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Physical review. B, Condensed matter

Gapped and gapless ground state of the one-dimensional spinless Holstein model

Hongxing Zheng, M. Avignon

Published Aug 15, 1998• 19 citations
Publisher

Abstract

An analytical approach has been developed to study the ground-state phase diagram of a one-dimensional spinless Holstein model and its gapped and gapless phases. We show that when the electron-phonon coupling constant ${g}^{2}/{\ensuremath{\omega}}_{0}$ decreases or the phonon frequency ${\ensuremath{\omega}}_{0}$ increases the lattice dimerization and the gap in the fermionic spectrum decreases gradually. At some critical value ${(g}^{2}/{\ensuremath{\omega}}_{0}{)}_{c}$ or ${\ensuremath{\omega}}_{0c},$ the lattice dimerization disappears and the system becomes gapless Luttinger liquid with phonon-renormalized velocity $v(q)$ and stiffness constant $K(q).$ Our numerical results agree quite well with the results of Monte Carlo simulations.

DOI
10.1103/physrevb.58.3704

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An analytical approach has been developed to study the ground-state phase diagram of a one-dimensional spinless Holstein model and its gapped and gapless phases. We show that when the electron-phonon coupling constant g2/\ensuremathω0{g}^{2}/{\ensuremath{\omega}}_{0}g2/\ensuremathω0​ decreases or the phonon frequency \ensuremathω0{\ensuremath{\omega}}_{0}\ensuremathω0​ increases the lattice dimerization and the gap in the fermionic spectrum decreases gradually. At some critical value (g2/\ensuremathω0)c{(g}^{2}/{\ensuremath{\omega}}_{0}{)}_{c}(g2/\ensuremathω0​)c​ or \ensuremathω0c,{\ensuremath{\omega}}_{0c},\ensuremathω0c​, the lattice dimerization disappears and the system becomes gapless Luttinger liquid with phonon-renormalized velocity v(q)v(q)v(q) and stiffness constant K(q).K(q).K(q). Our numerical results agree quite well with the results of Monte Carlo simulations.
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.
The effects of quantum lattice fluctuations on the Peierls transition and the optical conductivity in the one-dimensional Holstein model of spinless fermions have been studied by developing an analytical approach based on the unitary transformation method. We show that when the electron-phonon coupling constant decreases to a finite critical value, the Peierls dimerization is destroyed by the quantum lattice fluctuations. The dimerization gap is much more reduced by the quantum lattice fluctuations than the phonon order parameter. The calculated optical conductivity does not have the inverse-square-root singularity, but it has a peak above the gap edge, and there exists a significant tail below the peak. The peak of optical-conductivity spectrum is not directly corresponding to the dimerized gap. Our results of the phase diagram and the spectral-weight function agree with those of the density-matrix renormalization group and the exact diagonalization methods.
We study the nature of the ground state of a one-dimensional electron-phonon model for molecular crystals: The phonons are assumed dispersionless and couple to the local electronic density. We consider the half-filled-band sector and discuss the stability of the Peierls-dimerized ground state as a function of the phonon frequency ($\ensuremath{\omega}$), electron-phonon coupling constant ($\ensuremath{\lambda}$), and number of components of the electron spin (nnn). First, we discuss the properties of the model in the limiting cases of zero frequency and infinite frequency. We then perform a strong coupling expansion which maps the system onto a spinless fermion model with nearest-neighbor repulsion for both spinless and spin-\textonehalf{} electrons, but with different parameters in both cases. Finally, we perform a numerical study of the system using a Monte Carlo technique. We study the behavior of the order parameter and of correlation functions for various points in parameter space. We also perform a finite-size---scaling analysis of the numerical data. The conclusions of our study are the following: For the case of spinless electrons, there exists always a disordered phase for small coupling constant, and the system undergoes an infinite-order transition to a Peierls-dimerized state as the coupling constant increases beyond a critical value. The phase diagram is divided between a disordered and an ordered region by a line that connects the points ($\ensuremath{\omega}=0$, $\ensuremath{\lambda}=0$) and ($\ensuremath{\omega}=\ensuremath{\infty}$, $\ensuremath{\lambda}=\ensuremath{\infty}$). For the case of spin-\textonehalf{} electrons, the system is dimerized for arbitrary $\ensuremath{\lambda}$ and $\ensuremath{\omega}$ except in the limit $\ensuremath{\omega}=\ensuremath{\infty}$.
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.
Summary: The interplay of charge, spin and lattice degrees of freedom is studied for quasi-one-dimensional electron and spin systems coupled to quantum phonons. Special emphasis is put on the influence of the lattice dynamics on the Peierls transition. Using exact diagonalization techniques the ground-state and spectral properties of the Holstein model of spinless fermions and of a frustrated Heisenberg model with magneto-elastic coupling are analyzed on finite chains. In the non-adiabatic regime a (T = 0) quantum phase transition from a gapless Luttinger-liquid/spin-fluid state to a gapped dimerized phase occurs at a nonzero critical value of the electron/spin-phonon interaction. To study the nature of the spin-Peierls transition at finite temperatures for the infinite system, an alternative Green’s function approach is applied to the magnetostrictive XY model. With increasing phonon frequency the structure factor shows a remarkable crossover from soft-mode to central-peak behaviour. The results are discussed in relation to recent experiments on CuGeO3. 1
The one-dimensional Holstein model of spinless fermions interacting with dispersionless phonons is studied using a new variant of the density matrix renormalization group. By examining various low-energy excitations of finite chains, the metal-insulator phase boundary is determined precisely and agrees with the predictions of strong coupling theory in the antiadiabatic regime and is consistent with renormalization group arguments in the adiabatic regime. The Luttinger liquid parameters, determined by finite-size scaling, are consistent with a Kosterlitz-Thouless transition.