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arXiv:cond-mat/9405057 · Journal of Physics A Mathematical and General

The Fermi edge singularity and boundary condition changing operators

Ian K. Affleck, A W W Ludwig

Published Aug 21, 1994• 151 citations
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Abstract

The boundary conformal field theory approach to quantum impurity problems is used to study the Fermi edge singularity, occurring in the X-ray adsorption probability. The deep-hole creation operator, in the effective low-energy theory, changes the boundary condition on the conduction electrons. By a conformal mapping, the dimension of such an operator is related to the ground-state energy for a finite system with different boundary conditions at the two ends. The Fermi edge singularity is solved, using this method, for the Luttinger liquid including backscattering and for the multi-channel Kondo problem.

DOI
10.1088/0305-4470/27/16/007

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The boundary conformal field theory approach to quantum impurity problems is used to study the Fermi edge singularity, occurring in the X-ray adsorption probability. The deep-hole creation operator, in the effective low-energy theory, changes the boundary condition on the conduction electrons. By a conformal mapping, the dimension of such an operator is related to the ground-state energy for a finite system with different boundary conditions at the two ends. The Fermi edge singularity is solved, using this method, for the Luttinger liquid including backscattering and for the multi-channel Kondo problem.
We report on exact results for the low-temperature thermodynamics of a spin- 12\frac{1}{2}21​ magnetic impurity coupled to a one-dimensional interacting electron system. By using boundary conformal field theory, we show that there are only two types of critical behaviors consistent with the symmetries of the problem: either a local Fermi liquid or a theory with an anomalous response identical to that recently proposed by Furusaki and Nagaosa. Suppression of backscattering off the impurity leads to the same critical properties as for the two-channel Kondo effect.
Impurities are ubiquitous in condensed matter. Boundary Conformal Field Theory (BCFT) provides a powerful method to study a localized quantum impurity interacting with a gapless continuum of excitations. The results can also be implied to nanoscopic devices like quantum dots. In these lecture notes, I review this field, including the following topics: I. General Renormalization Group (RG) framework for quantum impurity problems: example of simplest Kondo model II. Multi-channel Kondo model III. Quantum Dots: experimental realizations of one and two channel Kondo models IV. Impurities in Luttinger liquids: point contact in a quantum wire V. Quantum impurity entanglement entropy VI. Y-junctions of Luttinger liquids VII. Boundary condition changing operators and the X-ray edge problem.
The problem of the Fermi-edge singularity in a one-dimensional Tomonaga-Luttinger liquid is reconsidered. The backward scattering of the conduction band electrons on the impuritylike hole in the valence band is analyzed by mapping the problem onto a Coulomb gas theory. For the case when the electron-electron interaction is repulsive, the obtained exponent of the one-dimensional Fermi-edge singularity appears to be different from the exponent found in previous studies. It is shown that the infrared physics of the Fermi-edge singularity in the presence of backward scattering and electron-electron repulsion resembles the physics of the Kondo problem. \textcopyright{} 1996 The American Physical Society.
We construct the boundary conformal field theory that describes the low-temperature behavior of the two-channel Anderson impurity model. The presence of an exactly marginal operator is shown to generate a line of stable fixed points parametrized by the charge valence nc{n}_{c}nc​ of the impurity. We calculate the exact zero-temperature entropy and impurity thermodynamics along the fixed line. We also derive the critical exponents of the characteristic Fermi edge singularities caused by time-dependent hybridization between conduction electrons and impurity. Our results suggest that in the mixed-valent regime ${(n}_{c}\ensuremath{\ne}0,1)$ the electrons participate in two competing processes, leading to frustrated screening of spin and channel degrees of freedom. By combining the boundary conformal field theory with the Bethe-Ansatz solution we obtain a complete description of the low-energy dynamics of the model.
We report the exact calculation of the Fermi-edge singularity exponent for correlated electrons in one dimension (Luttinger liquid). Focusing on the special interaction parameter g=1/2g=1/2g=1/2, the asymptotic long-time behavior can be obtained using the Wiener-Hopf method. The result confirms the previous assumption of an open boundary fixed point. In addition, a dynamic kkk-channel Kondo impurity is studied via Abelian bosonization for k=2k=2k=2 and k=4k=4k=4. It is shown that the corresponding orthogonality exponents are related to the orthogonality exponent in a Luttinger liquid.
We study the low-temperature properties of a spin-1/2 magnetic impurity coupled to a one-dimensional interacting electron system. Using a formalism by Affleck and Ludwig, with a scale invariant boundary condition replacing the impurity, we exploit boundary conformal field theory to deduce the impurity thermal and magnetic response. In the case of only forward electron scattering off the impurity, we predict the same critical scaling as for the two-channel Kondo effect for noninteracting electrons, but with a different Wilson ratio. Backward electron scattering off the impurity destabilizes this behavior and drives the system to a new fixed point. In the case of equal amplitudes for forward and backward scattering (Kondo interaction), we show that there are only two types of scaling behaviors consistent with the symmetries of the problem: either a local Fermi liquid or a critical theory with an anomalous specific heat. The latter case agrees with a recent poor-man's scaling result proposed by Furusaki and Nagaosa. \textcopyright{} 1996 The American Physical Society.