Phonon-induced rotation of the electronic nematic director in superconducting Bi2Se3

The doped topological insulator $A_{x}\\mathrm{Bi_{2}Se_{3}}$, with\n$A=\\{\\mathrm{Cu},\\mathrm{Sr},\\mathrm{Nb}\\}$, becomes a nematic superconductor\nbelow $T_{c}\\sim3-4\\,\\mathrm{K}$. The associated electronic nematic director is\ndescribed by an angle $\\alpha$ and is experimentally manifested in the\nelliptical shape of the in-plane critical magnetic field $H_{c2}$. Because of\nthe threefold rotational symmetry of the lattice, $\\alpha$ is expected to align\nwith one of three high-symmetry directions corresponding to the in-plane\nnearest-neighbor bonds, consistent with a $Z_{3}$-Potts nematic transition.\nHere, we show that the nematic coupling to the acoustic phonons, which makes\nthe nematic correlation length tend to diverge along certain directions only,\ncan fundamentally alter this phenomenology in trigonal lattices. Compared to\nhexagonal lattices, the former possesses a sixth independent elastic constant\n$c_{14}$ due to the fact that the in-plane shear strain doublet\n$(\\epsilon_{xx}-\\epsilon_{yy},-2\\epsilon_{xy})$ and the out-of-plane shear\nstrain doublet $(2\\epsilon_{yz},-2\\epsilon_{xz})$ transform as the same\nirreducible representation. We find that, when $c_{14}$ overcomes a threshold\nvalue, which is expected to be the case in doped $\\mathrm{Bi_{2}Se_{3}}$, the\nnematic director $\\alpha$ unlocks from the high-symmetry directions due to the\ncompetition between the quadratic phonon-mediated interaction and the cubic\nnematic anharmonicity. This implies the breaking of the residual in-plane\ntwofold rotational symmetry ($C_{2x}$), resulting in a triclinic phase. We\ndiscuss the implications of these findings to the structure of nematic domains\nand to the shape of the in-plane $H_{c2}$ in $A_{x}\\mathrm{Bi_{2}Se_{3}}$, and\nto presence of nodes inside the superconducting state.\n