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Physical review. B, Condensed matter

Long-range correlations in nonequilibrium quantum Lorentz gases

M. Yoshimura, T. R. Kirkpatrick

Published Sep 1, 1996• 7 citations
Publisher

Abstract

Spatial correlation functions in noninteracting disordered electronic systems maintained in a nonequilibrium steady state are studied using diagrammatic perturbation theory. The classical result obtained in a previous study is recovered for the correlation function of microscopic number densities of the moving particle and scatterer; it decays as 1/${\mathit{r}}^{2}$ in three dimensions. The number-density correlation function of moving particles is also calculated. We find that spatial correlations at second order in a concentration gradient decay linearly in three dimensions at low temperatures. The connection of this result with mesoscopic fluctuations is discussed. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.7109

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Physical review. B, Condensed matter•Apr 1, 1988•388 citations
Spatial correlation functions in noninteracting disordered electronic systems maintained in a nonequilibrium steady state are studied using diagrammatic perturbation theory. The classical result obtained in a previous study is recovered for the correlation function of microscopic number densities of the moving particle and scatterer; it decays as 1/r2{\mathit{r}}^{2}r2 in three dimensions. The number-density correlation function of moving particles is also calculated. We find that spatial correlations at second order in a concentration gradient decay linearly in three dimensions at low temperatures. The connection of this result with mesoscopic fluctuations is discussed. \textcopyright{} 1996 The American Physical Society.
The absence of a simple fluctuation-dissipation theorem is a major obstacle for studying systems that are not in thermodynamic equilibrium. We show that for a fluid in a nonequilibrium steady state characterized by a constant temperature gradient the commutator correlation functions are still related to response functions; however, the relation is to the bilinear response of products of two observables, rather than to a single linear response function as is the case in equilibrium. This modified fluctuation-response relation holds for both quantum and classical systems. It is both motivated and informed by the long-range correlations that exist in such a steady state and allows for probing them via response experiments. This is of particular interest in quantum fluids, where the direct observation of fluctuations by light scattering would be difficult. In classical fluids it is known that the coupling of the temperature gradient to the diffusive shear velocity leads to correlations of various observables, in particular temperature fluctuations, that do not decay as a function of distance, but rather extend over the entire system. We investigate the nature of these correlations in a fermionic quantum fluid and show that the crucial coupling between the temperature gradient and velocity fluctuations is the same as in the classical case. Accordingly, the nature of the long-ranged correlations in the hydrodynamic regime also is the same. However, as one enters the collisionless regime in the low-temperature limit the nature of the velocity fluctuations changes: they become ballistic rather than diffusive. As a result, correlations of the temperature and other observables are still singular in the long-wavelength limit, but the singularity is weaker than in the hydrodynamic regime.
The dependence of the three lowest order spatial correlation functions of a harmonically confined Bose gas on temperature and interaction strength is presented at equilibrium. Our analysis is based on a stochastic Langevin equation for the order parameter of a weakly interacting gas. Comparison of the predicted first order correlation functions to those of appropriate mean field theories demonstrates the potentially crucial role of density fluctuations on the equilibrium coherence length. Furthermore, the change in both coherence length and shape of the correlation function, from Gaussian to exponential, with increasing temperature is quantified. Moreover, the presented results for higher order correlation functions are shown to be in agreement with existing predictions. Experimentalists often define a 'quasi-condensate' via bimodal density fits, and this work reveals how consideration of local density-density correlations provides an alternative determination of this quantity. © 2006 The American Physical Society.
The fluctuation-dissipation theorem is applied to the study of the pair correlation functions of the system containing charged particles which is not invariant under spatial translation. The results are expressed in terms of the longitudinal response functions and involve correlation functions which depend only on screened short-range Coulomb interactions S0{S}_{0}S0​. In the case of the electron gas, the density-density correlation function is $S(\mathbf{r}{\mathbf{r}}^{\ensuremath{'}};\ensuremath{\omega})=[\ensuremath{\delta}(\mathbf{r}\ensuremath{-}\mathbf{R})\ensuremath{-}K(r{\mathbf{R}}^{\ensuremath{'}\ensuremath{'}};\ensuremath{\omega})v({\mathbf{R}}^{\ensuremath{'}\ensuremath{'}}\mathbf{R})]{S}_{0}(\mathbf{R}{\mathbf{R}}^{\ensuremath{'}};\ensuremath{\omega})[\ensuremath{\delta}({\mathbf{R}}^{\ensuremath{'}}\ensuremath{-}{\mathbf{r}}^{\ensuremath{'}})\ensuremath{-}v({\mathbf{R}}^{\ensuremath{'}}{\mathbf{R}}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}})\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}({\mathbf{R}}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}{\mathbf{r}}^{\ensuremath{'}};\ensuremath{-}\ensuremath{\omega})]$where KKK is the response function of the electron density to an external potential field, and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}$ its conjugate; vvv represents the Coulomb interaction; and the repeated spatial coordinates are integrated over the volume. It is shown that this expression implies the principle of superposition of dressed particles (the bare particles plus their associated screening clouds); for the calculations of the correlation functions, the dressed particles may be superposed without further consideration of the long-range parts of Coulomb interactions. Similar reductions are carried out for the various correlation functions in the electron-ion system, and the superposition principle is confirned. A possible extension of the results to a nonequilibrium stationarysystem is suggested and discussed. The interrelationship among various longitudinal response functions, which is characteristic of the inhomogeneous system, is discussed and clarified. Simplifications arising from the homogeneity and discussions from diagrammatic considerations are given in Appendices.
There is no simple fluctuation-dissipation theorem (FDT) for nonequilibrium systems. We show that for a fluid in a nonequilibrium steady state (NESS) characterized by a constant temperature gradient there is a generalized FDT that relates commutator correlation functions to the bilinear response of products of observables. This allows for experimental probes of the long-range correlations in such a system, quantum or classical, via response experiments. We also show that the correlations are not tied to thermal fluctuations but are intrinsic to the NESS and reflect a generalized rigidity.
Entanglement measures constitute powerful tools in the quantitative description of quantum many-body systems out of equilibrium. We study entanglement in the current-carrying steady state of a paradigmatic one-dimensional model of noninteracting fermions at zero temperature in the presence of a scatterer. We show that disjoint intervals located on opposite sides of the scatterer, and within similar distances from it, maintain volume-law entanglement regardless of their separation, as measured by their fermionic negativity and coherent information. The mutual information of the intervals, which quantifies the total correlations between them, follows a similar scaling. Interestingly, this scaling entails in particular that if the position of one of the intervals is kept fixed, then the correlation measures depend non-monotonically on the distance between the intervals. By deriving exact expressions for the extensive terms of these quantities, we prove their simple functional dependence on the scattering probabilities, and demonstrate that the strong long-range entanglement is generated by the coherence between the transmitted and reflected parts of propagating particles within the bias-voltage window. The generality and simplicity of the model suggest that this behavior should characterize a large class of nonequilibrium steady states.
The weakly interacting Bose gas in two dimensions is considered in the dilute limit n1/2{n}^{1/2}n1/2a\ensuremath{\ll}1, where n is the particle density and a is the range of the potential. The standard many-body perturbation theory for this system has two separate divergences: the first, associated with classical phase fluctuations, is responsible for the vanishing of the long-range order; the second is quantum mechanical and is connected with the vanishing of the scattering t matrix at long wavelengths and low energies. An earlier diagrammatic theory of Popov, which provides a consistent description of the system in the dilute limit, is rederived heuristically from a quasiparticle picture, and also using the renormalization group. It is shown that the superfluid transition temperature is Tc{T}_{c}Tc​\ensuremath{\approxeq}4\ensuremath{\pi}(${\ensuremath{\Elzxh}}^{2}$/2m)n/[ln ln(1/na2{\mathrm{na}}^{2}na2)], and the condition of validity of the dilute limit is ln ln(1/na2{\mathrm{na}}^{2}na2)\ensuremath{\gg}1. The connection to the dilute Bose gas in dimensions d>2 and the universal behavior beyond the extreme asymptotic low-density domain are also discussed.