Asymptotic expansions of Lambert series and related q-series

We study the symmetry resolved entanglement entropies in gapped integrable lattice models. We use the corner transfer matrix to investigate two prototypical gapped systems with a U (1) symmetry: the complex harmonic chain and the XXZ spin-chain. While the former is a free bosonic system, the latter is genuinely interacting. We focus on a subsystem being half of an infinitely long chain. In both models, we obtain exact expressions for the charged moments and for the symmetry resolved entropies. While for the spin chain we found exact equipartition of entanglement (i.e. all the symmetry resolved entropies are the same), this is not the case for the harmonic system where equipartition is effectively recovered only in some limits. Exploiting the gaussianity of the harmonic chain, we also develop an exact correlation matrix approach to the symmetry resolved entanglement that allows us to test numerically our analytic results. arXiv:1911.09588v2 [cond-mat.stat-mech]

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“…involved in the computation of the critical limit of (84). The asymptotic expression for u → 1 of θ 3 (z|u) is [89]…”

Section: Some Properties Of the Jacobi Theta Functionsmentioning

confidence: 99%

Symmetry resolved entanglement in gapped integrable systems: a corner transfer matrix approach

2020

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“…Here, we follow the notation used by Banerjee and Wilkerson who provide an asymptotic expansion of this series around q = 1 [31].…”

Section: Short-distance Expansionmentioning

confidence: 99%

“…Making use of the asymptotic expansion of the generalized Lambert series around q = 1 stated in theorem 2.2 of [31], we obtain from (58) for the Casimir free energy in the scalar case with Dirichlet boundary conditions…”

Section: Short-distance Expansionmentioning

confidence: 99%

Classical Casimir free energy for two Drude spheres of arbitrary radii: A plane-wave approach

Schoger

Ingold

2021

SciPost Phys. Core

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We derive an exact analytic expression for the high-temperature limit of the Casimir interaction between two Drude spheres of arbitrary radii. Specifically, we determine the Casimir free energy by using the scattering approach in the plane-wave basis. Within a round-trip expansion, we are led to consider the combinatorics of certain partitions of the round trips. The relation between the Casimir free energy and the capacitance matrix of two spheres is discussed. Previously known results for the special cases of a sphere-plane geometry as well as two spheres of equal radii are recovered. An asymptotic expansion for small distances between the two spheres is determined and analytical expressions for the coefficients are given.

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“…It is useful to define the dimensionless capacitance coefficients for the two spheres as c ij ≡ C ij /2π (R 1 + R 2 ). With the definitions above, the classical solutions using the method of images can be written in the following infinite Lambert series [21] like form…”

Section: (B) Capacitance Expressionsmentioning

confidence: 99%

“…Note that the voltages must be equal when the spheres are in contact, which allows for the cancellation of divergences. Using the sum of the Lambert series in terms of the q-digamma function [19,21] the capacitance coefficients can be written in closed-form as [12]…”

Section: (B) Capacitance Expressionsmentioning

confidence: 99%

Exact closed-form and asymptotic expressions for the electrostatic force between two conducting spheres

et al. 2021

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We present exact closed-form expressions and complete asymptotic expansions for the electrostatic force between two charged conducting spheres of arbitrary sizes. Using asymptotic expansions of the force, we confirm that even like-charged spheres attract each other at sufficiently small separation unless their voltages/charges are the same as they would be at contact. We show that for sufficiently large size asymmetries, the repulsion between two spheres increases when they separate from contact if their voltages or their charges are held constant. Additionally, we show that in the constant voltage case, this like-voltage repulsion can be further increased and maximized through an optimal lowering of the voltage on the larger sphere at an optimal sphere separation.

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Asymptotic expansions of Lambert series and related q-series

Cited by 18 publications

References 5 publications

Symmetry resolved entanglement in gapped integrable systems: a corner transfer matrix approach

Symmetry resolved entanglement in gapped integrable systems: a corner transfer matrix approach

Classical Casimir free energy for two Drude spheres of arbitrary radii: A plane-wave approach

Exact closed-form and asymptotic expressions for the electrostatic force between two conducting spheres

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