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Generalized bulk-interface correspondence for non-quantized spin transport

Jiayu Qiu, Hai Zhang

TL;DR

This work extends the bulk-interface correspondence to electronic systems with time-reversal symmetry and potentially nonconserved spin by defining a bulk spin conductance via a potential-current correlation and introducing two interface contributions: spin-drift conductance from interface modes and spin-torque conductance from spin generation near the interface. The authors develop a rigorous PV-trace framework, establish key trace-class and Helffer–Sjöstrand tools, and prove that the interface transport equals the bulk conductance difference, decomposed into drift and torque terms. When spin is conserved, the framework reduces to known BIC results based on the spin Chern number or Fu–Kane–Mele $\mathbb{Z}_2$ index, while in general TR-invariant, nonconserved-spin systems the BIC persists in a generalized form. The results provide a robust theoretical basis for spin transport phenomena at interfaces beyond quantized topological invariants and suggest new routes for spin-torque device design.

Abstract

This paper establishes a rigorous mathematical framework for a generalized bulk-interface correspondence (BIC) in electronic systems with possibly nonconserved spin charge, where the Hamiltonian and spin operator do not commute. We first introduce the bulk spin conductance as a character of the bulk medium, which is defined as a potential-current correlation function and is not quantized if the spin charge is nonconserved. Then we establish the principle of BIC, which states that the difference of bulk spin conductances across an interface equals the sum of two quantities associated with the spin transport along the interface: the spin-drift conductance, which captures spin transport carried by interface modes, and the spin-torque conductance, which accounts for spin generation near the interface due to the non-conservation of spin. Furthermore, when the spin charge is conserved, our result recovers the existing BIC based on the spin Chern number or Fu-Kane-Mele $\mathbb{Z}_2$ index. Our findings demonstrate that the principle of BIC is not restricted to systems with quantized characters and provides new insights into spin transport phenomena.

Generalized bulk-interface correspondence for non-quantized spin transport

TL;DR

This work extends the bulk-interface correspondence to electronic systems with time-reversal symmetry and potentially nonconserved spin by defining a bulk spin conductance via a potential-current correlation and introducing two interface contributions: spin-drift conductance from interface modes and spin-torque conductance from spin generation near the interface. The authors develop a rigorous PV-trace framework, establish key trace-class and Helffer–Sjöstrand tools, and prove that the interface transport equals the bulk conductance difference, decomposed into drift and torque terms. When spin is conserved, the framework reduces to known BIC results based on the spin Chern number or Fu–Kane–Mele index, while in general TR-invariant, nonconserved-spin systems the BIC persists in a generalized form. The results provide a robust theoretical basis for spin transport phenomena at interfaces beyond quantized topological invariants and suggest new routes for spin-torque device design.

Abstract

This paper establishes a rigorous mathematical framework for a generalized bulk-interface correspondence (BIC) in electronic systems with possibly nonconserved spin charge, where the Hamiltonian and spin operator do not commute. We first introduce the bulk spin conductance as a character of the bulk medium, which is defined as a potential-current correlation function and is not quantized if the spin charge is nonconserved. Then we establish the principle of BIC, which states that the difference of bulk spin conductances across an interface equals the sum of two quantities associated with the spin transport along the interface: the spin-drift conductance, which captures spin transport carried by interface modes, and the spin-torque conductance, which accounts for spin generation near the interface due to the non-conservation of spin. Furthermore, when the spin charge is conserved, our result recovers the existing BIC based on the spin Chern number or Fu-Kane-Mele index. Our findings demonstrate that the principle of BIC is not restricted to systems with quantized characters and provides new insights into spin transport phenomena.

Paper Structure

This paper contains 13 sections, 15 theorems, 144 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Setting and Main results
  4. Futher discussions
  5. Notations
  6. Preliminaries
  7. Some trace-class properties
  8. Almost-analytic extension, Helffer-Sjöstrand formula
  9. Vanishing spin-torque response in the bulk
  10. Bulk spin conductance: proof of Proposition \ref{['prop_spin_conduct_justified']}
  11. Reduction to Kubo formula: proof of Proposition \ref{['prop_kubo_formula']}
  12. Interface spin-torque conductance: proof of Proposition \ref{['prop_interface_spin_torque_justified']}
  13. Bulk-interface correspondence: proof of Theorem \ref{['thm_bic']}

Key Result

Proposition 1.6

For each switch function $\Lambda_2$, $\sigma_{\pm}^{\Lambda_2}$ in Definition def_spin_conducatance is well-defined as the the limit eq_spin_conduc_1 exists and is independent of the choice of switch function $\Lambda_1$, density function $\rho$ and the step size $k\in\mathbb{N}$.

Figures (2)

  • Figure 1: An interface model: two bulk mediums are joined along an interface. The bulk media are assumed to have rectangular lattice structures, and impurities are allowed near the interface.
  • Figure 2: (a) Transverse (Hall) Current in a periodic medium induced by an external potential along the longitudinal direction. (b) A thought experiment: an external potential is applied to the interface model, and the charge transport is analyzed within an imaginary box containing the interface.

Theorems & Definitions (27)

  • Definition 1.1: tight-binding operator
  • Definition 1.3: principal-value trace along $x_1$
  • Definition 1.4: switch functions
  • Definition 1.5: Bulk spin conductance
  • Proposition 1.6
  • Proposition 1.7
  • Definition 1.8: Interface spin-drifting conductance
  • Definition 1.9: Interface spin-torque conductance
  • Proposition 1.10
  • Theorem 1.11
  • ...and 17 more