Fermionic Condensation as the Universal Order Parameter for Mass Generation

Abstract

A fermion bilinear acquiring a non-zero vacuum expectation value and generating a mass gap underlies the Peierls instability, BCS superconductivity, the Nambu--Jona-Lasinio mechanism, and the Sachdev-Ye-Kitaev model. These mechanisms are usually discussed separately, and their structural similarities have been noted only in passing. We argue here that the correct universal object is not the bosonic order parameter but the fermionic condensate, and we show that the four mechanisms fall into two universality classes distinguished by a single dimensionless parameter, $r = \hat m_ϕ^2 / \hat y^2$, the ratio of the scalar mass to the Yukawa coupling. The classes are separated by a critical line $r_c(N_f)$ in the plane of $r$ and the fermion flavour number $N_f$; they correspond to two distinct infrared fixed points of the renormalisation-group flow and differ in their critical exponents. We derive the classification from a large-$N_f$ expansion followed by a functional renormalisation-group analysis using the Wetterich equation, which gives the finite-$N_f$ correction $\hat y_*^2(N_f) = 0.1125/N_f + 0.0023/N_f^2$, a $12.5\%$ enhancement over the leading-order prediction. The classification is predictive for systems whose class is not already known; for the organic conductor (TMTSF)$_2$PF$_6$ under pressure, we predict that both the spin-density-wave and superconducting phases lie in the fermion-bilinear class. We discuss the Standard Model Higgs as a borderline case in the scalar-primary class, and identify the composite-Higgs scenario as the particle-physics realisation of the fermion-bilinear class. A self-consistent Su--Schrieffer--Heeger calculation for polyyne chains realises the Peierls mechanism concretely, and we list five falsifiable predictions, two within reach of current techniques.