Spin-Valley Locked Kramers Doublets

• Spin–valley locked Kramers doublets are quantum eigenstates in solids where an electron’s spin is coupled with its valley index, creating protected twofold-degenerate energy levels.
• They manifest in materials like TMDCs, Dirac semimetals, and antiferromagnets, offering precise control over electronic, optical, and quantum information properties.
• The phenomenon is modeled using Hamiltonians that incorporate spin–orbit and Zeeman couplings, with experimental signatures identified through magnetospectroscopy and Hall effect measurements.

Spin-valley–locked Kramers doublets are quantum eigenstates in crystalline solids where the electron’s spin and valley (momentum-space extremum) indices are rigidly correlated, forming pairs that are protected by time-reversal or crystalline symmetry. This phenomenon generates distinct twofold-degenerate quantum levels, robust against many sources of decoherence, and plays a pivotal role in the emerging fields of valleytronics, quantum information, and topological matter. It is most prominent in materials where strong spin–orbit coupling and inversion-symmetry breaking cause each energy eigenstate’s spin degree of freedom to be “locked” to a specific valley, with the pair of locked states related by a symmetry (typically time reversal, $\mathcal{T}$, or, in some antiferromagnets, a crystal symmetry $C$). Such doublets not only underpin new regimes of elementary excitations, as in transition metal dichalcogenides (TMDCs), Dirac semimetals, and topological antiferromagnets, but also dictate fundamental transport, optical, and topological responses.

1. Formal Hamiltonians and Theoretical Foundations

A generic Hamiltonian capturing spin–valley–locked Kramers doublets can be constructed as

$H = H_0 + H_{\rm SO} + H_Z,$

where

• $H_0$ is the kinetic plus potential energy (e.g., $({\bf p}^2/2m^*) + V({\bf r})$ for a quantum dot),
• $H_{\rm SO} = \Delta_{\rm SO} \tau_z s_z$ encodes intrinsic spin–orbit coupling, with $\tau_z = \pm1$ for $K$/$K'$ valleys and $s_z = \pm1$ for spin up/down,
• $H_Z$ specifies Zeeman and valley-Zeeman couplings under magnetic fields: $$H_Z = \mu_B \bigl(g_s {\bf s}\cdot {\bf B} + g_{vl} \tau_z B_z\bigr)$$.

Time-reversal symmetry $\mathcal{T}$ acts as $$\mathcal{T} |\tau, s_z\rangle = |-\tau, -s_z\rangle$$, enforcing double degeneracy. The eigenstates forming a Kramers doublet have the general structure: $$\{ |K, \uparrow\rangle, \quad |K', \downarrow\rangle \},$$ with wavefunctions $$\Psi_{K,\uparrow}({\bf r}) = F({\bf r}) u_K({\bf r})|\uparrow\rangle$$ and $$\Psi_{K',\downarrow}({\bf r}) = F({\bf r}) u_{K'}({\bf r})|\downarrow\rangle$$.

Spin–valley locking can be generalized via crystal symmetry rather than $\mathcal{T}$, as in certain antiferromagnets where the little co-group $G^k$ and coset decomposition $G = \cup_m g_m G^k$ generate “$C$-paired” doublets.

2. Physical Realizations in Quantum Dots, Bulk, and Heterostructures

Transition Metal Dichalcogenides (TMDCs):

In monolayer and few-layer MoS$_2$, as shown by Krishnan et al., well-isolated Kramers doublets emerge in in-gap quantum dots with typical splittings $2\Delta_{\rm SO} \sim 100~\mu$eV. Magnetospectroscopy reveals a pronounced $g$-factor anisotropy: $g_\perp \approx 8.1$ (out-of-plane) and $g_\parallel \approx 0.6-0.8$ (in-plane). These values reflect Berry-curvature-induced orbital magnetism tied to the valley degree of freedom and a dominant intrinsic spin–orbit coupling (Krishnan et al., 2023).

Bulk Dirac Semimetals:

In BaMnSb$_2$, spin-valley locking is realized in both conduction and valence bands. The effective low-energy Hamiltonian,

$$H_{\rm eff}({\bf q}) = v_1 q_x \tau_x + v_2 q_y \tau_y + M \tau_z + \Delta_{\rm so} \tau_z \sigma_z,$$

produces eigenstates where spin is locked to valley. Kramers doublets such as $\{|K_+,\uparrow\rangle, |K_-,\downarrow\rangle\}$ are evidenced by ARPES and quantum Hall measurements, with a large SOC splitting $\sim0.35$ eV exceeding the Dirac gap (Liu et al., 2019).

Spin-Valley Kramers Doublets in Berry-curvature-free Regimes:

In buckled 2D hexagonal lattices, electric-field-induced transverse transport, not mediated by Berry curvature but through geometric backreflection phase shifts, generates spatial separation of Kramers pairs $|K,\uparrow\rangle$ and $|K',\downarrow\rangle$; the corresponding Hall conductances (valley and spin) can be controlled by external fields (Zeng, 2 Oct 2025).

Antiferromagnetic Systems:

A systematic group-theoretic classification identifies 12 elementary types of $C$-paired spin–valley–locked doublets in AFMs, depending on the magnetic point group and the crystal symmetry responsible for pairing. These patterns predict responses such as piezomagnetism and piezo-Hall effects, generalizing the Kramers doublet construction beyond time-reversal-symmetric systems (Hu et al., 2 Jul 2024).

Quantum Point Contacts and Valley Kondo Physics:

In bilayer graphene/WSe$_2$ heterostructures, proximity-induced Ising SOC splits the fourfold ground state into spin–valley–locked Kramers doublets, with the low-energy physics reflected in signatures such as the “0.7 anomaly” in conductance, a result of many-body valley-exchange within the doublet. Notably, these anomalies are insensitive to large in-plane magnetic fields, in contrast to conventional materials (Gerber et al., 9 Nov 2025).

3. Experimental Signatures and Spectroscopic Extraction

Zeeman Anisotropy and Magnetospectroscopy:

Measurement of the excitation spectra under different field orientations yields key physical parameters. In MoS$_2$ quantum dots, the Coulomb peak position shifts linearly with out-of-plane field $B_z$,

$$\Delta E_\perp(B_z) = \tfrac{1}{2} g_\perp \mu_B |B_z| + \Delta_{\rm SO},$$

whereas the in-plane shift follows

$$\Delta E_\parallel(B_\parallel) = \pm \tfrac{1}{2} \sqrt{(g_\parallel \mu_B B_\parallel)^2 + (2\Delta)^2},\;\; \Delta = \sqrt{\Delta_{\rm SO}^2 + \Delta_V^2}.$$

Linear fits extract $g_\perp, g_\parallel$, and allow estimation of the intervalley mixing scale $\Delta_V$.

Transport and Hall Phenomena:

In BaMnSb$_2$, quantum Hall measurements yield plateau heights corresponding to spin–valley degeneracy $s \approx 2$. Chiral surface states are observed as quantized, $T$-independent surface resistance plateaus, signatures of stacked QHE with spin-polarized channels due to spin–valley locking.

Electric Field–Driven Spin–Valley Hall Effects:

In buckled hexagonal lattices, channel-resolved transverse conductances $\sigma_{yx}^V$ (valley) and $\sigma_{yx}^S$ (spin) show odd/even dependence on perpendicular electric field $E_\perp$, revealing and enabling the control of the separation and manipulation of individual spin–valley–locked Kramers partners (Zeng, 2 Oct 2025).

Spectroscopic Table: Extraction Protocols

Material/System	Key Observables	Physical Parameter Extraction
MoS$_2$ QD	Zeeman splitting, $g$-factors	$g_\perp$, $g_\parallel$, $\Delta_{\rm SO}$, $\Delta_V$
BaMnSb$_2$	Hall plateaus, ARPES	Spin–valley degeneracy, SOC splitting
BLG/WSe$_2$ QPC	0.7 anomaly, field response	SOC gap $\Delta_{\rm SO}$, valley $g_v$
Buckled 2D hexagonal tunnel	Electric Hall effects	Transverse Hall conductance, Hall angles

4. Symmetry Principles: Time Reversal, Berry Curvature, and Crystal Constraints

The nature of the Kramers doublet depends on the symmetry underpinning their degeneracy and locking:

• Time-reversal ($\mathcal{T}$)–protected: Predominant in nonmagnetic TMDCs and Dirac semimetals, where $\mathcal{T}^2 = -1$ enforces degeneracy and imposes $$\mathcal{T}|K,\uparrow\rangle = |K',\downarrow\rangle$$.
• Crystal symmetry ($C$)–paired: In antiferromagnets, a crystal operation interchanges two valleys (or generic momenta), locking their spins in opposite directions via the coset decomposition of the magnetic point group. The sufficient conditions (as classified in (Hu et al., 2 Jul 2024)) are that the little co-group $G^k$ allows spin splitting and that a coset representative $g_m$ exists to pair and invert the spin at the symmetry-related valley.

Berry curvature, which encodes the anomalous velocity and orbital magnetic moment, is often concentrated in valleys and may be locked to spin, yielding $g$-factor anisotropy. However, mechanisms exist (such as electric field–induced backreflection phases) where spin–valley doublets and their transverse responses occur independently of Berry curvature (Zeng, 2 Oct 2025).

5. Quantum Information and Valleytronic Applications

Spin–valley–locked Kramers doublets provide a robust platform for quantum bits, as their two-level structure—protected by fundamental symmetries and well isolated from excited states—allows long-lived state preparation and coherent manipulation. The extracted splittings (e.g., $\sim 100~\mu$eV in MoS$_2$ dots (Krishnan et al., 2023), $\sim 0.35$ eV in BaMnSb$_2$ (Liu et al., 2019)) are suitable for both thermal robustness and experimental addressability.

Electrical and magnetic control, either through Berry curvature effects or via engineered local potentials, permit both single- and two-qubit operations, as well as measurement protocols (e.g., charge-sensor single-shot valley readout following spin–to–charge conversion as detailed in (Shandilya et al., 29 Oct 2024)).

In quantum point contacts, spin–valley doublets form the basis for many-body correlation effects, such as the 0.7 anomaly driven by valley-exchange in BLG/WSe$_2$ (Gerber et al., 9 Nov 2025).

Valley and spin Hall effects in buckled hexagonal systems, implemented without Berry curvature, allow pure electric manipulation and separation of spin–valley–locked doublets with full polarization. Hall angles reaching $|\theta_v| = |\theta_s| = 1$ signify completely separated Kramers partners at device edges, maintaining overall time-reversal symmetry (Zeng, 2 Oct 2025).

6. Generalization to Antiferromagnets and Pseudovector Locking

In magnetic materials with zero net magnetization, $C$-paired spin–valley (or, more generally, spin–momentum) locking arises from crystal (rather than time-reversal) symmetry. Little co-group and coset decomposition allow the exhaustive classification of all possible locking patterns, summarized as 12 elementary geometric-crystal combinations (e.g., Hex-L$_2^3$-Orth, Tri-P$_3^3$-Orth). This framework also captures responses where strain or electric field induces net magnetization (piezomagnetism) or anomalous Hall conductance (piezo-Hall), as demonstrated for specific AFMs such as RbV$_2$Te$_2$O (Hu et al., 2 Jul 2024).

The formalism generalizes to any pseudovector degree of freedom (e.g., Berry curvature, orbital moment), extending the physics of Kramers doublets and associated responses to broader classes of quantum materials.

7. Outlook and Future Prospects

Spin–valley–locked Kramers doublets serve as a unified framework to understand and harness quantum states protected by symmetry in diverse solid-state platforms. Their presence enables a range of applications:

• Coherence-protected quantum information carriers (qubits) in TMDCs and engineered quantum dots.
• Valleytronic devices exploiting robust, electrically controllable Hall responses in systems with or without Berry curvature contributions.
• Topological phases and many-body states emergent from the interplay of spin, valley, and crystal symmetry, as in chiral surface states of Dirac semimetals or piezomagnetic AFMs.

Central challenges and directions include experimental measurement of valley relaxation times at the single-particle level, coherent manipulation using all-electrical protocols, and material discovery guided by symmetry-based classifications. The systematic theoretical frameworks now allow identification of material classes and device architectures that optimize for robust spin–valley–locked Kramers doublets and their associated quantum functionalities.