Existence of spontaneous magnetic order for lattice spins coupled to itinerant interacting electrons with generic Rashba/Dresselhaus spin-orbit coupling in one and two dimensions

The Mermin-Wagner theorem, extended by Loss, Pedrocchi, and Leggett (Phys. Rev. Lett. 107, 107201 (2011)) to lattice spins coupled by an isotropic on-site interaction to itinerant, interacting electrons, forbids ferromagnetic, antiferromagnetic, and helical long-range order at any finite temperature in one and two dimensions, covering the RKKY regime and closing the two-dimensional gap left by Bruno's extension to long-ranged interactions. The proof, however, depends on spin conservation: Rashba/Dresselhaus spin-orbit interactions break spin-rotation symmetry and generate equilibrium spin currents that cut off the infrared fluctuations entering the Bogoliubov inequality, so for generic couplings with alpha unequal to beta the argument is inconclusive - order is neither demonstrated nor excluded - while at the fine-tuned point alpha = beta order is again rigorously excluded. The open problem is to determine, for this model class supplemented with generic Rashba/Dresselhaus spin-orbit coupling, whether spontaneous magnetic order of the lattice spins exists at finite temperature in one and two dimensions in the thermodynamic limit. Model-specific mean-field studies - helical nuclear order in quantum wires (with indirect experimental support) and conditional analyses of two-dimensional Rashba systems - have not settled the general question.

Background

In a seminal line of rigorous work, Hohenberg and Mermin and Wagner showed that a continuous symmetry cannot be spontaneously broken at any finite temperature in one- and two-dimensional (1D, 2D) systems with sufficiently short-ranged interactions; in particular, an isotropic Heisenberg magnet has no ferro- or antiferromagnetic long-range order for dimension $d\le 2$ . The proof uses the Bogoliubov inequality, a rigorous operator inequality bounding thermal averages through commutators with the Hamiltonian, applied to the continuity equation of a conserved spin density. Real magnetic moments in conductors, however, rarely interact directly: localized spins typically interact through the spin polarization they induce in itinerant electrons - the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction - which in two dimensions decays only as $1/R^2$ , outside the short-range hypothesis of the original theorem. Bruno strengthened the theorem to long-ranged interactions and excluded 1D RKKY order, but his decay condition (oscillatory interactions falling off faster than $1/R^{5/2}$ in 2D) left the two-dimensional RKKY case open. The physical setting of interest is nuclear spins in semiconductor nanostructures (GaAs or InAs wires and wells, carbon nanotubes), which couple to conduction electrons through the Fermi-contact hyperfine interaction; mean-field-level calculations had predicted that electron-electron interactions drive a ferromagnetic nuclear phase transition in an interacting two-dimensional electron gas (2DEG) at millikelvin temperatures, and helical nuclear order in 1D Luttinger liquids (interacting 1D conductors described by bosonized collective modes rather than by quasiparticles).

Loss, Pedrocchi, and Leggett closed the 2D gap: for a lattice of spins $\hat{\mathbf{I}}_j$ at sites $R_j$ coupled by an isotropic on-site spin-spin interaction to itinerant electrons - with the electronic Hamiltonian allowed to contain arbitrary spin-independent one-particle terms (lattice, disorder, confinement) and electron-electron interactions - they rigorously proved, via the Bogoliubov inequality, that neither ferromagnetic, antiferromagnetic, nor helical long-range order of the lattice spins is possible at any finite temperature in one and two dimensions, in the thermodynamic limit. This covers the RKKY regime (obtained as the lowest-order expansion in the on-site coupling) and thereby invalidates the isotropic 2D mean-field transition claim in the thermodynamic limit.

The theorem relies on spin conservation. Rashba and Dresselhaus spin-orbit interactions (SOI) - linear-in-momentum spin-momentum couplings generated by structural and bulk inversion asymmetry of the semiconductor, with coefficients $\alpha$ and $\beta$ - break spin-rotation symmetry explicitly. Then the commutator entering the Bogoliubov bound no longer vanishes; schematically $[[C_q, H_{SO}], C_q^\dagger]=4m(\alpha\,\hat{j}^y_x(q{=}0)+\beta\,\hat{j}^y_y(q{=}0))$ in terms of equilibrium spin-current operators, so a nonzero equilibrium spin current provides an intrinsic infrared cutoff of the $q$ -integral and the bound on the order parameter does not vanish. For generic $\alpha\ne\beta$ the argument is therefore inconclusive: magnetic order is neither demonstrated nor excluded. The spin currents were evaluated only perturbatively (free electrons at $T=0$ , then a perturbative expansion in the interaction, hyperfine coupling, and temperature), with the explicit caveat that they presumably remain nonzero beyond the perturbative regime. At the fine-tuned point $\alpha=\beta$ a gauge transformation removes the SOI for one spin component (the persistent spin helix symmetry), and order is rigorously excluded again; since $\alpha$ is electrically gate-tunable, this suggests electrical control of low-dimensional magnetism. The no-go holds only in the thermodynamic limit, so effective order in finite nanostructures at low temperature is not excluded by the theorem.

Problem Statement

For the class of models covered by the theorem of Loss, Pedrocchi, and Leggett - a lattice of localized spins $\hat{\mathbf{I}}_j$ at sites $R_j$ embedded in a continuum of itinerant electrons, coupled by an isotropic on-site spin-spin (hyperfine or exchange) interaction, with the electronic Hamiltonian allowed to contain arbitrary spin-independent one-particle terms (lattice, disorder, confinement) and electron-electron interactions, and optionally short-range Heisenberg interactions among the lattice spins - augmented with Rashba and Dresselhaus spin-orbit interactions of strengths $\alpha$ and $\beta$ at generic $\alpha\ne\beta$ (including the Rashba-only and Dresselhaus-only cases): determine whether spontaneous ferromagnetic, antiferromagnetic, or helical (spiral) long-range order of the lattice spins can occur at any finite temperature in the thermodynamic limit, in spatial dimension $d=1$ and $d=2$ .

The question is resolved for the generic $\alpha\ne\beta$ class in one of two ways: (i) the order is absent, under explicitly stated hypotheses that do not silently reduce to the known $\alpha=\beta$ or spin-independent cases; or (ii) the order is present, with its mechanism resting on the nonperturbative behavior of the equilibrium spin-current expectation values (beyond the perturbative regime, at finite temperature, and in the presence of interactions and disorder) and their effect on the small- $q$ spectrum of fluctuations, i.e., whether the spin-current-induced infrared cutoff suffices to stabilize order at finite temperature.

The question is posed for the generic $\alpha\ne\beta$ class, in both $d=1$ and $d=2$ , and not for a proper subset of that class.

References

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