Charge- 4e Superconductivity from Multicomponent Nematic Pairing: Application to Twisted Bilayer Graphene

Abstract

We show that unconventional nematic superconductors with multi-component order parameter in lattices with three-fold and six-fold rotational symmetries support a charge-4e vestigial superconducting phase above Tc. The charge-4e state, which is a condensate of four-electron bound states that preserve the rotational symmetry of the lattice, is nearly degenerate with a competing vestigial nematic state, which is non-superconducting and breaks the rotational symmetry. This robust result is the consequence of a hidden discrete symmetry in the Ginzburg-Landau theory, which permutes quantities in the gauge sector and in the crystalline sector of the symmetry group. We argue that random strain generally favors the charge-4e state over the nematic phase, as it acts as a random-mass to the former but as a random-field to the latter. Thus, we propose that twodimensional inhomogeneous systems displaying nematic superconductivity, such as twisted bilayer graphene, provide a promising platform to realize the elusive charge-4e superconducting phase.

Solvable model for a charge-$4e$ superconductor

“…We note that recently charge-4e superconducting states have been proposed to exist in twisted bilayer graphene [13], the pair-density-wave state of cuprate superconudctors [6], and in the putative Z 4 spin liquids [21]. It will be interesting to extend our results to these contexts, either perturbatively or using numerical methods such as quantum Monte Carlo in the Majorana basis.…”

“…It is straightforward to generalize these saddle point equations to Eqs. (12)(13)(14)(15) in the main text for the finite-dimensional model. One can also derive the Schwinger-Dyson equations by variation of the effective action (11) in the main text, where the pairing and spin symmetries of the considered saddle-point have been already accounted.…”

“…To analyze the pairing instability, we reinstate and cross-correlation ρ between J and ∆ 4e , and the pairing vertex Φ in the Schwinger-Dyson equations, Eqs. (12)(13)(14)(15). The SD equations for the self-energy and for the pairing vertex are nonlinear and, thus, complicated to solve in general case.…”

“…First, unlike charge-2e superconductivity that emerges for weak coupling via a logrithmically divergent contribution to pairing susceptibility [15], no such weak-coupling instability of electrons exists towards charge-4e superconductivity. For this reason, existing theories of charge-4e superconductivity usually assumes some underlying strong interactions between the electrons, and addresses the formation of charge-4e superconductivity within a bosonic theory describing the binding of two Cooper pairs and the condensing of the composite object [6,7,13,14,16]. As an emergent degrees of freedom, the Cooper pair order parameter breaks particle-number conservation as well as some other symmetries such as translational symmetry (known as a pair-density-wave state) [7,11], rotational symmetry [13,14], or certain internal symmetries [16].…”

“…For this reason, existing theories of charge-4e superconductivity usually assumes some underlying strong interactions between the electrons, and addresses the formation of charge-4e superconductivity within a bosonic theory describing the binding of two Cooper pairs and the condensing of the composite object [6,7,13,14,16]. As an emergent degrees of freedom, the Cooper pair order parameter breaks particle-number conservation as well as some other symmetries such as translational symmetry (known as a pair-density-wave state) [7,11], rotational symmetry [13,14], or certain internal symmetries [16]. Upon increasing temperature, charge-4e condensate emerges as a vestigial order [4,7,[17][18][19] via partial melting of the Cooper pair order parameter by restoring certain spatial or internal symmetries.…”