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

We predict a novel $(I_0,\phi_0)$-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 $I_S(\phi)$ resembles that of a $\phi_0$-junction differing from the latter due to a non-zero average $I_0(V) = \left< I_S(\phi)\right>_{\phi}$. The flux-dependent thermopower ${\mathcal S}(\Phi)$ of the system exhibits features similar to those of a $(I_0,\phi_0)$-junction and in certain limits it can reduce to either odd or even function of $\Phi$ in the agreement with a number of experimental observations.