Toeplitz Determinants: Some Applications, Theorems, and Conjectures

By M. Fisher, R. Hartwig. 2007

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Quantum quenches in the transverse field Ising chain: II. Stationary state properties (2012, methodology): “ the Fourier (Laurent) expansion logt(z) = X1 q=1 (logt) qzq; (logt) q= Z 2ˇ 0 dk 2ˇ logt(eik)e ikq: (A.2) The asymptotic behaviour of the determinant is then given by the strong Szego limit theorem [68], which states that det[T ‘] = E [t] e ‘(logt) 0 1 + O(‘1 2) ; (A.3) Quantum Quench in the Transv”
Entanglement entropy of excited states (2009, methodology): “Defining kF ≡ ∑n j=1(−1)j−1ϕj/2, the Fisher–Hartwig conjecture (that for this case with |λ| > 1, i.e. |Re(βj)| 1/2, has been proved by Basor [31]) is det|λ1 − Π| ∼ n∏ i j [ (2 − 2 cos(ϕi − ϕj))(−1) j−iβ(λ)2 × G(1 + β(λ))nG(1 − β(λ))n { (λ + 1) ( λ + 1 λ − 1 )−kF /π}L −nβ(λ) 2 ] , (24) doi:10.1088/”
Toeplitz Matrices and Toeplitz Determinants under the Impetus of the Ising Model: Some History and Some Recent Results (2012, background): “s to problems in analytic function theory, stationary stochastic processes and linear estimation theory in statistics. In addition to problems in probability theory and statistics, Fisher and Hartwig [FisHart1] present a long and varied list of applications of Toeplitz determinants to problems in st”