Exact solution of asymmetric diffusion with second-class particles of arbitrary size

Alcaraz, Francisco C.; Bariev, R. Z.

doi:10.1590/s0103-97332000000100003

Cited by 27 publications

(37 citation statements)

References 14 publications

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“…The condition (20) leads to the problem of evaluation the eigenvalues of the inhomogeneous transfer matrix (18). This can be done through the algebraic Bethe ansatz [43] or the coordinate Bethe ansatz (see [44] and [45] for example).…”

Section: The Mpamentioning

confidence: 99%

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

Lazo¹

2008

Braz. J. Phys.

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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.

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Section: The Mpamentioning

confidence: 99%

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

Lazo¹

2008

Braz. J. Phys.

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“…Motivated by this fact we decided to verify if we can solve the above quantum chains directly though a matrix product ansatz, without considering any time dependence as in the case of the dynamical matrix ansatz. Surprisingly, we were able to rederive all the results previously obtained though the Bethe ansatz for the asymmetric diffusion problem with one species of particles [15] or more [18,19]. Moreover, our derivation turns out to be quite simple and it is not difficult to extend it to many other quantum Hamiltonians related or not to stochastic particle dynamics [41].…”

Section: Introductionmentioning

confidence: 60%

“…n=1 In this case the problem is the same as that of section 3 and we obtain the energies given by (18). …”

Section: A Matrix Product Ansatz For the Generalized Diffusion Probmentioning

confidence: 99%

“…More recently [39], [40] it has been shown that this ansatz can also be formulated in the problem of asymmetric diffusion of two types of particles. The validity of the ansatz was confirmed in the regions where the model is known to be exactly integrable through the Bethe ansatz [3,18]. Motivated by this fact we decided to verify if we can solve the above quantum chains directly though a matrix product ansatz, without considering any time dependence as in the case of the dynamical matrix ansatz.…”

Section: Introductionmentioning

confidence: 99%

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The exact solution of the asymmetric exclusion problem with particles of asrbitrary size: matrix product ansatz

Alcaraz¹,

Lazo²

2003

Braz. J. Phys.

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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 Schrödinger 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. I IntroductionThe representation of interacting stochastic particle dynamics in terms of quantum spin systems produced interesting and fruitful interchanges between the areas of equilibrium and nonequilibrium statistical mechanics. The connection between these areas follows from the similarity between the master equation describing the time-fluctuations on the nonequilibrium stochastic problem and the quantum fluctuations of the equilibrium quantum spin chains [1]- [20].Unlike the area of nonequilibrium interacting systems, where very few models are fully solvable, there exists a huge family of quantum chains appearing in equilibrium problems that are exactly integrable. The machinery that allows the exact solutions of these quantum chains comes from the Bethe ansatz in its several formulations (see [21]-[24] for reviews). The above mentioned mathematical connection between equilibrium and nonequilibrium revealed that some quantum chains related to interacting stochastic problems are exactly solvable through the Bethe ansatz. The simplest example is the problem of asymmetric diffusion of hard-core particles on the one dimensional lattice (see [16,17,20] for reviews). The time fluctuations of this last model are governed by a time evolution operator that coincides with the exact integrable anisotropic Heisenberg chain, or the so called, XXZ quantum chain, in its ferromagnetically ordered regime. A generalization of this stochastic problem where exact integrability is also known [25]-[27] is the case where there are N (N = 1, 2, ...) classes of particles hierarchically ordered and diffusing asymmetrically on the lattice. The quantum chain related to this problem is known in the literature as the anisotropic Sutherland model [28...

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“…The Lie algebra gl 3 is the central extension of sl3 by the 3 × 3 identiy matrix. In terms of quantum groups, this corresponds to a central extension by the element k (1,1,1) It was shown in [13] that the following element is central:…”

Section: Glmentioning

confidence: 99%

Stochastic duality of ASEP with two particle types via symmetry of quantum groups of rank two

Kuan¹

2016

J. Phys. A: Math. Theor.

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We study two generalizations of the asymmetric simple exclusion process with two types of particles. Particles of type 1 can jump over particles of type 2, while particles of type 2 can only influence the jump rates of particles of type 1. We prove that the processes are self-dual and explicitly write the duality function. As an application, an expression for the r-th moment of the exponentiated current is written in terms of r-particle evolution.The construction and proofs of duality are accomplished using symmetry of the quantum groups Uq(gl 3 ) and Uq(sp 4 ), with each node in the Dynkin diagram corresponding to a particle type, and the number of edges corresponding to the jump rates.

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Exact solution of asymmetric diffusion with second-class particles of arbitrary size

Cited by 27 publications

References 14 publications

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

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

The exact solution of the asymmetric exclusion problem with particles of asrbitrary size: matrix product ansatz

Stochastic duality of ASEP with two particle types via symmetry of quantum groups of rank two

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