Polarons and bipolarons in strongly interacting electron-phonon systems

Abstract

The Holstein Hubbard and Holstein t-J models are studied for a wide range of phonon frequencies, electron-electron and electron-phonon interaction strengths on finite lattices with up to ten sites by means of direct Lanczos diagonalization. Previously the necessary truncation of the phononic Hilbert space caused serious limitations to either very small systems (four or even two sites) or to weak electron-phonon coupling, in particular in the adiabatic regime. Using parallel computers we were able to investigate the transition from 'large' to 'small' polarons in detail. By resolving the low-lying eigenstates of the Hamiltonian and by calculating the spectral function we can identify a polaron band in the strong-coupling case, whose dispersion deviates from the free-particle dispersion at low and intermediate phonon frequencies.For two electrons (holes) we establish the existence of bipolaronic states and discuss the formation of a bipolaron band. For the 2D Holstein t-J model we demonstrate that the formation of hole-polarons is favoured by strong Coulomb correlations. Analyzing the hole-hole correlation functions we find that hole binding is enhanced as a dynamical effect of the electron-phonon interaction.

Spectral properties of the 2D Holstein polaron

“…Needless to say, that we have carefully to check for the convergence of both the ground-state energy,E 0 (M ), and the phonon distribution, |c m | 2 (M ) = s |c m,s K=0 | 2 , as a function of the maximal number of phonons retained [13]. Implementing an improved Lanczos algorithm on parallel computers, we are able to calculate the ground-state properties of systems with a total dimension of about 10 7 .…”

“…Since the Hilbert space associated with the phonons is infinite even for a finite system, we apply a truncation procedure [22,13] restricting ourselves to phononic…”

“…In the weak-coupling (adiabatic) limit, one expects (from scaling arguments) a quasi-free electron behaviour for D > 1, while in a 1D system the carrier state becomes polaronic at arbitrary small λ ("large" polaron). Previous exact diagonalization (ED) work has concentrated on the 1D case [8][9][10][11][12]21,[13][14][15][16][17], where, however, the calculations were limited to either very small clusters, rather weak EP coupling (λ < 1) or to the adiabatic limit (ω o = 0). On the other hand, the most interesting effects will be expected if the characteristic electronic and phononic energy scales are not well separated (λ ∼ 1; α ∼ 1).…”

Path Integrals in the Physics of Lattice Polarons

“…The lattice bipolaron, mostly with Holstein interaction, has been studied by a variety of other numerical methods: exact diagonalization [50,118], variational [57,58,60], density matrix renormalization group [66], diagrammatic quantum Monte Carlo [44], and Lang-Firsov quantum Monte Carlo [47,119].…”

“…There exist several other QMC techniques, in particular the path-integral QMC without phonon integration [40,41], Fourier QMC [21,22], diagrammatic QMC [7,8,42,43,44,45] and Lang-Firsov QMC [46,47,48]. Non-QMC classes of methods include exact diagonalization [49,50,51,52,53,54], variational calculations [13,55,56,57,58,59,60,61,62,63] and the density-matrix renormalization group [64,65,66]. Despite proliferation of methods, most of them have been applied to the two major polaron models: the ionic crystal model of Fröhlich [5] and molecular crystal model of Holstein [67].…”