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Journal of Physics C Solid State Physics

The Peierls instability at T=0-a dimensional regularisation approach in the 1/N expansion

D. Schmeltzer

Published Jul 30, 1987• 7 citations
Publisher

Abstract

The effect of quantum fluctuations on the Peierls instability is investigated for two models, the Su-Schrieffer-Heeger model and the molecular model introduced by Holstein. The author finds for both models that when the ionic mass M not=0 the beta function has the form beta ( lambda R )=-(1-(1/N)f( lambda R / alpha R )) lambda R 2 / pi with alpha R being an invariant quantity alpha R =M R lambda R exp(-2 pi / lambda R ). This allows the study of the crossover behaviour M to 0 to M to infinity . In the limit M to 0 he obtains exactly the Gross-Neveau model (M=O) which has a Peierls gap for spin 1 / 2 (N=2). In the limit M to infinity the mean field Peierls gap is found. The molecular model with M=0 is different from the Gross-Neveau model (M=0). It is similar to the non-linear sigma model with the beta -function: beta ( lambda R )=-(1-2/N) lambda R 2 / pi . In this case there is no Peierls gap for spin 1 / 2 (N=2) or for spinless (N=1) electrons. The analysis is performed in a 1/N expansion. The renormalisation group method is used to compute the beta -function with an anisotropic parameter alpha R which controls the crossover behaviour. The renormalisation functions are obtained by a dimensional regularisation method in 2- in dimensions where simple poles and in (1/ in ) are cancelled.

DOI
10.1088/0022-3719/20/21/012

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The effect of quantum fluctuations on the Peierls instability is investigated for two models, the Su-Schrieffer-Heeger model and the molecular model introduced by Holstein. The author finds for both models that when the ionic mass M not=0 the beta function has the form beta ( lambda R )=-(1-(1/N)f( lambda R / alpha R )) lambda R 2 / pi with alpha R being an invariant quantity alpha R =M R lambda R exp(-2 pi / lambda R ). This allows the study of the crossover behaviour M to 0 to M to infinity . In the limit M to 0 he obtains exactly the Gross-Neveau model (M=O) which has a Peierls gap for spin 1 / 2 (N=2). In the limit M to infinity the mean field Peierls gap is found. The molecular model with M=0 is different from the Gross-Neveau model (M=0). It is similar to the non-linear sigma model with the beta -function: beta ( lambda R )=-(1-2/N) lambda R 2 / pi . In this case there is no Peierls gap for spin 1 / 2 (N=2) or for spinless (N=1) electrons. The analysis is performed in a 1/N expansion. The renormalisation group method is used to compute the beta -function with an anisotropic parameter alpha R which controls the crossover behaviour. The renormalisation functions are obtained by a dimensional regularisation method in 2- in dimensions where simple poles and in (1/ in ) are cancelled.
We study the nature of the ground state of the Su-Schrieffer-Heeger model for electron-phonon interactions in one dimension in the half-filled-band case. We consider the cases of spinless electrons (n=1n=1n=1) and spin-\textonehalf{} electrons (n=2n=2n=2), and discuss the stability of the Peierls-dimerized ground state as a function of the ionic mass and electron-phonon coupling constant. We first consider the zero-mass limit of the theory and extend our results to finite mass using renormalization-group arguments. For spinless electrons, it is found that quantum fluctuations destroy the long-range dimerization order for the small electron-phonon coupling constant if the ionic mass is finite. For spin-\textonehalf{} electrons, the system is dimerized for an arbitrary coupling constant and phonon frequency. Renormalization-group trajectories show that the low-energy behavior of the system is governed by the zero-mass limit of the theory, an nnn-component Gross-Neveu model. Monte Carlo simulations are performed for the model at finite phonon frequencies, and the results are compared with the static limit result. We study in particular the set of parameters appropriate for polyacetylene and find a 15% reduction in the phonon order parameter due to fluctuations of the phonon field. A finite-size scaling analysis of the numerical data for the cases n=1and2n=1 \mathrm{and} 2n=1and2 is performed, which confirms the results obtained from the renormalization-group analysis.
The effect of quantum fluctuations of the phonon field on the ground-state character of one-dimensional electron-phonon systems is studied with a numerical simulation technique. Two different electron-phonon models are studied, both for spinless and spin-\textonehalf{} electrons, in the half-filled-band case. In both models, it is found that in the spinless case quantum fluctuations destroy the long-range dimerization order for small electron-phonon coupling constant $\ensuremath{\lambda}$. For the spin-\textonehalf{} case, the results are consistent with the Peierls transition occurring at $\ensuremath{\lambda}=0$.
The one-dimensional Hubbard model is studied using a real-space renormalisation-group block method previously applied to other quantum systems. The authors primarily consider the half-filled band for which the ground state is found to be insulating for all non-zero values of the Hubbard interaction, in agreement with the exact solution. When the method is applied to first order in perturbation, they find that the insulating gap for small U/t behaves like exp (-At 2 /U 2 ) where U is the Hubbard interaction and t is the nearest-neighbour transfer integral. When the method is extended to second order, the authors find that for small U/t the gap behaves like (U/t) alpha with alpha approximately=11.2. The exact asymptotic behaviour of the gap ( approximately (U/t) 1/2 exp(-At/U)) is not recovered, but a good semiquantative agreement with the exact solution is obtained for both the gap and the ground state energy. Extensions to non-half-filled bands and to higher dimensions are discussed. The validity of the renormalisation group method is also discussed.
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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
We use the density matrix renormalization group method to investigate the role of longitudinal quantized phonons on the Peierls transition in the spin-Peierls model. For both the XY and Heisenberg spin-Peierls model we show that the staggered phonon order parameter scales as sqrt[lambda] (and the dimerized bond order scales as lambda) as lambda-->0 (where lambda is the electron-phonon interaction). This result is true for both linear and cyclic chains. Thus, we conclude that the Peierls transition occurs at lambda=0 in these models. Moreover, for the XY spin-Peierls model we show that the quantum predictions for the bond order follow the classical prediction as a function of inverse chain size for small lambda. We therefore conclude that the zero lambda phase transition is of the mean-field type.