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Physical review. B, Condensed matter

Quantum lattice fluctuations in the ground state of an XY spin-Peierls chain

H. Zheng

Published Dec 1, 1997• 18 citations
Publisher

Abstract

An analytical approach, based on the unitary transformation method, has been developed to study the effect of quantum lattice fluctuations on the ground state of an $\mathrm{XY}$ spin-Peierls chain, which is equivalent to the spinless Su-Schrieffer-Heeger model in half-filling after the Jordan-Wigner transformation. We show that when the spin-phonon coupling constant ${\ensuremath{\alpha}}^{2}/4K$ decreases or the phonon frequency ${\ensuremath{\omega}}_{\ensuremath{\pi}}$ increases, the lattice dimerization and the gap in the fermion spectrum decrease gradually. At some critical value $({\ensuremath{\alpha}}^{2}{/4K)}_{c}$ or ${\ensuremath{\omega}}_{\ensuremath{\pi}c},$ the system becomes gapless and the lattice dimerization disappears. This can be attributed to the fact that the ground state fails to develop the spin-Peierls long-range order because of the quantum lattice fluctuations.

DOI
10.1103/physrevb.56.14414

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Physical Review Letters•Aug 9, 1982•115 citations
An analytical approach, based on the unitary transformation method, has been developed to study the effect of quantum lattice fluctuations on the ground state of an XY\mathrm{XY}XY spin-Peierls chain, which is equivalent to the spinless Su-Schrieffer-Heeger model in half-filling after the Jordan-Wigner transformation. We show that when the spin-phonon coupling constant \ensuremathα2/4K{\ensuremath{\alpha}}^{2}/4K\ensuremathα2/4K decreases or the phonon frequency \ensuremathω\ensuremathπ{\ensuremath{\omega}}_{\ensuremath{\pi}}\ensuremathω\ensuremathπ​ increases, the lattice dimerization and the gap in the fermion spectrum decrease gradually. At some critical value (\ensuremathα2/4K)c({\ensuremath{\alpha}}^{2}{/4K)}_{c}(\ensuremathα2/4K)c​ or \ensuremathω\ensuremathπc,{\ensuremath{\omega}}_{\ensuremath{\pi}c},\ensuremathω\ensuremathπc​, the system becomes gapless and the lattice dimerization disappears. This can be attributed to the fact that the ground state fails to develop the spin-Peierls long-range order because of the quantum lattice fluctuations.
Using a one-dimensional antiferromagnetic Heisenberg chain model with spin-phonon interaction, the effects of quantum lattice fluctuations on the phase diagram and correlation functions in the ground state of the spin-Peierls system are investigated by developing an analytical approach based on the unitary transformation method. To study the nonadiabatic case, the Green's function method is used to implement the perturbation treatment. We show that when the spin-phonon coupling constant decreases or phonon frequency increases, the lattice dimerization and the spin dimerization gap decrease gradually. At some finite critical value, the Peierls dimerization is destroyed by the quantum lattice fluctuations and the system becomes gapless. The inverse-square-root singularity of the density of states at the gap edge in the adiabatic case disappears because of the nonadiabatic effect, which is consistent with the measurement of the optical absorption spectrum in quasi-one-dimensional spin-Peierls systems. The dimerization gap is much more reduced by the quantum lattice fluctuations than the spin dimer order parameter is. This could result in a true spin dimerization gap that is smaller than the activation energy of the optical absorption spectrum.
Using the bosonization technique and Gaussian wave functional method, we investi gate the low-energy excitations of a spin-Peierls chain where the spins interact with quantum phonons. The results show that the energy gap of the one- and two- particle excitation states decrease gradually when spin-phonon coupling constant decreases or the phonon frequency increases. A quantum phase transition from a gapless spin-fluid phase to a gapped dimerized phase occurs at a nonzero critica l value of the spin-phonon coupling. Our results indicate that the two-particle excitation spectrum is much more sensitive to spin-phonon coupling constant than the one-particle excitation spectrum.
The effect of quantum lattice fluctuations on the dimerized ground state is studied in the half-filled-band Takayama-Lin-Liu-Maki model. The nonadiabatic effect due to finite phonon frequency ${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$\ensuremath{\gtrsim}0 is treated through an energy-dependent electron-phonon scattering function \ensuremath{\delta}(${\mathit{k}}^{\ensuremath{'}}$,k) introduced by means of a unitary transformation. This leads to a weakening of the effective (adiabatic) potential stabilizing the dimerized state and results in four-fermion interaction. By decoupling of the electronic correlations we show that our approach gives a good description of the continuous variation of the dimerization as functions of the dimensionless coupling constant g2{\mathit{g}}^{2}g2 and the phonon frequency ${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$ when the ratio ${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$/\ensuremath{\pi}t (t is the electron hopping integral) is not large. Our results show that, at least in the weak-coupling limit, quantum lattice fluctuations change the functional dependence of the dimerization parameter mp{\mathit{m}}_{\mathit{p}}mp​ on the coupling constant even if the ratio ${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$/\ensuremath{\pi}t is small (but finite). In the spin-1/2 case our result is the same as that predicted by Fradkin and Hirsch. But in the spinless case we still predict a long-range dimerization order even if the coupling is weak and ${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$/\ensuremath{\pi}t is finite. In the large ${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$ (antiadiabatic) limit we show that our effective Hamiltonian becomes an n-component Gross-Neveu model and the ratio mp{\mathit{m}}_{\mathit{p}}mp​(${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$=\ensuremath{\infty})/mp{\mathit{m}}_{\mathit{p}}mp​(${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$=0) is equal to 1/cosh(\ensuremath{\pi}/2ng2{\mathit{ng}}^{2}ng2) (n=1: spinless; n=2: spin-1/2). By using the same input parameters as those of some previous authors we get a 7.2% reduction of the dimerization parameter mp{\mathit{m}}_{\mathit{p}}mp​ compared with the adiabatic value. \textcopyright{} 1996 The American Physical Society.
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)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 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.
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 $\ensuremath{\lambda}=0$.