Origin of fermion masses without spontaneous symmetry breaking

Using large scale Monte Carlo calculations in a simple three dimensional lattice fermion model, we establish the existence of a second order quantum phase transition between a massless fermion phase and a massive one, both of which have the same symmetries. This shows that fermion masses can arise due to dynamics without the need for spontaneous symmetry breaking. Universality suggests that this alternate origin of the fermion mass should be of fundamental interest.

Figure 1

Two possible scenarios for the phase diagram of lattice models that show the origin of a fermion mass without spontaneous symmetry breaking. The massless and the massive fermion phases have the same symmetries. Previous studies in four space-time dimensions found results consistent with scenario A, while our work in three space-time dimensions is consistent with scenario B with a second order quantum critical point at $U_c$.

Figure 2

An example of a monomer configuration $[n]$ showing free fermion bags on a two dimensional lattice. The filled circles represent monomers and the connected regions without monomers form free fermion bags.

Figure 3

Plots of ${\rho }_m$ and ${\chi }_1$ as a function of $U$ for various values of $L$. The susceptibility shows a peak and the average monomer density shows a sharp rise at the phase boundary ($U\sim 1$).

Figure 4

Plot of $R_1$ as a function of $L$ for various values of $U$ near the critical region. The solid lines are fits to the form $1/L^{1+\eta }$ where $\eta$ values are given in Table 1, except at $U=1.03$ where the solid line has the form $\exp (-0.07L)$ suggesting the fermions are already massive.

Figure 5

Plots of $R_1$ as a function of $U$ for various lattice sizes showing peaks. The values of the peaks $R_{1,p}$ and their locations $U_{1,p}$ are also marked. These are determined by approximating the function to be a quadratic near the maximum.

Figure 6

Plots of $R_{1,p}$ and $R_{2p}$ as a function of $L$ (left figure) and $U_{1,p}$ and $U_{2,p}$ as a function of $L$. The solid lines represent fits to the form $R_{a,p}=b_a/L^{1+\eta }$ and $U_{a,p}=U_c+d_a/L^{\nu }$ with $U_c=0.943$ fixed. The dashed line is a fit including correction to scaling of the form $R_{2,p}=b_2/L^{1+\eta }+c_2/L^{1+\eta +\omega }$, where $\omega \approx 1$.

Figure 7

Evidence for universal scaling in our large lattice data with $U_c=0.943$, $\eta =1.08$, and $\nu =1.30$. Our data may also be consistent with $U_c=0.945$, $\eta =1$, and $\nu =1$ expected from large $N$ analysis (shown in the inset), but only after including corrections to scaling.