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arXiv:cond-mat/0504017 · Physical Review B

Infrared catastrophe and tunneling into strongly correlated electron systems: Perturbative x-ray edge limit

Kelly R. Patton, Michael R. Geller

Published Sep 9, 2005• 4 citations
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Abstract

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.

DOI
10.1103/physrevb.72.125108

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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.
We develop a nonperturbative method to calculate the electron propagator in low-dimensional and strongly correlated electron systems. The method builds on our earlier work using a Hubbard-Stratonovich transformation to map the tunneling problem to the x-ray edge problem, which accounts for the infrared catastrophe caused by the sudden introduction of a new electron into a conductor during a tunneling event. Here we use a cumulant expansion to include fluctuations about this x-ray edge limit. We find that the dominant effect of electron-electron interaction at low energies is to correct the noninteracting Green's function by a factor ${e}^{\ensuremath{-}S∕\ensuremath{\hbar}}$, where SSS can be interpreted as the Euclidean action for a density field describing the time-dependent charge distribution of the newly added electron. Initially localized, this charge distribution spreads in time as the electron is accommodated by the host conductor, and during this relaxation process action is accumulated according to classical electrostatics with a screened interaction. The theory applies to lattice or continuum models of any dimensionality, with or without translational invariance. In one dimension the method correctly predicts a power-law density of states for electrons with short-range interaction and no disorder, and when applied to the solvable Tomonaga-Luttinger model, the exact density of states is obtained.
A general nonperturbative theory of the low-energy electron propagator is developed and used to calculate the single-particle density of states in a variety of systems. This method involves the decoupling of the electron-electron interaction through a Hubbard-Stratonovich transformation, followed by a saddle-point approximation of the remaining functional integral. The final expression is found to be the tunneling analog of the infrared catastrophe that occurs in the x-ray edge problem; here, the host system responds to the potential produced by the abrupt addition of an electron during a tunneling event. This response can lead to a suppression in the tunneling density of states near the Fermi energy. This method is adaptable to lattice or continuum models of any dimensionality, with or without translational invariance. When applied, the exact density of states is obtained for the Tomonaga-Luttinger model, and the pseudogap of a fractional quantum Hall fluid is recovered.
In previous work [K. R. Patton and M. R. Geller, Phys. Rev. B 72, 125108 (2005)] we have proposed that the non-Fermi-liquid spectral properties in a variety of low-dimensional and strongly correlated electron systems are caused by the infrared catastrophe, and we used an exact functional integral representation for the interacting Green's function to map the tunneling problem onto the x-ray edge problem, plus corrections. The corrections are caused by the recoil of the tunneling particle, and, in systems where the method is applicable, are not expected to change the qualitative form of the tunneling density of states (DOS). Qualitatively correct results were obtained for the DOS of the one-dimensional electron gas and two-dimensional Hall fluid when the corrections to the x-ray edge limit were neglected and when the corresponding Nozi\`eres-De Dominicis integral equations were solved by resummation of a divergent perturbation series. Here we reexamine the x-ray edge limit for these two models by solving these integral equations exactly, finding the expected modifications of the DOS exponent in the one-dimensional case but finding no changes in the DOS of the two-dimensional Hall fluid with short-range interaction. Our analysis provides an exact solution of the Nozi\`eres-De Dominicis equation for the two-dimensional electron gas in the lowest Landau level.
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.
We calculate the exponents of the threshold singularities in the photoemission spectrum of a deep core hole and its x-ray absorption spectrum in the framework of a systematic many-body theory of slave bosons and pseudofermions (for the empty and occupied core level). In this representation, photoemission and x-ray absorption can be understood on the same footing; no distinction between orthogonality catastrophe and excitonic effects is necessary. We apply the conserving slave particle T-matrix approximation (CTMA), recently developed to describe both Fermi and non-Fermi liquid behavior in systems with strong local correlations, to the x-ray problem as a test case. The numerical results for both photoemission and x-ray absorption are found to be in agreement with the exact infrared power law behavior in the weak as well as in the strong coupling regions. We point out a close relation of the CTMA with the parquet equation approach of Nozi\`eres et al.