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arXiv:1011.1993 · Physical Review Letters

Gradient Catastrophe and Fermi-Edge Resonances in Fermi Gas

Eldad Bettelheim, Y. Kaplan, Paul B. Wiegmann

Published Apr 20, 2011• 7 citations
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Abstract

Any smooth spatial disturbance of a degenerate Fermi gas inevitably becomes sharp. This phenomenon, called the gradient catastrophe, causes the breakdown of a Fermi sea to multiconnected components characterized by multiple Fermi points. We argue that the gradient catastrophe can be probed through a Fermi-edge singularity measurement. In the regime of the gradient catastrophe the Fermi-edge singularity problem becomes a nonequilibrium and nonstationary phenomenon. We show that the gradient catastrophe transforms the single-peaked Fermi-edge singularity of the tunneling (or absorption) spectrum to a sequence of multiple asymmetric singular resonances. An extension of the bosonic representation of the electronic operator to nonequilibrium states captures the singular behavior of the resonances.

DOI
10.1103/physrevlett.106.166804

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Any smooth spatial disturbance of a degenerate Fermi gas inevitably becomes sharp. This phenomenon, called the gradient catastrophe, causes the breakdown of a Fermi sea to multiconnected components characterized by multiple Fermi points. We argue that the gradient catastrophe can be probed through a Fermi-edge singularity measurement. In the regime of the gradient catastrophe the Fermi-edge singularity problem becomes a nonequilibrium and nonstationary phenomenon. We show that the gradient catastrophe transforms the single-peaked Fermi-edge singularity of the tunneling (or absorption) spectrum to a sequence of multiple asymmetric singular resonances. An extension of the bosonic representation of the electronic operator to nonequilibrium states captures the singular behavior of the resonances.
A semiclassical wave packet propagating in a dissipationless Fermi gas inevitably enters a "gradient catastrophe" regime, where an initially smooth front develops large gradients and undergoes a dramatic shock-wave phenomenon. The nonlinear effects in electronic transport are due to the curvature of the electronic spectrum at the Fermi surface. They can be probed by a sudden switching of a local potential. In equilibrium, this process produces a large number of particle-hole pairs, a phenomenon closely related to the orthogonality catastrophe. We study a generalization of this phenomenon to the nonequilibrium regime and show how the orthogonality catastrophe cures the gradient catastrophe, by providing a dispersive regularization mechanism.
We formulate the problem of the Fermi edge singularity in non-equilibrium states of a Fermi gas as a matrix Riemann–Hilbert problem with an integrable kernel. This formulation is the most suitable for studying the singular behavior at each edge of non-equilibrium Fermi states by means of the method of steepest descent, and also reveals the integrable structure of the problem. We supplement this result by extending the familiar approach to the problem of the Fermi edge singularity via the bosonic representation of the electronic operators to non-equilibrium settings. It provides a compact way to extract the leading asymptotes.
The Fermi-edge singularity and the Anderson orthogonality catastrophe describe the universal physics which occurs when a Fermi sea is locally quenched by the sudden switching of a scattering potential, leading to a brutal disturbance of its ground state. We demonstrate that the effect can be seen in the controllable domain of ultracold trapped gases by providing an analytic description of the out-of-equilibrium response to an atomic impurity, both at zero and at finite temperature. Furthermore, we link the transient behavior of the gas to the decoherence of the impurity, and to the degree of the non-Markovian nature of its dynamics.
The nonperturbative solution to the problem of threshold singularities for a (${\mathrm{\ensuremath{\mu}}}_{1}$,${\mathrm{\ensuremath{\mu}}}_{2}$) nonequilibrium Fermi sea is obtained using the determinantal method of Ohtaka and Tanabe. The critical exponents of the absorption power-law behavior we find agree with those estimated from the perturbative treatment of the problem given in papers I and II. A family of possibly diverging singularities is found at energies ${\mathrm{\ensuremath{\mu}}}_{2}$+n(${\mathrm{\ensuremath{\mu}}}_{1}$-${\mathrm{\ensuremath{\mu}}}_{2}$), for n\ensuremath{\ge}1.
The tunneling density of states exhibits anomalies (cusps, algebraic suppressions, and pseudogaps) at the Fermi energy in a wide variety of low-dimensional and strongly correlated electron systems. We argue that in many cases these spectral anomalies are caused by an infrared catastrophe in the screening response to the sudden introduction of a new electron into the system during a tunneling event. A nonperturbative functional integral method is introduced to account for this effect, making use of methods developed for the x-ray edge singularity problem. The formalism is applicable to lattice or continuum models of any dimensionality, with or without translational invariance. An approximate version of the technique is applied to the 1D electron gas and the 2D Hall fluid, yielding qualitatively correct results.