The Bethe ansatz as a matrix product ansatz

Alcaraz, Francisco C.; Lazo, Matheus J.

doi:10.1088/0305-4470/37/1/l01

Cited by 44 publications

(76 citation statements)

References 35 publications

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“…An open problem is the relationship of the MPA to the integrability of the bulk Hamiltonian and hence to the extension of the MPA approach to dynamical observables. Work on eigenfunctions [22,23] and recent exact results by Eisler for the density matrix with bulk Lindblad terms [24] hint at this possibility.…”

Section: The Stationary Lindblad Equation (5) Can Thus Be Split Into mentioning

confidence: 96%

Exact Matrix Product Solution for the Boundary-Driven LindbladXXZChain

2013

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We demonstrate that the exact non-equilibrium steady state of the one-dimensional Heisenberg XXZ spin chain driven by boundary Lindblad operators can be constructed explicitly with a matrix product ansatz for the non-equilibrium density matrix where the matrices satisfy a quadratic algebra. This algebra turns out to be related to the quantum algebra Uq[SU (2)]. Coherent state techniques are introduced for the exact solution of the isotropic Heisenberg chain with and without quantum boundary fields and Lindblad terms that correspond to two different completely polarized boundary states. We show that this boundary twist leads to non-vanishing stationary currents of all spin components. Our results suggest that the matrix product ansatz can be extended to more general quantum systems kept far from equilibrium by Lindblad boundary terms.PACS numbers: 03.65. Yz, 75.10.Pq, The non-equilibrium behaviour of open quantum systems has become accessible through recent advances in artificially assembled nanomagnets consisting of just a few atoms [1] or in the study of quasi one-dimensional spin chain materials like SrCuO 2 where many transport characteristics are measurable experimentally [2,3]. In particular, it is desirable to understand the interplay between many-body bulk properties (e.g. magnon excitations or magnetization currents in quantum spin systems) and local pumping (applied to the boundary of a system) driving the system constantly out of equilibrium. A good starting point is provided by the anisotropic Heisenberg modelof coupled spins. The pure quantum version of this model is exactly solvable by the Bethe ansatz. Interestingly, within linear response theory, i.e., close to equilibrium, it was found that at finite temperature a diffusive contribution to the Drude weight appears [5][6][7], which is at variance with the long-held belief that integrability protects the ballistic nature of transport phenomena. Unfortunately the Bethe-ansatz fails in the more relevant context of open far-from-equilibrium systems where these questions can be addressed directly in terms of the Lindblad Master equationfor the reduced density matrix ρ associated to the chain (here and below we set = 1). The dissipative termswith the Lindblad operators D L,R acting locally at the open ends of the quantum chain (see below) describe the coupling to external reservoirs that drive a current through the system and thus keep the system in a permanent non-equilibrium steady state. Indeed, using dissipative dynamics for the preparation of quantum states is becoming a promising field of research [9, 10].Significant progress has been achieved very recently in two remarkable papers by Prosen [11,12] who observed that the exact stationary density matrix for the XXZ chain with one specific pair of Lindblad boundary terms can be constructed explicitly in matrix product operator form [13] by a matrix product ansatz (MPA) somewhat reminiscent of the matrix product ansatz of Derrida et al. [14] for the stationary distribution of purely classical stochas...

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Section: The Stationary Lindblad Equation (5) Can Thus Be Split Into mentioning

confidence: 96%

Exact Matrix Product Solution for the Boundary-Driven LindbladXXZChain

2013

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“…They reduced the problem to an algebraic set of equations for the different matrices A i , and quite a lot of nontrivial systems could be solved that way. Although the matrix product ansatz version of the algebraic Bethe ansatz by Alcaraz and Lazo [90] was actually inspired by the approach of Derrida et al, the solutions are quite different, as the last ones give rise to matrices with bond dimensions scaling linearly in the systems size as opposed to exponentially. In line with this article, let us illustrate their approach for obtaining exact solutions for steady states of stochastic cellular automata.…”

Section: Non-equilibrium Steady States As Matrix Product Statesmentioning

confidence: 99%

Diagonalizing Transfer Matrices and Matrix Product Operators: A Medley of Exact and Computational Methods

Haegeman

Verstraete

2017

Annu. Rev. Condens. Matter Phys.

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Transfer matrices and matrix product operators play an ubiquitous role in the field of many body physics. This paper gives an ideosyncratic overview of applications, exact results and computational aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and of new results. Topics discussed are exact solutions of transfer matrices in equilibrium and non-equilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators.

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“…The Bethe Ansatz (Dhar 1987) provides spectral information about the evolution (Markov) operator (Gwa and Spohn 1992;Schütz 1993;Kim 1995) which can be used to derive large deviation functions (Derrida and Lebowitz 1998;Derrida and Appert 1999;Derrida and Evans 1999). The exact relation between these two techniques is still a matter of investigation (Alcaraz et al 1994;Stinchcombe and Schütz 1995;Alcaraz and Lazo 2003).…”

Section: Introductionmentioning

confidence: 99%

Bethe ansatz calculation of the spectral gap of the asymmetric exclusion process

Golinelli¹,

Mallick²

2004

J. Phys. A: Math. Gen.

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We present a new derivation of the spectral gap of the totally asymmetric exclusion process on a half-filled ring of size L by using the Bethe Ansatz. We show that, in the large L limit, the Bethe equations reduce to a simple transcendental equation involving the polylogarithm, a classical special function. By solving that equation, the gap and the dynamical exponent are readily obtained. Our method can be extended to a system with an arbitrary density of particles.

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The Bethe ansatz as a matrix product ansatz

Cited by 44 publications

References 35 publications

Exact Matrix Product Solution for the Boundary-Driven LindbladXXZChain

Exact Matrix Product Solution for the Boundary-Driven LindbladXXZChain

Diagonalizing Transfer Matrices and Matrix Product Operators: A Medley of Exact and Computational Methods

Bethe ansatz calculation of the spectral gap of the asymmetric exclusion process

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