Narrow Results

We reformulate the Kerov–Kirillov–Reshetikhin (KKR) map in the combinatorial Bethe ansatz from paths to rigged configurations by introducing local energy distribution in crystal base theory. Combined with an earlier result on the inverse map, it completes the crystal interpretation of the KKR bijection for . As an application, we solve an integrable cellular automaton, a higher spin generalization of the periodic box-ball system, by an inverse scattering method and obtain the solution of the initial value problem in terms of the ultradiscrete Riemann theta function.

Bethe ansatz goes back to 1931, when H. Bethe invented it to solve some one-dimensional models, such as XXX spin chain, proposed by W. Heisenberg in 1928. Although it is a very powerful method to compute eigenvalues and eigenvectors of the corresponding Hamiltonian, it can be applied only for very specific boundary conditions: periodic boundary ones, and so-called open-diagonal boundary ones. After reviewing this method, we will present a generalization of it that applies also to open-triangular boundary conditions. This short note presents only the basic ideas of the technique, and does not attend to give a general overview of the subject. Interested readers should refer to the original papers and references therein.

We describe an integrable model, related to the Gaudin magnet, and its relation to the matrix model of Brézin, Itzykson, Parisi and Zuber. Relation is based on Bethe ansatz for the integrable model and its interpretation using orthogonal polynomials and saddle point approximation. Large-N limit of the matrix model corresponds to the thermodynamic limit of the integrable system. In this limit (functional) Bethe ansatz is the same as the generating function for correlators of the matrix models.

A study of the one-loop dilatation operator in the scalar sector of SYM is presented. The dilatation operator is analyzed from the point of view of Hamiltonian matrix models. A Lie algebra underlying operator mixing in the planar large-N limit is presented, and its role in understanding the integrability of the planar dilatation operator is emphasized. A classical limit of the dilatation operator is obtained by considering a contraction of this Lie algebra, leading to a new way of constructing classical limits for quantum spin chains. An infinite tower of local conserved charges is constructed in this classical limit purely within the context of the matrix model. The deformation of these charges and their relation to the charges of the spin chain is also elaborated upon.

We present exact solutions of the Schrödinger equation with spherically symmetric octic potential. We give closed-form expressions for the energies and the wave functions as well as the allowed values of the potential parameters in terms of a set of algebraic equations.

We propose a Q-system for the $A_m^{(1)}$ quantum integrable spin chain. We also find compact determinant expressions for all the Q-functions, both for the rational and trigonometric cases.

A new discrete Lax operator involving discrete canonical variable is introduced which generate new integrable system, and is analyzed in the light of the new concept of canonical Bäcklund transformation and classical r-matrix. The generating function of the transformation is explicitly deduced. The second half of the paper deals with the quantization problem where an explicit form of the Bethe equations are deduced.

We present a method for computing the nonperturbative mass-gap in the theory of bosonic membranes in flat background space–times. The analysis is extended to the study of membranes coupled to background fluxes as well. The computation of mass-gaps is carried out using a matrix regularization of the membrane Hamiltonians. The mass gap is shown to be naturally organized as an expansion in a "hidden" parameter, which turns out to be being the related to the dimensionality of the background space. We then proceed to develop a large N perturbation theory for the membrane/matrix-model Hamiltonians around the quantum/mass corrected effective potential. The same parameter that controls the perturbation theory for the mass gap is also shown to control the Hamiltonian perturbation theory around the effective potential. The large N perturbation theory is then translated into the language of quantum spin chains and the one-loop spectra of various bosonic matrix models are computed by applying the Bethe ansatz to the one-loop effective Hamiltonians for membranes in flat space–times. The spin chains corresponding to the large N effective Hamiltonians for the relevant matrix models are generically not integrable. However, we are able to find large integrable subsectors for all the spin chains of interest. Moreover, the continuum limits of the spin chains are mapped to integrable Landau–Lifschitz models even if the underlying spin chains are not integrable. Apart from membranes in flat space–times, the recently proposed matrix models (arXiv:hep-th/0607005) for noncritical membranes in plane wave type space–times are also analyzed within the paradigm of quantum spin chains. The bosonic sectors of all the models proposed in arXiv:hep-th/0607005 are diagonalized at the one-loop level and an intriguing connection between the existence of supersymmetric vacua and one-loop integrability is also presented.

In this paper, we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang–Baxter and boundary Yang–Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equations and the braid group is briefly discussed. A short review on quantum groups as well as the quantum inverse scattering method (algebraic Bethe ansatz) is also presented.

Recently, Kazakov, Gromov and Vieira applied the discrete Hirota dynamics to study the finite size spectra of integrable two dimensional quantum field theories. The method has been tested from large values of the size L down to moderate values using the SU(2) × SU(2) principal chiral model as a theoretical laboratory. We continue the numerical analysis of the proposed nonlinear integral equations showing that the deep ultraviolet region L → 0 is numerically accessible. To this aim, we introduce a relaxed iterative algorithm for the numerical computation of the low-lying part of the spectrum in the U(1) sector. We discuss in detail the systematic errors involved in the computation. When a comparison is possible, full agreement is found with previous thermodynamical Bethe ansatz computations.

We propose exact S-matrices for the AdS₃/CFT₂ duality between type IIB strings on AdS₃×S³×M₄ with M₄ = S³×S¹ or T⁴ and the corresponding two-dimensional conformal field theories. We fix the two-particle S-matrices on the basis of the symmetries su(1|1) and su(1|1)×su(1|1). A crucial justification comes from the derivation of the all-loop Bethe ansatz matching exactly the recent conjecture proposed by Babichenko et al. [J. High Energy Phys.1003, 058 (2010), arXiv:0912.1723 [hep-th]] and Ohlsson Sax and Stefanski, Jr. [J. High Energy Phys.1108, 029 (2011), arXiv:1106.2558 [hep-th]].

$XXX$ spin chain with spin $s=-1$ appears as an effective theory of Quantum Chromodynamics. It is equivalent to lattice nonlinear Schroediger’s equation: interacting chain of harmonic oscillators [bosonic]. In thermodynamic limit each energy level is a scattering state of several elementary excitations [lipatons]. Lipaton is a fermion: it can be represented as a topological excitation [soliton] of original [bosonic] degrees of freedom, described by the group $Z_2$. We also provide the CFT description (including local quenches) and Yang–Yang thermodynamics of the model.

The missing label for basis vectors of $SU(3)$ representations corresponding to the reduction $SU(3)\supset SO(3)$ can be provided by the eigenvalues of $SO(3)$ scalars in the enveloping algebra of $su(3)$. There are only two such independent elements of degrees three and four. It is shown how the one of degree four can be diagonalized using the analytical Bethe ansatz.

In this paper, we apply two-dimensional supersymmetric gauge theories to directly construct a new Bethe ansatz for the wave functions of the q-boson hopping model, and then derive the q-boson algebras from this ansatz.

A model is introduced for two reduced BCS systems which are coupled through the transfer of Cooper pairs between the systems. The model may thus be used in the analysis of the Josephson effect arising from pair tunneling between two strongly coupled small metallic grains. At a particular coupling strength the model is integrable and explicit results are derived for the energy spectrum, conserved operators, integrals of motion, and wave function scalar products. It is also shown that form factors can be obtained for the calculation of correlation functions. Furthermore, a connection with perturbed conformal field theory is made.

We study the exact solution for a two-mode model describing coherent coupling between atomic and molecular Bose–Einstein condensates (BEC), in the context of the Bethe ansatz. By combining an asymptotic and numerical analysis, we identify the scaling behaviour of the model and determine the zero temperature expectation value for the coherence and average atomic occupation. The threshold coupling for production of the molecular BEC is identified as the point at which the energy gap is minimum. Our numerical results indicate a parity effect for the energy gap between ground and first excited state depending on whether the total atomic number is odd or even. The numerical calculations for the quantum dynamics reveals a smooth transition from the atomic to the molecular BEC.

The shortcomings and advantages of the generalized self-consistent field (GSCF) approach, which includes the electron–hole order parameter and the average spin s within the one-dimensional repulsive Hubbard model, are analyzed by comparison of the ground state properties with the corresponding Bethe ansatz results in an entire parameter space of interaction strength U/t ≥ 0, magnetic field h ≥ 0 and electron concentration 0 ≤ n ≤ 1. The GSCF spectral characteristics are derived and criteria are found for the stability of incommensurate magnetic phase with the wave number 0 < q < π. The GSCF theory displays a simple relationship for the double occupancy D⁽⁺⁾ in terms of n, s and , where D⁽⁺⁾ underestimates electron correlations at weak and intermediate ranges and overestimates correlations at large interaction strengths. Beyond some critical U/t and n≠1 the GSCF D⁽⁺⁾ for spatially homogeneous state vanishes, while the exact D for all n at h=0 decreases gradually as U/t increases. At n ≠ 1 the GSCF chemical potential μ⁽⁺⁾ overestimates electron correlations everywhere and variation μ⁽⁺⁾versusn displays electron instability toward the phase separation in the vicinity of n=1. Exactly at n=1 the GSCF μ⁽⁺⁾versusU/t always underestimates electron correlation and as U/t → ∞ at h=0 we have μ⁽⁺⁾ → 0, while the Bethe ansatz result gives μ → 2t. In the limiting cases U/t → 0 and U/t → ∞ for all h ≥ 0 and 0 ≤ n ≤ 1 the GSCF ground state energy is exact, which is necessary for formulation of converging perturbation procedure about mean field solution in the entire parameter space.

The Bethe Ansatz can be treated using two approaches: The Co-ordinate Bethe Ansatz (CBA) and The Algebraic Bethe Ansatz (ABA). In this paper the relation between the solutions given by these two methods is addressed. The considerations are based on the finite Heisenberg magnet with N=8 nodes and for r=4 reversed spins s=1/2. Both methods provide consistent solutions in the entire Brillouin zone (BZ). Moreover there is a special point — a border of the BZ — for both approaches, in which the solutions are not well-defined.

We introduce a Monte–Carlo simulation approach to thermodynamic Bethe ansatz (TBA). We exemplify the method on one-particle integrable models, which include a free boson and a free fermions systems along with the scaling Lee–Yang model (SLYM). It is confirmed that the central charges and energies are correct to a very good precision, typically 0.1% or so. The advantage of the method is that it allows the calculation of all the dimensions and even the particular partition function.

We study the one-dimensional S = 1/2 Heisenberg model with an uniform and a staggered magnetic fields, using the dynamical density-matrix renormalization group (DDMRG) technique. The DDMRG enables us to investigate the dynamical properties of chain with lengths up to a few hundreds, and the results are numerically exact in the same sense as 'exact diagonalization' results are. Thus, we can analyze the low-energy spectrum almost in the thermodynamic limit. In this work, we calculate the dynamical spin structure factor and demonstrate the performance of the DDMRG method applying the open-end boundary conditions as well as the periodic boundary conditions.