Current-phase relation and flux-dependent thermoelectricity in Andreev interferometers

We predict a novel $(I_0,{\varphi }_0)$-junction state of multi-terminal Andreev interferometers that emerges from an interplay between long-range quantum coherence and non-equilibrium effects. Under non-zero bias $V$ the current-phase relation $I_S(\varphi )$ resembles that of a ${\varphi }_0$-junction differing from the latter due to a non-zero average $I_0(V) = \left< I_S(\phi)\right>_{\phi}$\mathrm{.}Theflux-dependentthermopower${\mathcal S}(\Phi)$ofthesystemexhibitsfeaturessimilartothoseofa$(I_0,\phi_0)$-junction and in certain limits it can reduce to either odd or even function of $\mathrm{\Phi }$ in the agreement with a number of experimental observations.