Matrix coordinate Bethe Ansatz: applications to XXZ and ASEP models

Crampé, Nicolas; Ragoucy, É.; Simon, Damien

doi:10.1088/1751-8113/44/40/405003

Cited by 53 publications

(67 citation statements)

References 36 publications

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“…The coordinate version of the Ansatz which we presented in section III B 1, where the particles are treated as plane waves, relies on the number of particles being fixed, and breaks down in the open case. Variants of the coordinate Bethe Ansatz have been used successfully to build excited eigenstates of the system for some special cases of the boundary parameters [34,35,38] (in particular with triangular boundary matrices), and, in conjunction with numerical analysis, to find the relaxation speed of the system (i.e. the gap of the Markov matrix) [36,37,123], as well as the asymptotic large deviation function of the current inside of the Gaussian phases [124].…”

Section: Brief Overview Of the Asep's Family Treementioning

confidence: 99%

The physicist's companion to current fluctuations: one-dimensional bulk-driven lattice gases

Lazarescu¹

2015

J. Phys. A: Math. Theor.

105

156

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One of the main features of statistical systems out of equilibrium is the currents they exhibit in their stationary state: microscopic currents of probability between configurations, which translate into macroscopic currents of mass, charge, etc. Understanding the general behaviour of these currents is an important step towards building a universal framework for non-equilibrium steady states akin to the Gibbs-Boltzmann distribution for equilibrium systems. In this review, we consider onedimensional bulk-driven particle gases, and in particular the asymmetric simple exclusion process (ASEP) with open boundaries, which is one of the most popular models of one-dimensional transport. We focus, in particular, on the current of particles flowing through the system in its steady state, and on its fluctuations. We show how one can obtain the complete statistics of that current, through its large deviation function, by combining results from various methods: exact calculation of the cumulants of the current, using the integrability of the model ; direct diagonalisation of a biased process in the limits of very high or low current ; hydrodynamic description of the model in the continuous limit using the macroscopic fluctuation theory (MFT). We give a pedagogical account of these techniques, starting with a quick introduction to the necessary mathematical tools, as well as a short overview of the existing works relating to the ASEP. We conclude by drawing the complete dynamical phase diagram of the current. We also remark on a few possible generalisations of these results.

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Section: Brief Overview Of the Asep's Family Treementioning

confidence: 99%

The physicist's companion to current fluctuations: one-dimensional bulk-driven lattice gases

Lazarescu¹

2015

J. Phys. A: Math. Theor.

105

156

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“…This approach was later completed by the identification of a second reference state [8]. Several other methods (coordinate Bethe ansatz with elements of matrix product ansatz [13,14], q-Onsager algebra [9,10] etc.) led to the same constraint as a necessary condition to characterize the spectrum in terms of (polynomial) solutions of a T -Q equation.…”

Section: Introductionmentioning

confidence: 99%

The open XXX spin chain in the SoV framework: scalar product of separate states

Kitanine¹,

Maillet²,

Niccoli³

et al. 2017

J. Phys. A: Math. Theor.

105

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We consider the XXX open spin-1/2 chain with the most general non-diagonal boundary terms, that we solve by means of the quantum separation of variables (SoV) approach. We compute the scalar products of separate states, a class of states which notably contains all the eigenstates of the model. As usual for models solved by SoV, these scalar products can be expressed as some determinants with a non-trivial dependance in terms of the inhomogeneity parameters that have to be introduced for the method to be applicable. We show that these determinants can be transformed into alternative ones in which the homogeneous limit can easily be taken. These new representations can be considered as generalizations of the well-known determinant representation for the scalar products of the Bethe states of the periodic chain. In the particular case where a constraint is applied on the boundary parameters, such that the transfer matrix spectrum and eigenstates can be characterized in terms of polynomial solutions of a usual T -Q equation, the scalar product that we compute here corresponds to the scalar product between two off-shell Bethe-type states. If in addition one of the states is an eigenstate, the determinant representation can be simplified, hence leading in this boundary case to direct analogues of algebraic Bethe ansatz determinant representations of the scalar products for the periodic chain.

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“…the reader is referred to [27]. on the spirit of the standard coordinate Bethe ansatz [1,2,3] generalized to the coordinate matrix product ansatz picture for the ASEP [56]. In this approach higher decay modes are understood as excitations upon the NESS which is understood as a vacuum.…”

Section: The First (Leading Decay Mode) Orbitalmentioning

confidence: 99%

Strongly correlated non-equilibrium steady states with currents – quantum and classical picture

Buča

Prosen

2018

Eur. Phys. J. Spec. Top.

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In this minireview we will discuss recent progress in the analytical study of current-carrying nonequilibrium steady states (NESS) that can be constructed in terms of a matrix product ansatz. We will focus on one-dimensional exactly solvable strongly correlated cases, and will study both quantum models, and classical models which are deterministic in the bulk. The only source of classical stochasticity in the timeevolution will come from the boundaries of the system. Physically, these boundaries may be understood as Markovian baths, which drive the current through the system. The examples studied include the open XXZ Heisenberg spin chain, the open Hubbard model, and a classical integrable reversible cellular automaton, namely the Rule 54 of Bobenko et al. [Commun. Math. Phys. 158, 127 (1993)] with stochastic boundaries. The quantum NESS can be at least partially understood through the Yang-Baxter integrability structure of the underlying integrable bulk Hamiltonian, whereas for the Rule 54 model NESS seems to come from a seemingly unrelated integrability theory. In both the quantum and the classical case, the underlying matrix product ansatz defining the NESS also allows for construction of novel conservation laws of the bulk models themselves. In the classical case, a modification of the matrix product ansatz also allows for construction of states beyond the steady state (i.e., some of the decay modes -Liouvillian eigenvectors of the model). We hope that this article will help further the quest to unite different perspectives of integrability of NESS (of both quantum and classical models) into a single unified framework. arXiv:1710.08319v1 [cond-mat.stat-mech]

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Matrix coordinate Bethe Ansatz: applications to XXZ and ASEP models

Cited by 53 publications

References 36 publications

The physicist's companion to current fluctuations: one-dimensional bulk-driven lattice gases

The physicist's companion to current fluctuations: one-dimensional bulk-driven lattice gases

The open XXX spin chain in the SoV framework: scalar product of separate states

Strongly correlated non-equilibrium steady states with currents – quantum and classical picture

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