= Solution
Momentum-resolved photoemission and directional tunneling can map the magnitude and nodes of the cuprate gap. Low-temperature <heat capacity>, thermal transport and penetration-depth measurements constrain the low-energy <quasiparticle> spectrum. These probes do not all determine the relative sign of the gap. A <phase-sensitive test of d-wave pairing> uses Josephson <interference> between differently oriented crystal regions. In the <tricrystal half-flux experiment>, a superconducting film crosses three designed grain boundaries; a frustrated ring can produce a half-flux signal detected with a scanning <SQUID>. Unfrustrated geometries supply controls. This probes phase structure beyond merely finding a V-shaped tunneling spectrum.<Pairing symmetry and flux quantization in a tricrystal superconducting ring by C. C. Tsuei and others>[1994, tricrystal geometry and flux measurement]{ref}
For the ideal nearest-neighbor <square lattice> at one <electron> per site including both spins, particle-hole symmetry puts the <chemical potential> at zero. The two requested cuts are
$$
\varepsilon(k_x,0)=-2t(1+\cos k_xa),\qquad
\varepsilon(k_x,k_x)=-4t\cos k_xa.
$$
The first runs from $-4t$ at the zone center to zero at $k_xa=\pm\pi$; the diagonal cut crosses zero at $k_xa=\pm\pi/2$. The <Fermi surface> obeys $\cos k_xa+\cos k_ya=0$, giving the diamond $|k_x|+|k_y|=\pi/a$ in the <first Brillouin zone>.
\Image[/past-exam-of-the-superconductivity-and-quantum-coherence-course-of-the-university-of-cambridge/2012/paper-1-nodal-square-lattice.png]
{title=Half-filled square-lattice band cuts, diamond Fermi surface and anisotropic nodal quasiparticle contours}
Near the specified node define $\delta\mathbf k=\mathbf k-\mathbf k_0$ and
$$
q_\parallel=(\delta k_x+\delta k_y)/\sqrt2,\qquad
q_\perp=(\delta k_y-\delta k_x)/\sqrt2.
$$
Expanding $\cos(\pi/2+\delta k a)=-a\delta k+O((a\delta k)^3)$ gives
$$
\varepsilon_{\mathbf k}\simeq2ta(\delta k_x+\delta k_y)=\alpha q_\parallel,\qquad
\Delta_{\mathbf k}\simeq\Delta_0a(\delta k_y-\delta k_x)=\beta q_\perp,
$$
where $\alpha=2\sqrt2ta$ and $\beta=\sqrt2\Delta_0a$. Therefore
$$
\boxed{E^2\simeq\alpha^2q_\parallel^2+\beta^2q_\perp^2.}
$$
There are four such nodes. An <energy> contour about one node encloses <momentum> area $\pi E^2/(\alpha\beta)$. Counting all four nodes and two independent spin <quasiparticles>, with one state per $(2\pi)^2$ of <momentum> area per unit real-space area, gives the positive-excitation count and <nodal quasiparticle density of states in a two-dimensional d-wave superconductor>
$$
\frac{\mathcal N(E)}{\mathcal A}=\frac{2E^2}{\pi\alpha\beta},\qquad
\boxed{\frac{g_{\rm qp}(E)}{\mathcal A}=\frac{4E}{\pi\alpha\beta}\quad(E>0).}
$$
This counts thermally excitable Bogoliubov modes, including particle-like and hole-like excitations, not the electron-addition spectral density alone. That distinction prevents a factor-of-two ambiguity. It is a per-layer result; dividing by the layer spacing gives the corresponding <volume> density in the independent-layer approximation.
The thermal <energy> density above its zero-temperature value follows from Fermi occupation:
$$
\frac{U(T)-U(0)}{\mathcal A}
=\frac4{\pi\alpha\beta}\int_0^\infty\frac{E^2\,dE}{e^{E/(k_BT)}+1}
=\frac{4I_2}{\pi\alpha\beta}(k_BT)^3,
\qquad I_2=\frac32\zeta(3)\simeq1.80.
$$
Thus
$$
\boxed{\frac C{\mathcal A}=\frac{12I_2}{\pi\alpha\beta}k_B^3T^2
\simeq\frac{21.6}{\pi\alpha\beta}k_B^3T^2.}
$$
This clean nodal contribution assumes $k_BT$ small compared with the scale over which <linearization> fails; impurity states and other excitations can add further terms.
In the <vortex> state, each flux quantum occupies area of order $\pi R^2=\Phi_0/B$, so $R\propto B^{-1/2}$. Outside a core of radius $\xi$, the phase flow has magnitude $v_s(r)\simeq\kappa/(2\pi r)$ in the simple circular-cell estimate, with $\kappa=h/M_{\rm pair}$ and condensed-pair <mass> $M_{\rm pair}$. Its cell average is
$$
\langle|v_s|\rangle\simeq\frac1{\pi R^2}\int_\xi^R\frac\kappa{2\pi r}\,2\pi r\,dr
=\frac{\kappa(R-\xi)}{\pi R^2}\propto\sqrt B.
$$
The \b[signed vector average can vanish] around a <vortex>; the relevant quantity is a typical magnitude or Doppler shift, not a net transport velocity. Flow shifts nodal energies by order $\hbar\mathbf v_s\cdot\mathbf k_F$. Because the zero-field nodal <density of states> is linear in <energy>, averaging the magnitude of these shifts produces a residual <density of states> proportional to $\sqrt B$. The resulting <Volovik effect> gives
$$
\boxed{C(T,B)=\gamma(B)T+\cdots,\qquad \gamma(B)\propto\sqrt B.}
$$
This is the clean semiclassical nodal regime, at <temperatures> below the field-induced Doppler scale and in an appropriate mixed-state field range. Core states, disorder, <Zeeman splitting> and screening can modify it. Its replacement by a square root of applied field additionally assumes the mean induction tracks that field away from the lower-critical-field onset; it is not an exact all-field law.
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