Matheus Jatkoske Lazo

Exact solutions of exactly integrable quantum chains by a matrix product ansatz

Most of the exact solutions of quantum one-dimensional Hamiltonians are obtained thanks to the success of the Bethe ansatz on its several formulations. According to this ansatz, the amplitudes of the eigenfunctions of the Hamiltonian are given by a sum of permutations of appropriate plane waves. In this paper, alternatively, we present a matrix product ansatz that asserts that those amplitudes are given in terms of a matrix product. The eigenvalue equation for the Hamiltonian defines the algebraic properties of the matrices defining the amplitudes. The consistency of the commutativity relations among the elements of the algebra implies the exact integrability of the model. The matrix product ansatz we propose allows an unified and simple formulation of several exact integrable Hamiltonians. In order to introduce and illustrate this ansatz we present the exact solutions of several quantum chains with one and two global conservation laws and periodic boundaries such as the XXZ chain, spin-1 Fateev–Zamolodchikov model, Izergin–Korepin model, Sutherland model, t–J model, Hubbard model, etc. Formulation of the matrix product ansatz for quantum chains with open ends is also possible. As an illustration we present the exact solution of an extended XXZ chain with z-magnetic fields at the surface and arbitrary hard-core exclusion among the spins.

The Bethe ansatz as a matrix product ansatz

The Bethe ansatz in its several formulations is a common tool for the exact solution of one-dimensional quantum Hamiltonians. This ansatz asserts that several eigenfunctions of the Hamiltonians are given in terms of a sum of permutations of plane waves. We present results that induce us to expect that, alternatively, the eigenfunctions of all the exact integrable quantum chains can also be expressed by a matrix product ansatz. In this ansatz several components of the eigenfunctions are obtained through the algebraic properties of properly defined matrices. This ansatz allows an unified formulation of several exact integrable Hamiltonians. We show how to formulate this ansatz for a large family of quantum chains such as the anisotropic Heisenberg model, Fateev–Zamolodchikov model, Izergin–Korepin model, Sutherland model, t–J model, Hubbard model, etc.

The Exact Solution of the Asymmetric Exclusion Problem With Particles of Arbitrary Size: Matrix Product Ansatz

The exact solution of the asymmetric exclusion problem and several of its generalizations is obtained by a matrix product ansatz. Due to the similarity of the master equation and the Schr ¨ odinger equation at imaginary times the solution of these problems reduces to the diagonalization of a one dimensional quantum Hamiltonian. Initially, we present the solution of the problem when an arbitrary mixture of molecules, each of then having an arbitrary size (s = 0;1;2;:::) in units of lattice spacing, diffuses asymmetrically on the lattice. The solution of the more general problem where we have the diffusion of particles belonging to N distinct classes of particles (c = 1;:::;N ), with hierarchical order and arbitrary sizes, is also presented. Our matrix product ansatz asserts that the amplitudes of an arbitrary eigenfunction of the associated quantum Hamiltonian can be expressed by a product of matrices. The algebraic properties of the matrices defining the ansatz depend on the particular associated Hamiltonian. The absence of contradictions in the algebraic relations defining the algebra ensures the exact integrability of the model. In the case of particles distributed in N > 2 classes, the associativity of this algebra implies the Yang-Baxter relations of the exact integrable model.

Generalization of the matrix product ansatz for integrable chains

We present a general formulation of the matrix product ansatz for exactly integrable chains on periodic lattices. This new formulation extends the matrix product ansatz present in our previous articles (F C Alcaraz and M J Lazo 2004 J. Phys. A: Math. Gen. 37 L1–L7 and F C Alcaraz and M J Lazo 2004 J. Phys. A: Math. Gen. 37 4149–82).

Integrable inhomogeneous spin chains in generalized Lunin-Maldacena backgrounds

We obtain through a Matrix Product Ansatz the exact solution of the most general inhomogeneous spin chain with nearest neighbor interaction and with U(1)2 and U(1)3 symmetries. These models are related to the one loop mixing matrix of the Leigh-Strassler deformed N = 4 SYM theory, dual to type IIB string theory in the generalized Lunin-Maldacena backgrounds, in the sectors of two and three kinds of fields, respectively. The solutions presented here generalizes the results obtained by the author in a previous work for homogeneous spins chains with U(1)N symmetries in the sectors of N = 2 and N = 3.

Exactly solvable interacting vertex models

We introduce and solve a special family of integrable interacting vertex models that generalizes the well known six-vertex model. In addition to the usual nearest neighbour interactions among the vertices, there exist extra hard-core interactions among pairs of vertices at larger distances. The associated row-to-row transfer matrices are diagonalized by using the recently introduced matrix product ansatz. Like for the relation of the six-vertex model with the XXZ quantum chain, the row-to-row transfer matrices of these new models are also the generating functions of an infinite set of commuting conserved charges. Among these charges we identify the integrable generalization of the XXZ chain that contains hard-core exclusion interactions among the spins. These quantum chains have already appeared in the literature. The present paper explains their integrability.

The electromagnetic mass in the Born-Infeld theory

The relativistic electromagnetic mass?energy relation is shown to follow Born?Infeld nonlinear electrodynamics for an arbitrary charge distribution if the electromagnetic four-momentum of the system in motion is defined by the Lorentz transformation of its rest state value, as in Maxwell electrodynamics.

Asymmetric exclusion model with several kinds of impurities

We formulate a new integrable asymmetric exclusion process with N ? 1 = 0, 1, 2, ... kinds of impurities and with hierarchically ordered dynamics. The model we proposed displays the full spectrum of the simple asymmetric exclusion model plus new levels. The first excited state belongs to these new levels and displays unusual scaling exponents. We conjecture that, while the simple asymmetric exclusion process without impurities belongs to the KPZ universality class with dynamical exponent , our model has a scaling exponent . In order to check the conjecture, we solve numerically the Bethe equation with N = 3 and N = 4 for the totally asymmetric diffusion and found the dynamical exponents and in these cases.

The Matrix Product Ansatz for integrable U(1)N models in Lunin-Maldacena backgrounds

We obtain through a Matrix Product Ansatz (MPA) the exact solution of the most general N-state spin chain with U(1)N symmetry and nearest neighbour interaction. In the case N = 6 this model contain as a special case the integrable SO(6) spin chain related to the one loop mixing matrix for anomalous dimensions in N = 4 SYM, dual to type IIB string theory in the generalised Lunin-Maldacena backgrounds. This MPA is construct by a map between scalar fields and abstract operators that satisfy an appropriate associative algebra. We analyses the Yang-Baxter equation in the N = 3 sector and the consistence of the algebraic relations among the matrices defining the MPA and find a new class of exactly integrable model unknown up to now.