Current Vertices and Ward-Consistent Response

The one-particle Green function alone does not determine a conserving response. An external probe also couples through a vertex, and the vertex must change consistently with the self-energy. The Ward identity makes this statement exact: it ties charge conservation to a finite difference of the full inverse propagator. The many-body response construction and its conservation proof are given in Baym and Kadanoff 1961, pp. 291–295.

Required background. Dyson Equations and Self-Energy defines $G^{-1}$ and $\Sigma$ ; Densities, Currents, and Nonrelativistic Ward Identities derives the operator conservation law; Sources, Linear Response, and Kubo Formulae fixes the response convention; and Spurions, Local Counterterms, and Symmetry Response supplies the background-field variation used below.

The electromagnetic vertex

Couple a nonrelativistic system to an external gauge field $A_\mu=(\phi,\mathbf A)$ using $D_\mu=\partial_\mu-iA_\mu$ . The exact amputated vertex is the response of the inverse propagator,

\Gamma^\mu(p+q,p) =\frac{\delta G^{-1}(p+q,p)}{\delta A_\mu(q)}\bigg|_{A=0},

with the overall sign fixed by the coupling convention. Gauge invariance gives the Ward–Takahashi identity

q_\mu\Gamma^\mu(p+q,p) =G^{-1}(p+q)-G^{-1}(p).

For a bare quadratic dispersion, $G_0^{-1}(p)=p_0-\xi_{\mathbf p}$ , this reduces to the bare density and current vertices. For an interacting propagator, derivatives of $\Sigma$ necessarily appear in the small- $q$ vertex.

Taking a uniform dynamic limit gives

\Gamma^0(p,p)=\frac{\partial G^{-1}(p)}{\partial p_0} =1-\frac{\partial\Sigma(p)}{\partial p_0},

whereas a static momentum limit gives the current vertex

\boldsymbol\Gamma(p,p)=-\boldsymbol\nabla_{\mathbf p}G^{-1}(p) =\boldsymbol\nabla_{\mathbf p}\xi_{\mathbf p} +\boldsymbol\nabla_{\mathbf p}\Sigma(p)

in the stated sign convention. The two limits encode different physics and need not be interchangeable.

Linear response with a dressed vertex

The paramagnetic current correlator has the schematic form

K_{\rm para}^{\mu\nu}(q) =-\int_p G(p+q)\Gamma^\mu(p+q,p)G(p)\gamma^\nu(p,p+q),

where $\gamma^\nu$ is the probe vertex on the other leg. For spatial response, minimal coupling also generates a diamagnetic or contact term. The physical kernel is

K^{ij}=K_{\rm para}^{ij}+K_{\rm dia}^{ij}.

Dropping the contact term violates gauge invariance even for free particles. Combining the Ward identity with the contact contribution yields transversality, $q_\mu K^{\mu\nu}=0$ , and the associated $f$ -sum rule.

Consistent vertices from the self-energy

If the self-energy is a functional of $G$ , its variation generates the irreducible kernel

I(12;34)=\frac{\delta\Sigma(1,2)}{\delta G(3,4)}.

Differentiating Dyson’s equation in the external field then gives a Bethe–Salpeter equation for $\Gamma$ with the same $I$ . This is the operational meaning of a response consistent with the self-energy. A dressed $G$ combined with a bare vertex generally violates the Ward identity unless the self-energy is correspondingly trivial or the selected limit makes the correction vanish.

Static and dynamic checks

Different constraints test different limit orders:

the static density limit connects $\chi_{nn}(q\to0,0)$ to $-\partial n/\partial\mu$ ;
the dynamic uniform limit enforces the response to a spatially constant scalar potential;
the longitudinal current response satisfies continuity and the $f$ -sum;
in a Galilean-invariant continuum, interactions cannot renormalize the total current-to-momentum ratio, although they do renormalize quasiparticle velocities.

A calculation can satisfy one check and fail another. State which response, analytic continuation, and limit order were tested.

Spectral Moments and Many-Body Sum Rules supplies response checks, while Polarization, the Lindhard Function, and Particle–Hole Continua provides the bare-vertex benchmark.

Common pitfalls

Inferring response from a dressed bubble. Replacing $G_0$ by $G$ changes the propagators but not the required vertex. Conservation requires the compatible pair.

Forgetting the diamagnetic term. The paramagnetic bubble alone does not give the full electromagnetic response.

Setting $q=0$ too early. The Ward identity relates finite differences. Prematurely imposing a uniform limit can erase the distinction between compressibility and dynamic response.

Exercises

Obtain the zero-transfer density vertex

Take $q=(\Omega,\mathbf0)$ in the Ward identity and let $\Omega\to0$ .

Solution

$\Omega\Gamma^0(p+\Omega,p)=G^{-1}(p_0+\Omega,\mathbf p)-G^{-1}(p_0,\mathbf p)$ . Dividing by $\Omega$ and taking the limit gives $\Gamma^0=\partial_{p_0}G^{-1}=1-\partial_{p_0}\Sigma$ .

Test a constant self-energy

Suppose $\Sigma$ is independent of frequency and momentum. Does the bare vertex satisfy the Ward identity?