Ward identities for the Anderson impurity model: derivation via functional methods and the exact renormalization group

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 thermodynamic particle number and spin susceptibilities of the impurity. Our proof underlines the relation of the Yamada-Yosida identities to the U(1) x U(1) symmetry associated with particle number and spin conservation in a magnetic field.


I. INTRODUCTION
In quantum field theory symmetries and the associated conservation laws imply Ward identities, which are exact relations between different types of Green functions or vertex functions. 1 The constraints imposed by Ward identities can be very useful to devise accurate approximation schemes which do not violate conservation laws. In this work we shall present a non-perturbative derivation of the Ward identities relating the coefficients in the low-frequency expansion of the retarded self-energy Σ(ω + i0) of the Anderson impurity model (AIM) to certain thermodynamic susceptibilities. A perturbative derivation of these identities has first been given by Yamada and Yosida. 2 For particle-hole symmetric filling and in the limit of an infinite bandwidth of the conduction electron dispersion with flat density of states these identities read Re Σ(ω + i0) = U 2 + 1 −χ c +χ s 2 ω + O(ω 2 ), (1.1) Here, U is the on-site interaction at the impurity site, ∆ is the imaginary part of the hybridization function in the limit of an infinite bandwidth of the conduction electron band, andχ c andχ s are dimensionless particle number (charge) and spin susceptibilities, which will be defined in Eqs. (4.34) and (4.35) below. A generalization of Eqs. (1.1) and (1.2) to the AIM out of equilibrium can be found in Ref. [3]. Yamada and Yosida obtained the above identities by comparing the coefficients in the perturbation series of both sides of Eqs. (1.1) and (1.2) to all orders in powers of U/∆. An alternative derivation using diagrammatic techniques can be found in the book by Hewson. 4 Unfortunately, in both approaches the close relation of the above identities to the U (1) × U (1) symmetry associated with particle number and spin conservation of the AIM in a magnetic field is not manifest. Although powerful functional methods for deriving Ward identities non-perturbatively are well known, 1 apparently there exists no derivation of Eqs. (1.1) and (1.2) in the literature using these functional methods. In this work we shall present such a non-perturbative proof of Eqs. (1.1) and (1.2) by combining standard functional techniques 1 with certain exact relations between derivatives of the selfenergy with respect to frequency, chemical potential and magnetic field which we derive within the framework of the exact renormalization group. 5,6 To set the stage for our calculation and to define our notation, let us recall that the single-site Anderson impurity model is defined in terms of the following second quantized Hamiltonian, whereĉ kσ annihilates a non-interacting conduction electron with momentum k, energy dispersion ǫ k and spin projection σ, while the operatord σ annihilates a localized correlated d-electron with atomic energy E d and spin projection σ. The hybridization between the d-electrons and the conduction electrons is characterized by the hybridization energy V k , and h is the Zeemann energy associated with an external magnetic field. Since we are only interested in the correlation functions of the impurity, we integrate over the conduction electrons using the functional integral representation of the model. 7 The generating functional G c [ σ , j σ ] of the connected Green functions can then be represented as the following ratio of fermionic functional integrals, with the Euclidean action given by Here, ω = 1 β ω denotes summation over fermionic Matsubara frequencies iω and β 0 dτ denotes integration over imaginary time, where β is the inverse temperature. The non-interacting Green function is is the energy of a localized d-electron with spin projection σ relative to the chemical potential µ, and the spindependent hybridization function is given by The Fourier transform of the Grassmann fields d σ (τ ) in frequency space is defined by The functional G c [ σ , j σ ] in Eq. (1.4) depends on Grassmann sources σ and j σ and we have introduced the following notation for the source terms,

II. FUNCTIONAL WARD IDENTITIES
A. U (1) Ward identities due to particle number and spin conservation in a magnetic field The Euclidean action for the correlated impurity given in Eq. (1.5) is invariant under independent global U (1) transformations of the fields for a given spin projection. This symmetry implies that the generating functional G c [ σ , j σ ] satisfies certain functional differential equations, so-called functional Ward identities. By taking functional derivatives of these relations, we shall derive the Yamada-Yosida identities for the self-energy. Following Refs. [1,6], we perform a local gauge transformation on the fermion fields in the (imaginary) time domain, where α σ (τ ) are arbitrary real functions. The interaction part S U of our action S = S 0 +S U is invariant under these transformations, so that where ∆ σ (τ ) = ω e −iωτ ∆ σ (iω). Using the invariance of the functional integral representation (1.4) of G c [ σ , j σ ] with respect to a change of the integration variables d σ → d ′ σ ,d σ →d ′ σ , and expanding to linear order in the gauge factors α σ (τ ) we obtain the desired functional Ward identity. 1,6 For our purpose it is convenient to express the "current terms" of this Ward identity via the where on the right-hand side it is understood that the sources σ and j σ should be calculated as functions of the field averagesd σ and d σ by solving the equations After Fourier transformation to frequency space and some rearrangements analogous to those in Refs. [6,8] we obtain the functional Ward identity whereω is an external bosonic Matsubara frequency. Summing both sides of this functional equation over σ ′ we obtain the functional Ward identity due to the U (1) symmetry associated with particle number conservation, To obtain the analogous U (1) Ward identity associated with conservation of the spin projection along the axis of the magnetic field, we multiply Eq. (2.5) by σ ′ before summing over σ ′ , which yields B. SU (2) Ward identity due to spin conservation In the absence of a magnetic field our action (1.5) is invariant under arbitrary rotations in spin space. To derive the corresponding SU (2) Ward identity, we perform a local rotation in spin space, where the SU (2) matrix U (τ ) can be written as Here, σ = [σ x , σ y , σ z ] is the vector of Pauli matrices and α(τ ) is a time-dependent three-component vector.
Expanding to linear order in α(τ ) we obtain, after the same manipulations as in Sec. II A, the following SU (2) Ward identity, where the superscript i = x, y, z labels the three components of the vector operator σ. Together with the particle number conservation Ward identity (2.6) these equations are equivalent to the four Ward identities where σ, σ ′ ∈ {↑, ↓} are now fixed spin projections. Note that in the absence of a magnetic field the Green function and the self-energy are independent of the spin quantum number σ, so that we may write G σ (iω) = G(iω) and Σ σ (iω) = Σ(iω). For i = z we have σ z σσ ′ = σ ′ δ σ,σ ′ so that Eq. (2.10) reduces to the zero-field limit of the U (1) Ward identity (2.7) associated with the conservation of the spin component in the direction of the magnetic field.

A. Particle number conservation
For the AIM in a magnetic field, the first terms in the functional Taylor expansion of the generating functional where Γ 0 is proportional to the interaction correction to the grand canonical potential, Σ σ (iω) is the exact irreducible self-energy, and U is the exact effective interaction. The term in the second line of our functional Ward identity (2.6) then yields To extract the Ward identity for the self-energy from our functional Ward identity (2.6), we take the second functional derivative δ δdωσ δ δdω+ω,σ of both sides of Eq. (2.6) and set then all fields equal to zero, which yields To obtain the right-hand side of this Ward identity, we have used which follows from the tree expansion relating the connected Green functions generated by G c to the irreducible vertices. 6,7 In the limit of vanishing bosonic frequencȳ ω → 0 the Ward identity (3.3) reduces to where we have defined σ,σ ′ (iω, iω ′ ; iω ′ , iω). (3.6) The above identities are valid for arbitrary hybridization functions ∆ σ (iω). Of special interest is the limit of infinite bandwidth of the conduction electron band with flat density of states, where the general expression for ∆ σ (iω) given in Eq. (1.8) is given by Here, the hybridization energy ∆ is assumed to be independent of the chemical potential and the magnetic field. In this limit is then ambiguous because the δ-function in the first term is multiplied by the sign-function associated with the hybridization function in G 2 σ ′ (iω ′ ). To properly define this term one should use the Morris-Lemma, 9,10 which states that the product of the delta-function with an arbitrary function f (Θ(x)) of the Θ-function should be defined via (3.9) We conclude that in the infinite bandwidth limit (3.10) where ξ σ = ξ 0,σ + Σ σ (i0) is the true excitation energy of a localized electron with spin projection σ, and is the spectral density of the localized d-electron with spin σ at vanishing energy. Substituting Eq. (3.10) into (3.5) we obtain the corresponding Ward identity in the infinite bandwidth limit, 11 If we (incorrectly) replace ρ σ ′ (0) → ρ σ (0) in the second line of Eq. (3.12), we arrive at Eq. (5.71) of Ref. [4].

B. Spin conservation
The Ward identity for the self-energy associated with the conservation of the spin projection in the direction of the magnetic field can be derived analogously from the corresponding functional Ward identity (2.7). By comparing the functional Ward identity (2.6) due to particle number conservation with the corresponding U (1) Ward identity (2.7) due to spin conservation, we conclude that in the spin case the Ward identities for the self-energy can be obtained from those of the particle number case by simply replacing (3.14) In particular, the spin-analogue of the particle number Ward identity (3.5) is Combining this equation with the corresponding equation for the particle number, Eq. (3.5), we obtain the two Ward identities In the limit of an infinite bandwidth these equations can be written in analogy to Eq. (3.12) as Finally, we note that in the absence of a magnetic field the SU (2) functional Ward identity (2.10) does not imply any further independent relation for the self-energy, since in this case we just have Σ σ (iω) = Σ(iω). However, taking higher order functional derivatives of Eq. (2.10) we may obtain symmetry relations involving higher order vertices, which are beyond the scope of this work.

IV. PROOF OF THE YAMADA-YOSIDA IDENTITIES
A. Relations between derivatives of the self-energy The above Ward identities can now be used to derive exact relations between the derivatives of the self-energy with respect to frequency, the chemical potential and the magnetic field. In the book by Hewson 4 one can find a derivation of these identities using diagrammatic arguments. Here, we show that the correct relations can be obtained quite simply within the framework of the exact renormalization group (also known as the functional renormalization group). 5,6 As a special case of the general exact renormalization group equation for the irreducible self-energy of an interacting Fermi system derived in Refs. [6,12,13] we find for the self-energy of the AIM where we have used again the notation (3.6) for the effective interaction, and where Λ is some flow parameter (cutoff) appearing in the Gaussian part of the action. All Green functions and vertices in Eq. (4.1) implicitly depend on the parameter Λ. The so-called single-scale propagator is defined bẏ The exact renormalization group flow equation (4.1) is valid for any choice of the flow parameter Λ. In particular, we may choose the chemical potential µ as a flow parameter. 14 Then Λ = µ and hence so that the single-scale propagator is simplẏ In this chemical potential cutoff scheme the exact renormalization group flow equation (4.1) reduces to To further manipulate this expression, let us note that for ω = 0 the integrand of the hybridization function given in Eq. (1.8) is non-singular, such that For ω = 0, however, we have to be more careful. Defining the spectral density the hybridization function ∆ σ (iω) can be rewritten as Using the well-known formula where P is the principle value, we see that ∆ σ (z) has a branch cut along the real axis. More explicitly, defining we obtain where the term −2∆δ(ω) arises from the discontinuity across the branch cut. In the limit of a flat band of infinite width discussed above, g(ǫ) = ∆ for all ǫ and ∂∆ σ (iω)/∂µ = 0, so that Eq. (4.11) reduces to Eq. (3.8).
Using the identity (4.11) we may write our exact renormalization group flow equation (4.5) in the form Comparing the right-hand side of this exact relation with the right-hand side of the particle number Ward identity (3.5) we conclude that for all frequencies we have the following exact identity, Next, let us choose in our exact renormalization group flow equation (4.1) the magnetic field as a flow parameter (Λ = h). The relevant single-scale propagator is theṅ and hence Noting that Eq. (4.15) can also be written as Comparing this with the U (1) spin Ward identity (3.15), we conclude that Of particular interest are the above relations for ω → 0, because in this limit the ω-derivative determines the wave-function renormalization factor Z σ via Taking the limit ω → 0 in Eqs. (4.13) and (4.18) and using the fact that due to antisymmetry the effective interaction at vanishing frequencies has the form we obtain Adding and subtracting these equations we obtain Eq. (4.23) is a corrected version of Eq. (5.80) of Ref. [4].

B. Yamada-Yosida identities
To make contact with the work of Yamada and Yosida 2 , we now assume a flat density of states and take the limit of an infinite width of the conduction electron band, D → ∞. Then there is a simple exact relation between the occupation numbers n σ of the impurity level and the self-energies Σ σ (i0) at vanishing frequency, 4 Taking derivatives of this expression with respect to µ and h we obtain Hence, Substituting these relations into the Ward identities (4.23) and (4.24) we obtain for the ω-derivative of the self-energy and for the effective interaction at zero energy For h → 0 the self-energy and the density of states are independent of the spin projection, so that we may write Σ σ (iω) = Σ(iω), ρ σ (0) = ρ(0), and Γ σ,−σ = Γ ⊥ . As in Ref. [2], we now focus on the particle-hole symmetric case, where ρ(0) = 1/(π∆) in the flat band infinite bandwidth limit considered here. Averaging both sides of Eqs. (4.30) and (4.31) over both spin projections, we obtain the Ward identities where the dimensionless particle number (charge) and spin susceptibilities are defined bỹ The identities (4.32) and (4.33) have first been obtained by Yamada and Yosida, 2 who showed that both sides of these equations have identical series expansions in powers of U/∆. Such a perturbative proof relies on the assumption that there are no non-analytic terms which are missed by the series expansions. Our proof of Eqs. (4.32) and (4.33) given above shows more clearly that these identities are a direct consequence of the U (1) × U (1) symmetry associated with particle number and spin conservation of the AIM in a magnetic field. Moreover, our derivation of Eqs. (4.32) and (4.33) is non-perturbative, because it relies only on the symmetries of the AIM and the associated functional Ward identities, and on the exact renormalization group flow equation for the irreducible self-energy. Using relation (4.19), we see that the identity (4.32) implies that the wave-function renormalization factor of the AIM can be expressed in terms of the susceptibilities as The other Ward identity (4.33) allows us to express the imaginary part of the retarded self-energy to the susceptibilities. Therefore we recall that the skeleton equation for the self-energy of the AIM (which is a consequence of the Dyson-Schwinger equation, which in turn follows within a functional integral approach from the invariance of the functional integral under infinitesimal shifts of the fields 1,6 ) implies the following exact expression for the imaginary part of the self-energy of the symmetric AIM in the infinite bandwidth limit, 4 Im Σ(ω + i0) = − Γ ⊥ π∆ On the imaginary frequency axis, the low-frequency expansion of the self-energy of the symmetric AIM in the infinite bandwidth limit is therefore Σ(iω) = U 2 + 1 −χ c +χ s 2 iω + i (χ c −χ s ) 2 8∆ ω 2 sgn ω + analytic terms O(ω 2 ). (4.39) Taking the different notations for the dimensionless susceptibilities into account, 15 Eq. (4.39) agrees with the expressions derived by Yamada and Yosida. 2

V. CONCLUSIONS
In this work 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 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 nonperturbative proof relies on exact Ward identities and on an exact renormalization group flow equation for the irreducible self-energy. From our derivation it is obvious that the Yamada-Yosida identities are a direct consequence of the U (1) × U (1) symmetry of the Anderson impurity model associated with the conservation of the particle number and the total spin component in the direction of an external magnetic field.
We have also presented a thorough derivation of the general functional Ward identities of the AIM due to particle number and spin conservation. Furthermore, we have shown that various identities relating the derivatives of the self-energy with respect to frequency, chemical potential, and magnetic field can be obtained by combining the Ward identities with exact renormalization group flow equations for the self-energy.