The single polaron of the Rashba–Holstein model — one electron on a two-dimensional square lattice with Rashba spin–orbit coupling and local coupling to dispersionless quantum phonons — has been exactly characterized by variational numerics only down to phonon frequency ωE/t=0.1\omega_E/t=0.1, where the ground state shows smooth crossovers between a delocalized polaron and a small polaron. Semiclassical theory in the strict adiabatic limit (ωE=0\omega_E=0) instead predicts, for any nonzero spin–orbit coupling, two transitions and a wide intermediate large-polaron regime. Whether that intermediate phase survives quantum phonon fluctuations at ωE/t≪0.1\omega_E/t\ll 0.1 is unsettled. The problem asks for a controlled determination of the ground-state phase structure of this model across the (spin–orbit coupling, electron–phonon coupling, phonon frequency) parameter space, with the deeply adiabatic regime as the decisive region. The standard analytic tools for this model — weak-coupling perturbation theory (whose predicted mass enhancement diverges as the spin–orbit coupling vanishes), the Lang–Firsov strong-coupling expansion, and the Momentum Average approximation — have documented failure modes in parts of this space, as detailed in the Background.

Background

A polaron forms when an electron in a crystal distorts the surrounding lattice and drags that distortion along as it moves, lowering its energy and inflating its effective mass. The Holstein model is the minimal lattice description: a single electron band whose every site carries a dispersionless harmonic oscillator (an Einstein phonon of frequency ωE\omega_E) coupled locally to the electron density. The dimensionless adiabatic ratio ωE/t\omega_E/t, with tt the nearest-neighbour hopping, sets how quantum the lattice is: real materials with light electrons and heavy ions typically have ωE/t≪1\omega_E/t\ll 1 (the adiabatic regime), which is precisely where both exact numerics and analytic approximations become hardest, because many phonon quanta participate and the required Hilbert space grows explosively.

Rashba spin–orbit coupling (RSOC) arises in two-dimensional systems lacking structural inversion symmetry and locks the electron spin to its momentum. On a square lattice with nearest-neighbour hopping the single-polaron Hamiltonian is

H=−t∑⟨i,j⟩,α(ciα†cjα+h.c.)+iVS∑i(ci†σxci+y^−ci†σyci+x^+h.c.)−gωE∑i,αciα†ciα(ai+ai†)+ωE∑iai†aiH=-t\sum_{\langle i,j\rangle,\alpha}(c^\dagger_{i\alpha}c_{j\alpha}+\mathrm{h.c.})+iV_S\sum_i(c^\dagger_i\sigma_x c_{i+\hat y}-c^\dagger_i\sigma_y c_{i+\hat x}+\mathrm{h.c.})-g\omega_E\sum_{i,\alpha}c^\dagger_{i\alpha}c_{i\alpha}(a_i+a_i^\dagger)+\omega_E\sum_i a_i^\dagger a_i

with spin index α=↑,↓\alpha=\uparrow,\downarrow, Pauli matrices σx,y\sigma_{x,y}, Rashba strength VSV_S, and dimensionless electron–phonon coupling λ≡2g2ωE/(4πt)\lambda\equiv 2g^2\omega_E/(4\pi t). RSOC splits the bare tight-binding band into two Rashba bands; for VS>0V_S>0 the lower band has four degenerate minima away from the Brillouin-zone centre, the bare effective mass decreases as 1/1+VS2/(2t2)1/\sqrt{1+V_S^2/(2t^2)}, and the density of states at the band bottom is strongly enhanced, D(E0)=2/(πVS)D(E_0)=\sqrt{2}/(\pi V_S), because one curvature direction of each of the four minima softens as VS→0V_S\to 0 (its curvature scales as VS2/t2V_S^2/t^2, so the band bottom becomes quasi-one-dimensional at small Rashba coupling).

Three earlier lines of work set the stage. Weak-coupling perturbation theory for continuum (parabolic-band) Rashba systems found that RSOC enhances the phonon-induced mass renormalization. The Momentum Average (MA) approximation — a systematic, improvable resummation of the electron self-energy in powers of the hopping, whose hierarchy and accuracy limits were analyzed by Berciu and Goodvin — applied to the tight-binding Rashba–Holstein polaron found instead that RSOC effectively weakens the electron–phonon coupling, so the polaron mass decreases with VSV_S at intermediate and strong coupling. Finally, the strict adiabatic limit ωE=0\omega_E=0 was treated semiclassically (a static, self-consistently determined classical lattice distortion) by Grimaldi, who found that for any nonzero VSV_S the ground state undergoes two transitions as λ\lambda grows — from a delocalized electron to a large polaron (a state whose spatial extent spans many lattice sites, with localization length inversely proportional to the Rashba coupling), then to a small polaron (self-trapped on one site) — so that an intermediate large-polaron regime occupies a wide region of the (VS,λ)(V_S,\lambda) plane.

Li, Covaci, Berciu, Baillie and Marsiglio (Phys. Rev. B 83, 195104 (2011)) settled the exact finite-frequency picture above ωE/t=0.1\omega_E/t=0.1 using Trugman's method — an exact variational expansion in phonon-number sectors combined with Lanczos diagonalization and extrapolated to the thermodynamic limit. They confirmed the MA conclusions (mass increases with VSV_S at weak coupling, decreases at intermediate and strong coupling), found only smooth crossovers and no intermediate phase down to ωE/t=0.1\omega_E/t=0.1, and could not exclude a large-polaron regime below that frequency, though they judged it unlikely. They also documented the failure modes of the analytic tools: MA is quantitatively inaccurate at low ωE\omega_E; the Lang–Firsov strong-coupling expansion (a unitary transformation that decouples electron and phonon operators exactly in the t=0t=0 limit) degrades with increasing VSV_S because the effective bandwidth grows; and weak-coupling perturbation theory predicts a mass enhancement m∗/mSO=1+2 λt/VSm^*/m_{\rm SO}=1+\sqrt{2}\,\lambda t/V_S that diverges as VS→0V_S\to 0, with an uncharted domain of validity. The question is physically relevant because strong-spin–orbit-coupled materials of current interest typically have phonon frequencies small compared to the electronic bandwidth, i.e. ωE/t≪1\omega_E/t\ll 1.

Problem Statement

For the single polaron described by the Rashba–Holstein Hamiltonian on the two-dimensional square lattice,

H=−t∑⟨i,j⟩,α(ciα†cjα+h.c.)+iVS∑i(ci†σxci+y^−ci†σyci+x^+h.c.)−gωE∑i,αciα†ciα(ai+ai†)+ωE∑iai†ai,H=-t\sum_{\langle i,j\rangle,\alpha}(c^\dagger_{i\alpha}c_{j\alpha}+\mathrm{h.c.})+iV_S\sum_i(c^\dagger_i\sigma_x c_{i+\hat y}-c^\dagger_i\sigma_y c_{i+\hat x}+\mathrm{h.c.})-g\omega_E\sum_{i,\alpha}c^\dagger_{i\alpha}c_{i\alpha}(a_i+a_i^\dagger)+\omega_E\sum_i a_i^\dagger a_i,

with λ≡2g2ωE/(4πt)\lambda\equiv 2g^2\omega_E/(4\pi t), determine the ground-state structure over the parameter space (VS/t,λ,ωE/t)(V_S/t,\lambda,\omega_E/t). The structure at ωE/t≥0.1\omega_E/t\ge 0.1 is already resolved by the variational numerics cited below; the decisive unresolved region — where the deliverable specified below is required — is the deeply adiabatic regime ωE/t≪0.1\omega_E/t\ll 0.1. The common objective is a controlled, quantified characterization of this ground state.

Establish whether, for nonzero spin–orbit coupling and intermediate electron–phonon coupling, an intermediate large-polaron regime — predicted by the semiclassical strict-adiabatic (ωE=0\omega_E=0) treatment to be the ground state over a wide region of the (VS,λ)(V_S,\lambda) plane — survives quantum phonon fluctuations. The answer is one of three mutually exclusive alternatives: (i) it appears as a genuine phase (nonanalytic ground-state behavior); (ii) it appears as a sharp crossover that persists and narrows as ωE/t→0\omega_E/t\to 0; or (iii) the exact ground state instead exhibits only the single smooth delocalized-to-small-polaron crossover already resolved at ωE/t=0.1\omega_E/t=0.1. The deliverable is the ground-state energy and effective-mass map over (VS,λ)(V_S,\lambda) at the adiabatic phonon frequencies ωE/t≤0.1\omega_E/t\le 0.1, obtained by exact or systematically controlled computation with stated convergence and error estimates. The completion standard is joint: the map alone does not resolve the question, and a verdict among (i)–(iii) must rest on an explicit operational criterion applied over the whole map — the scaling of the crossover width (or of an equivalent order parameter) with ωE\omega_E, which is what separates the persisting-and-narrowing crossover (ii) from the single smooth crossover (iii), and, whenever a genuine phase (i) is claimed, a nonanalyticity test of the ground-state energy in λ\lambda. The map at adiabatic frequencies, the applied criterion, and the resulting verdict are all required.

References

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