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The t-J model approximates the low-energy sector of a repulsive Hubbard model with $U\gg t$ near one electron per site. Large on-site repulsion suppresses double occupancy. Virtual hopping out of and back into that low-energy sector produces antiferromagnetic superexchange with $J=4t^2/U$. One conventional effective Hamiltonian iswith projected electron operator ${\tilde{c}}_{i\sigma }=c_{i\sigma }(1-n_{i\bar{\sigma }})$. At half filling, nearest-neighbor hopping cannot move an electron without entering a forbidden doubly occupied state, leaving the leading spin-exchange model. Doped holes can move and couple the charge dynamics to these spin correlations. Additional terms, including three-site processes at order $t^2/U$, are omitted in this simplified Hamiltonian.

The exchange term energetically favors a nearest-neighbor spin singlet. With $b_{ij}^{†}=(c_{i\uparrow }^{†}{c}_{j\downarrow }^{†}-c_{i\downarrow }^{†}{c}_{j\uparrow }^{†})/\sqrt{2}$ in the no-double-occupancy subspace, ${\mathbf{S}}_i\cdot {\mathbf{S}}_j-n_i{n}_j/4=-b_{ij}^{†}{b}_{ij}$. A square-lattice pairing ansatz can therefore use bond singlets with opposite signs along the two axes. This d-wave bond order hasIt changes sign under a quarter-turn and vanishes along the zone diagonals. It is an even-parity, spin-singlet pairing pattern, not a statement of triplet pairing or necessarily broken time reversal. Phase-sensitive interference experiments distinguish this sign structure from a merely anisotropic nonnegative gap in cuprate superconductors.^[ref]

The model captures strong local repulsion, antiferromagnetism and mobile doped carriers. Its interacting phase diagram is not solved by writing down the Hamiltonian: a mean-field d-wave ansatz is a proposed approximation, not a proof that every parameter choice has that exact ground state or a complete microscopic explanation of every high-temperature superconductor.

The superconducting gap function is a momentum-dependent pairing amplitude $\Delta (\mathbf{k})$, including a phase as well as a magnitude. A fully gapped isotropic state and a state with symmetry-required nodes can have different low-temperature responses. Heat capacity, thermal transport, London penetration depth and tunnelling spectra probe the low-energy quasiparticle spectrum. Activated behavior suggests a nonzero minimum gap; power laws can indicate nodes. For example, clean line nodes give a penetration-depth correction linear in temperature, whereas disorder can change the apparent power.

Such magnitude-sensitive probes require care: impurities, several bands, anisotropic full gaps and surface states can imitate or modify simple expectations. A node does not by itself establish a particular sign pattern, and the absence of a coherence peak is not a complete symmetry determination. The strongest identification combines momentum-sensitive spectra with phase-sensitive superconducting gap experiments.

For an illustrative $d_{x^2-y^2}$ state, $\Delta (\phi )\propto \cos 2\phi$ changes sign under a $90^{\circ }$ rotation. A Josephson junction to a conventional s-wave reference can turn that sign into an intrinsic phase shift. In a corner DC SQUID with junctions on orthogonal crystal faces, one branch can acquire an extra $\pi$ relative to the other. For equal junctions and negligible loop inductance,Changing ${\delta }_{\mathrm{int}}$ from zero to $\pi$ shifts the interference pattern by half a flux period. The 1993 YBCO–Pb experiment compared phase-coherent junction geometries and found evidence for that sign reversal, while explicitly discussing SQUID asymmetry and trapped vortices. Those controls matter: an arbitrary shifted trace is not by itself a proof of pairing symmetry.^[ref]

A tricrystal half-flux experiment arranges grain-boundary junctions so that a sign-changing order parameter produces an odd total $\pi$ shift around a ring. Fluxoid consistency then favors a half-integer flux state, with the detailed spontaneous flux depending on loop inductance and junction strengths. Choosing comparison geometries without that frustration and controlling trapped external flux separates the pairing-phase effect from ordinary flux trapping.^[ref]

The conclusion is strongest when the same candidate symmetry explains nodes, crystal-axis dependence and these phase-sensitive interference results. The phase-sensitive observation distinguishes a sign-changing gap from a merely anisotropic positive gap; no single temperature power law supplies that distinction.

A ring crossing three appropriately oriented grain-boundary junctions tests a superconducting pairing sign pattern. An odd total intrinsic $\pi$ offset frustrates integer phase matching and can favor a half-flux state. Actual enclosed flux also depends on loop inductance and junction currents; the measurement requires controls against ordinary trapped flux.^[ref]