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Spin transport and lack of quantisation in the $A\mathrm{II}$ class on the honeycomb structure

Luca Fresta, Giovanna Marcelli

Abstract

We investigate spin transport in a class of two-dimensional $A\mathrm{II}$ insulators on the honeycomb structure, the Kane-Mele model being an emblematic example in this class. We derive the spin conductivity by the linear response à la Kubo and show that it is well-defined and independent of the choice of the spin current. For models that do not conserve the spin, we demonstrate that the deviation of the spin conductivity from the quantised value is, at worst, quadratic in the spin-non-conserving terms, thus improving previous results. Additionally, we show that the leading-order corrections are actually quadratic for some models in the class, demonstrating that the spin conductivity is not universally quantised. Consequently, our results show that, in general, there is no direct connection between the spin conductivity and the Fu-Kane-Mele index.

Spin transport and lack of quantisation in the $A\mathrm{II}$ class on the honeycomb structure

Abstract

We investigate spin transport in a class of two-dimensional insulators on the honeycomb structure, the Kane-Mele model being an emblematic example in this class. We derive the spin conductivity by the linear response à la Kubo and show that it is well-defined and independent of the choice of the spin current. For models that do not conserve the spin, we demonstrate that the deviation of the spin conductivity from the quantised value is, at worst, quadratic in the spin-non-conserving terms, thus improving previous results. Additionally, we show that the leading-order corrections are actually quadratic for some models in the class, demonstrating that the spin conductivity is not universally quantised. Consequently, our results show that, in general, there is no direct connection between the spin conductivity and the Fu-Kane-Mele index.

Paper Structure

This paper contains 16 sections, 28 theorems, 237 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The $\mathrm{AII}$ class on the honeycomb structure
  3. Honeycomb structure and Hilbert space
  4. The $A\mathrm{II}^{\hexagon}$ class
  5. Spin transport in $\mathrm{AII}^{\hexagon}$
  6. Trace per unit volume
  7. Linear response à la Kubo
  8. Spin conductivity
  9. Spin-non-conserving contributions
  10. Phase transition and lack of quantisation
  11. The extended Kane--Mele model
  12. Phase diagram of the extended Kane--Mele model
  13. Lack of quantisation in the $\mathrm{AII}^{\hexagon}$ class
  14. Proof of Theorem \ref{['prop:imaginary-time-repr']}
  15. Grand canonical formalism
  16. ...and 1 more sections

Key Result

Theorem 1.1

Figures (4)

  • Figure 1: The honeycomb structure.
  • Figure 2: Plot of the energy bands $\mathcal{E}^{w}_{l,\sigma}(k_1,k_2)$ along the line $(\frac{2 \pi}{3}, k_2)$ at $t= 2/3$, $\lambda_{\mathrm{SO}} = 3\sqrt{3}/5$, $\lambda_{\mathrm{R}} = 0$ and for $w>\lambda_{\mathrm{SO}}$ and $w = \lambda_{\mathrm{SO}}$, respectively on the left and on the right. The blue and the yellow bands correspond to $\sigma = +$ and $\sigma= -$ respectively.
  • Figure 3: Plot of $w_c^{\pm}/|\lambda_{\mathrm{SO}}|$ as a function of $\lambda_{\mathrm{R}}/|\lambda_{\mathrm{SO}}|$.
  • Figure 4: Pictorial representation of the annulus $\widetilde{C}_{L}$ inside of $C_{L}$.

Theorems & Definitions (87)

  • Theorem 1.1
  • Definition 2.1: Bloch--Floquet Transform
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4: $\mathrm{AII^{\hexagon}}$ class
  • Definition 3.1: Trace per unit volume
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • ...and 77 more