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PhyslibAlpha.CondensedMatter.TightBindingChain.MaxCurrentVariances

The variances of the maximal current state

i. Overview

In the maximal current state of the open chain with `N` sites, write `θ = π / (N + 1)`. The energy and position variances are

`Var H = 4 t² sin² θ (N - 1) / (N + 1)`, `Var X = a² (((N + 1)² + 2) / 12 - 1 / (2 sin² θ))`.

The position variance rests on one sum over the `(N + 1)`-th roots of unity, `∑ j < N + 1, (j - (N + 1) / 2)² cos (2 j θ) = (N + 1) / (2 sin² θ)`, obtained by summation by parts. Hence `C_Nava²` has the closed form `2 (N - 1) / ((N + 1) cos² θ) · (((N + 1)² + 2) / 6 · sin² θ - 1)`.

ii. Key results

  • `variance_openHamiltonian_maxCurrentState` : `Var H = 4 t² sin² θ (N - 1) / (N + 1)`.
  • `variance_position_maxCurrentState` : `Var X = a² (((N + 1)² + 2) / 12 - 1 / (2 sin² θ))`.
  • `CNava_sq_eq` : the closed form of `C_Nava²`.

iii. Table of contents

  • A. Sums over the roots of unity
  • B. The energy variance
  • C. The position variance
  • D. The closed form of `C_Nava`

iv. References

* https://www.damtp.cam.ac.uk/user/tong/aqm/aqmtwo.pdf. [ref: tong_statistical_physics]

A. Sums over the roots of unity

B. The energy variance

C. The position variance

D. The closed form of `C_Nava`

3 declarations

theorem

Energy Variance of the Maximal Current State is 4t2(N−1)sin⁡2θN+1\frac{4 t^2 (N - 1) \sin^2 \theta}{N + 1}

For an open tight-binding chain TT with NN sites and hopping parameter tt, let ωmax\omega_{\text{max}} be the maximal current vector state and HopenH_{\text{open}} be the open Hamiltonian observable. The variance of the Hamiltonian in this state is given by Varωmax(Hopen)=4t2(N−1)sin⁡2θN+1,\text{Var}_{\omega_{\text{max}}}(H_{\text{open}}) = \frac{4 t^2 (N - 1) \sin^2 \theta}{N + 1}, where θ=πN+1\theta = \frac{\pi}{N + 1}.

theorem

Position Variance Var(X)\text{Var}(X) of the Maximal Current State

For an open tight-binding chain TT with NN sites and lattice spacing aa, let XX be the position observable and ωmax\omega_{\text{max}} be the maximal current vector state. The variance of the position observable in this state is given by: Varωmax(X)=a2((N+1)2+212−12sin⁡2(πN+1))\text{Var}_{\omega_{\text{max}}}(X) = a^2 \left( \frac{(N + 1)^2 + 2}{12} - \frac{1}{2 \sin^2 \left( \frac{\pi}{N + 1} \right)} \right)

theorem

Closed form of the squared uncertainty constant CNava2C_{\text{Nava}}^2 in the maximal current state

For an open tight-binding chain with NN sites and hopping parameter t≠0t \neq 0, if N≥2N \geq 2, then the squared uncertainty constant CNava2C_{\text{Nava}}^2 in the maximal current state is given by the closed form: CNava2=2(N−1)(N+1)cos⁡2θ((N+1)2+26sin⁡2θ−1)C_{\text{Nava}}^2 = \frac{2 (N - 1)}{(N + 1) \cos^2 \theta} \left( \frac{(N + 1)^2 + 2}{6} \sin^2 \theta - 1 \right) where θ=πN+1\theta = \frac{\pi}{N + 1}.