Diagrammatics at Finite Density

Finite density changes diagrammatics through the state, not through new local ultraviolet physics. Propagators acquire occupation-dependent pole prescriptions, loops admit particle–hole kinematics, and condensates or broken symmetries require enlarged propagator bases. Vacuum divergences should be renormalized with vacuum data before medium contributions are interpreted. The separation between Fermi-surface kinematics and local ultraviolet renormalization is central to Shankar 1994, §§II–III, pp. 135–151.

Required background. Dyson Equations and Self-Energies fixes propagator and self-energy conventions; Second-Quantized Bosons and Fermions fixes statistics; Chemical Potentials and Finite-Density Ensembles fixes $K=H-\mu N$ .

Helpful background. Imaginary Time and Matsubara Frequencies supplies the thermal sum representation.

Medium propagators

For a free fermion with $\xi_{\mathbf p}=\varepsilon_{\mathbf p}-\mu$ , the retarded propagator is always

G_0^R(\omega,\mathbf p)=\frac1{\omega-\xi_{\mathbf p}+i0}.

The zero-temperature time-ordered propagator instead knows whether the state is occupied:

G_0^T(\omega,\mathbf p) =\frac1{\omega-\xi_{\mathbf p}+i0\,\operatorname{sgn}\xi_{\mathbf p}} =\frac{1-n_{\mathbf p}}{\omega-\xi_{\mathbf p}+i0} +\frac{n_{\mathbf p}}{\omega-\xi_{\mathbf p}-i0},

where $n_{\mathbf p}=\theta(-\xi_{\mathbf p})$ . The second term is often called a hole or medium insertion. It is not a new interaction; it records the chosen Fermi sea.

At temperature $T$ , Matsubara sums or lesser/greater functions replace the step by $f(\xi)$ . A contour calculation and a Matsubara calculation must agree after analytic continuation if their frequency origins and thermal factors match.

Vacuum and medium separation

For a loop $I(\mu,T)$ , write

I(\mu,T)=I_{\rm vac}+\Delta I_{\rm med}(\mu,T).

$I_{\rm vac}$ contains the ultraviolet divergence already present at zero density and is renormalized by vacuum masses, scattering amplitudes, or other matching data. Occupation factors restrict $\Delta I_{\rm med}$ to momenta near or inside the occupied region and often make it ultraviolet finite. Introducing a density-dependent counterterm for a divergence that belongs to $I_{\rm vac}$ would obscure the matching.

For a contact interaction between two fermion species, the first-order self-energy is

\Sigma_\uparrow^{(1)}=C_0 \int\frac{\mathrm d^d p}{(2\pi)^d}f(\xi_{\mathbf p\downarrow}) =C_0 n_\downarrow.

This Hartree term is a medium contribution. For identical spinless fermions the $s$ -wave contact vanishes by antisymmetry; for Coulomb matter Hartree and neutralizing-background terms must be treated together.

The particle–hole bubble

The retarded free density polarization obtained from a Matsubara sum is

\Pi_0^R(\mathbf q,\omega) =g\int\frac{\mathrm d^d p}{(2\pi)^d} \frac{f(\xi_{\mathbf p})-f(\xi_{\mathbf p+\mathbf q})} {\omega+\xi_{\mathbf p}-\xi_{\mathbf p+\mathbf q}+i0},

where $g$ is the declared internal degeneracy. Its imaginary part is nonzero when an occupied state can be moved to an unoccupied state with the supplied $(\omega,\mathbf q)$ . This is the particle–hole continuum.

The numerator is as important as the denominator. Replacing occupation factors by one reproduces a vacuum-looking loop and destroys Pauli blocking. Conversely, inserting Fermi functions into a vacuum counterterm double counts medium physics.

Cuts, power counting, and resummation

At finite density, small denominators can arise from nearly on-shell particle–hole pairs even when the microscopic coupling is weak. Their phase space determines whether a loop is enhanced. In a Fermi liquid, forward and Cooper channels require separate counting; a generic vacuum loop order does not by itself rank diagrams near the Fermi surface.

An infinite series may be required for a physical reason:

particle–particle ladders resum a large scattering length or Cooper logarithm;
particle–hole bubbles generate screening and collective density modes;
anomalous propagators resum a broken-symmetry saddle.

Each resummation selects a subset of diagrams. It must state what prevents omitted vertex or crossed diagrams from contributing at the same order. A geometric series is algebraically exact for its chosen kernel but not necessarily a controlled approximation to the many-body theory.

Condensates and Nambu doubling

For a Bose condensate, separate the zero-momentum expectation value and fluctuations, taking the thermodynamic and source limits in the declared order. Linear terms vanish only when the saddle equation is satisfied. For paired fermions, use a Nambu matrix propagator; factors of $1/2$ compensate the doubled basis, and anomalous signs depend on the spinor convention.

Vacuum subtraction, medium occupation, and Nambu doubling are distinct bookkeeping operations. Combining them without a written convention is a common source of missing factors and false ultraviolet dependence.

Common pitfalls

Using a time-ordered medium propagator as a retarded one. The retarded $i0$ does not change across the Fermi surface; the time-ordered one does.

Renormalizing medium pieces with new arbitrary parameters. Short-distance counterterms are fixed by the underlying matching unless the medium opens a genuinely new operator or scale.

Resumming because a series can be summed. State the enhanced kinematics or control parameter and test omitted diagrams.

Exercises

Perform the Matsubara sum

Show that

T\sum_{\nu_n} \frac1{i\nu_n-\xi_{\mathbf p}} \frac1{i\nu_n+i\omega_m-\xi_{\mathbf p+\mathbf q}}

equals the numerator difference in the polarization divided by $i\omega_m+\xi_{\mathbf p}-\xi_{\mathbf p+\mathbf q}$ .

Solution

Partial fractions reduce the product to the difference of two simple fermionic sums. Using $T\sum_n(i\nu_n-\xi)^{-1}=f(\xi)$ up to the common convergence prescription gives

\frac{f(\xi_{\mathbf p})-f(\xi_{\mathbf p+\mathbf q})} {i\omega_m+\xi_{\mathbf p}-\xi_{\mathbf p+\mathbf q}}.

Continuation $i\omega_m\to\omega+i0$ yields the retarded bubble.

Separate a tadpole

Explain why the nonrelativistic vacuum Hartree loop vanishes in a normal-ordered zero-density theory but gives $C_0n$ in matter.