Microscopic origin of the coupling and phonon-frequency scalings in the next-nearest-neighbour Holstein model, and the fate of the single-polaron mass divergence

When next-nearest-neighbour (NNN) hopping $t_2=\beta t$ is added to the Holstein small-polaron model, exact numerics show the polaron effective-mass curve $m^*(g)$ nearly collapses onto the standard nearest-neighbour result if the phonon frequency is rescaled by the low-energy effective-bandwidth factor ( $1+4\beta$ in 1D and 3D, $1+2\beta$ in 2D) and the dimensionless electron-phonon coupling by a heuristic factor $B=1+4\beta$ in 1D and $B=1+\beta$ in 3D. Both mappings were found empirically by Chandler, Prosko and Marsiglio (Sci. Rep. 6, 32591, 2016), who state that the physical motivation for $B$ 'is still rather unclear' and that the frequency prescription rests on their physical judgement rather than proof. The problem is to replace these fitted scalings with first-principles understanding: determine whether $B$ and its dimension-dependence follow from the microscopic Hamiltonian, with a stated validity domain, establish whether the bandwidth-based frequency rescaling is the physically correct comparison between models with different hopping ranges (and where it fails), and assess whether such lattice-structure effects can help reconcile the strong electron-phonon coupling inferred for real superconductors with the absence of divergent single-polaron masses.

Background

The Holstein model (Holstein, 1959) is the minimal lattice model for an electron interacting with crystal vibrations: an electron hops between nearest-neighbour sites with amplitude $t$ , every site carries a local, dispersionless optical phonon ('Einstein oscillator') of frequency $\omega_E$ , and the electron couples linearly to oscillator on its own site with dimensionless strength $g$ . The electron dressed by its phonon cloud is a polaron; its key property is the effective mass $m^*$ , defined from the curvature of the lowest polaron band at the band minimum. At strong coupling the local lattice distortion self-traps the electron and $m^*$ grows roughly exponentially with coupling — the small-polaron crossover — and the ratio $\omega_E/t$ (the adiabatic ratio, small for real materials whose phonon energies are far below electronic bandwidths) controls how sharp this crossover is.

Because even rather modest coupling produces very large single-polaron masses, while several conventional superconductors show strong electron-phonon coupling with no polaronic signature in their normal state, the question of what lattice effects modify the single-polaron mass is of long-standing interest. Chakraborty, Das and Chakrabarti (2011) showed that adding next-nearest-neighbour (NNN) hopping $t_2$ to the one-dimensional Holstein model can substantially decrease $m^*$ at strong coupling. Chandler, Prosko and Marsiglio (2016) extended this to one, two, and three dimensions (square and simple-cubic lattices, with NNN hops to the diagonal neighbours), using weak-coupling perturbation theory plus a modified Trugman method — a numerically exact variational scheme that builds polaron eigenstates from a systematic local basis and works directly in the thermodynamic limit. Writing $\beta=t_2/t$ , they established two results that define the present problem.

First, NNN hopping leaves the total band width unchanged but alters the curvature of the electronic dispersion at the band minimum; measuring the bandwidth in a small window around $k=0$ (the 'low-energy effective bandwidth') gives $W_{\mathrm{NNN}}/W_{\mathrm{NN}}\approx 1+4\beta$ in 1D and 3D and $\approx 1+2\beta$ in 2D. Arguing that the phonon-to-electronic-energy ratio should be held fixed, they rescale $\omega_E$ by these factors — and the sign of the NNN effect on $m^*$ reverses (with the rescaling, positive $t_2$ increases $m^*$ in 1D and 3D, opposite to the unrescaled conclusion of the 2011 work). The prescription is suggested by approximate perturbation theory, but the authors support it only by stating they 'feel' it 'is the more physically correct procedure'; its correctness outside weak coupling is unproven.

Second, rescaling the dimensionless coupling $g\to gB$ with $B=1+4\beta$ in 1D and $B=1+\beta$ in 3D maps the NNN $m^*(g)$ curves onto the nearest-neighbour ones almost perfectly, including through the sharp crossover regime in 3D — but $B$ was found by trial, and 'the physical reason for this effect is not known'. Notably, in 1D $B$ coincides with the band-curvature factor $1+4\beta$ that drives the frequency rescaling, while in 3D it does not, so no single obvious mechanism explains both.

The stake is stated by the authors: even with these scalings, for small $t_2/t$ the crossover to very large $m^*$ remains in the moderate-coupling regime, so 'it remains difficult to reconcile the fairly strong coupling attributed to some real metals/superconductors with the diverging effective mass predicted for a single polaron at the same coupling strength'. They also note that weak-coupling perturbation theory agrees with the exact numerics over a 'surprisingly limited' range of coupling, even in 3D, and their analytic approximations (their Table II) fail quantitatively in 2D and 3D — leaving no controlled analytic description bridging weak and strong coupling for this model family.

Problem Statement

For the Holstein model with an additional next-nearest-neighbour (NNN) hopping $t_2=\beta t$ (in 2D and 3D the NNN hops connect the diagonal neighbours of the square and simple-cubic lattices, respectively),

$\hat{H}=\sum_{\langle i,j\rangle}(-t)\,\hat{c}^{\dagger}_{i}\hat{c}_{j}+\sum_{\langle\langle i,j\rangle\rangle}(-t_{2})\,\hat{c}^{\dagger}_{i}\hat{c}_{j}+\hbar\omega_{E}\sum_{i}\hat{a}^{\dagger}_{i}\hat{a}_{i}+\hbar g\omega_{E}\sum_{i}(\hat{a}^{\dagger}_{i}+\hat{a}_{i})\,\hat{c}^{\dagger}_{i}\hat{c}_{i},$

where $\langle i,j\rangle$ runs over nearest-neighbour pairs, $\langle\langle i,j\rangle\rangle$ over diagonal next-nearest-neighbour pairs, $\omega_E$ is the frequency of the local (Einstein) phonon, and $g$ the dimensionless electron-phonon coupling. The corresponding free-electron dispersions are $\epsilon(k)=-2t\cos(ka)-2t_{2}\cos(2ka)$ in 1D, $\epsilon(\mathbf{k})=-2t(\cos k_{x}a+\cos k_{y}a)-4t_{2}\cos(k_{x}a)\cos(k_{y}a)$ in 2D, and $\epsilon(\mathbf{k})=-2t(\cos k_{x}a+\cos k_{y}a+\cos k_{z}a)-4t_{2}[\cos(k_{x}a)\cos(k_{y}a)+\cos(k_{x}a)\cos(k_{z}a)+\cos(k_{y}a)\cos(k_{z}a)]$ in 3D (Table I of Chandler, Prosko and Marsiglio, 2016); for $\beta>-1/4$ the band minimum, and hence the polaron ground state, is at zero crystal momentum, which is the regime considered. For this model, exact numerics show that the polaron effective-mass curve $m^*(g)$ is mapped nearly perfectly back onto the standard nearest-neighbour Holstein result when (a) the phonon frequency is rescaled by the low-energy effective-bandwidth factor $1+4\beta$ in 1D and 3D and $1+2\beta$ in 2D, and (b) the dimensionless coupling is rescaled $g\to gB$ with $B=1+4\beta$ in 1D and $B=1+\beta$ in 3D — both mappings discovered empirically. The problem, as explicitly left open by that work, is:

(1) Determine whether the heuristic coupling scaling $B$ has a microscopic origin derivable from the Hamiltonian and, if so, derive it together with its dimension-dependence ( $B=1+4\beta$ in 1D but $B=1+\beta$ in 3D, and no analogue reported in 2D); delimit the regimes of coupling strength $g$ , adiabatic ratio $\omega_E/t$ , and $\beta$ in which it is asymptotically valid and those in which it fails.

(2) Establish whether rescaling the phonon frequency by the low-energy effective-bandwidth change is the physically correct way to compare models with different hopping ranges — currently justified only by approximate weak-coupling perturbation theory and the authors' stated physical judgement — determine its limits of validity, or supply the correct comparison principle if this one is not.

(3) As the overarching physical question that (1) and (2) serve: determine whether lattice-structure effects of this kind remove or defer the single-polaron effective-mass divergence at moderate coupling, thereby reconciling the strong electron-phonon couplings inferred for real metals and conventional superconductors with the absence of polaronic masses in their normal states.

The problem requires all three sub-questions: (1) and (2) call for a first-principles account with a stated validity domain, and the overarching question (3) calls for a principled account of how hopping range and related lattice effects shift or soften the small-polaron crossover.

References

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