Strain-induced Berry phase in chiral superconductors

Abstract

We study the topology of the order parameter in the intermediate phase between the superconducting and time-reversal symmetry breaking transitions of a $p_x+ip_y$ superconductor under strain. The application of in-plane strain reduces the underlying crystal symmetry and lifts the degeneracy of the critical temperature between the $p_x$ and $p_y$ orbitals, resulting in a Dirac cone structure in the thermodynamic phase diagram. When the strain is varied adiabatically along a closed path enclosing the Dirac cone, the order parameter acquires a Berry phase of $π$, which originates from a half rotation of the superconducting gap function. This half rotation leaves a topological signature in the superfluid stiffness tensor, making it directly observable through the geometry of vortices and the upper critical field.

Figures (6)

• Figure 1: Theoretical phase diagram of a $p_x\pm i p_y$ superconductor under strain. The vertical axis denotes temperature $T$ relative to the critical temperature $T_c^0$ in the absence of strain, and the horizontal axes $\epsilon_3\equiv (\epsilon_{xx}-\epsilon_{yy})/2$ and $\epsilon_1\equiv \epsilon_{xy}$ represent components of the in-plane strain tensor associated with tension and shear, respectively. The phase diagram exhibits a Dirac cone structure: the region inside the upper cone (silver) corresponds to the metallic phase, while the lower cone (blue) corresponds to the $p_x\pm ip_y$ superconductor. When the strain is varied adiabatically in a loop that encloses the Dirac cone, the gap function rotates by $\pi$.
• Figure 2: Evolution of the gap function as the strains $\epsilon_3$ and $\epsilon_1$ are varied adiabatically. The gap function is depicted as a $p$ orbital inside a constant $k_z$ slice of the Fermi sea (peach disc), and its nodes are indicated by crosses. The two lobes and nodes of the gap are distinguished by their colors. The directions of the Fermi velocity at the gap maxima and nodes are indicated with arrows. When the strain is varied adiabatically in a loop that encloses the Dirac point at the origin (black dot), the gap function only performs half a rotation, indicating the presence of a branch cut in the phase diagram (wavy line).
• Figure 3: (a) Phase of $K_3 + iK_1$ on the $\epsilon_3$–$\epsilon_1$ plane, showing $2\pi$ winding around the origin. (b) Two closed paths in strain space: a loop that winds once around the origin (red) and a contractible loop (blue). Their corresponding trajectories on the $K_3$–$K_1$ plane are shown in (c) and (d), respectively. The red path winds once around the origin, whereas the blue path backtracks to its starting position and is topologically trivial. The black dot marks the origin, and the cross indicates the starting point of the loops. Parameter values are $\Lambda = 10$, $N_0(0)V = 0.3$, $C = 3.5$, and $\epsilon_0=0$.
• Figure 4: Evolution of the vortex profile as strain is varied along the two paths in Fig. \ref{['fig: stiffness']}(b), with (a) corresponding to the red path and (b) to the blue path. The vortex is represented by an ellipse whose semimajor axis is indicated by an arrow. The initial, intermediate, and final configurations are shown in dotted, dashed, and solid lines, respectively. Note that the initial and final ellipses overlap, and the arrows also coincide in (b). The cross indicates the initial, arbitrarily chosen orientation of the semimajor axis, and the colored curve shows how the semimajor axis evolves as the strain is varied. In (a), the semimajor axis flips orientation, whereas in (b) it returns to its original configuration. Here, $\xi_0$ denotes the initial length of the semimajor axis.
• Figure 5: Angles $\theta_{\text{max}}$ at which the in-plane upper critical field $B_{c2}$ is maximal as the strain is varied along the two paths in Fig. \ref{['fig: stiffness']}(b). At the end of the evolution, the two maxima exchange positions for the red path, whereas they return to their original locations for the blue path.