Abstract
Using functional methods and the exact renormalization group we derive Ward identities for the Anderson impurity model. In particular, we present a non-perturbative proof of the Yamada–Yosida identities relating certain coefficients in the low-energy expansion of the self-energy to the thermodynamic particle number and spin susceptibilities of the impurity. Our proof underlines the relation of the Yamada–Yosida identities to the U(1) × U(1) symmetry associated with the particle number and spin conservation in a magnetic field.
Highlights
In quantum field theory symmetries and the associated conservation laws imply Ward identities, which are the exact relations between different types of Green’s functions or vertex functions [1]
We have used modern functional methods to give a non-perturbative proof of the Yamada–Yosida identities, which express the coefficients in the low-frequency expansion of the self-energy of the Anderson impurity model (AIM) in terms of thermodynamic susceptibilities
In contrast to the derivation of these relations given by Yamada and Yosida [2], which is based on a series expansion in powers of the interaction, our non-perturbative proof relies on exact Ward identities and on an exact renormalization group flow equation for the irreducible self-energy
Summary
In quantum field theory symmetries and the associated conservation laws imply Ward identities, which are the exact relations between different types of Green’s functions or vertex functions [1]. An alternative derivation using diagrammatic techniques can be found in the book by Hewson [4] In both approaches the close relation of the above identities to the U (1) × U (1) symmetry associated with the particle number and spin conservation of the AIM in a magnetic field is not manifest. We present such a non-perturbative proof of equations (1.1) and (1.2) by combining standard functional techniques [1] with certain exact relations between the derivatives of the self-energy with respect to frequency, chemical potential and magnetic field which we derive within the framework of the exact renormalization group [5, 6].
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