Semiclassical excitation spectrum of an integrable discrete spin chain

Manuel Gessner, V. M. Bastidas, Tobias Brandes +1 more

We study the excitation spectrum of a family of transverse-field spin chain models with variable interaction range and arbitrary spin $S$, which in the case of $S=1/2$ interpolates between the Lipkin-Meshkov-Glick and the Ising model. For any finite number $N$ of spins, a semiclassical energy manifold is derived in the large-$S$ limit employing bosonization methods, and its geometry is shown to determine not only the leading-order term but also the higher-order quantum fluctuations. Based on a multiconfigurational mean-field ansatz, we obtain the semiclassical backbone of the quantum spectrum through the extremal points of a series of one-dimensional energy landscapes---each one exhibiting a bifurcation when the external magnetic field drops below a threshold value. The obtained spectra become exact in the limit of vanishing or very strong external, transverse magnetic fields. Further analysis of the higher-order corrections in $1/\sqrt{2S}$ enables us to analytically study the dispersion relations of spin-wave excitations around the semiclassical energy levels. Within the same model, we are able to investigate quantum bifurcations, which occur in the semiclassical $(S\ensuremath{\gg}1)$ limit, and quantum phase transitions, which are observed in the thermodynamic $(N\ensuremath{\rightarrow}\ensuremath{\infty})$ limit.

Antal Jevicki, N. Papanicolaou

Hans C. Fogedby

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 E n =16JS 3 sin 2 (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.

Vladyslav O. Cheranovskii

Anastasia Doikou

Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...

Jonathan P. Keating, Noah Linden, H. J. Wells