Bethe Ansatz

We consider the most general three-state spin chain with U(1)^3 symmetry and nearest neighbour interaction. Our model contains as a special case the spin chain describing the holomorphic three scalar sector of the three parameter complex deformation of N=4 SYM, dual to type IIB string theory in the generalized Lunin-Maldacena backgrounds discovered by Frolov. We formulate the coordinate space Bethe ansatz, calculate the S-matrix and determine for which choices of parameters the S-matrix fulfills the Yang-Baxter equations. For these choices of parameters we furthermore write down the R-matrix. We find in total four classes of integrable models. In particular, each already known model of the above type is nothing but one in a family of such models.

We present the solution of the SU(N)⊗SU(M) Anderson impurity model using the Bethe-Ansatz. We first explain what extensions to the formalism were required for the solution. Subsequently we determine the ground state and derive the thermodynamics over the full range of temperature and fields. We identify the different regimes of valence fluctuation at high temperatures, followed by moment formation or

We described the recently introduced method of algebraic bosonization of the (1 + 1)-dimensional Luttinger systems by discussing in detail the specific case of the Calogero–Sutherland model, and mentioning the treatment of the hard-core Bose gas. We also compare our findings with the exact Bethe Ansatz results.

This is a historical note. Bethe Ansatz solvable models are considered, for example XXZ Heisenberg anti-ferromagnet and Bose gas with delta interaction. Periodic boundary conditions lead to Bethe equation. The square of the norm of Bethe wave function is equal to a determinant of linearized system of Bethe equations (determinant of matrix of second derivatives of Yang action). The proof was first published in Communications in Mathematical Physics, vol 86, page 391 in l982. Also domain wall boundary conditions for 6 vertex model were discovered in the same paper [see Appendix D]. These play an important role for algebraic combinatorics: alternating sign matrices, domino tiling and plane partition. Many publications are devoted to six vertex model with domain wall boundary conditions.

A recently introduced one-dimensional two-particle Bose-Hubbard model with a single impurity
[J. M. Zhang et al., Phys. Rev. Lett. 109, 116405 (2012)] is studied on finite lattices. The model
possesses a discrete reflection symmetry and we demonstrate that all eigenstates odd under this
symmetry can be obtained with a generalized Bethe ansatz if periodic boundary conditions are
imposed. Furthermore, we provide numerical evidence that this holds true for open boundary
conditions as well. The model exhibits backscattering at the impurity site—which usually destroys
integrability—yet there exists an integrable subspace. We investigate the non-integrable even sector
numerically and find a class of states which have almost the Bethe ansatz form. These weakly
diffractive states correspond to a weak violation of the non-local Yang-Baxter relation which is
satisfied in the odd sector. We bring up a method based on the Prony algorithm to check whether
a numerically obtained wave function is in the Bethe form or not, and if so, to extract parameters
from it. This technique is applicable to a wide variety of other lattice models.

We investigate bound states in the one-dimensional two-particle Bose-Hubbard model with an attractive (V>0) impurity potential. This is a one-dimensional, discrete analogy of the hydrogen negative ion (H−) problem. There are several different types of bound states in this system, each of which appears in a specific region. For given V, there exists a (positive) critical value Uc1 of U (the on-site atom-atom interaction), below which the ground state is a bound state. Interestingly, close to the critical value (U≲Uc1), the ground state can be described by the Chandrasekhar-type variational wave function, which was initially proposed for H−. For U>Uc1, the ground state is no longer a bound state. However, there exists a second (larger) critical value Uc2 of U, above which a molecule-type bound state is established and stabilized by the repulsion. We have also tried to solve for the eigenstates of the model using the Bethe ansatz. The model possesses a global Z2 symmetry (parity) which allows classification of all eigenstates into even and odd states. It is found that all states with odd parity have the Bethe form, but none of the states in the even-parity sector. This allows us to identify analytically two odd-parity bound states, which appear in the parameter regions −2V<U<−V and −V<U<0, respectively. Remarkably, the latter one can be embedded in the continuum spectrum with appropriate parameters. Moreover, in part of these regions, there exists an even-parity bound state accompanying the corresponding odd-parity bound state with almost the same energy.

We revisit the one-dimensional attractive Hubbard model by using the Bethe-ansatz-based density-functional theory and density-matrix renormalization method. The ground-state properties of this model are discussed in details for different fillings and different confining conditions in weak-to-intermediate coupling regime. We investigate the ground-state energy, energy gap, and pair-binding energy and compare them with those calculated from the canonical Bardeen-Cooper-Schrieffer approximation. We find that the Bethe-ansatz-based density-functional theory is computationally easy and yields an accurate description of the ground-state properties for weak-to-intermediate interaction strength, different fillings, and confinements. In order to characterize the quantum phase transition in the presence of a harmonic confinement, we calculate the thermodynamic stiffness, the density-functional fidelity, and fidelity susceptibility, respectively. It is shown that with the increase in the number of particles or attractive interaction strength, the system can be driven from the Luther-Emery-type phase to the composite phase of Luther-Emery-type in the wings and insulatinglike in the center.

We develop some useful techniques for integrating over Higgs branches in supersymmetric theories with 4 and 8 supercharges. In particular, we define a regularized volume for hyperkähler quotients. We evaluate this volume for certain ALE and ALF spaces in terms of the hyperkähler periods. We also reduce these volumes for a large class of hyperkähler quotients to simpler integrals. These quotients include complex coadjoint orbits, instanton moduli spaces on 4 and ALE manifolds, Hitchin spaces, and moduli spaces of (parabolic) Higgs bundles on Riemann surfaces. In the case of Hitchin spaces the evaluation of the volume reduces to a summation over solutions of Bethe Ansatz equations for the non-linear Schrödinger system. We discuss some applications of our results.

The exact perturbation approach is used to derive the elementary correlation lengths xii and related mass gaps mi of the two-dimensional dilute AL lattice model in regimes 1 and 2 for L odd from the Bethe ansatz solution. In regime 2 the A3 model is the E8 lattice realisation of the two-dimensional Ising model in a magnetic field at T=Tc. The calculations for the A3 model in regime 2 start from the eight thermodynamically significant string types found in previous numerical studies. These string types are seen to be consistent in the ordered high field limit. The eight masses obtained reduce with the approach to criticality to the E8 masses predicted by Zamolodchikov, thus providing a further direct lattice determination of the E8 mass spectrum.

We present the solution of the SU(N)⊗SU(M) Anderson impurity model using the Bethe-Ansatz. We first explain what extensions to the formalism were required for the solution. Subsequently we determine the ground state and derive the thermodynamics over the full range of temperature and fields. We identify the different regimes of valence fluctuation at high temperatures, followed by moment formation or

We study the Mott insulating phase of the one-dimensional Hubbard model using a local-density approximation (LDA) that is based on the Bethe Ansatz (BA). Unlike conventional functionals, the BA-LDA has an explicit derivative discontinuity. We demonstrate that as a consequence of this discontinuity the BA-LDA yields the correct Mott gap, independently of the strength of the correlations. A convenient analytical formula for the Mott gap in the thermodynamic limit is also derived. We find that in one-dimensional quantum systems the contribution of the discontinuity to the full gap is more important than that of the band-structure gap, and discuss some consequences this finding has for electronic-structure calculations.