Neutral order parameters in metallic criticality ind=2+1from a hairy electron star

Abstract

We use holography to study the spontaneous condensation of a neutral order parameter in a $(2+1)$-dimensional field theory at zero temperature and finite density, dual to the electron-star background of Hartnoll and Tavanfar. An appealing feature of this field theory is the emergence of an IR Lifshitz fixed point with a finite dynamical critical exponent $z$, which is due to the strong interaction between critical bosonic degrees of freedom and a finite density of fermions (metallic quantum criticality). We show that under some circumstances the electron-star background develops a neutral scalar hair whose holographic interpretation is that the boundary field theory undergoes a quantum phase transition, with a Berezinski-Kosterlitz-Thouless character, to a phase with a neutral order parameter. Including the backreaction of the bulk neutral scalar on the background, we argue that the two phases across the quantum critical point have different $z$, a novelty that exists in certain quantum phase transitions in condensed matter systems. We also analyze the system at finite temperature and find that the phase transition becomes, as expected, second order. Embedding the neutral scalar into a higher form, a variety of interesting phases could potentially be realized for the boundary field theory. Examples which are of particular interest to condensed matter physics include an antiferromagnetic phase where a vector condenses and breaks the spin symmetry, a quadrupole nematic phase which involves the condensation of a symmetric traceless tensor breaking rotational symmetry, or different phases of a system with competing order parameters.