Screened Peierls instability in a nearly-free-electron quasi-one-dimensional metal: nonperturbative equilibrium, wave-vector selection, and finite-temperature theory

In a quasi-one-dimensional nearly-free-electron metal with long-range Coulomb interaction, linear screening would suppress any weak lattice distortion at the Peierls wave number $2k_F$, because the one-dimensional static dielectric function is logarithmically singular there. Yeats and Kroemer (Phys. Rev. B 14, 3335, 1976) showed, within a zero-temperature harmonic perturbation theory that treats screening self-consistently and nonlinearly, that a Peierls instability nevertheless exists but requires a dimensionless electron-phonon parameter $\gamma(q)$ to exceed unity, need not occur at $q=2k_F$ (for $\gamma$ barely above 1 the energy minimum can move to $q<2k_F$, opening a gap below the Fermi level), and need not involve a phonon softening. Their own analysis flagged the nonperturbative regime in which the predicted gap amplitude violates $V\ll\xi$ so that anharmonicity presumably limits the realized gap, the possible shift of the energy minimum to $q<2k_F$, and the conjectured non-BCS temperature dependence of the gap; the $T=0$, harmonic, single-harmonic, Hartree-plus-local-field scope of their treatment in addition leaves open the quantitative boundary between $q=2k_F$ and $q<2k_F$ minima, the role of exchange and correlation beyond transplanted three-dimensional local-field factors, the classification of instabilities at $q>2k_F$ and of multi-harmonic distortions, and the rest of the finite-temperature behavior, including the transition temperature and the collective modes of the screened distorted state. No later work has completed this program within the source model; the problem remains open.

Background

A one-dimensional (1D) metal with one partially filled band is, in the classic Peierls argument, unstable at low temperature toward a static lattice distortion with wave number $q=2k_F$ (twice the Fermi wave number $k_F$): the distortion mixes electron states at $k_F$ and $-k_F$, opens an energy gap at the Fermi level, and lowers the electronic energy by an amount that grows logarithmically faster than the elastic cost, producing a commensurate or incommensurate charge-density wave (CDW), a static modulation of the electron density coupled to the lattice distortion. This textbook instability, however, is usually derived for noninteracting electrons.

In a real quasi-one-dimensional (quasi-1D) metal — a chain-like conductor with essentially 1D electronic motion but a three-dimensional (3D) Coulomb field, characterized by a chain cross section $A$ — the conduction electrons screen the lattice distortion. The obstacle is sharp: the static Lindhard dielectric function (the random-phase-approximation dielectric function of a free-electron gas) of a strictly 1D electron gas diverges logarithmically at $q=2k_F$, so within linear response any externally imposed weak perturbation at $2k_F$ would be screened away entirely. Whether a Peierls instability survives long-range Coulomb interaction at all, at which wave number, and with what gap therefore cannot be answered by linear-response (Kohn-anomaly/phonon-softening) arguments alone; the response must be treated self-consistently and nonlinearly in the distortion amplitude.

Yeats and Kroemer (1976) carried out this program at $T=0$ for a nearly-free-electron model (electrons with parabolic dispersion and effective mass $m^{*}$, weakly perturbed by the lattice) in the Hartree (random-phase) approximation for the electron-electron interaction. Their central quantities, which pin the model, are: the amplitude $V$ of the screened single-harmonic distortion potential (the full Peierls gap is $2V$); the electronic scale $\xi=\hbar^{2}q^{2}/(2m^{*})$; a dimensionless Coulomb strength $\alpha(q)=2\phi_{q}m^{*}/(\pi A\hbar^{2}q)$, where $\phi_{q}$ is the Fourier transform of the electron-electron interaction; a dimensionless electron-phonon coupling $\gamma(q)=2|g_{q}|^{2}/(\phi_{q}\hbar\omega_{q}N/\Omega)$, where $g_{q}$ is the electron-phonon matrix element, $N$ the number of distorting ions, and $\Omega$ the crystal volume (for point ions of charge $Ze$ and mass $M$ with bare phonon frequency $\omega_{q}$ this reduces to $\gamma(q)=\omega_{\mathrm{pl}}^{2}/\omega_{q}^{2}$, the squared ratio of the bare ionic plasma frequency to the bare phonon frequency); the nonlinear static dielectric function

$$\epsilon_{q}=1+\alpha(q)\,\ln\left|\frac{2\xi\delta}{V^{2}}+1\right|,$$

where $\delta(k,q)$ is the degenerate second-order perturbation energy shift of the distorted-band states; and the screened phonon frequency

$$\omega_{s}^{2}(q)=\omega_{q}^{2}\left[1-\gamma(q)\right]+\omega_{q}^{2}\gamma(q)/\epsilon_{q}.$$

Their $T=0$ conclusions: (i) an instability at $q\le 2k_F$ requires $\gamma(q)>1$ — a finite-coupling threshold, reflecting that the screened Peierls instability is a finite-amplitude instability, not an infinitesimal one; (ii) the renormalized phonon frequency need not soften to zero for the instability to occur; (iii) when $\gamma(2k_F)$ exceeds 1 only slightly, graphical energy minimization shows the minimum can occur at $q<2k_F$, opening a gap below the Fermi level rather than at it (so the screened Peierls state can remain metallic); (iv) harmonic energy minimization often predicts $V\gtrsim\xi$, outside the domain $V\ll\xi$ of the perturbative treatment, so the actual gap is probably limited by anharmonic lattice couplings; and (v) the screened-instability energy contains a leading $\ln^{2}$ term (from Hartree overcounting combined with nonlinear screening) that is absent in unscreened treatments, so the gap is not expected to have the BCS-type (Bardeen-Cooper-Schrieffer mean-field) temperature dependence familiar from the unscreened Peierls instability.

What the source did not do defines the open problem. The anharmonicity statement is an inference, not a calculation from an anharmonic lattice model. The $q<2k_F$ conclusion rests on graphical evaluation for two special phonon spectra (Einstein and Debye) at one fixed value $\alpha(2k_F)=5$, with no analytic map of the $(\gamma,\alpha,\omega_{q})$ region where the $q<2k_F$ minimum is global; instabilities at $q>2k_F$ were argued to be unlikely in the examples examined but not excluded by any general argument, and the single-harmonic ansatz omits higher harmonics and Umklapp-coupled components (mixing with reciprocal-lattice vectors) that a real lattice would allow. Exchange and correlation beyond the Hartree term were treated only by transplanting local-field factors (the standard $G(q)$ corrections to the Coulomb kernel, $\phi_{q}\to\phi_{q}[1-G(q)]$, taken from well-known approximations developed for the three-dimensional electron gas) with no justification for quasi-1D systems. No finite-temperature free energy, no transition temperature, and no collective-mode (amplitude- and phase-mode) spectrum of the screened distorted state were derived.

The neighboring literature clarifies what is and is not already known. Izuyama and Saitoh (1975) treated long-range Coulomb effects on the Peierls instability within linear response. Mazumdar and Dixit (1983) and Hirsch (1983) studied Coulomb effects on Peierls dimerization in lattice models with on-site and nearest-neighbor interactions (Peierls-Hubbard and Holstein-type models), finding enhancement or suppression of the distortion depending on the interaction range and phonon frequency — a different model class with short-range interactions, no nonlinear screening of a continuum distortion potential, no wave-vector selection, and no finite-temperature theory. Barisic and Tutis (1986), in the closest continuum treatment, computed the $T=0$ Peierls gap of spinless fermions with long-range Coulomb forces in a bosonized (rewritten in terms of collective density-fluctuation rather than electron variables) generalized Tomonaga model within a self-consistent harmonic approximation, finding that to leading order the gap and the mean-field transition temperature are governed by the short-range part of the interaction — again harmonic, restricted to $q=2k_F$, and without anharmonicity or band-structure nonlinear screening. Modern exact numerical work (Hohenadler, Assaad, and Fehske, 2012) maps the phase diagram of half-filled 1D electron-phonon models from local (Holstein) to long-range (Frohlich) coupling but omits electron-electron Coulomb screening; first-principles-informed studies of real quasi-1D systems (Jeckelmann et al., 2016, on In/Si(111)) find first-order, finite-amplitude Peierls transitions; and the density-functional analysis of Johannes and Mazin (2008) argues that the ideal Peierls construction is fragile in real materials. None of these works determines, within the source's continuum nearly-free-electron model, the nonperturbative gap and displacement, the wave-vector selection map, the beyond-local-field exchange-correlation corrections, or the finite-temperature behavior of the screened Peierls state.

Problem Statement

Determine the complete equilibrium and finite-temperature theory of the Peierls instability in the nearly-free-electron model of a quasi-one-dimensional metal when the electron-electron Coulomb interaction is retained and screening is treated self-consistently and nonlinearly in the distortion amplitude — going beyond the harmonic, perturbative ($V\ll\xi$), single-harmonic, Hartree-plus-local-field, zero-temperature treatment of Yeats and Kroemer (1976). The model is pinned to the source: electrons of effective mass $m^{*}$ in a single parabolic band along the chain with Fermi wave number $k_F$; Coulomb interaction $\phi_{q}$ across a chain cross section $A$ (three-dimensional field, one-dimensional band structure); longitudinal ionic displacements with bare phonon frequency $\omega_{q}$ and electron-phonon matrix element $g_{q}$, defining a dimensionless electron-phonon coupling $\gamma(q)=2|g_{q}|^{2}/(\phi_{q}\hbar\omega_{q}N/\Omega)$ (with $N$ the number of distorting ions and $\Omega$ the crystal volume), which for point ions of charge $Ze$ and mass $M$ reduces to $\gamma(q)=\omega_{\mathrm{pl}}^{2}/\omega_{q}^{2}$, the squared ratio of the bare ionic plasma frequency $\omega_{\mathrm{pl}}$ to the bare phonon frequency; a screened single-harmonic distortion potential of amplitude $V$ at wave number $q$, with gap $2V$, electronic scale $\xi=\hbar^{2}q^{2}/(2m^{*})$, and $\alpha(q)=2\phi_{q}m^{*}/(\pi A\hbar^{2}q)$. The four linked sectors, which jointly constitute the complete theory of the screened Peierls state, are:

(1) Nonperturbative equilibrium. For given $(\gamma(q),\alpha(q),\omega_{q})$ including anharmonic lattice interactions, determine the energy-minimizing screened potential amplitude $V$ and ionic displacement, in particular in the regime where harmonic minimization predicts $V\gtrsim\xi$; establish whether and how anharmonicity limits the realized gap and displacement.

(2) Wave-vector selection and classification of instabilities. Map the region of parameter space in which the global energy minimum occurs at $q=2k_F$ versus $q<2k_F$ (gap at versus below the Fermi level) as a function of $\gamma$, $\alpha$, and the phonon dispersion, replacing the source's graphical examples; and determine by general argument, not example, whether instabilities at $q>2k_F$, or multi-harmonic and Umklapp-coupled distortions, can ever be the leading instability.

(3) Exchange and correlation. Determine how exchange-correlation effects beyond the Hartree approximation with transplanted three-dimensional local-field factors modify $\gamma(q)$, $\alpha(q)$, the total energy change, and the conclusions of sectors (1) and (2) — in particular whether a quasi-1D treatment changes them qualitatively or only quantitatively.

(4) Finite temperature. Derive the transition temperature of the screened Peierls instability, the temperature dependence of its gap (in particular, whether the source's conjecture of non-BCS behavior is correct), and the collective-mode (amplitude- and phase-mode) spectrum of the screened distorted state, with nonlinear screening and anharmonicity both retained.

The four sectors jointly constitute the complete theory of the screened Peierls state, and a correction to a source conjecture within one sector does not remove the others from the target. Where the source does not fix them, the anharmonic ionic potential of sector (1) and the multi-harmonic or Umklapp-coupled ansatz of sector (2) may be any standard, explicitly stated form, provided its coefficients and conventions are declared and the source's harmonic, single-harmonic limit is recovered.

The solution must be anchored: in the limits where the source's treatment applies ($V/\xi\ll 1$, $T=0$), it must reproduce the published results (see References item 1), namely the $\gamma(q)>1$ necessary condition for instability at $q\le 2k_F$; the nonlinear dielectric function $\epsilon_{q}=1+\alpha(q)\ln|2\xi\delta/V^{2}+1|$ (with $\delta=\delta(k_{F},q)$ the twofold-degenerate second-order perturbation energy shift of the distorted-band states) and screened phonon frequency $\omega_{s}^{2}(q)=\omega_{q}^{2}[1-\gamma(q)]+\omega_{q}^{2}\gamma(q)/\epsilon_{q}$; and the existence, for $\gamma(2k_F)$ barely above unity, of a $q<2k_F$ minimum within that perturbative regime, which anchors the existence of such a minimum there but not its global selection outside the regime — and in the unscreened limit it must recover the standard unscreened Peierls results (exponentially small gap, BCS-like mean-field relation between the zero-temperature gap and the transition temperature). The solution may legitimately overturn a source conjecture (for example, the non-BCS temperature dependence, or the preference of the global energy minimum for $q<2k_F$ outside the perturbative anchor regime) provided the anchor limits are still reproduced and the correction is established within the pinned model or a demonstrably equivalent regularization (for numerical work, a lattice regularization that preserves the low-energy nearly-free-electron physics and long-range Coulomb tail).

References

1. R. Yeats and H. Kroemer, "Peierls instability in a nearly-free-electron model, including nonlinear screening", Physical Review B 14, 3335 (1976). DOI: 10.1103/PhysRevB.14.3335, https://doi.org/10.1103/PhysRevB.14.3335. (Source paper.)
2. G. A. Toombs, "Quasi-one-dimensional conductors", Physics Reports 40, 181-240 (1978). DOI: 10.1016/0370-1573(78)90149-7, https://doi.org/10.1016/0370-1573(78)90149-7.
3. M. D. Johannes and I. I. Mazin, "Fermi surface nesting and the origin of charge density waves in metals", Physical Review B 77, 165135 (2008). DOI: 10.1103/PhysRevB.77.165135; arXiv:0708.1744, https://arxiv.org/abs/0708.1744.
4. T. Izuyama and M. Saitoh, "Effects of long range Coulomb interaction on the Peierls instability", Solid State Communications 16, 549-552 (1975). DOI: 10.1016/0038-1098(75)90421-4, https://doi.org/10.1016/0038-1098(75)90421-4.
5. S. Mazumdar and S. N. Dixit, "Coulomb effects on one-dimensional Peierls instability: The Peierls-Hubbard model", Physical Review Letters 51, 292-295 (1983). DOI: 10.1103/PhysRevLett.51.292, https://doi.org/10.1103/PhysRevLett.51.292.
6. J. E. Hirsch, "Effect of Coulomb interactions on the Peierls instability", Physical Review Letters 51, 296-299 (1983). DOI: 10.1103/PhysRevLett.51.296, https://doi.org/10.1103/PhysRevLett.51.296.
7. S. Barisic and E. Tutis, "Effect of the Coulomb interaction on the Peierls gap", Journal of Physics C: Solid State Physics 19, 6303-6310 (1986). DOI: 10.1088/0022-3719/19/32/006, https://doi.org/10.1088/0022-3719/19/32/006.
8. M. Hohenadler, F. F. Assaad, and H. Fehske, "Effect of electron-phonon interaction range for a half-filled band in one dimension", Physical Review Letters 109, 116407 (2012). DOI: 10.1103/PhysRevLett.109.116407; arXiv:1205.0612, https://arxiv.org/abs/1205.0612.
9. E. Jeckelmann, S. Sanna, W. G. Schmidt, E. Speiser, and N. Esser, "Grand canonical Peierls transition in In/Si(111)", Physical Review B 93, 241407 (2016). DOI: 10.1103/PhysRevB.93.241407, https://doi.org/10.1103/PhysRevB.93.241407.