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We investigate equilibrium properties, including structure of the orderparameter, superflow patterns, and thermodynamics of low-temperature surfacephases of layered d_{x^2-y^2}-wave superconductors in magnetic field. At zeroexternal magnetic field, time-reversal symmetry and continuous translationalsymmetry along the edge are broken spontaneously in a second order phasetransition at a temperature $T^*\approx 0.18 T_c$, where $T_c$ is thesuperconducting transition temperature. At the phase transition there is a jumpin the specific heat that scales with the ratio between the edge length $D$ andlayer area ${\cal A}$ as $(D\xi_0/{\cal A})\Delta C_d$, where $\Delta C_d$ isthe jump in the specific heat at the d-wave superconducting transition and$\xi_0$ is the superconducting coherence length. The phase with broken symmetryis characterized by a gauge invariant superfluid momentum ${\bf p}_s$ thatforms a non-trivial planar vector field with a chain of sources and sinks alongthe edges with a period of approximately $12\xi_0$, and saddle pointdisclinations in the interior. To find out the relative importance oftime-reversal and translational symmetry breaking we apply an external fieldthat breaks time-reversal symmetry explicitly. We find that the phasetransition into the state with the non-trivial ${\bf p}_s$ vector field keepsits main signatures, and is still of second order. In the external field, thesaddle point disclinations are pushed towards the edges, and thereby a chain ofedge motifs are formed, where each motif contains a source, a sink, and asaddle point. Due to a competing paramagnetic response at the edges, the phasetransition temperature $T^*$ is slowly suppressed with increasing magneticfield strength, but the phase with broken symmetry survives into the mixedstate.

Spontaneously broken translational symmetry at edges of high-temperature superconductors: thermodynamics in magnetic field