An electronic nematic phase can be classified by a spontaneously broken discrete rotational symmetry of a host lattice. In a square lattice, there are two distinct nematic phases. The parallel nematic phase breaks
x and
y symmetry, while the diagonal nematic phase breaks the diagonal
(x+y) and anti-diagonal $(x\ensuremath{-}y)$ symmetry. We investigate the interplay between the parallel and diagonal nematic orders using mean field theory. We found that the nematic phases compete with each other, while they coexist in a finite window of parameter space. The quantum critical point between the diagonal nematic and isotropic phases exists, and its location in a phase diagram depends on the topology of the Fermi surface. We discuss the implication of our results in the context of neutron scattering and Raman spectroscopy measurements on ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$.
We investigate the interplay between superconductivity and nematic order in a two-dimensional electron gas with competing pairing and quadrupolar forward-scattering interactions. The model includes both
s-wave and
d-wave superconducting channels. We compute the mean-field free energy density and determine the phase diagrams as functions of interaction strengths and temperature by solving a set of coupled self-consistent equations. At zero-temperature, we find that the nematic order competes strongly with
d-wave superconductivity, leading to a direct first-order phase transition, while its interplay with
s-wave pairing allows for a coexistence phase characterized by an anisotropic Fermi surface with a uniform superconducting gap. At finite-temperatures, quadrupolar interactions promote the emergence of additional superconducting components, giving rise to regimes where
s-wave,
d-wave, and nematic orders coexist. Our results highlight the role of symmetry and interaction strength in shaping the phase structure and provide a minimal framework to describe intertwined nematic and superconducting phases in correlated electron systems.
Systems with strong electron-phonon couplings typically exhibit various forms of charge order, while strong electron-electron interactions lead to magnetism. We use determinant quantum Monte Carlo calculations to solve a model on a square lattice with a caricature of these interactions. In the limit where electron-electron interactions dominate it has antiferromagnetic order, while where electron-phonon coupling dominates there is columnar valence-bond-solid (VBS) order. We find an intervening phase that hosts coexisting nematic and antiferromagnetic orders. We have also found evidence of a Landau-forbidden continuous quantum phase transition with an emergent
O(4) symmetry between the VBS and the nematic antiferromagnetic phases.