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Write the superconducting order parameters as ${\psi }_j=|{\psi }_j|e^{i{\theta }_j}$, using a field of pair electric charge $q=-2e$. For a specified path across the superconducting weak link, defineHere $\mathbf{A}$ is the magnetic vector potential. To check electromagnetic gauge invariance, use a scalar function $\chi$ and transformThe change of ${\theta }_2-{\theta }_1$ is $q({\chi }_2-{\chi }_1)/\hslash$, exactly cancelled by the change of the line integral. Thus $\varphi$ is unchanged although each local quantum phase changes. This is the same charge/sign bookkeeping discussed in gauge-coupling sign in Ginzburg-Landau superconductivity. Different physical paths can enclose magnetic flux, so fixing the path matters.

For stationary reservoirs with equal internal pair chemical potentials and a time-independent vector potential, the local Schrödinger equation gives ${\dot{\theta }}_j=-q{U}_j/\hslash$ after dropping the common internal term. With electric potentials $U_j$ and $V=U_2-U_1$, subtraction gives $\dot{\varphi }=-qV/\hslash =2eV/\hslash$. For a general circuit, $V$ denotes the corresponding measured electrochemical voltage and the electromagnetic contribution must be treated consistently. A bare difference of phases without the vector potential term is insufficient in an arbitrary gauge.

High-temperature superconductivity is a historical name for superconductivity above earlier metallic benchmarks. The original Bednorz-Müller experiment operated at tens of kelvins, requiring deep cooling; cuprate superconductors opened a new materials family.^[ref]

A higher superconducting transition temperature enlarges the available operating range, but useful transport also needs a sufficient superconducting critical current at the operating temperature and magnetic field. In a ceramic, connections between grains can limit the whole specimen even when individual grains support superconductivity. This connects the materials problem to superconducting weak links and the Josephson effect. The parallel-path model of a superconducting ceramic below isolates why connectivity matters.

A superconducting weak link is a connection between regions with superconductivity whose coupling is weak enough that a substantial difference of quantum phase can develop across the connection. An insulating tunnel barrier is one realization; a narrow conducting constriction can also couple the regions. The current-phase relation of a superconducting weak link depends on the connection. A sinusoid is the leading tunnel-junction approximation, not a universal law for every superconducting weak link. ^[ref]