The Fisher-Hartwig conjecture and Toeplitz eigenvalues

Basor, Estelle; Morrison, Kent E.

doi:10.1016/0024-3795(94)90187-2

Cited by 90 publications

(106 citation statements)

References 13 publications

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“…This logarithmic divergence is a direct consequence of a general theorem on the spectra of Toeplitz matrices [12], which in our case reduces to the statement that in the limit N → ∞ all except for O(ln N) of the eigenvalues of T N [g] migrate toward the points 1 and −1, where, trivially, we have e(1, 1) = e(1, −1) = 0. We now make an observation relating to (2.25) that will have important consequences for the way we shall proceed.…”

Section: Bipartite Entanglement In the Xy Modelmentioning

confidence: 71%

“…Since the symbol g(θ) has absolute value one, a theorem on the spectrum of Toeplitz matrices [12] -the same theorem mentioned in section 2 -states that the eigenvalues of T N [g] are all inside the unit circle and approach the image of g in the limit N → ∞. It follows that all the eigenvalues of ( 1].…”

Section: U(n ) Symmetrymentioning

confidence: 99%

“…Furthermore, since the Fourier coefficients defined in equation (2.10) are real, there exists an appropriate V ∈ SO(2N) that block-diagonalizes C N : 12) where the ν j are real and, for reasons that will become apparent later, lie in the interval [−1, 1]. Now, let us define 13) which are Fermi operators, i.e.…”

Section: Bipartite Entanglement In the Xy Modelmentioning

confidence: 99%

“…A brief look to table 1 in appendix C shows that this is a necessary condition for all the other compact groups too. Since A is a real symmetric matrix, its eigenvalues Λ k are real and therefore 12) where the φ k s are the eigenvectors of A. We now need to diagonalize A; as for the unitary group, this can be done explicitly.…”

Section: O + (2n ) Symmetrymentioning

confidence: 99%

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Random Matrix Theory and Entanglement in Quantum Spin Chains

Keating

Mezzadri

2004

Commun. Math. Phys.

126

248

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We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians -those that are related to quadratic forms of Fermi operators -between the first N spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as N → ∞. This is shown to grow logarithmically with N . The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlevé type. In some cases these solutions can be evaluated to all orders using recurrence relations.

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Section: Bipartite Entanglement In the Xy Modelmentioning

confidence: 71%

Section: U(n ) Symmetrymentioning

confidence: 99%

Section: Bipartite Entanglement In the Xy Modelmentioning

confidence: 99%

Section: O + (2n ) Symmetrymentioning

confidence: 99%

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Random Matrix Theory and Entanglement in Quantum Spin Chains

Keating

Mezzadri

2004

Commun. Math. Phys.

126

248

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“…We here consider genuinely complex-valued symbols, in which case the overall picture is less complete. Papers [12], [15], [19] describe the limiting behavior of the eigenvalues of T n (a) if a is a rational function, while papers [1] and [28] are devoted to the asymptotic eigenvalue distribution in the case of non-smooth symbols. In [25] and [28], it is in particular shown that if a ∈ L ∞ and the essential range R(a) does…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotics of Individual Eigenvalues of a Class of Large Hessenberg Toeplitz Matrices

Bogoya

Böttcher

Grudsky

2012

Recent Progress in Operator Theory and Its Applications

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Abstract. We study the asymptotic behavior of individual eigenvalues of the n-by-n truncations of certain infinite Hessenberg Toeplitz matrices as n goes to infinity. The generating function of the Toeplitz matrices is supposed to be of the form a(t) = t, where α is a positive real number but not an integer and f is a smooth function in H ∞ . The classes of generating functions considered here and in a recent paper by Dai, Geary, and Kadanoff are overlapping, and in the overlapping cases, our results imply in particular a rigorous justification of an asymptotic formula which was conjectured by Dai, Geary, and Kadanoff on the basis of numerical computations.MSC 2010. Primary 47B35. Secondary 15A15, 15A18, 47N50, 65F15

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Toeplitz Determinants with One Fisher–Hartwig Singularity

Ehrhardt

Silbermann

1997

Journal of Functional Analysis

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The Fisher-Hartwig conjecture and Toeplitz eigenvalues

Cited by 90 publications

References 13 publications

Random Matrix Theory and Entanglement in Quantum Spin Chains

Random Matrix Theory and Entanglement in Quantum Spin Chains

Asymptotics of Individual Eigenvalues of a Class of Large Hessenberg Toeplitz Matrices

Toeplitz Determinants with One Fisher–Hartwig Singularity

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