Aspects of the inverse problem for the Toda chain

Kozłowski, Karol K.

doi:10.1063/1.4848778

Cited by 7 publications

(6 citation statements)

References 40 publications

(53 reference statements)

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“…Note that a reconstructions of local operators in the lattice Toda model have been achieved in [174] in terms of a set of quantum separate variables defined by a change of variables in terms of the original Sklyanin's quantum separate variables. Recent analysis of this reconstruction problem for the lattice Toda model appear also in [175] and [176].…”

Section: Discussionmentioning

confidence: 99%

On the Form Factors of Local Operators in the Bazhanov–Stroganov and Chiral Potts Models

Grosjean

Maillet

Niccoli

2014

Ann. Henri Poincaré

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We consider general cyclic representations of the 6-vertex Yang-Baxter algebra and analyze the associated quantum integrable systems, the Bazhanov-Stroganov model and the corresponding chiral Potts model on finite size lattices. We first determine the propagator operator in terms of the chiral Potts transfer matrices and we compute the scalar product of separate states (including the transfer matrix eigenstates) as a single determinant formulae in the framework of Sklyanin's quantum separation of variables. Then, we solve the quantum inverse problem and reconstruct the local operators in terms of the separate variables. We also determine a basis of operators whose form factors are characterized by a single determinant formulae. This implies that the form factors of any local operator are expressed as finite sums of determinants. Among these form factors written in determinant form are in particular those which will reproduce the chiral Potts order parameters in the thermodynamic limit. The results presented here are the generalization to the present models associated to the most general cyclic representations of the 6-vertex Yang-Baxter algebra of those we derived for the lattice sine-Gordon model.

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Section: Discussionmentioning

confidence: 99%

On the Form Factors of Local Operators in the Bazhanov–Stroganov and Chiral Potts Models

Grosjean

Maillet

Niccoli

2014

Ann. Henri Poincaré

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“…Expressions for the norms or correlation functions for various models solvable by the quantum separation of variables method have been established, e.g. in the works [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.…”

Section: An Opening Discussionmentioning

confidence: 99%

“…The resolution of the inverse problem for the Toda chain has been pioneered by Babelon [8,9] in 2002 and further developed in the works [105,139]. These results, along with the unitarity of the separation of variables transform U lead to multiple integral representations for the form factors.…”

Section: Multiple Integral Representationsmentioning

confidence: 99%

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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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“…(5.13) Thus, ϕ (−) y N cannot be identically zero. Regarding to ψ (−) y N , as in [18], one may compute the x N → ∞, x a+1 − x a → +∞ of ψ (−) y N (x N ) staring from its Mellin-Barnes integral representation by pushing the various integration contours slightly in the upper-half plane. This shows that the function is non-vanishing in this asymptotic regime, and hence is non-identically zero.…”

Section: Proof -mentioning

confidence: 99%

On the Separation of Variables for the Modular XXZ Magnet and the Lattice Sinh-Gordon Models

Derkachov

Kozłowski

Manashov

2019

Ann. Henri Poincaré

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We construct the generalised Eigenfunctions of the entries of the monodromy matrix of the N-site modular XXZ magnet and show, in each case, that these form a complete orthogonal system in L 2 (R N ). In particular, we develop a new and simple technique, allowing one to prove the completeness of such systems. As a corollary of out analysis, we prove the Bystko-Teschner conjecture relative to the structure of the spectrum of the B(λ)-operator for the odd length lattice Sinh-Gordon model.

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Aspects of the inverse problem for the Toda chain

Cited by 7 publications

References 40 publications

On the Form Factors of Local Operators in the Bazhanov–Stroganov and Chiral Potts Models

On the Form Factors of Local Operators in the Bazhanov–Stroganov and Chiral Potts Models

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

On the Separation of Variables for the Modular XXZ Magnet and the Lattice Sinh-Gordon Models

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