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arXiv:cond-mat/9604054 · Physical Review Letters

Density Matrix Renormalization Group Applied to the Ground State of theXYSpin-Peierls System

Laurent G. Caron, S. Moukouri

Published May 20, 1996• 72 citations
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Abstract

We use the density matrix renormalization group (DMRG) to map out the ground state of an XY spin chain coupled to dispersionless phonons of frequency $\ensuremath{\omega}$. We confirm the existence of a critical spin-phonon coupling ${\ensuremath{\alpha}}_{c}\ensuremath{\propto}{\ensuremath{\omega}}^{0.7}$ for the onset of the spin gap bearing the signature of a Kosterlitz-Thouless transition. We also observe a classical-quantum crossover when the spin-Peierls gap $\ensuremath{\Delta}$ is of order $\ensuremath{\omega}$. In the classical regime $\ensuremath{\Delta}>\ensuremath{\omega}$, the mean-field parameters are strongly renormalized by nonadiabatic corrections. This is the first application of the DMRG to phonons.

DOI
10.1103/physrevlett.76.4050

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We use the density matrix renormalization group (DMRG) to map out the ground state of an XY spin chain coupled to dispersionless phonons of frequency \ensuremathω\ensuremath{\omega}\ensuremathω. We confirm the existence of a critical spin-phonon coupling \ensuremathαc\ensuremath∝\ensuremathω0.7{\ensuremath{\alpha}}_{c}\ensuremath{\propto}{\ensuremath{\omega}}^{0.7}\ensuremathαc​\ensuremath∝\ensuremathω0.7 for the onset of the spin gap bearing the signature of a Kosterlitz-Thouless transition. We also observe a classical-quantum crossover when the spin-Peierls gap \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ is of order \ensuremathω\ensuremath{\omega}\ensuremathω. In the classical regime \ensuremathΔ>\ensuremathω\ensuremath{\Delta}>\ensuremath{\omega}\ensuremathΔ>\ensuremathω, the mean-field parameters are strongly renormalized by nonadiabatic corrections. This is the first application of the DMRG to phonons.
The density-matrix renormalization-group method is used to investigate the spin-Peierls transition for Heisenberg spins coupled to quantized phonons. We use a phonon spectrum that interpolates between a gapped, dispersionless (Einstein) limit to a gapless, dispersive (Debye) limit. A variety of theoretical probes are used to determine the quantum phase transition, including energy gap crossing, a finite-size scaling analysis, bond-order autocorrelation functions, and bipartite quantum entanglement. All these probes indicate that in the antiadiabatic phonon limit a quantum phase transition of the Berezinskii-Kosterlitz-Thouless type is observed at a nonzero spin-phonon coupling, gc{g}_{\text{c}}gc​. An extrapolation from the Einstein limit to the Debye limit is accompanied by an increase in gc{g}_{\text{c}}gc​ for a fixed optical $(q=\ensuremath{\pi})$ phonon gap. We therefore conclude that the dimerized ground state is more unstable with respect to Debye phonons with the introduction of phonon-dispersion renormalizing the effective spin-lattice coupling for the Peierls-active mode. We also show that the staggered spin-spin and phonon displacement order parameters are unreliable means of determining the transition.
The density matrix renormalization group method is used to investigate the Peierls transition for the extended Hubbard model coupled to quantized phonons. Following our earlier work on spin-Peierls systems, we use a phonon spectrum that interpolates between a gapped, dispersionless (Einstein) limit and a gapless, dispersive (Debye) limit to investigate the entire frequency range. A variety of theoretical probes are used to determine the quantum phase transition, including energy gap crossing, a finite-size scaling analysis, and bipartite quantum entanglement. All these probes indicate that a transition of Berezinskii-Kosterlitz-Thouless type is observed at a nonzero electron-phonon coupling gc{g}_{c}gc​ for a nonvanishing electron-electron interaction. An extrapolation from the Einstein limit to the Debye limit is accompanied by an increase in gc{g}_{c}gc​ for a fixed optical ($q=\ensuremath{\pi}$) phonon gap. We therefore conclude that the dimerized ground state is more unstable with respect to Debye phonons, with the introduction of phonon dispersion renormalizing the effective electron-lattice coupling for the Peierls-active mode. By varying the Coulomb interaction UUU, we observe a generalized Peierls transition, intermediate between the uncorrelated (U=0U=0U=0) and spin-Peierls ($U\ensuremath{\rightarrow}\ensuremath{\infty}$) limits, where UUU is the Hubbard Coulomb parameter. Using the extended Hubbard model with Debye phonons, we investigate the Peierls transition in trans-polyacetylene and show that the transition is close to the critical regime.
The spin-Peierls instability in a spin- XY chain coupled to dispersionless phonons of frequency has been studied in the non-adiabatic limit. We have chosen the Lang-Firsov variational wavefunction for the phonon subsystem to obtain an effective spin Hamiltonian. The effective spin Hamiltonian is then solved in the framework of the mean-field approximation and bond operator technique. We observed two types of phase transitions, one is from the spin liquid phase to the dimerized phase and the other is from the dimerized phase to the antiferromagnetic phase as we vary the spin-phonon coupling from a very low value. The variation of lattice distortion, dimerized order parameter and energy gap with spin-phonon coupling parameter have also been investigated here.
Using a new version of the density-matrix renormalization group we determine the phase diagram of a model of an antiferromagnetic Heisenberg spin chain where the spins interact with quantum phonons. A quantum phase transition from a gapless spin-fluid state to a gapped dimerized phase occurs at a nonzero value of the spin-phonon coupling. The transition is in the same universality class as that of a frustrated spin chain, to which the model maps in the diabatic limit. We argue that realistic modeling of known spin-Peierls materials should include the effects of quantum phonons.
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.