XXZ chain with a boundary

Jimbo, Michio; Kedem, Rinat; Kojima, Takeo; Konno, Hitoshi; Miwa, Tetsuji

doi:10.1016/0550-3213(95)00062-w

Cited by 132 publications

(277 citation statements)

References 34 publications

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“…The boundary interaction is specfied by the reflection matrix K for lattice models [6], and by the boundary S-matrix for massive quantum theories [7]. It is shown in [8] that the space of states of the boundary XXZ model can be described in terms of vertex operators associated with the bulk XXZ model [3]. The explicit bosonic formulae of the boundary vacuum of the boundary XXZ model were obtained by using the bosonization of the vertex operators [8].…”

Section: Introductionmentioning

confidence: 99%

Difference Equations for the Higher Rank XXZ Model With a Boundary

Kojima

Quano

2000

Int. J. Mod. Phys. A

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Section: Introductionmentioning

confidence: 99%

Difference Equations for the Higher Rank XXZ Model With a Boundary

Kojima

Quano

2000

Int. J. Mod. Phys. A

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“…Such integrals arise quite naturally in concrete applications. For instance, it is well known that correlation functions in quantum integrable models are described by N-fold multiple integrals [89,90,91,98] or series thereof [97]. Usually, for reasons stemming from the physics of the underlying model, one is interested in the large-N behaviour of these integrals and, in particular, in the constant term arising in their asymptotics.…”

Section: β-Ensembles With Non-varying Weightsmentioning

confidence: 99%

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Borot¹,

Guionnet²,

Kozłowski³

2016

Mathematical Physics Studies

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This paper develops a method to carry out the large-N asymptotic analysis of a class of Ndimensional integrals arising in the context of the so-called quantum separation of variables method. We push further ideas developed in the context of random matrices of size N, but in the present problem, two scales 1/N α and 1/N naturally occur. In our case, the equilibrium measure is N α -dependent and characterised by means of the solution to a 2×2 Riemann-Hilbert problem, whose large-N behaviour is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-N behaviour of the free energy explicitly up to o(1). The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales 1/N α and 1/N, thus waiving the analyticity assumptions often used in random matrix theory.1 gborot@mpim-bonn.mpg.de 2 guionnet@math.mit.edu 3 karol.kozlowski@ens-lyon.fr 2 An opening discussionThe present work develops techniques enabling one to carry out the large-N asymptotic analysis of a class of multiple integrals that arise as representations for the correlation functions in quantum integrable models solvable by the quantum separation of variables. We shall refer to the general class of such integrals as the sinh model: [9,59,60,82,106,105,139,146]. The expressions obtained there are either directly of the form given above or are amenable to this form (with, possibly, a change of the integration contour from R N to C N , with C a curve in C) upon elementary manipulations. Furthermore, a degeneration of z N [W] arises as a multiple integral representation for the partition function of the six-vertex model subject to domain wall boundary conditions [93]. In the context of quantum integrable systems, the number N of integrals defining z N is related to the number of sites in a model (as, e.g. in the case of the compact or non-compact XXZ chains or the lattice regularisations of the Sinh or Sine-Gordon models) or to the number of particles (as, e.g. in the case of the quantum Toda chain). From the point of view of applications, one is mainly interested in the thermodynamic limit of the model, which is attained by sending N to +∞. For instance, in the case of an integrable lattice discretisation of some quantum field theory, one obtains in this way an exact and non-perturbative description of a quantum field theory in 1 + 1 dimensions and in finite volume. This limit, at the level of z N [W], translates itself in the need to extract the large N-asymptotic expansion of ln z N [W] up to o(1). It is, in fact, the constant term in the expansion of ln zwith W ′ some deformation of W that gives rise to the correlation functions of the underlying quantum field theory in finite volume. These applications to physics constitute the first motivation for our analysis. From the purely mathematical side, the motivation of our works stems ...

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“…The latter one gives rise to the KCG top Hamiltonian (36). Another wide class of models embedded recently into this scheme [49] includes the so called quasi-solvable models; some of them are physically relevant.…”

Section: Integrable Models With Anticommuting Variables (Fermions)mentioning

confidence: 99%

“…The constructions of the quantum group invariant spin systems [30] uses the RE [31,32,42,54]. Recent field theoretical applications of the RE can be found in [23][24][25][26][27]36].…”

Section: Reflection Equations and Their Covariancementioning

confidence: 99%

Yang-Baxter equation and reflection equations in integrable models

Kulish

1996

Low-Dimensional Models in Statistical Physics and Quantum Field Theory

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The definitions of the main notions related to the quantum inverse scattering methods are given. The Yang-Baxter equation and reflection equations are derived as consistency conditions for the factorizable scattering on the whole line and on the half-line using the ZamolodchikovFaddeev algebra. Due to the vertex-IRF model correspondence the face model analogue of the ZF-algebra and the IRF reflection equation are written down as well as the Z 2 -graded and colored algebra forms of the YBE and RE.

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XXZ chain with a boundary

Cited by 132 publications

References 34 publications

Difference Equations for the Higher Rank XXZ Model With a Boundary

Difference Equations for the Higher Rank XXZ Model With a Boundary

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Yang-Baxter equation and reflection equations in integrable models

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