The Fermi Gas and Fermi-Surface Kinematics

At zero temperature, Pauli exclusion fills every one-particle state inside a Fermi sea. The boundary $\epsilon_{\mathbf k}=\mu$ is a codimension-one Fermi surface, so arbitrarily low-energy particle–hole excitations live in thin normal shells around an extended set of momenta rather than near an isolated point.

Required background. Use second-quantized fermions and finite-density ensembles. Helpful background. Fourier and Plancherel conventions fix state counting.

Fermi-surface state counting and density of states

For degeneracy $g$ and isotropic dispersion $\epsilon_k=k^2/(2m)$,

$$n=g\int_{|\mathbf k|<k_F}\frac{\mathrm d^dk}{(2\pi)^d} =\frac{gS_{d-1}}{d(2\pi)^d}k_F^d, \qquad E_F=\frac{k_F^2}{2m},$$

where $S_{d-1}=2\pi^{d/2}/\Gamma(d/2)$. In $d=3$, $n=gk_F^3/(6\pi^2)$. The total density of states per volume is

$$N(\epsilon)=g\int\frac{\mathrm d^dk}{(2\pi)^d}\, \delta(\epsilon-\epsilon_k), \qquad N(0)\equiv N(E_F)=\frac{gS_{d-1}k_F^{d-1}}{(2\pi)^dv_F},$$

with $v_F=k_F/m$. Declaring whether $g$ is included prevents the ubiquitous factor-of-two error. The low-temperature specific heat is $C_V/V=(\pi^2/3)N(0)T+O(T^3)$ for a smooth three-dimensional density of states.

Normal and tangential kinematics

Write $\mathbf k=\mathbf k_F+\hat{\mathbf n}k_\perp+\mathbf k_\parallel$. Near a smooth surface,

$$\xi_{\mathbf k}=\epsilon_{\mathbf k}-\mu =v_Fk_\perp+\frac{k_\parallel^2}{2m_c}+\cdots.$$

Energy constrains the normal displacement directly, while tangential momentum labels the many low-energy patches. A particle just outside and a hole just inside the surface can carry small energy but momentum ranging from nearly zero to order $2k_F$. Their continuum at fixed $(\mathbf q,\omega)$ follows the condition $\omega=\epsilon_{\mathbf k+\mathbf q}-\epsilon_{\mathbf k}$ with one state occupied and the other empty.

This geometry explains why forward scattering and back-to-back Cooper scattering survive low-energy phase-space restrictions. It also explains why a van Hove point, nesting, or a band touching requires a separate scaling analysis: the assumption of a smooth surface with nonzero $v_F$ has failed. Shankar develops this kinematic organization in Shankar 1994, §§ II–IV, pp. 133–154.

Exercises

Derive $N(0)$ for a spin-$1/2$ three-dimensional gas and check it against $\partial n/\partial E_F$.

Solution

With $g=2$, $n=k_F^3/(3\pi^2)$. Since $\mathrm dk_F/\mathrm dE_F=m/k_F$, $\partial n/\partial E_F=(k_F^2/\pi^2)(m/k_F)=mk_F/\pi^2$. The general formula gives the same total DOS; the per-spin value is half of it.

References

• Ramamurti Shankar, “Renormalization-Group Approach to Interacting Fermions,” Reviews of Modern Physics 66 (1994) 129–192, §§ II–IV, doi:10.1103/RevModPhys.66.129.