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Physical Review Letters

Field-theoretical approach to the dynamical many-polaron system in one spatial dimension

Eduardo C. Marino

Published Dec 30, 1985• 10 citations
Publisher

Abstract

It is shown that the Hamiltonian describing the acoustic many-polaron dynamics in one spatial dimension is identical, except for the cutoff procedure, to that of an exactly solvable field theory. Cutoff independent quantities, such as the spectrum of excitations and dynamical correlation functions, are computed. A critical coupling is found above which there is a condensation of polaron pairs and the system exhibits a polaronic superconductivity. Below the critical coupling, the spectrum contains only collective excitations of polarons and the system has a metallic behavior.

DOI
10.1103/physrevlett.55.2991

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It is shown that the Hamiltonian describing the acoustic many-polaron dynamics in one spatial dimension is identical, except for the cutoff procedure, to that of an exactly solvable field theory. Cutoff independent quantities, such as the spectrum of excitations and dynamical correlation functions, are computed. A critical coupling is found above which there is a condensation of polaron pairs and the system exhibits a polaronic superconductivity. Below the critical coupling, the spectrum contains only collective excitations of polarons and the system has a metallic behavior.
This chapter presents the energy spectrum and the momentum distribution of the Frohlich–Toyozawa–Luttinger liquid. The one-dimensional many-electron problem has generated renewed interest after the discovery of quasi-one dimensional solids such as carbon nanotubes, quantum wires, and other one-dimensional systems. Tomonaga was the first to solve the one-dimensional interacting fermion problem exactly using bosonization and certain approximations regarding the commutator of the density operators. Marino considered a one-dimensional acoustic many-polaron Hamiltonian within the framework of the Luttinger model and showed that for a certain range of the coupling constants the system would exhibit a metallic behavior because the spectrum contains gapless collective excitations of polaronic plasmons. Rao et al. have exactly solved the one-dimensional interacting electron problem within the framework of Luttinger model incorporating both the electron-optical-phonon interaction and the electron-acoustic-phonon interaction along with the electron-electron interaction to obtain the energy spectrum.
A dynamical mean-field theory of the small polaron problem is presented, which becomes exact in the limit of infinite dimensions. The ground-state properties and the one-electron spectral function are obtained for a single electron interacting with Einstein phonons by a mapping of the lattice problem onto a polaronic impurity model. The one-electron propagator of the impurity model is calculated through a continued fraction expansion, at both zero and finite temperature, for any electron-phonon coupling and phonon energy. In contrast to the ground-state properties, such as the effective polaron mass, which show a continuous behavior as the coupling is increased, spectral properties exhibit a sharp qualitative change at low enough phonon frequency: beyond a critical coupling, one energy gap and then more open in the density of states at low energy, while the high-energy part of the spectrum is broad and can be qualitatively explained by a strong coupling adiabatic approximation. As a consequence, narrow and coherent low-energy subbands coexist with an incoherent featureless structure at high energy. The subbands denote the formation of quasiparticle polaron states. Also, divergencies of the self-energy may occur in the gaps. At finite temperature such an effect triggers an important damping and broadening of the polaron subbands. On the other hand, in the large phonon frequency regime such a separation of energy scales does not exist and the spectrum always has a multipeaked structure.
The transport dynamics of a large polaron has been studied in a one-dimensional system whose underlying vibrational spectrum is acoustic in character. Such a polaron moves as a heavy quasiparticle with transport relaxation time, $\ensuremath{\tau}$, determined mainly by collisions with thermally ambient phonons. Concrete results show that $\ensuremath{\tau}$ decreases monotonically with increasing temperature, in sharp contrast to the previously treated problem of a large polaron in a one-dimensional Einstein lattice.
We have studied one dimensional acoustic phonon linearly coupled to the electron density. The phonon spectrum is linear with cutoff at the Debye frequency, it is straightforward modification of three dimensional optical phonons. Phonon field results in substituting the dynamical coupling for the screened coulomb interaction. We have shown that the electron-phonon coupling quantitatively changed the phase diagram of Luttinger liquid with a single impurity. The existence of slow and fast polaron modes with different weights in different regimes resulted in different renormalization group flow for a weak scatterer and a weak tunneling link. The resulting phase diagram depending on the parameters of the problem, regimes corresponding to purely metallic or purely insulating behavior and an intermediate regime with two stable fixed points and one unstable, finite conductance fixed point. we have found that the direction of renormalization group flow change upon varying either the relative strength of the electron-electron and electron-phonon coupling or the ratio of Fermi to sound velocities. The obtained results were found in good agreement with previously obtained results.