Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. Some history and some recent results

Deift, Percy; Its, Alexander; Krasovsky, I. V.

doi:10.48550/arxiv.1207.4990

Cited by 16 publications

(35 citation statements)

References 141 publications

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“…In the framework of this successful ideology, we can find two classes of block Toeplitz matrices [1]. On one hand, those which can be reduced to a pure Toeplitz one, with scalar symbol, where the Fisher-Hartwig theorem [2] is the key to obtain an asymptotic expansion of our determinant.…”

Section: Introductionmentioning

confidence: 99%

Entanglement in fermionic chains with finite-range coupling and broken symmetries

et al. 2015

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We obtain a formula for the determinant of a block Toeplitz matrix associated with a quadratic fermionic chain with complex coupling. Such couplings break reflection symmetry and/or charge conjugation symmetry. We then apply this formula to compute the Rényi entropy of a partial observation to a subsystem consisting of contiguous sites in the limit of large size. (2008)]. A striking feature of our formula for the entanglement entropy is the appearance of a term scaling with the logarithm of the size. This logarithmic behavior originates from certain discontinuities in the symbol of the block Toeplitz matrix. Equipped with this formula we analyze the entanglement entropy of a Dzyaloshinski-Moriya spin chain and a Kitaev fermionic chain with long-range pairing.

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Section: Introductionmentioning

confidence: 99%

Entanglement in fermionic chains with finite-range coupling and broken symmetries

et al. 2015

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“…Starting from the seminal works of Onsager and Kaufman on the Ising model whose mathematical needs led to the birth of the Strong Szegö Theorem in the theory of Toeplitz matrices (see e.g. [10] for more on the history of the matter), the evaluation of the constant terms in the asymptotics of different correlation and distribution functions of the random matrix theory and of the theory of solvable statistical mechanics models has always been a great challenge in the field. In addition to the Strong Szegö Theorem we mention the work of Tracy [36], where the "constant problem" related to the Ising model was solved.…”

Section: Introductionmentioning

confidence: 99%

Connection Problem for the Sine-Gordon/Painlevé III Tau Function and Irregular Conformal Blocks: Fig. 1.

Its

Lisovyy

Tykhyy

2014

Int Math Res Notices

106

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Abstract. The short-distance expansion of the tau function of the radial sine-Gordon/Painlevé III equation is given by a convergent series which involves irregular c = 1 conformal blocks and possesses certain periodicity properties with respect to monodromy data. The long-distance irregular expansion exhibits a similar periodicity with respect to a different pair of coordinates on the monodromy manifold. This observation is used to conjecture an exact expression for the connection constant providing relative normalization of the two series. Up to an elementary prefactor, it is given by the generating function of the canonical transformation between the two sets of coordinates.

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“…Note that the FH singularity is just a zero of the weight function ω (ϕ) of the model for ϕ = π. Using the extended formula [17], we obtain:…”

Section: The Large N Limit Without Scaling Of Couplings (ν B Fixed)mentioning

confidence: 99%

Multiple phases in a generalized Gross-Witten-Wadia matrix model

Russo,

Tierz

2020

Preprint

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We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large N results are obtained by using Szegö theorem with a Fisher-Hartwig singularity. In the large N (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space.

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Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. Some history and some recent results

Cited by 16 publications

References 141 publications

Entanglement in fermionic chains with finite-range coupling and broken symmetries

Entanglement in fermionic chains with finite-range coupling and broken symmetries

Connection Problem for the Sine-Gordon/Painlevé III Tau Function and Irregular Conformal Blocks: Fig. 1.

Multiple phases in a generalized Gross-Witten-Wadia matrix model

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