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arXiv:cond-mat/0408516 · Physical Review B

Many-polaron effects in the Holstein model

Sanjoy Datta, Arnab Das, Sudhakar Yarlagadda

Published Jun 30, 2005• 26 citations
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Abstract

We derive an effective polaronic interaction Hamiltonian, exact to second order in perturbation, for the spinless one-dimensional Holstein model. The small parameter is given by the ratio of the hopping term $(t)$ to the polaronic energy $({g}^{2}{\ensuremath{\omega}}_{0})$ in the whole region of validity for our perturbation; however, the exception is the regime of extreme antiadiabaticity $(t∕{\ensuremath{\omega}}_{0}\ensuremath{\leqslant}0.1)$ and small electron-phonon coupling $(g<1)$ where the small parameter is $t∕{\ensuremath{\omega}}_{0}$. We map our polaronic Hamiltonian onto a next-to-nearest-neighbor interaction anisotropic Heisenberg spin model. By studying the mass gap and the power-law exponent of the spin-spin correlation function for our Heisenberg spin model, we analyze the Luttinger liquid to charge-density wave transition at half filling in the effective polaronic Hamiltonian. We calculate the structure factor at all fillings and find that the spin-spin correlation length decreases as one deviates from half filling. We also extend our derivation of polaronic Hamiltonian to $d$ dimensions.

DOI
10.1103/physrevb.71.235118

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We derive an effective polaronic interaction Hamiltonian, exact to second order in perturbation, for the spinless one-dimensional Holstein model. The small parameter is given by the ratio of the hopping term (t)(t)(t) to the polaronic energy (g2\ensuremathω0)({g}^{2}{\ensuremath{\omega}}_{0})(g2\ensuremathω0​) in the whole region of validity for our perturbation; however, the exception is the regime of extreme antiadiabaticity (t∕\ensuremathω0\ensuremath⩽0.1)(t∕{\ensuremath{\omega}}_{0}\ensuremath{\leqslant}0.1)(t∕\ensuremathω0​\ensuremath⩽0.1) and small electron-phonon coupling (g<1)(g<1)(g<1) where the small parameter is t∕\ensuremathω0t∕{\ensuremath{\omega}}_{0}t∕\ensuremathω0​. We map our polaronic Hamiltonian onto a next-to-nearest-neighbor interaction anisotropic Heisenberg spin model. By studying the mass gap and the power-law exponent of the spin-spin correlation function for our Heisenberg spin model, we analyze the Luttinger liquid to charge-density wave transition at half filling in the effective polaronic Hamiltonian. We calculate the structure factor at all fillings and find that the spin-spin correlation length decreases as one deviates from half filling. We also extend our derivation of polaronic Hamiltonian to ddd dimensions.
Employing a largely unbiased variational exact diagonalization technique, we analyze the consequences of longer-ranged electron hopping and electron-phonon interaction on polaron formation in one dimension. Having at our disposal the accurate ground-state energy and wave function, we calculate and discuss various physical quantities, such as the renormalized band structure, effective mass, wave-function renormalization factor, phonon dressing, and Drude weight, characterizing the properties of the polaronic quasiparticle. We demonstrate that the electron-phonon coupling affects the relative strength of the nearest neighbor (NN) and next nearest neighbor (NNN) hopping processes in a dynamic way. Most notably we observe that the minimum of the polaron band, occurring at a finite momentum for large negative ratio between NN and NNN transfer, jumps to zero momentum as the electron-phonon coupling exceeds a critical one, thereby causing a rather sharp polaron transition in the one-dimensional extended Holstein model. The signatures of this transition are seen in the effective mass and polaron mobility, and therefore should be easily detectable by transport measurements.
The Holstein model of spinless fermions interacting with dispersionless phonons in one dimension is studied by a Green's function Monte Carlo technique. The ground-state energy, first fermionic excited state, density wave correlations, and mean lattice displacements are calculated for lattices of up to 16 sites, for one fermion per two sites, i.e., a half-filled band. Results are obtained for values of the fermion hopping parameter of t=0.1\ensuremath{\omega}, \ensuremath{\omega}, and 10\ensuremath{\omega}, where \ensuremath{\omega} is the phonon frequency. At a finite fermion-phonon coupling g there is a transition from a metallic phase to an insulating phase in which there is charge-density-wave order. Finite-size scaling is found to hold in the metallic phase and is used to extract the coupling dependence of the Luttinger liquid parameters, ${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$ and ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, the velocity of charge excitations, and the correlation exponent, respectively. For free fermions (g=0) and for strong coupling (g2{\mathit{g}}^{2}g2\ensuremath{\gg}t\ensuremath{\omega}) our results agree well with known analytic results. \textcopyright{} 1996 The American Physical Society.
Allowing a single electron to hop to next-nearest neighbours (NNN) in addition to the closest atomic sites in the Holstein model, a modified Trugman method is applied to exactly calculate the effect on the polaronic effective mass in one, two, and three dimensions, building on the previous study of the one-dimensional NNN Holstein model. We also present perturbative calculations and a heuristic scaling factor for the coupling strength and ion frequency to nearly map the NNN Holstein model back onto the original Holstein model. When account is taken of the modified electronic bandwidth near the electron energy, we find that including NNN hopping effectively increases the polaron effective mass.
We derive an effective Hamiltonian for the one-dimensional Hubbard-Holstein model, valid in a regime of both strong electron-electron (e-e) and electron-phonon (e-ph) interactions and in the nonadiabatic limit ($t/{\ensuremath{\omega}}_{0}\ensuremath{\le}1$), by using a nonperturbative approach. We obtain the phase diagram at quarter filling by employing a modified Lanczos method and studying various density-density correlations. The spin-spin AF (antiferromagnetic) interactions and nearest-neighbor repulsion, resulting from the e-e and the e-ph interactions, respectively, are the dominant terms (compared to hopping) and compete to determine the various correlated phases. As e-e interaction (U/t)(U/t)(U/t) is increased, the system transits from an AF cluster to a correlated singlet phase through a discontinuous transition at all strong e-ph couplings $2\ensuremath{\le}g\ensuremath{\le}3$ considered. At higher values of U/tU/tU/t and moderately strong e-ph interactions ($2\ensuremath{\le}g\ensuremath{\le}2.6$), the singlets break up to form an AF order and then to a paramagnetic order all in a single sublattice; whereas at larger values of ggg (>>>2.62.62.6), the system jumps directly to the spin disordered charge-density-wave (CDW) phase.
The one-dimensional Holstein model and its generalizations have been studied extensively to understand the effects of electron-phonon interaction. The half-filled case is of particular interest, as it describes a transition from a metallic phase with a spin gap due to attractive backscattering to a Peierls insulator with charge-density-wave order. Our quantum Monte Carlo results support the existence of a metallic phase with dominant power-law charge correlations, as described by the Luther-Emery fixed point. We demonstrate that for Holstein and also for purely fermionic models the spin gap significantly complicates finite-size numerical studies, and explains inconsistent previous results for Luttinger parameters and phase boundaries. On the other hand, no such complications arise in spinless models. The correct low-energy theory of the spinful Holstein model is argued to be that of singlet bipolarons with a repulsive, mutual interaction. This picture naturally explains the existence of a metallic phase, but also implies that gapless Luttinger liquid theory is not applicable.
The renormalization group technique is applied to one-dimensional electron-phonon Hubbard models at half filling and zero temperature. For the Holstein-Hubbard model, the results of one-loop calculations are congruent with the phase diagram obtained by quantum Monte Carlo simulations in the (U,gph)(U,{g}_{\mathrm{ph}})(U,gph​) plane for the phonon-mediated interaction gph{g}_{\mathrm{ph}}gph​ and the Coulomb interaction UUU. The incursion of an intermediate phase between a fully gapped charge-density-wave state and a Mott antiferromagnet is supported along with the growth of its size with the molecular phonon frequency ${\ensuremath{\omega}}_{0}$. We find additional phases enfolding the base boundary of the intermediate phase. A Luttinger liquid line is found below some critical ${U}^{*}\ensuremath{\approx}{g}_{\mathrm{ph}}^{*}$, followed at larger $U\ensuremath{\sim}{g}_{\mathrm{ph}}$ by a narrow region of bond-order-wave ordering which is either charge or spin gapped depending on UUU. For the Peierls-Hubbard model, the region of the (U,gph)(U,{g}_{\mathrm{ph}})(U,gph​) plane with a fully gapped Peierls-bond-order-wave state shows a growing domination over the Mott gapped antiferromagnet as the Debye frequency ${\ensuremath{\omega}}_{D}$ decreases. A power-law dependence ${g}_{\mathrm{ph}}\ensuremath{\sim}{U}^{2\ensuremath{\eta}}$ is found to map out the boundary between the two phases, whose exponent is in good agreement with the existing quantum Monte Carlo simulations performed when a finite nearest-neighbor repulsion term VVV is added to the Hubbard interaction.