A novel quasi-exactly solvable spin chain with nearest-neighbors interactions
arXiv:0704.3046 · doi:10.1016/j.nuclphysb.2007.07.001
Abstract
In this paper we study a novel spin chain with nearest-neighbors interactions depending on the sites coordinates, which in some sense is intermediate between the Heisenberg chain and the spin chains of Haldane-Shastry type. We show that when the number of spins is sufficiently large both the density of sites and the strength of the interaction between consecutive spins follow the Gaussian law. We develop an extension of the standard freezing trick argument that enables us to exactly compute a certain number of eigenvalues and their corresponding eigenfunctions. The eigenvalues thus computed are all integers, and in fact our numerical studies evidence that these are the only integer eigenvalues of the chain under consideration. This fact suggests that this chain can be regarded as a finite-dimensional analog of the class of quasi-exactly solvable Schroedinger operators, which has been extensively studied in the last two decades. We have applied the method of moments to study some statistical properties of the chain's spectrum, showing in particular that the density of eigenvalues follows a Wigner-like law. Finally, we emphasize that, unlike the original freezing trick, the extension thereof developed in this paper can be applied to spin chains whose associated dynamical spin model is only quasi-exactly solvable.
38 pages, 8 figures
References in corpus (21)
- The Bethe-Ansatz for N=4 Super Yang-Mills
- Long-Range PSU(2,2|4) Bethe Ansaetze for Gauge Theory and Strings
- Yang-Baxter equation in spin chains with long range interactions
- Lattice Integrable Systems of Haldane-Shastry Type
- On Models with Inverse-Square Exchange
- Deformations of N=4 SYM and integrable spin chain models
- Integrable Systems for Particles with Internal Degrees of Freedom
- Exact Spectrum of SU(n) Spin Chain with Inverse-Square Exchange
- Spectrum of a spin chain with inverse square exchange
- Yang-Mills Correlation Functions from Integrable Spin Chains
- Global properties of the spectrum of the Haldane-Shastry spin chain
- A class of N-body problems with nearest- and next-to-nearest neighbour interactions
- Exact partition function of SU(m|n) supersymmetric Haldane-Shastry spin chain
- Haldane-Shastry spin chains of BC_N type
- Variational Mean Field approach to the Double Exchange Model
- A_N-type Dunkl operators and new spin Calogero-Sutherland models
- New spin Calogero-Sutherland models related to B_N-type Dunkl operators
- Integrable Spin Chains with U(1)^3 symmetry and generalized Lunin-Maldacena backgrounds
- Solvable scalar and spin models with near-neighbors interactions
- Exchange operator formalism for N-body spin models with near-neighbors interactions
- Affine Toda-Sutherland Systems