For the large-radius Holstein polaron in a one-dimensional continuum molecular chain (an electron of effective mass mm coupled to dispersionless optical phonons of frequency ω0\omega_0 with momentum-independent coupling), Lakhno (2014) constructed a translation-invariant (delocalized) strong-coupling solution with ground-state energy E≈−0.2037 ma02ℏ2g4ℏ2ω02E \approx -0.2037\,\frac{m a_0^2}{\hbar^2}\frac{g^4}{\hbar^2\omega_0^2}, i.e. E~=−0.10185 x2\tilde E = -0.10185\,x^2 in the paper's scaled units (E~=E/ℏω0\tilde E = E/\hbar\omega_0 and x=ℏω0/ν (g/ℏω0)2x = \sqrt{\hbar\omega_0/\nu}\,(g/\hbar\omega_0)^2, where ν\nu is the nearest-neighbor hopping energy of the underlying lattice), below Holstein's localized self-trapped estimate E~=−0.08333 x2\tilde E = -0.08333\,x^2, and concluded that spontaneous breaking of translational symmetry does not occur. The paper itself left the verification of this energy by exact numerical methods, such as quantum Monte Carlo for a quantum molecular chain, as an open problem, and the exact strong-coupling solution of the continuum Hamiltonian is unknown. Later literature neither verifies nor refutes the estimate: no independent quantitative check of it has been published, and Klimin and Devreese (2012) proved that the underlying Gross-Tulub functional, in its Fröhlich-polaron applications, has an incomplete recoil energy and is not a proven variational upper bound. The open problem is to determine the exact ground-state energy of the one-dimensional continuum Holstein polaron, in particular its strong-coupling asymptotics, by an exact method, and thereby decide whether the translation-invariant estimate is accurate and consistent with the variational principle, and whether the delocalized state is energetically favored.

Background

A polaron is a charge carrier dressed by the lattice distortion it induces. In the Holstein molecular-crystal model (Holstein, 1959), a single electron hops along a chain of sites and couples locally to an Einstein optical phonon (a dispersionless harmonic oscillator of frequency ω0\omega_0) at each site, with coupling matrix element gg; the physics is governed by the dimensionless coupling g/ℏω0g/\hbar\omega_0 and the adiabaticity ratio ν/ℏω0\nu/\hbar\omega_0, where ν\nu is the nearest-neighbor hopping matrix element. At strong coupling (g/ℏω0≫1g/\hbar\omega_0 \gg 1) two regimes are distinguished: the small-radius polaron (ν≪ℏω0\nu \ll \hbar\omega_0), whose exact strong-coupling solution (Tyablikov; Lang-Firsov form) is a translation-invariant superposition over sites, and the large-radius polaron (ν≫ℏω0\nu \gg \hbar\omega_0), whose continuum limit is the Hamiltonian H=−ℏ22md2dx2+∑kVk(akeikx+ak†e−ikx)+∑kℏω0 ak†ak,Vk=gN,H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \sum_k V_k\left(a_k e^{ikx} + a^\dagger_k e^{-ikx}\right) + \sum_k \hbar\omega_0\, a^\dagger_k a_k, \quad V_k = \frac{g}{\sqrt{N}}, where ak†,aka^\dagger_k, a_k are phonon creation and annihilation operators, the sums run over the NN phonon momenta of the first Brillouin zone of the chain, m=ℏ2/(2νa02)m = \hbar^2/(2\nu a_0^2), a0a_0 is the lattice constant, and NN the number of sites. Holstein's strong-coupling treatment of this continuum problem used the adiabatic (Born-Oppenheimer) product ansatz Ψ=ψ(x)Φ({qk})\Psi = \psi(x)\Phi(\{q_k\}), in which the electron wave function ψ(x)\psi(x) is separated from the phonon coordinates; the energy minimum is then reached at a localized ψ(x)=±(2r cosh⁡x−x0r)−1\psi(x) = \pm\left(\sqrt{2r}\,\cosh\frac{x-x_0}{r}\right)^{-1} with radius r=ℏ3ω0/(mg2a0)r = \hbar^3\omega_0/(m g^2 a_0) and energy EH=−16ℏ2mr2∝−g4E_H = -\frac{1}{6}\frac{\hbar^2}{m r^2} \propto -g^4. Because the minimum occurs at an arbitrary center x0x_0, the classical solution breaks the translational symmetry of the Hamiltonian; upon quantizing around it, a zero-frequency Goldstone mode (a massless excitation that normally accompanies spontaneous breaking of a continuous symmetry) appears in the phonon spectrum.

This symmetry breaking, however, contradicts the fact that the total momentum of the electron-phonon system commutes with the Hamiltonian, so exact eigenstates can be chosen as delocalized momentum eigenstates. Tulub's translation-invariant strong-coupling method (the Gross-Tulub approach), originally developed for the Fröhlich polaron, eliminates the electron coordinate by a Heisenberg canonical transformation, works with Gaussian (squeezed) phonon states, and yields an energy functional of phonon displacement amplitudes fkf_k that includes the electron recoil through a function D(s)D(s). Lakhno (2014) applied this scheme to the continuum Holstein Hamiltonian above and, minimizing with a Gaussian fkf_k, obtained ETI≈−0.2037 ma02ℏ2g4ℏ2ω02E_{\mathrm{TI}} \approx -0.2037\,\frac{m a_0^2}{\hbar^2}\frac{g^4}{\hbar^2\omega_0^2}, i.e. E~=−0.10185 x2\tilde E = -0.10185\,x^2 in the scaled units of the paper, where E~=E/ℏω0\tilde E = E/\hbar\omega_0 and x=ℏω0/ν (g/ℏω0)2x = \sqrt{\hbar\omega_0/\nu}\,(g/\hbar\omega_0)^2 is the paper's scaled coupling variable, below Holstein's −0.08333 x2-0.08333\,x^2. He concluded that the ground state is delocalized at all couplings with an analytic energy E(g)E(g) (a smooth crossover near gc≈1.86 (ℏ/ma02ω0)1/4g_c \approx 1.86\,(\hbar/m a_0^2\omega_0)^{1/4} rather than a phase transition), that the adiabatic product ansatz is invalid for delocalized states, that renormalized phonon frequencies lie above the bare ω0\omega_0, and that no Goldstone mode exists.

Several rigorous results bear on the qualitative claims: Gerlach and Löwen proved the absence of phonon-induced localization for the free optical polaron (Phys. Rev. B 37, 8042 (1988)) and, with Löwen's companion proof of the absence of phase transitions in Holstein systems (Phys. Rev. B 37, 8661 (1988)), reviewed the analyticity of polaron ground-state energies in the coupling constant (Rev. Mod. Phys. 63, 63 (1991)); these theorems, which Lakhno cites, support delocalization and analyticity but give no quantitative energy for the one-dimensional continuum model.

On the quantitative side, exact numerical benchmarks exist only for neighboring models: diagrammatic quantum Monte Carlo (QMC) for the three-dimensional Fröhlich polaron (Prokof'ev and Svistunov, 1998), and continuous-time or path-integral QMC for the discrete lattice Holstein polaron (Kornilovitch, 1998; Hohenadler, Evertz and von der Linden, 2005). No exact-method computation for the one-dimensional continuum Holstein Hamiltonian, and no comparison of any exact result with the translation-invariant energy, has been published. Moreover, the variational status of the Gross-Tulub machinery is contested: Klimin and Devreese (2012) analyzed the same functional as used for the Fröhlich polaron and bipolaron and proved that its recoil energy is incomplete (it omits the Porsch–Röseler boundary term that arises when a phonon-momentum cutoff is imposed), so energies derived from it have not been shown to be variational upper bounds on the exact ground-state energy. Whether Lakhno's Holstein estimate inherits this defect is exactly what an exact benchmark would reveal: if the exact energy lies above −0.10185 x2-0.10185\,x^2, the translation-invariant estimate is not a valid variational bound; if it lies below, the estimate's quantitative accuracy can be assessed.

Problem Statement

Determine, by exact non-variational means (for example quantum Monte Carlo simulation of the quantum molecular chain, or a rigorous analysis), the ground-state energy E(g)E(g) of the one-dimensional continuum Holstein polaron: a single electron of effective mass mm in a continuum chain coupled to dispersionless optical phonons of frequency ω0\omega_0 with momentum-independent coupling, H=−ℏ22md2dx2+∑kgN(akeikx+ak†e−ikx)+∑kℏω0 ak†ak,H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \sum_k \frac{g}{\sqrt{N}}\left(a_k e^{ikx} + a^\dagger_k e^{-ikx}\right) + \sum_k \hbar\omega_0\, a^\dagger_k a_k, where ak†,aka^\dagger_k, a_k are phonon creation and annihilation operators and the sums run over the NN phonon momenta of the first Brillouin zone of the underlying lattice of constant a0a_0; this continuum model is the large-radius limit (ν≫ℏω0\nu \gg \hbar\omega_0) of the lattice Holstein chain, with hopping matrix element ν=ℏ2/(2ma02)\nu = \hbar^2/(2ma_0^2). The required deliverable is the strong-coupling limit g/ℏω0≫1g/\hbar\omega_0 \gg 1 of this energy, in which the exact solution is not known; finite-coupling values of E(g)E(g) are optional and lie outside the completion boundary. The answer must deliver: (i) the strong-coupling asymptotic behavior of the exact ground-state energy in the source paper's scaled units (E~=E/ℏω0\tilde E = E/\hbar\omega_0 and x=ℏω0/ν (g/ℏω0)2x = \sqrt{\hbar\omega_0/\nu}\,(g/\hbar\omega_0)^2), so that its leading coefficient can be compared with Holstein's localized self-trapped estimate E~=−0.08333 x2\tilde E = -0.08333\,x^2 and Lakhno's translation-invariant estimate E~=−0.10185 x2\tilde E = -0.10185\,x^2, with controlled statistical and systematic errors and a controlled continuum (large-chain) limit; and (ii) an explicit adjudication of the source paper's central claim: whether the translation-invariant estimate is consistent with the variational principle (an upper bound on the exact energy) and quantitatively accurate, and correspondingly whether the delocalized translation-invariant state is energetically favored over the localized product state in the exact quantum theory, so that the claimed absence of spontaneous translational-symmetry breaking stands or falls. The source frames exactly this as the open comparison 'which can be made only for a quantum molecular chain with the use, for example, of a quantum Monte Carlo method'; extensions to discrete lattices, higher dimensions, DNA-scale molecular systems and other electron-phonon models that the paper acknowledges as unexplored are follow-on directions beyond this problem's completion boundary.

References

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