Fermion masses through four-fermion condensates

TL;DR

The paper investigates a 4D lattice model in which fermions gain mass through four-fermion condensates without forming fermion bilinear condensates, posing a non-perturbative mass-generation mechanism outside spontaneous symmetry breaking. Using the fermion bag approach, it analyzes the phase diagram between a massless PMW phase and a massive PMS phase, revealing a surprisingly narrow intermediate FM phase where SU(4) symmetry is broken and bilinear condensates can form. Finite-size scaling of monomer density and susceptibilities suggests two second-order-like transitions bounding the intermediate phase, though the width is much smaller than in earlier studies, leaving open the possibility of a direct PMW-PMS transition in extended models. These findings support the viability of mass generation via four-fermion dynamics and motivate further exploration of direct PMW-PMS transitions and their continuum implications, including a composite-fermion perspective on mass terms.

Abstract

Fermion masses can be generated through four-fermion condensates when symmetries prevent fermion bilinear condensates from forming. This less explored mechanism of fermion mass generation is responsible for making four reduced staggered lattice fermions massive at strong couplings in a lattice model with a local four-fermion coupling. The model has a massless fermion phase at weak couplings and a massive fermion phase at strong couplings. In particular there is no spontaneous symmetry breaking of any lattice symmetries in both these phases. Recently it was discovered that in three space-time dimensions there is a direct second order phase transition between the two phases. Here we study the same model in four space-time dimensions and find results consistent with the existence of a narrow intermediate phase with fermion bilinear condensates, that separates the two asymptotic phases by continuous phase transitions.

Figures (6)

• Figure 1: An illustration of a fermion bag configuration. The sites with monomers are marked with filled circles. Connected sites without monomers form fermion bags.
• Figure 2: An illustration of a $\nu=1$ topological fermion bag configuration. The Dirac matrix $W_{\cal B}$ associated with this bag will contain at least one zero mode. This connection between topology and zero modes is analogous to the index theorem of the massless Dirac operator in non-Abelian gauge theories.
• Figure 3: An illustration of a symmetry fluctuation of a fermion bag when other fermion bags are sufficiently far apart. The fermion bag in the center of the figure on the left has been translated by one unit to the right and shown in the figure on the right. Such a change in a fermion bag is referred to as a symmetry fluctuation and the sites affected during the fluctuation are shown with a different color in the right figure.
• Figure 4: The monomer density (left) and the condensate susceptibility $\chi_1$ (right) plotted as a function of $U$ in the intermediate coupling region for various lattice sizes. There is no sign of a first order transition, but the rapid growth of the susceptibility suggests an intermediate phase with spontaneous breaking of the $SU(4)$ symmetry.
• Figure 5: The plots on the left show $2\chi_1/L^4$ and $2\chi_2/L^4$ as a function of $L$ at $U=1.67$ (squares) and $1.75$ (circles). Also $\chi_1$ is higher than $\chi_2$. The plot on the right shows the condensate $\Phi = \langle O^0_{ab}(x) \rangle$ as a function of $U$. We see the intermediate FM phase extends roughly from $1.60 \leq U \leq 1.81$.