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Progress of Theoretical Physics Supplement

Application of the t-Adaptive Density Matrix Renormalization Group Method to the Quantum Frenkel-Kontorova Model

Jiaxiang Wang, Bambi Hu, XiaoqunWang XiaoqunWang

Published Jan 1, 2007• 2 citations
Publisher

Abstract

In this paper, the t-adaptive density matrix renormalization group method has been applied to study the dynamics of the one-dimensional quantum Frenkel-Kontorova model. It is found that as the quantum fluctuations increase, the local excitations will become unstable. The unstableness will lead to the dominance of the phonon excitations as the elementary excitations of the system in the deep quantum regime.

DOI
10.1143/ptps.166.95

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In this paper, the t-adaptive density matrix renormalization group method has been applied to study the dynamics of the one-dimensional quantum Frenkel-Kontorova model. It is found that as the quantum fluctuations increase, the local excitations will become unstable. The unstableness will lead to the dominance of the phonon excitations as the elementary excitations of the system in the deep 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.
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.
The density matrix renormalization group (DMRG) method and its applications to finite temperatures and two-dimensional systems are reviewed. The basic idea of the original DMRG method, which allows precise study of the ground-state properties and low-energy excitations, is presented for models which include long-range interactions. The DMRG scheme is then applied to the diagonalization of the quantum transfer matrix for one-dimensional systems, and a reliable algorithm at finite temperatures is formulated. Dynamic correlation functions at finite temperatures are calculated from the eigenvectors of the quantum transfer matrix with analytical continuation to the real frequency axis. An application of the DMRG method to two-dimensional quantum systems in a magnetic field is demonstrated and reliable results for quantum Hall systems are presented.
The Density Matrix Renormalization Group (DMRG) has become a powerful numerical method that can be applied to low-dimensional strongly correlated fermionic and bosonic systems. It allows for a very precise calculation of static, dynamical and thermodynamical properties. Its field of applicability has now extended beyond Condensed Matter, and is successfully used in Statistical Mechanics and High Energy Physics as well. In this article, we briefly review the main aspects of the method. We also comment on some of the most relevant applications so as to give an overview on the scope and possibilities of DMRG and mention the most important extensions of the method such as the calculation of dynamical properties, the application to classical systems, inclusion of temperature, phonons and disorder, field theory, time-dependent properties and the ab initio calculation of electronic states in molecules.