= Solution
One orbital per primitive cell supplies two states per <wave vector> when <spin> is included. A monovalent metal therefore half-fills this band. Nearest-neighbor hopping on the bipartite rectangular lattice has $E(\mathbf k+\mathbf Q)=2E_0-E(\mathbf k)$ for $\mathbf Q=(\pi/a,\pi/b)$, fixing the half-filled <Fermi energy> at
$$
\boxed{E_F=E_0.}
$$
For $t_x=2t_y>0$, the two <Fermi surface> obey
$$
\boxed{\cos(k_xa)=-\tfrac12\cos(k_yb),\qquad
k_xa=\pm\arccos[-\tfrac12\cos(k_yb)].}
$$
They are corrugated open sheets in an extended-zone description, with $|k_xa|$ ranging from $\pi/3$ to $2\pi/3$. They are not a small circular pocket centered at the band bottom. The same $\mathbf Q$ maps one Fermi line onto its nested partner. A density-wave or Peierls-like modulation at that <wave vector> can open additional gaps at the <Fermi energy>. The unperturbed single-orbital band does not acquire those new gaps without a symmetry-breaking modulation or another coupling; ordinary lattice Bragg effects are already built into its dispersion.
For the three-dimensional tetragonal version set $t_x=t_y=t$, $t_z=t/2$, $a=b<c$. The corresponding half-filled surface is
$$
\boxed{E_F=E_0,\qquad \cos(k_xa)+\cos(k_ya)+\tfrac12\cos(k_zc)=0.}
$$
It is a corrugated periodic surface, not a sphere. At $k_z=\pi/c$ its regular closed contour is electron-like around $\Gamma$, with $\cos(k_xa)+\cos(k_ya)=1/2$; at $k_z=0$ the closed contour is hole-like around $M$, with the sum $-1/2$. The topology switches through the square-lattice separatrix at $k_zc=\pm\pi/2$. Drawing one simple weakly warped cylindrical pocket across every section misses that feature of the half-filled cosine band.
Angle-resolved photoemission can directly map the dispersion and occupied <Fermi surface>. Angular quantum-oscillation measurements test its closed extremal cross sections and cyclotron masses; momentum-density probes such as positron annihilation or Compton scattering give a complementary bulk test. The different techniques constrain different features rather than merely providing interchangeable pictures.
\Image[/past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2015-tetragonal-half-filled-fermi-surface.png]
{title=Half-filled tetragonal cosine-band Fermi surface and its symmetry-related closed orbits}
{description=Original surface of cos x plus cos y plus one half cos z equal to zero. An electron contour at kz c equal to pi and a hole contour at kz c equal to zero have equal closed-orbit areas by particle-hole symmetry.}
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