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arXiv:cond-mat/0510601 · Physical Review B

Infrared catastrophe and tunneling into strongly correlated electron systems: Exact x-ray edge limit for the one-dimensional electron gas and two-dimensional Hall fluid

Kelly R. Patton, Michael R. Geller

Published Jun 7, 2006• 1 citation
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Abstract

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.

DOI
10.1103/physrevb.73.245306

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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.
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.
We study a model of a quantum Hall system with the electrons confined to a narrow, linear channel. The system is mapped to a one-dimensional (1D) system which in the low-energy approximation has the form of a Luttinger liquid with different interactions between particles of equal and of opposite chiralities. In a previous paper (Part I) [M. Horsdal and J. M. Leinaas, Phys. Rev. B 76, 195321 (2007)], we studied this mapping at the microscopic level, and examined how the interaction in the two-dimensional (2D) system renormalizes 1D Luttinger liquid parameters. We follow up this study here and derive the low-energy form of the 2D electron correlation function by applying the mapping from the 1D correlation function. The derived expression has a natural separation in a bulk part, which is not modified by interactions, and an edge part, which is modified. We study the effect of interactions on the electron density and on the asymptotic form of the correlation function. The asymptotic parameter is found to be consistent with the expected value from the Luttinger liquid theory, but deviations from the asymptotic form are found in a rather wide intermediate interval. We suggest some questions of interest for further study.
We describe the relationship between quantum Hall edge states and the one-dimensional Luttinger liquid model. The Luttinger liquid model originated from studies of one-dimensional Fermi systems, however, it results that many ideas inspired by such a model can find applications to phenomena occurring even in higher dimensions. Quantum Hall systems which essentially are correlated two-dimensional electronic systems in a strong perpendicular magnetic field have an edge. It turns out that the quantum Hall edge states can be described by a one-dimensional Luttinger model. In this work, we give a general background of the quantum Hall and Luttinger liquid physics and then point out the relationship between the quantum Hall edge states and its one-dimensional Luttinger liquid representation. Such a description is very useful given that the Luttinger liquid model has the property that it can be bosonized and solved. The fact that we can introduce a simpler model of noninteracting bosons, even if the quantum Hall edge states of electrons are interacting, allows one to calculate exactly various quantities of interest. One such quantity is the correlation function which, in the asymptotic limit, is predicted to have a power law form. The Luttinger liquid model also suggests that such a power law exponent should have a universal value. A large number of experiments have found the quantum Hall edge states to show behavior consistent with a Luttinger liquid description. However, while a power law dependence of the correlation function has been observed, the experimental values of the exponent appear not to be universal. This discrepancy might be due to various correlation effects between electrons that sometimes are not easy to incorporate within a standard Luttinger liquid model.
We describe the relationship between quantum Hall edge states and the one-dimensional Luttinger liquid model. The Luttinger liquid model originated from studies of one-dimensional Fermi systems, however, it results that many ideas inspired by such a model can find applications to phenomena occurring even in higher dimensions. Quantum Hall systems which essentially are correlated two-dimensional electronic systems in a strong perpendicular magnetic field have an edge. It turns out that the quantum Hall edge states can be described by a one-dimensional Luttinger model. In this work, we give a general background of the quantum Hall and Luttinger liquid physics and then point out the relationship between the quantum Hall edge states and its one-dimensional Luttinger liquid representation. Such a description is very useful given that the Luttinger liquid model has the property that it can be bosonized and solved. The fact that we can introduce a simpler model of noninteracting bosons, even if the quantum Hall edge states of electrons are interacting, allows one to calculate exactly various quantities of interest. One such quantity is the correlation function which, in the asymptotic limit, is predicted to have a power law form. The Luttinger liquid model also suggests that such a power law exponent should have a universal value. A large number of experiments have found the quantum Hall edge states to show behavior consistent with a Luttinger liquid description. However, while a power law dependence of the correlation function has been observed, the experimental values of the exponent appear not to be universal. This discrepancy might be due to various correlation effects between electrons that sometimes are not easy to incorporate within a standard Luttinger liquid model.