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30 Citations
Résumé:

We give a complete classification of integrable Markovian boundary conditions for the asymmetric simple exclusion process with two species (or classes) of particles. Some of these boundary conditions lead to non-vanishing particle currents for each species. We explain how the stationary state of all these models can be expressed in a matrix product form, starting from two key components, the Zamolodchikov-Faddeev and Ghoshal-Zamolodchikov relations. This statement is illustrated by studying in detail a specific example, for which the matrix Ansatz (involving 9 generators) is explicitly constructed and physical observables (such as currents, densities) calculated.

Commentaires: 19 pages; typos corrected, more details on the Matrix Ansatz algebra. Réf Journal: J. Phys. A: Math. Theor. 48 (2015) 175002

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22 Citations
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We study the one-dimensional totally asymmetric simple exclusion process in contact with two reservoirs including also a fugacity at one boundary. The eigenvectors and the eigenvalues of the corresponding Markov matrix are computed using the modified algebraic Bethe ansatz, method introduced recently to study the spin chain with non-diagonal boundaries. We provide in this case a proof of this method.

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Ref Arxiv: 1408.5357
DOI: 10.1088/1742-5468/2014/11/P11032
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42 Citations
Résumé:

We study the matrix ansatz in the quantum group framework, applying integrable systems techniques to statistical physics models. We start by reviewing the two approaches, and then show how one can use the former to get new insight on the latter. We illustrate our method by solving a model of reaction-diffusion. An eigenvector for the transfer matrix for the XXZ spin chain with non-diagonal boundary is also obtained using a matrix ansatz.

Commentaires: 44 pages

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Ref Arxiv: 1307.4023
DOI: 10.3842/SIGMA.2014.014
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6 Citations
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We propose the notion of integrable boundary in the context of discrete quad-graph systems. The equation characterizing the boundary must satisfy a compatibility equation with the one characterizing the bulk that we called the three-dimensional (3D) boundary consistency. In comparison to the usual 3D consistency condition which is linked to a cube, our 3D boundary consistency condition lives on half of a rhombic dodecahedron. We provide a list of integrable boundaries associated to each quad-graph bulk equation of the classification obtained by Adler, Bobenko and Suris. Then, the use of the term 'integrable boundary' is justified by the facts that there are Bäcklund transformations and a zero curvature representation for systems with a boundary satisfying our condition. We discuss the three-leg form of boundary equations and hence obtain associated discrete Toda models with a boundary. Finally, the connection between the 3D boundary consistency and the set-theoretical reflection equation is established.

Commentaires: 24 pages, 10 figures

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Ref Arxiv: 1309.6165
DOI: 10.3842/SIGMA.2013.072
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88 Citations
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We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al. The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the length of the chain. We prove this result for the chains with small length. We obtain also an off-shell equation (i.e. satisfied without the Bethe equations) formally similar to the ones obtained in the periodic case or with diagonal boundaries.

Commentaires: Journal: SIGMA 9 (2013), 072, 12 pages

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DOI: 10.1088/1751-8113/46/40/405001
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6 Citations
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We classify all Hamiltonians with rank 1 symmetry, acting on a periodic three-state spin chain, and solvable through (generalisation of) the coordinate Bethe ansatz (CBA). We obtain in this way four multi-parametric extensions of the known 19-vertex Hamiltonians (such as Zamolodchikov-Fateev, Izergin-Korepin, Bariev Hamiltonians). Apart from the 19-vertex Hamiltonians, there exists 17-vertex and 14-vertex Hamiltonians that cannot be viewed as subcases of the 19-vertex ones. In the case of 17-vertex Hamiltonian, we get a generalization of the genus 5 special branch found by Martins, plus three new ones. We get also two 14-vertex Hamiltonians. We solve all these Hamiltonians using CBA, and provide their spectrum, eigenfunctions and Bethe equations. A special attention is made to provide the specifications of our multi-parametric Hamiltonians that give back known Hamiltonians.

Commentaires: 30 pages; web page http://www.coulomb.univ-montp2.fr/3Ham

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Ref Arxiv: 1210.7143
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41 Citations
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We study the XX model for quantum spins on the star graph with three legs (i.e., on a Y-junction). By performing a Jordan-Wigner transformation supplemented by the introduction of an auxiliary space we find a Kondo Hamiltonian of fermions, in the spin 1 representation of su(2), locally coupled with a magnetic impurity. In the continuum limit our model is shown to be equivalent to the 4-channel Kondo model coupling spin-1/2 fermions with a spin-1/2 impurity and exhibiting a non-Fermi liquid behavior. We also show that it is possible to find a XY model such that - after the Jordan-Wigner transformation - one obtains a quadratic fermionic Hamiltonian directly diagonalizable.