Quantum spin-tensor Hall effect protected by pseudo time-reversal symmetry

TL;DR

This work introduces quantum spin-tensor Hall (QSTH) insulators that are protected by a unique pseudo-time-reversal symmetry and host a quantized rank-2 spin-tensor Hall conductivity, while rank-0 and rank-1 responses vanish. It develops a pseudospin-1 honeycomb model (and a square-lattice toy model) showing a bulk gap, robust pTRS-protected edge states, and a $\ ext{Z}_2$ Wilson-loop invariant that signals a nontrivial QSTH phase. Through Kubo formula calculations, the authors demonstrate a universal $\sigma_{xy}^{zz}=q/(4\pi)$, mapping the rank-2 response to a pseudo-spin current akin to the Kane–Mele QSH framework, and they relate QSTH to QSH via projections in an appropriate basis. The results broaden the Hall effect family, offering a new route toward spin-tensor electronics and atomtronics with potential realization in cold-atom lattices and beyond.

Abstract

The celebrated family of the Hall effect plays a fundamental role in modern physics. Starting from the anomalous Hall effect (AHE) and the quantum AHE (QAHE) with broken time-reversal symmetry (TRS) to their spinful generalizations, including spin Hall effect (SHE) and quantum SHE (QSHE) protected by TRS, they reveal rich transport and topological phenomena. However, in larger-spin $S$ ($S>1/2$) systems, besides charge current and spin current, there arise higher-rank spin-tensor currents. Recent work has uncovered an interesting spin-tensor Hall effect with spin-tensor currents in these larger-spin systems. Taking a step further, this work discovers a new class of topological states of matter dubbed \textit{quantum spin-tensor Hall} (QSTH) insulators with broken TRS, and their nontrivial topology is protected by a unique \textit{pseudo-TRS}. Most strikingly, QSTH insulators exhibit a quantized rank-2 spin-tensor Hall conductivity, whereas both charge (rank-0) and spin (rank-1) conductivities vanish. We also fully characterize their topological properties and highlight the physical interpretations via the underlying connections to QSHE. Our work enriches the family of the famous Hall effects and sheds light on the intriguing topological state of matter in larger-spin systems. It further offers new avenues toward spin-tensor-tronics and low-power atomtronics.

Figures (6)

• Figure 1: The family of Hall effects. From left to right, we have the classical and quantum versions. From top to bottom, we show how they generalize along the spin degrees of freedom. This work completes the puzzle by introducing quantum spin-tensor Hall effect (QSTHE) on the bottom right.
• Figure 2: (a) Illustration of the honeycomb lattice with A/B sublattices. Solid lines and dashed arrows correspond to the nearest-neighbor $t$ and next-nearest-neighbor $t'$ hopping. When the particles hop around the dashed arrows, the particles with pseudospin $\tau = \pm 1$ acquire an accumulated phase $\pi/2$, and the particles with pseudospin $\tau =0$ acquire an accumulated phase $-\pi/2$. (b) The phase diagram concerning $t'$ and the staggered sublattice potential $\lambda_v$. NI is a trivial normal insulator, and QSTH indicates the quantum spin tensor Hall state. (c) and (d) The representative energy spectra in QSTH (${\lambda _v} = 0.2$) and NI (${\lambda _v} = 1.2$) phases, when $t'=0.2$. We set $u=1$ and choose $t=1$ as the unit of energy.
• Figure 3: (a) and (b) The energy spectra on strips with zigzag boundary conditions and the Berry phase for $\mathbb{Z}_2$ invariant in topological QSTH phase. The red (blue) curve highlights the helical edge states at one (the other) edge, and we set ${\lambda _v} = 0.2$ as in Fig. \ref{['fig_band']}(c). (c) and (d) Similar to (a) and (b), but for trivial NI phase when ${\lambda _v} = 2$. Common parameters are $t = 1,t' = 0.2$ and $u = 1$.
• Figure 4: (a) Hall conductivity at different ranks computed from the bulk using the Kubo formula for the honeycomb lattice. All those, including Hall conductivity (HC) for rank-0 charge current, spin-Hall conductivity (SHC) for rank-1 spin current, and spin-tensor-Hall conductivity (STHC) for rank-2 spin-tensor current, are plotted across both NI and QSTH phases. Only in the nontrivial QSTH phase is a quantized STHC observed. (b) Energy spectra for a strip in QSTH phase on a square lattice under periodic boundary conditions along $x$ but open boundary conditions along $y$. The parameter $m_z$ is set to be $m_z=3.2$. (c) Similar to (b) but for the model on a square lattice in NI phase with $m_z=7.2$. (d) Similar to (a) but for the model on a square lattice.
• Figure B1: (a) Illustration of edge states on a stripe in QSTH phase. (b) and (d) The density distribution of the helical edge states on one edge. (c) and (e) The density distribution of the helical edge states on the other edge. ${\left| {{\psi _{\pm v,0,s}}} \right|^2}$ and ${\left| {{\psi _{\pm v,+,s}}} \right|^2}$ denote the particle density for different spin components (${s = 1,0, - 1}$) of the edge states. The number of lattice sites along $y$ is $N_y=100$.