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arXiv:cond-mat/9904287 · Europhysics Letters (EPL)

Ground-state wave function of the quantum Frenkel-Kontorova model

Bingcheng Hu, Baowen Li

Published Jun 1, 1999• 13 citations
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Abstract

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.

DOI
10.1209/epl/i1999-00315-2

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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.
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.
The commensurate Frenkel Kontorova (FK) model is studied using path-integral molecular dynamics (PIMD). We focus on the highly discrete case, in which the embedding potential has a much greater maximum curvature than the harmonic potential connecting two particles in the FK chain. When efficient sampling methods are used, the dynamical interpretation of adiabatic PIMD appears to represent quite accurately the true time correlation functions of this highly correlated many-body system. We have found that the discrete, quantum FK model shows different behavior than its continuum version. The spectral density does not show the characteristic omega-2Theta(omega-omegac) cusp of the continuum solution in the pinned phase (m>m(c)). We also identify a dynamical quantum hysteresis in addition to the regular classical hysteresis when an external force is applied to the FK chain. In the unpinned phase (m<or=m(c)), we find a linear response damping coefficient which is finite and only weakly dependent on temperature T at small values of T.
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​.
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 present a simple and complete variational approach to the one-dimensional quantum Frenkel-Kontorova model. Dirac's time-dependent variational principle is adopted together with a Hartree-type many-body trial wave function for the atoms. The single-particle state is assumed to have the Jackiw-Kerman form. We obtain an effective classical Hamiltonian for the system, which is simple enough for a complete numerical solution for the static ground state of the model. Numerical results show that our simple approach captures the essence of the quantum effects first observed in quantum Monte Carlo studies.
We consider the ground-state phase transition from broken to restored phases of the one-dimensional quantum discrete ${\ensuremath{\varphi}}^{4}\ensuremath{-}{\ensuremath{\varphi}}^{2}$ model as a function of two parameters: the strength of the quantum fluctuations (measured by a dimensionless Planck constant $\overline{\ensuremath{\Elzxh}}$), and the classical dimensionless coupling parameter \ensuremath{\gamma}, which characterizes the strength of interaction of the neighboring atoms. We introduce several distinct variational methods based on self-consistent phonons and soliton wave functions, and compare their predictions for the phase diagram with the (numerically exact) results of previous quantum Monte Carlo simulations. Our calculations show that in the region of weak coupling $(\ensuremath{\gamma}<1),$ both the ``tunneling soliton'' and the ``two-level'' approaches provide good approximations to the true result. In the intermediate coupling region $({\ensuremath{\gamma}}_{c}>\ensuremath{\gamma}>1),$ the method of self-consistent phonons gives a satisfactory description of phase transition. We also compare and contrast our results with previously published approximate methods, and discuss problems for further investigation.