Spin-Tensor Hall Effect

• Spin-Tensor Hall Effect is defined by the emergence of higher-rank (tensor) spin currents from multipole spin operators in systems with spin S > 1/2.
• The effect features robust, quantized rank‑2 spin–tensor conductivities (e.g. σₓᵧ^(zz) = q/(4π)) and topologically protected edge states enabled by pseudo time-reversal symmetry.
• Experimental realizations using ultracold atoms and engineered lattices are paving the way toward spin–tensor–tronics and advanced quantum information applications.

The Spin-Tensor Hall Effect (STHE) generalizes the conventional spin Hall effect (SHE) to accommodate higher-order spin transport—beyond vector (rank‑1) spin currents—by introducing tensorial spin currents originating from multipole spin operators in systems with spin quantum number $S > 1/2$. This extension emerges from both fundamental considerations in solid-state systems and the design of synthetic quantum materials, including ultracold atomic gases and engineered lattices, where spin–tensor–momentum couplings or symmetry-protected lattice models drive novel transverse responses inaccessible to spin-1/2 electrons. Recent theoretical and experimental advances have established the STHE as both a probe of quantum geometry and a candidate for realizing robust, tunable, and topologically protected spin–tensor transport phenomena. The effect is central to the concept of "spin–tensor–tronics," in which information is encoded, manipulated, and detected not simply via spin polarization but through tensorial spin observables.

1. Fundamental Principles: From the Spin Hall Effect to Tensor Spin Currents

The SHE involves the generation of a transverse spin current in response to an applied charge current, realized via spin–orbit coupling and typically described as a rank‑2 tensor $j^s_{ik}$: the flow component in real space ($i$) and the spin polarization axis ($k$). Extensions of this framework to higher-spin systems naturally lead to spin–tensor Hall effects, where the spin current is promoted to a higher-rank tensor—specifically, for spin‑1 systems, a rank‑3 tensor $j^s_{ijk}$ or transport of quantities such as quadrupole spin moments. The defining physical signature of STHE is the emergence of dissipationless transverse spin–tensor currents, even when both the charge and conventional spin Hall currents vanish (Sinova et al., 2014, Hou et al., 2020, Wu et al., 19 Oct 2025).

For a spin-1 system, the relevant rank‑2 spin–tensor operator, $N_{zz}$, is given by

$N_{zz} = F_z^2 - \frac{2}{3},$

and the spin–tensor current operator is

$J_2^{zz} = \frac{1}{2}\{P_2, \hat{v}\},$

where $P_2 = \hbar N_{zz}$ and $\hat{v}$ is the velocity operator (Hou et al., 2020, Wu et al., 19 Oct 2025). This generalization underpins the novel transport behavior whereby spin–tensor observables, rather than spin polarization vectors, are carried perpendicularly to the driving electric field.

2. Microscopic Models and Symmetry Protection

Recent theoretical developments have yielded explicit tight-binding models supporting quantum spin–tensor Hall (QSTH) states on both honeycomb and square lattices for pseudospin‑1 fermions. These models are characterized by the following features (Wu et al., 19 Oct 2025):

• The bulk insulating phase with vanishing charge (rank‑0) and spin (rank‑1) Hall conductivities.
• Nonzero, quantized rank‑2 spin–tensor Hall conductivity:

$\sigma_{xy}^{zz} = \frac{q}{4\pi},$

where $q$ is the carrier charge.

• Robust edge states that carry spin–tensor current exclusively, realized as helical modes with distinct rank‑2 spin content.

The topological protection of the QSTH phase is not afforded by conventional time-reversal symmetry (TRS), which is explicitly broken in these systems. Instead, a pseudo-TRS (pTRS) operation, constructed from a rank‑2 spin–tensor operator combined with complex conjugation, guarantees a Kramers-like degeneracy for edge modes and quantization of the spin–tensor Hall response:

$$\Xi \hat{h}(\boldsymbol{k}) \Xi^{-1} = \hat{h}(-\boldsymbol{k}), \qquad \Xi = e^{-i\sigma_0 \otimes N_{yz} \mathcal{K}},$$

where $N_{yz}$ is a canonical spin–tensor, $\mathcal{K}$ denotes complex conjugation, and $\sigma_0$ acts in sublattice space. The operator satisfies $\Xi^2 = -I_A$, with $I_A$ the anti-diagonal identity (Wu et al., 19 Oct 2025). This pseudo-symmetry underlies the definition of a $\mathbb{Z}_2$ topological invariant for the occupied bands, formally paralleling the classification of quantum spin Hall insulators in spin-1/2 systems.

3. Quantized Spin–Tensor Hall Conductivity and Edge Physics

The QSTH phase exhibits strictly vanishing conventional Hall responses while the rank‑2 contribution is quantized and robust to perturbations that respect pTRS. The conductivities can be summarized as: $$\sigma_{xy} = 0,\qquad \sigma_{xy}^z = 0,\qquad \sigma_{xy}^{zz} = \frac{q}{4\pi}$$ (Wu et al., 19 Oct 2025).

Numerical studies of strip geometries with zigzag and armchair edges reveal that the QSTH state supports two edge modes per edge: one localized in the spin‑0 manifold, the other an equal superposition of spin $+1$ and $-1$ (i.e., $|↑⟩ \propto |1⟩ + |−1⟩$ and $|↓⟩ = |0⟩$). The pTRS operator $\Xi$ acts as a Kramers degeneracy, ensuring their orthogonality and topological protection.

In a projected “pseudospin‑1/2” subspace, the QSTH Hamiltonian reduces to a Kane-Mele-type (quantum spin Hall) model, with the rank‑2 current operator mapping onto the standard spin current operator (plus a term proportional to the identity, vanishing by symmetry). This establishes a direct correspondence with the conventional quantum spin Hall effect, but for tensor currents: $$P_s N_{zz} P_s^{-1} = \frac{1}{2} \sigma_z - \frac{1}{6}\sigma_0$$ where $P_s$ is the projection (Wu et al., 19 Oct 2025). Thus, the quantized spin–tensor Hall effect is a higher-rank generalization of the established QSHE.

4. Experimental Realizations and Detection

The experimental detection of QSTH and more generally STHE presents significant challenges since conventional probes target charge or spin-1 observables. However, the implementation of engineered spin–tensor–momentum couplings in ultracold atomic gases provides a tractable platform. Schemes based on three-level (spin-1) atoms employing Raman-induced couplings have been proposed. By selectively coupling $|+1\rangle$–$|0\rangle$ and $|0\rangle$–$|-1\rangle$ subspaces (with independent phase control), the necessary Hamiltonian structure for QSTH can be realized (Hou et al., 2020). Edge accumulation of tensor spin (e.g., via population imbalance in $N_{zz}$) can be detected with state-selective imaging. Robustness to disorder or parameter inhomogeneity is theoretically guaranteed by the quantization and topological origin of the tensor current.

In solid-state systems, possible candidate materials are those supporting emergent spin-1 excitations or multipole moments on a lattice with suitable symmetry; however, unambiguous detection of strictly tensorial edge currents remains an open experimental challenge.

5. Quantum Geometry and Relationship to Other Transverse Effects

The QSTH effect is fundamentally tied to quantum geometric properties of the Bloch bands. Its quantized Hall response originates from generalized Berry–Chern-type invariants formulated with tensorial spin operators, and is thus robust to local perturbations that respect the pseudo-TRS. The connection between QSTH and QSHE is explicit: under projection onto a relevant subspace, the tensor Hall response is proportional to a spin Hall response, implying that the QSTH indexes a higher element of the “Hall hierarchy.” At the same time, the STHE is distinct from other known effects such as the anomalous Hall effect or quantum anomalous Hall effect, as its response vanishes for both charge and dipolar spin currents while remaining quantized for the tensor channel (Wu et al., 19 Oct 2025).

This places QSTH as a higher-rank member in the topological classification of band insulators, with an enriched set of edge and bulk invariants.

6. Implications and Outlook for Spin–Tensor–Tronics

The discovery of QSTH points to new device paradigms—termed "spin–tensor–tronics"—in which information processing leverages tensorial spin currents for enhanced robustness, multiplexed channels, or functionality not possible with vector spin currents alone. The quantized nature of $\sigma_{xy}^{zz}$ suggests intrinsic immunity to disorder and thermal fluctuations for edge transport. Furthermore, the protection via pseudo-TRS rather than TRS may allow integration with magnetically ordered systems, avoiding the need for time-reversal invariance typically required in QSHE devices.

Future research directions include the exploration of higher-spin ($S>1$) lattice models with even richer topological multipole responses, practical realization in synthetic quantum matter systems, and the development of detection techniques sensitive to tensor spin observables. Elucidation of their role in strong correlation, ultrafast, or low-dimensional regimes remains a frontier.

7. Table: Quantum Hall Effects Hierarchy in Spin Systems

Effect	System	Protected Quantity	Quantized Conductivity
Quantum Hall Effect	Charge (rank‑0)	Charge current	$\sigma_{xy} = q^2/h$
Quantum Spin Hall Effect	Spin-1/2 (rank‑1)	Spin current	$\sigma_{xy}^z = q/(2\pi)$
Quantum Spin-Tensor Hall Effect	Spin-1 (rank‑2)	Spin-tensor current ($N_{zz}$)	$\sigma_{xy}^{zz} = q/(4\pi)$

This hierarchy illustrates the progression from conventional to multipolar Hall effects, highlighting the unique rank, protected observable, and topological response of QSTH (Wu et al., 19 Oct 2025, Hou et al., 2020).

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The QSTH insulator exemplifies an emergent topological phase distinct from both quantum spin Hall and quantum anomalous Hall insulators, with a quantized rank‑2 spin–tensor Hall conductivity protected by pseudo time-reversal symmetry, strictly vanishing charge and spin Hall responses, helical edge states carrying tensor current, and promising implications for tensor-based quantum technologies (Wu et al., 19 Oct 2025, Hou et al., 2020).