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arXiv:1405.2901 · Journal of the Physical Society of Japan

Quantum Phase Transition in One-Dimensional Commensurate Frenkel–Kontorova Model

Yongjun Ma, Jiaxiang Wang, Xinye Xu, Qi Wei, Sabre Kais

Published Dec 1, 2014• 3 citations
arXivHTMLPDF

Abstract

In this paper, we have studied the one-dimensional commensurate quantum Frenkel-Kontorova model by a density-matrix renormalization group (DMRG) algorithm. The focus has been on its properties of the entanglement, the coordinate correlation, the ground state energy and the energy gap between the ground state and the first excited one. It is demonstrated that a quantum phase transition (QPT) between the pinned and the sliding phases takes place as the quantum fluctuation, measured by an effective Planck constant $\tilde\hbar$, increases to a threshold value $\tilde \hbar_c$.

DOI
10.7566/jpsj.83.124603

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In this paper, we have studied the one-dimensional commensurate quantum Frenkel-Kontorova model by a density-matrix renormalization group (DMRG) algorithm. The focus has been on its properties of the entanglement, the coordinate correlation, the ground state energy and the energy gap between the ground state and the first excited one. It is demonstrated that a quantum phase transition (QPT) between the pinned and the sliding phases takes place as the quantum fluctuation, measured by an effective Planck constant ℏ~\tilde\hbarℏ~, increases to a threshold value ℏ~c\tilde \hbar_cℏ~c​.
In this paper, the one-dimensional incommensurate quantum Frenkel-Kontorova model is investigate by a density-matrix renormalization group algorithm. Special attention is given to the entanglement and the ground state energy. The energy gap between the ground state and the first excited is also calculated. From all the numerical results, we have observed an obvious property changes from the pinned state to the sliding one as the quantum fluctuation is increased. But no expected quantum critical point can be justified by the present data.
We study analytically and numerically the one-dimensional quantum Frenkel-Kontorova chain in the regime where the classical model is located in the pinned phase characterized by the gaped phonon excitations and devil's staircase. By extensive quantum Monte Carlo simulations, we show that for the effective Planck constant Planck smaller than the critical value Planck(c) the quantum chain is in the pinned instanton glass phase. In this phase, the elementary excitations have two branches: phonons, separated from zero energy by a finite gap, and instantons that have an exponentially small excitation energy. At Planck = Planck(c) the quantum phase transition takes place and for Planck > Planck(c) the pinned instanton glass is transformed into the sliding phonon gas with gapless phonon excitations. This transition is accompanied by the divergence of the spatial correlation length and appearance of sliding modes at Planck > Planck(c).
By using the density-matrix renormalization group technique, the incommensurate quantum Frenkel-Kontorova model is investigated numerically. It is found that when the quantum fluctuation is strong enough, the ggg function featured by a sawtooth map in the depinned state will show a different kind of behavior, similar to a standard map, but with reduced magnitude. The related position correlations are studied in detail, which leads to a potentially interesting application to the recently well-explored phase transitions in cold atoms loaded in optical lattices.
We study the ground state quantum phase transition by means of entanglement in the one-dimensional asymmetric Hubbard model with open boundary conditions. The local entanglement between the middle two sites and the rest of the system, and the block entanglement between the left and right portions of the system, are calculated using the density-matrix renormalization group (DMRG) method. We find that the entanglement shows interesting scaling and singular behavior around the phase transition line.
The wave function of an incommensurate ground state for a one-dimensional discrete sine-Gordon model—the Frenkel-Kontorova (FK) model—at zero temperature is calculated by the quantum Monte Carlo method. It is found that the ground state wave function crosses over from an extended state to a localized state when the coupling constant exceeds a certain critical value. So, although the quantum fluctuation has smeared out the breaking of analyticity transition as observed in the classical case, the remnant of this transition is still discernible in the quantum regime.