Quantum Heisenberg Chain Research Articles

Solution to three-magnon problem for S=1/2 periodic quantum spin chains with elliptic exchange

The method of solving the three-particle quantum elliptic Calogero–Moser problem is applied to the description of three-magnon wave functions for the S=1/2 quantum Heisenberg chains with the exchange interaction given by the Weierstrass d function. The Bethe-like algebraic equations for the three-magnon case are presented in the explicit form.

Quantum Heisenberg Chain with Long-Range Ferromagnetic Interactions at Low Temperatures

A modified spin-wave theory is applied to the one-dimensional quantum Heisenberg model with long-range ferromagnetic interactions. ( H =- J ∑ i < j ( r i j ) - p S i · S j ). Low-temperature properties of this model are investigated. The susceptibility and the specific heat are calculated; the relation between their behaviors and strength of the long-range interactions is obtained. This model includes both the Haldane-Shastry model and the nearest-neighbor Heisenberg model; the corresponding results in this paper are in agreement with the solutions of both the models. It is shown that there exists an ordering transition at a finite temperature for 1< p <2 where the model has longer-range interactions than the HS model. The critical temperature is estimated.

AN INTRODUCTION TO YANGIAN SYMMETRIES

We review some aspects of the quantum Yangians as symmetry algebras of two-dimensional quantum field theories. The plan of these notes is the following: 1 — The classical Heisenberg model: •Non-Abelian symmetries; •The generators of the symmetries and the semi-classical Yangians; •An alternative presentation of the semi-classical Yangians; (Digression on Poisson-Lie groups. 2 — The quantum Heisenberg chain: •Non-Abelian symmetries and the quantum Yangians; •The transfer matrix and an alternative presentation of the Yangians.

Haldane Gap in the Spin-1/2Double Chain Anisotropic Heisenberg Antiferromagnet

The double chain Heisenberg antiferromagnet with anisotropy is studied using the mapping onto the coupled anisotropic nonlinear σ model. This model consists of two anisotropic antiferromagnetic quantum Heisenberg chains with half-integer spin coupled with each other via the ferromagnetic interchain coupling J '. For small J ', our analysis suggests that J ' is an irrelevant parameter and consequently the two chains behave independently and no gap is generated. On the other hand, for large enough J ', the two chains behave as a single Heisenberg chain with integer spin which has a gap in the excitation spectrum. The expected behavior in the isotropic limit is also discussed.

Haldane Gap in the Spin-1/2 Double Chain Heisenberg Antiferromagnet -Numerical Diagonalization and Projector Monte Carlo Study-

The properties of the spin-1/2 double chain Heisenberg antiferromagnet are studied by means of the numerical diagonalization and the projector Monte Carlo simulation. The model consists of the two spin 1/2 antiferromagnetic quantum Heisenberg chains coupled with each other via the ferromagnetic interchain coupling J ′ . It is shown that the first excited state has a gap for large enough J ′ . This gap continues to the Haldane gap in the limit J ′ →∞, where spins on the different chains form triplets on each site. On the other hand, the gap decreases rapidly between J ′ =1.0 and 0.6 and is undetectable for J ′ <0.6 within the present accuracy.

Bethe ansatz solution of the Fateev-Zamolodchikov quantum spin chain with boundary terms

We solve the Fateev-Zamolodchikov quantum spin chain (i.e., the spin-1 XXZ quantum Heisenberg chain) with a class of boundary terms by the quantum inverse scattering method. For a particular choice of boundary terms, the model has the quantum symmetry U q [SU (2)].

The HeisenbergXXZ Hamiltonian with Dzyaloshinsky-Moriya interactions

The quantum Heisenberg chain with Dzyaloshinsky-Moriya interactions is solved by relating it to theXXZ Hamiltonian with a certain type of boundary conditions. Several properties of the ground state are derived which agree with the intuition derived from related soluble classical models. Implications to the model of known results from the theory of conformal invariance, as well as generalizations to higher spin, are briefly discussed.

Low-temperature behavior of the correlation length and the susceptibility of the ferromagnetic quantum Heisenberg chain.

The renormalization-group recursion relations for the spin-S nearest-neighbor quantum Heisenberg ferromagnet recently derived by Chakravarty and the present author are analyzed in one dimension. It is found that the susecptibility \ensuremath{\chi} is given at low temperatures T by T\ensuremath{\chi}=${C}_{\ensuremath{\chi}}^{(0)}$(${\mathrm{JS}}^{2}$/T) [1+${C}_{\ensuremath{\chi}}^{(1)}$\ensuremath{\eta}+O(${\ensuremath{\eta}}^{2}$)], where J is the exchange coupling and the dimensionless expansion parameter \ensuremath{\eta} is given by \ensuremath{\eta}=${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}1}$(T/${\mathrm{JS}}^{3}$${)}^{1/2}$. For the correlation length \ensuremath{\xi} we find (\ensuremath{\xi}/a)=${C}_{\ensuremath{\xi}}^{(0)}$(${\mathrm{JS}}^{2}$/T) [1+${C}_{\ensuremath{\xi}}^{(1)}$\ensuremath{\eta}+O(${\ensuremath{\eta}}^{2}$)], where a is the lattice spacing. The values of the coefficients ${C}_{\ensuremath{\chi}}^{(j)}$ and ${C}_{\ensuremath{\xi}}^{(j)}$ (j=0,1), have been determined by fitting the above expressions with Monte Carlo data obtained for the S=1/2 Heisenberg chain at temperatures as low as T/J=0.01.

Hidden local gauge invariance in integrable magnetic chains

It is shown that local gauge transformations preserve the integrability of one-dimensional quantum Heisenberg chains. Abelian U(1) gauge transformations associated to z-rotations appear in the XXZ model which is worked out in detail. The exact energy spectrum derived by the Bethe ansatz turns out to be gauge-invariant whereas the eigenvectors are explicitly gauge-dependent. Isotropic XXX chains exhibit SU(2) ⊗ Z2 gauge invariance properties and anisotropic XYZ chains possess discrete Z2 ⊗ Z2 gauge invariance.

Variational approach to the spin-1 anisotropic Heisenberg chain.

A variational procedure with a large number of parameters is used to study the ground state of the spin-1 quantum Heisenberg chain. The different phases are identified from the symmetries and degeneracies of the ground-state wave function. Irrespective of the number of variational parameters, in this approximation the isotropic antiferromagnetic point is always on the boundary between the antiferromagnetic and planar phases. The singlet Heisenberg phase is not found.

Renormalization group treatment for spin waves in the randomly disordered Heisenberg chain

We calculate local densities of states in the randomly disordered binary quantum Heisenberg chain using a generalization of a recently developed approach based on renormalization group ideas. It involves decimating alternate spins along the chain in such a way as to obtain recursion relations to describe the renormalized set of Green's function equations of motion. The densities of states are richly structured, indicating that the method takes into account compositional fluctuations of arbitrary range.

The spectrum of the continuous isotropic quantum Heisenberg chain: quantum solitons as magnon bound states

The continuous Heisenberg chain is quantised semiclassically. The spectrum consists of unit spin bosons (magnons) and integer spin bosons (solitons) with dispersion laws E=JS(ka)2 and En=16JS3 sin2(ka/4S)/n (n=1...) respectively. It is proposed that the n=1 soliton is the magnon and that the spectrum is exact, including all quantum corrections. For S=1/2 the spectrum agrees with results for the discrete chain and it is inferred that the soliton is a multi-magnon bound state.