conjugate states by dual eigenfunctions to calculate correlation functions for integrable systems.

Norms of bound states

The formula for the norms of the Bethe wave functions in the form of a Jacobian plays an important role in the computation of the correlation functions. In the present paper this formula is generalized to the case of bound states.

Fermionic perturbation theory for the statistical mechanics of the nonlinear Schr�dinger model

A fermionic perturbation theory is developed for the statistical mechanics of the nonlinear Schrodinger model. The theory is based on an interacting-fermion picture of the Bethe wave function. The inner product of the Bethe wave function is explicitly evaluated, and a simple graphical representation of it is given. The basic equations obtained for the free energy agree with those of Yang and Yang. In particular, the present theory gives a clear-cut meaning to the ɛ function of Yang and Yang: It represents a fermion energy at finite temperatures.

Traitement unifié du modèle Heisenberg–Ising et du système de fermions correspondant

It is shown that both the Heisenberg–Ising model and the corresponding fermion system yield the same recursion relation for the coefficients of the decomposition of any given eigenstate. The application of the Pauli exclusion principle to the latter model makes it possible to unify the treatment of the two systems. The problem is then reduced to constructing symmetric and antisymmetric Bethe wave function solutions of the recursion relation. Numerical results for the fermion system as a function of both the filling of the band and the interaction strength are presented.

Calculation of norms of Bethe wave functions

A class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered. Eigenfunctions of the Hamiltonians are known for these models. Norms of these eigenfunctions in the finite box are calculated in the present paper. These models include in particular the quantum nonlinear Schrodinger equation and the HeisenbergXXZ model.

Polynomial conservation laws in (1 + 1)-dimensional classical and quantum field theory

The quantum-mechanical implications of the inverse-scattering-transform method and its relationship to the structure of Bethe's ansatz are discussed in the context of the nonlinear Schroedinger equation (many-body problem) associated with the classical (quantum) field theory L = (i/2) phi*partial/sub 0/phi - vertical-barpartial/sub 1/phivertical-bar/sup 2/ - cvertical-barphivertical-bar/sup 4/. We review the transformation of the classical problem to action and angle variables and the derivation of an infinite number of polynomial conservation laws. The values of the conserved constants are given by the moments of the classical action variable. It is suggested that there exists a corresponding set of conserved polynomial operators in the quantum field theory and that they reflect the conservation of velocity content which characterizes the solution of the many-body scattering problem (Bethe's ansatz). This implies that the quantized action variable is just the occupation-number density operator in the asymptotic momentum- (velocity-) parameter space of Bethe's ansatz, and that Bethe's wave functions are eigenstates of all conserved operators with eigenvalues given by the moments of the N-particle distribution in asymptotic momentum space. These statements are verified for the first four operators, including one which has not previously been studied.

Boundary Energy of a Bose Gas in One Dimension

By the superposition of Bethe's wave functions, using the Lieb's solution for the system of identical bosons interacting in one dimension via a $\ensuremath{\delta}$-function potential, we construct the wave function of the corresponding system enclosed in a box by imposing the boundary condition that the wave function must vanish at the two ends of an interval. Coupled equations for the energy levels are derived, and approximately solved in the thermodynamic limit in order to calculate the boundary energy of this Bose gas in its ground state. The method of superposition is also applied to the analogous problem of the Heisenberg-Ising chain (not the ring).