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Physical Review Letters

Dynamics of sliding charge-density waves in 4-ε dimensions

Onuttom Narayan, Daniel S. Fisher

Published Jun 15, 1992• 101 citations
Publisher

Abstract

The critical behavior of sliding charge-density waves just above threshold is analyzed. In 4-\ensuremath{\epsilon} dimensions, the exponent \ensuremath{\zeta}, defined through ${\mathit{I}}_{\mathrm{CDW}}$\ensuremath{\sim}(V-${\mathit{V}}_{\mathit{T}}$${)}^{\mathrm{\ensuremath{\zeta}}}$, is found to be 1-\ensuremath{\epsilon}/6+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). Two different correlation lengths are found, with exponents \ensuremath{\nu}=1/2 exactly and ${\ensuremath{\nu}}_{\mathrm{FS}}$=2/d+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). These results compare favorable with recent numerical results in two and three dimensions.

DOI
10.1103/physrevlett.68.3615

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The critical behavior of sliding charge-density waves just above threshold is analyzed. In 4-\ensuremath{\epsilon} dimensions, the exponent \ensuremath{\zeta}, defined through ICDW{\mathit{I}}_{\mathrm{CDW}}ICDW​\ensuremath{\sim}(V-{\mathit{V}}_{\mathit{T}}$${)}^{\mathrm{\ensuremath{\zeta}}}, is found to be 1-\ensuremath{\epsilon}/6+O(\ensuremathϵ2{\mathrm{\ensuremath{\epsilon}}}^{2}\ensuremathϵ2). Two different correlation lengths are found, with exponents \ensuremath{\nu}=1/2 exactly and \ensuremathνFS{\ensuremath{\nu}}_{\mathrm{FS}}\ensuremathνFS​=2/d+O(\ensuremathϵ2{\mathrm{\ensuremath{\epsilon}}}^{2}\ensuremathϵ2). These results compare favorable with recent numerical results in two and three dimensions.
Charge-density waves in the Hubbard chain are studied by means of finite-temperature quantum Monte Carlo simulations and Lanczos diagonalizations for the ground state. We present results both for the charge susceptibilities and for the charge structure factor at densities $\ensuremath{\rho}=1/6$ and 1/3; for $\ensuremath{\rho}=1/2$ (quarter filled) we only present results for the charge structure factor. The data are consistent with a 4kF{4k}_{F}4kF​ instability dominating over the 2kF{2k}_{F}2kF​ one, at least for sufficiently large values of the Coulomb repulsion U. This can only be reconciled with the Luttinger liquid analyses if the amplitude of the 2kF{2k}_{F}2kF​ contribution vanishes above some ${U}^{*}(\ensuremath{\rho}).$
The critical behavior of charge-density waves (CDWs) in the pinned phase is studied for applied fields increasing toward the threshold field, using recently developed renormalization-group techniques and simulations of automaton models. Despite the existence of many metastable states in the pinned state of the CDW, the renormalization-group treatment can be used successfully to find the divergences in the polarization and the correlation length, and, to first order in an \ensuremath{\epsilon}=4-d expansion, the diverging time scale. The automaton models studied are a charge-density wave model and a ``sandpile'' model with periodic boundary conditions; these models are found to have the same critical behavior, associated with diverging avalanche sizes. The numerical results for the polarization and the diverging length and time scales in dimensions d=2,3 are in agreement with the analytical treatment. These results clarify the connections between the behavior above and below threshold: the characteristic correlation lengths on both sides of the transition diverge with different exponents. The scaling of the distribution of avalanches on the approach to threshold is found to be different for automaton and continuous-variable models.
Large-scale numerical simulations are used to study the elastic dynamics of two-dimensional vortex lattices driven on a disordered medium in the case of weak disorder. We investigate the so-called elastic depinning transition by decreasing the driving force from the elastic dynamical regime to the state pinned by the quenched disorder. Similarly to the plastic depinning transition, we find results compatible with a second-order phase transition, although both depinning transitions are very different from many viewpoints. We evaluate three critical exponents of the elastic depinning transition. β = 0.29 ± 0.03 is found for the velocity exponent at zero temperature, and from the velocity–temperature curves we extract the critical exponent δ−1 = 0.28 ± 0.05. Furthermore, in contrast with charge-density waves, a finite-size scaling analysis suggests the existence of a unique diverging length at the depinning threshold with an exponent ν = 1.04 ± 0.04, which controls the critical force distribution, the finite-size crossover force distribution and the intrinsic correlation length. Finally, a scaling relation is found between velocity and temperature with the β and δ critical exponents both independent with regard to pinning strength and disorder realizations.
The zero-field, two-point correlation function of an nnn-vector system in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions is calculated to order ${\ensuremath{\epsilon}}^{2}$ for T> TcT>~{T}_{c}T> Tc​. The scaling function is obtained as a closed, cutoff-independent integral. As $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$ at fixed wave vector qqq, the leading variation is ${{\stackrel{^}{E}}_{1}}^{n,d}(q){t}^{1\ensuremath{-}\ensuremath{\alpha}}+{{\stackrel{^}{E}}_{2}}^{n,d}(q)t$, where $\ensuremath{\alpha}$ is the specific-heat exponent; thence the maximum in the scattering above Tc{T}_{c}Tc​ is located, in good agreement with high-TTT series-expansion estimates.
Large-scale numerical simulations are used to study the elastic dynamics of two-dimensional vortex lattices driven on a disordered medium in the case of weak disorder. We investigate the so-called elastic depinning transition by decreasing the driving force from the elastic dynamical regime to the state pinned by the quenched disorder. Similarly to the plastic depinning transition, we find results compatible with a second-order phase transition, although both depinning transitions are very different from many viewpoints. We evaluate three critical exponents of the elastic depinning transition. β = 0.29 ± 0.03 is found for the velocity exponent at zero temperature, and from the velocity–temperature curves we extract the critical exponent δ −1 = 0.28 ± 0.05. Furthermore, in contrast with charge-density waves, a finite-size scaling analysis suggests the existence of a unique diverging length at the depinning threshold with an exponent ν = 1.04 ± 0.04, which controls the critical force distribution, the finite-size crossover force distribution and the intrinsic correlation length. Finally, a scaling relation is found between velocity and temperature with the β and δ critical exponents both independent with regard to pinning strength and disorder realizations.