Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms

Nepomechie, Rafael I.; Ravanini, Francesco

doi:10.1088/0305-4470/36/45/003

Cited by 104 publications

(173 citation statements)

References 13 publications

Supporting

Mentioning

162

Contrasting

Order By: Relevance

“…Part of the conclusions were achieved with the help of a numerical study of the spectrum for finite values of L [14]. More recently, new results have been obtained in ref.…”

Section: S (λ) T (S)mentioning

confidence: 95%

Bethe ansatz for the XXX-S chain with non-diagonal open boundaries

2005

View full text Add to dashboard Cite

We consider the algebraic Bethe ansatz solution of the integrable and isotropic XXX-S Heisenberg chain with non-diagonal open boundaries. We show that the corresponding K-matrices are similar to diagonal matrices with the help of suitable transformations independent of the spectral parameter. When the boundary parameters satisfy certain constraints we are able to formulate the diagonalization of the associated double-row transfer matrix by means of the quantum inverse scattering method. This allows us to derive explicit expressions for the eigenvalues and the corresponding Bethe ansatz equations. We also present evidences that the eigenvectors can be build up in terms of multiparticle states for arbitrary S.

show abstract

“…Part of the conclusions were achieved with the help of a numerical study of the spectrum for finite values of L [14]. More recently, new results have been obtained in ref.…”

Section: S (λ) T (S)mentioning

confidence: 95%

Bethe ansatz for the XXX-S chain with non-diagonal open boundaries

2005

View full text Add to dashboard Cite

show abstract

“…However, such equations were derived at a subset of the exceptional points, those with ǫ = 1 [12][13][14]16] and those with ǫ = −1 in the case of L odd [16]. It was further noticed that one needs two sets of Bethe Ansatz equations to describe the complete spectrum [15,16].…”

Section: Tmentioning

confidence: 99%

Magic in the spectra of the XXZ quantum chain with boundaries at and

et al. 2005

View full text Add to dashboard Cite

We show that from the spectra of the Uq(sl(2)) symmetric XXZ spin-1/2 finite quantum chain at ∆ = −1/2 (q = e πi/3 ) one can obtain the spectra of certain XXZ quantum chains with diagonal and non-diagonal boundary conditions. Similar observations are made for ∆ = 0 (q = e πi/2 ). In the finite-size scaling limit the relations among the various spectra are the result of identities satisfied by known character functions. For the finite chains the origin of the remarkable spectral identities can be found in the representation theory of one and two boundaries Temperley-Lieb algebras at exceptional points. Inspired by these observations we have discovered other spectral identities between chains with different boundary conditions.

show abstract

“…However, for generic values of the PASEP parameters (4) can also be satisfied by choosing k = −L/2. For this choice of k we infer from [18] that for even L there is an isolated level with energy E 0 = 0, the ground state energy of the PASEP. Furthermore, all excited levels are given by…”

mentioning

confidence: 95%

Bethe Ansatz Solution of the Asymmetric Exclusion Process with Open Boundaries

2005

View full text Add to dashboard Cite

We derive the Bethe ansatz equations describing the complete spectrum of the transition matrix of the partially asymmetric exclusion process with the most general open boundary conditions. For totally asymmetric diffusion we calculate the spectral gap, which characterizes the approach to stationarity at large times. We observe boundary induced crossovers in and between massive, diffusive and KPZ scaling regimes. The partially asymmetric simple exclusion process (PASEP) describes the asymmetric diffusion of particles along a one-dimensional chain with L sites. It is one of the most studied models of non-equilibrium statistical mechanics, see [1,2] for recent reviews. This is in part due to the fact that is one of the simplest lattice gas models, but also because of its applicability to molecular diffusion in zeolites [3], bioploymers [4], traffic flow [5] and other one-dimensional complex systems [6].At large times the PASEP exhibits a relaxation towards a nonequilibrium stationary state. An interesting feature of the PASEP is the presence of boundary induced phase transitions [7]. In particular, in an open system with two boundaries at which particles are injected and extracted with given rates, the bulk behaviour in the stationary state is strongly dependent on the injection and extraction rates. Over the last decade many stationary state properties of the PASEP with open boundaries have been determined exactly [1,2,[8][9][10][11]. On the other hand, much less is known about its dynamics. This is in contrast to the PASEP on a ring for which exact results using Bethe's ansatz have been available for a long time [12,13]. For open boundaries there have been several studies of dynamical properties based mainly on numerical and phenomenological methods [14,15]. In this Letter we employ Bethe's ansatz to obtain exact results for the approach to stationarity at large times in the PASEP with open boundaries. Upon varying the boundary rates, we find crossovers in massive regions, with dynamic exponents z = 0, and between massive and scaling regions with diffusive (z = 2) and KPZ (z = 3/2) behaviour.The dynamical rules of the PASEP are as follows. At any given time t each site is either occupied by a particle or empty and the system evolves subject to the following rules. In the bulk of the system (i = 2, . . . , L − 1) a particle attempts to hop one site to the right with rate p and one site to the left with rate q. The hop is executed unless the neighbouring site is occupied, in which case nothing happens. On the first and last sites these rules are modified. If site i = 1 is empty, a particle may enter the system with rate α. If on the other hand site 1 is occupied by a particle, the latter will leave the system with rate γ. Similarly, at i = L particles are injected and extracted with rates δ and β respectively. With every site i we associate a Boolean variable τ i , indicating whether a particle is present (τ i = 1) or not (τ i = 0). The state of the system at time t is then characterized by the probability distribu...

show abstract

Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms

Cited by 104 publications

References 13 publications

Bethe ansatz for the XXX-S chain with non-diagonal open boundaries

Bethe ansatz for the XXX-S chain with non-diagonal open boundaries

Magic in the spectra of the XXZ quantum chain with boundaries at and

Bethe Ansatz Solution of the Asymmetric Exclusion Process with Open Boundaries

Contact Info

Product

Resources

About