Adiabatic–nonadiabatic crossover of small-polaron hopping in the one-dimensional Holstein model

The one-dimensional Holstein polaron — an electron hopping with amplitude $t$ on a lattice whose sites carry local quantum oscillators of frequency $ω_0$ coupled with strength $g$ — has a numerically well-mapped phase diagram, but its translational dynamics in the transitory regime of small pinned polarons ( $ε_p = g^2/ω_0 ≫ ω_0$ with $ω_0 ≲ t ≲ g$ , inside the intermediate 'white' region $1 ≲ t/ω_0 ≲ 20$ ) still lacks a controlled quantitative theory. There the semiclassical adiabatic hopping estimate $t_pol^A ≈ π^{-1} g e^{-ε_p/ω_0}$ and the nonadiabatic Lang–Firsov estimate $t_LF = t e^{-ε_p/ω_0}$ share the same exponential while differing in prefactor and in the kinetic (nonadiabatic) corrections each neglects; Barišić and Barišić (2008) state explicitly that the applicability of their adiabatic formula across the whole $g ≳ t$ regime 'still remains to be verified', and the semiclassical moving-coordinates formalism behind it is itself restricted to small curvature $κ(s) ≪ 1$ of the adiabatic minimal-energy path. The problem is to determine the small-polaron hopping energy, effective mass, and low-frequency dispersion in this mixed transitory region, where static quantities vary slowly while the dispersion changes dramatically, and to establish the validity domain and corrections of the semiclassical hopping formulae with a treatment that stays controlled where $κ(s)$ is not small.

Background

The Holstein model is the minimal lattice model of local electron–phonon coupling: a single electron hops between nearest-neighbor sites of a one-dimensional lattice with amplitude $t$ , while each site carries a dispersionless quantum oscillator (a 'phonon') of frequency $ω_0$ that the electron displaces with coupling strength $g$ . The electron drags a cloud of lattice deformation with it — the polaron — and the competition between delocalization (kinetic energy $t$ ) and lattice distortion (binding energy $ε_p = g^2/ω_0$ in the atomic limit $t = 0$ ; the dimensionless adiabatic coupling is $λ = ε_p/t$ ) produces a sequence of crossovers rather than true phase transitions in one dimension. Two limits are classic. When the phonons are fast compared with the electron ( $t/ω_0 ≪ 1$ , the antiadiabatic regime), the electron moves by detaching from its deformation cloud and re-forming it on the neighboring site: this is the Lang–Firsov small polaron, whose band is narrowed to the hopping $t_LF = t e^{-ε_p/ω_0}$ , a nonadiabatic process. When the phonons are slow ( $t/ω_0 ≫ 1$ , the adiabatic regime), the lattice can be treated as a static classical deformation: the polaron is a self-consistent lattice well, and on a discrete lattice a small (single-site-scale) polaron can translate only by tunneling through the Peierls–Nabarro barrier, the energy barrier between equivalent lattice-deformation minima shifted by one site. Barišić and Barišić (Eur. Phys. J. B 64, 1 (2008)) organized the full 1D phase diagram into five regimes — weak coupling, large adiabatic polarons, small adiabatic polarons, small nonadiabatic (Lang–Firsov) polarons, and a transitory regime of small pinned polarons — filling the gaps between limiting analytic results with their relevant coherent state method (RCSM), a numerical scheme that computes low-frequency polaron spectra and wave functions across the whole parameter space. For the adiabatic regimes they used a semiclassical 'moving set of coordinates', a reaction-path quantization borrowed from molecular physics in which one coordinate $s$ runs along the minimal-energy path of the adiabatic deformation and normal modes $Q_α$ describe fluctuations perpendicular to it, with curvature $κ(s)$ ; the formalism is valid only for $κ(s) ≪ 1$ and develops singularities otherwise (for small polarons, $κ(s)$ is already singular at the minima of the adiabatic potential for $λ ≳ 1.4$ ). For the small-polaron translational dynamics they compared the nonadiabatic Lang–Firsov hopping with semiclassical (WKB) adiabatic estimates: for $t ≳ g$ the adiabatic hopping $∼ e^{-ε_p/ω_0} e^{t^2/2g^2}$ dominates, whereas for $g ≳ t$ the adiabatic estimate $t_pol^A ≈ π^{-1} g e^{-ε_p/ω_0}$ differs from the Lang–Firsov result only in the prefactor, so the true character of the hop in this window is mixed; their exactly solvable two-site model gives the leading nonadiabatic correction $t_pol^A ∼ g e^{-ε_p/ω_0}(1 − (π/8)(ω_0/t) + O(ω_0/g))$ , which breaks the adiabatic treatment for $t ≲ ω_0$ . The uncontrolled part of the diagram is precisely the intermediate 'white' region $1 ≲ t/ω_0 ≲ 20$ : there, static properties such as the binding energy vary slowly with parameters, but the dispersion changes dramatically — the effective mass changes by two orders of magnitude between $g/ω_0 = 3$ and $g/ω_0 = 3.5$ at $t/ω_0 = 5$ — and no single analytic approximation applies, because adiabatic and nonadiabatic contributions and lattice-coarsening effects are inextricably mixed. Numerically this regime has been mapped by several converged methods, but analytically it remains open: in particular, the validity of the semiclassical adiabatic hopping formulae in the mixed regime $g ≳ t$ was never verified on the lattice, and the semiclassical formalism that produces them is uncontrolled exactly where its path curvature is not small.

Problem Statement

For the one-dimensional Holstein Hamiltonian (electron hopping $t$ , local phonon frequency $ω_0$ , coupling $g$ , atomic-limit binding energy $ε_p = g^2/ω_0$ , $λ = ε_p/t$ ), establish a controlled quantitative description of the translational dynamics of the small pinned polaron in the strong-coupling regime $ε_p ≫ ω_0$ at intermediate adiabaticity $ω_0 ≲ t ≲ g$ — the transitory regime inside the 'white' region $1 ≲ t/ω_0 ≲ 20$ of the phase diagram of Barišić and Barišić, where adiabatic and nonadiabatic contributions to the hopping are of comparable importance. Concretely: (1) determine the polaron hopping energy $t_pol$ , equivalently the bandwidth and effective mass of the lowest polaron band, throughout this regime; and (2) establish the validity domain of the semiclassical adiabatic estimate $t_pol^A ≈ π^{-1} g e^{-ε_p/ω_0}$ (Eq. (48) of the source) across the whole $g ≳ t$ range (the range appearing in the source's explicit open question, which is wider than the transitory window of item (1)), including its leading nonadiabatic kinetic corrections, of which the two-site result $t_pol^A ∼ g e^{-ε_p/ω_0}(1 − (π/8)(ω_0/t) + …)$ is the local prototype — the verification the source explicitly leaves open. Any successful treatment must remain controlled in the part of this regime where the curvature $κ(s)$ of the adiabatic minimal-energy path is not small, since the moving-set-of-coordinates semiclassical formalism develops singularities there. These requests serve one objective: an expert-checkable quantitative theory of the polaron's translational dynamics in the one regime of the 1D Holstein phase diagram where neither the adiabatic nor the nonadiabatic limit applies.

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