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 Schrodinger 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.
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 Schrodinger 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.
The exact solution of the asymmetric exclusion problem with N distinct classes of particles (c = 1, 2, ..., N), with hierarchical order is presented. In this model the particles (size 1) are located at lattice points, and diffuse with equal asymmetric rates, but particles in a class c do not distinguish those in the classes c' > c from holes (empty sites). We generalize and solve exactly this model by considering the molecules in each distinct class c = 1, 2, ..., N with sizes s c (s c = 0, 1, 2, ...), in units of the lattice spacing. The solution is derived via a Bethe ansatz of nested type.
The exact solution of the asymmetric exclusion problem with first- and scond-class particles is presented. In this model the particles (size 1) of both classes are located at lattice points, and diffuse with equal asymmetric rates, but particles in the first class do not distinguish those in the second class from holes (empty sites). We generalize and solve exactly this model by considering molecules in the first and second class with sizes s1 and s2 (s1, s2 = 0, 1, 2, ...), in units of lattice spacing, respectively. The solution is derived by a Bethe ansatz of nested type. We give a simple pedagogical presentation of the Bethe ansatz solution of the problem which can easily be followed by a reader not specialized in exactly integrable models.
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.
A generalization of the simple exclusion asymmetric model is introduced. In this model an arbitrary mixture of molecules with distinct sizes s=0,1,2, ..., in units of lattice space, diffuses asymmetrically on the lattice. A related surface growth model is also presented. Variations of the distribution of the molecules sizes may change the excluded volume almost continuously. We solve the model exactly through the Bethe ansatz and the dynamical critical exponent z is calculated from the finite-size corrections of the mass gap of the related quantum chain. Our results show that for an arbitrary distribution of molecules, the dynamical critical behavior is on the Kardar-Parizi-Zhang universality.
Recently it was shown that the eigenfunctions for the the asymmetric exclusion problem and several of its generalizations as well as a huge family of quantum chains, like the anisotropic Heisenberg model, Fateev- Zamolodchikov model, Izergin-Korepin model, Sutherland model, t-J model, Hubbard model, etc, can be expressed by a matrix product ansatz. Differently from the coordinate Bethe ansatz, where the eigenvalues and eigenvectors are plane wave combinations, in this ansatz the components of the eigenfunctions are obtained through the algebraic properties of properly defined matrices. In this work, we introduce a formulation of a matrix product ansatz for the six-vertex model with periodic boundary condition, which is the paradigmatic example of integrability in two dimensions. Remarkably, our studies of the six-vertex model are in agreement with the conjecture that all models exactly solved by the Bethe ansatz can also be solved by an appropriated matrix product ansatz.
Within the Matrix Product Formalism we have already introduced a multi-species exclusion process ( Phys. Rev. E , 59 (1999) 25, cond-mat/9809193), in which different particles hop with different rates and fast particles stochastically overtake slow ones. In this letter we show that on an open chain, the master equation of this process can be exactly solved via the coordinate Bethe ansatz. It is shown that the N -body S -matrix of this process is factorized into a product of two-body S -matrices, which in turn satisfy the quantum Yang-Baxter equation (QYBE). This solution is, to our knowledge, a new solution of QYBE.