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arXiv:2512.18449

Beyond spin-1/2: Multipolar spin-orbit coupling in noncentrosymmetric crystals with time-reversal symmetry

Masoud Bahari, Kristian Mæland, Carsten Timm, Björn Trauzettel

cond-mat.mes-hallcond-mat.mtrl-scicond-mat.str-elcond-mat.supr-conquant-phSubmitted Dec 20, 2025• Announced Dec 23, 2025
v1 · Dec 20, 2025v2 · Feb 4, 2026v3 · Mar 10, 2026
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Abstract

We develop a general multipolar theory of strong spin-orbit coupling for large total angular momentum $j$ in time-reversal-symmetric, noncentrosymmetric crystals. Using a $j\in\{1/2,3/2,5/2\}$ multiplet basis appropriate for heavy-element \textit{p}- and \textit{d}-bands, we systematically construct all symmetry-allowed spin-orbit coupling terms up to fifth order in momentum and generalize the usual spin texture to a total-angular-momentum texture. For $j>1/2$, multipolar spin-orbit coupling qualitatively reshapes Fermi surfaces and makes the topology of Bloch states band dependent. This leads to anisotropic high-$j$ textures that go beyond a single Rashba helix. We classify these textures by their total-angular-momentum vorticity $W_{n}$ for every energy band and identify distinct $|W_{n}|=1,2,5$ phases. We show that their crossovers generate enhanced and nonmonotonic current-induced spin-polarization responses, namely the Edelstein effect, upon tuning the chemical potential. Our results provide a symmetry-based framework for analyzing and predicting multipolar spin-orbit coupling, total-angular-momentum textures, and spintronic responses in heavy-element materials without an inversion center.

Journal reference
Phys. Rev. B 113, 165137 (2026)
DOI
10.1103/2jln-lctf
Comments
(20 pages, 7 figures, 11 tables)

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We develop a general multipolar theory of strong spin-orbit coupling for large total angular momentum jjj in time-reversal-symmetric, noncentrosymmetric crystals. Using a j∈{1/2,3/2,5/2}j\in\{1/2,3/2,5/2\}j∈{1/2,3/2,5/2} multiplet basis appropriate for heavy-element \textit{p}- and \textit{d}-bands, we systematically construct all symmetry-allowed spin-orbit coupling terms up to fifth order in momentum and generalize the usual spin texture to a total-angular-momentum texture. For j>1/2j>1/2j>1/2, multipolar spin-orbit coupling qualitatively reshapes Fermi surfaces and makes the topology of Bloch states band dependent. This leads to anisotropic high-jjj textures that go beyond a single Rashba helix. We classify these textures by their total-angular-momentum vorticity WnW_{n}Wn​ for every energy band and identify distinct ∣Wn∣=1,2,5|W_{n}|=1,2,5∣Wn​∣=1,2,5 phases. We show that their crossovers generate enhanced and nonmonotonic current-induced spin-polarization responses, namely the Edelstein effect, upon tuning the chemical potential. Our results provide a symmetry-based framework for analyzing and predicting multipolar spin-orbit coupling, total-angular-momentum textures, and spintronic responses in heavy-element materials without an inversion center.
Spin-orbit coupling exists in materials in general. However, it entangles the spin and orbital degrees of freedom and complicates the model. Thus, theorists usually neglect the effects induced by spin-orbit coupling first and consider spin-orbit coupling as perturbation next. The non-perturbative effects brought up by spin-orbit coupling are thus often less studied or overlooked.\nOn the other hand, the majority in the study of interacting topological order focusing on the general structure of theories and made significant advances by leaving material details behind. It is thus important to find possible microscopic models that could realize the new phases in laboratories and benefits from the progress of theories to make experimental predictions.\nIn this thesis, we study the physical effects due to strong spin-orbit coupling from the perspective of searching new quantum orders and the non-trivial responses.\n(i) The first project, we propose the nontrivial dipolar-octupolar(DO) doublets on the pyrochlore lattice. By studying the most general symmetry allowed model at the localized and the itinerant limit for DO doublets, we found two 3D symmetry enriched topological orders and topological insulator correspondingly. (ii) In the second project, we analyze the 2D model descending from the localized limit of DO doublets on pyrochlore. The discrete onsite symmetry and space group symmetry could lead to a symmetry-enriched topological order with symmetry fractionalization pattern that cannot emerge from a spin model with continuous spin rotational symmetry. The non-trivial symmetry fractionalization pattern contributes to the striking numerical signal that can help identifying the topological order. (iii) In the third project, we develop a theory to understand the high-energy Raman signal in Sr2IrO4.
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Magnetic order with vanishing net magnetization can produce non-relativistic spin-split bands, broadly categorized into even-parity altermagnets and odd-parity ppp-wave magnets. Here, we introduce a spin-bond theory that unifies these seemingly distinct phenomena into a single algebraic framework. We demonstrate that non-relativistic spin textures are fundamentally governed by two components of the electronic bond: unitary spin phases and Hermitian spin amplitudes. The unitary sector generates odd-parity p-wave and emergent spin-orbit-like textures, while the Hermitian sector generates even-parity spin fields, including the uniform ΓΓΓ-split and bond-structured altermagnetic limits. Beyond unifying known phases, our theory uncovers a mixed non-commuting regime that emerges when the unitary and Hermitian sectors fail to commute, revealing an underlying non-commuting spin-bond structure. This regime generates a non-coplanar spin texture characterized by an even-in-momentum transverse spin polarization, providing a direct spectroscopic fingerprint for spin- and angle-resolved photoemission spectroscopy. Furthermore, we establish that this synthetic spin-orbit coupling can be dynamically tuned by geometrically controlling the non-commutation of the bond sectors. By providing a microscopic foundation for such tuning, our theory paves the way for advanced applications, including field-free spin qubits.
This dissertation contains a collection of topics on the spin-orbit coupling of cold atoms and semiconductors. We first consider the effects of an optically induced spin-orbit interaction on a system of Bosonic atoms. The spin-orbit term couples the emergent pseudo-spin- 1/2 degree of freedom to momentum. The single particle energy spectrum has a low energy band with two minima at non-zero momentum. At low temperatures, a many-body system will condense into these minima. In the presence of vanishing interactions, the ground state wavefunction is found to be N00N state of "left" and "right"-moving state. An experimentally observable signature of the condensate is predicted that can be observed using time-of-flight imaging. We then predict a bulk manifestation of the spin-Hall effect in an inhomoge- neous spin-orbit-coupled system. The phenomena is predicted in the framework of the spin diffusion equations generalized to include arbitrary Rashba and linear and cubic Dresselhaus terms. This framework shows that a bulk spin-density wave with a wavevector oriented perpendicular to an applied electric field will induce a charge- density wave characterized by a π/2-phase shift and a non-monotonic time-varying amplitude. The optimal values of spin-orbit coupling for observation of the effect are determined. We propose a scheme for using atom interferometry to measure weak time- dependent accelerations. This proposal uses an ensemble of dilute trapped bosons with two pseudo-spin states coupled to a synthetic magnetic field, but with opposite effective charges. The synthetic field acts to couple spin to momentum continuously, which continuously imparts the acceleration on the phase of the internal states. We use time reversal pulses to reduce noise. The sensitivity of such a system is estimated to be S ∼ 10−7 m/s^2/√Hz Finally, we consider the semiclassical dynamics of a trapped spin-orbit coupled system. We find non-linear dynamics parameterized by the value of the initial displacement of the trap, and the anisotropy of the spin-orbit energy spectrum. We show that the dynamics can give a Berry's phase, and propose an experiment to measure this phase. We then propose a generalization of the 4-level scheme that allows for spin-orbit coupling described by a vector potential that is proportional to the angular momentum operator.
Spin-orbit (SO) coupling is an important ingredient in many recently discovered phenomena such as the spin-Hall effect and topological insulators. Of particular interest is topological superconductivity, with its potential application in topological quantum computation. The absence of disorder in ultracold atomic systems makes them ideal for quantum computation applications; however, the SO coupling schemes proposed thus far are experimentally impractical owing to large spontaneous emission rates in the alkali fermions. In this paper, we develop a scheme to generate Rashba SO coupling with a low spontaneous emission extension to a recent experiment. We show that this scheme generates a Fermi surface spin texture for 40K^{40}\mathrm{K}40K atoms, which is observable in time-of-flight measurements. The chiral spin texture, together with conventional sss-wave interactions, leads to topological superconductivity and non-Abelian Majorana quasiparticles.
Intrinsic spin angular-momentum coupling of an electron has a relativistic quantum origin with the coupling arising from charged orbits, which does not carry over to charge-neutral atoms. Here, we propose a mechanism of spontaneous generation of spin angular-momentum coupling with spinor atomic bosons loaded into p-orbital bands of a two-dimensional optical lattice. This spin angular-momentum coupling originates from many-body correlations and spontaneous symmetry breaking in a superfluid, with the key ingredients attributed to spin-channel quantum fluctuations and an approximate rotation symmetry. The resultant spin angular-momentum intertwined superfluid has Dirac excitations. In the presence of a chemical potential difference for adjacent sites, it provides a bosonic analogue of a symmetry-protected-topological insulator. Through a dynamical mean-field calculation, this novel superfluid is found to be a generic low-temperature phase, and it gives way to Mott localization only at strong interactions and even-integer fillings. We show the temperature to reach this order is accessible with present experiments.