Critical scaling in the $N=1$ Thirring Model in $(2+1)d$
Simon Hands, Jude Worthy
TL;DR
The paper addresses whether the $N=1$ Thirring model in $2+1$ dimensions hosts a UV-stable fixed point at a quantum critical point separating a symmetric phase from a broken one. It employs Wilson-kernel domain wall fermions on a $16^3\times L_s$ lattice, with a partial $L_s\to\infty$ extrapolation to control symmetry-preserving effects, and fits an RG-inspired equation of state to extract critical exponents. The resulting exponents $\delta=1.30(36)$, $\beta_m=2.43(15)$, $\nu=1.88(13)$, and $\eta=1.61(4)$ differ from earlier Shamir/staggered results and align qualitatively with Schwinger-Dyson predictions for a nontrivial fixed point; preliminary $N=2$ results further suggest the presence of a strong-coupling critical point. Overall, the work supports a strongly interacting quantum critical point in this model and demonstrates the importance of fermion formulation choices for capturing the correct universality class.
Abstract
The Thirring model in 2+1$d$ with $N$ Dirac flavors can exhibit spontaneous U($2N)\to$U($N)\otimes$U($N$) breaking through fermion - antifermion condensation in the limit $m\to0$. With no small parameter in play the symmetry-breaking dynamics is strongly-interacting and quantitative work requires a fermion formulation accurately capturing global symmetries. We present simulation results for $N=1$ obtained with Wilson kernel domain wall fermions on $16^3\times L_s$, with $L_s=24,\ldots,120$. The $L_s\to\infty$ extrapolation of the bilinear condensate $\langle\barψψ\rangle$ as a function of coupling and bare mass is fitted to an empirical equation of state; the resulting critical exponents are significantly altered from previously obtained values, and for the first time resemble those emerging from analytic predictions based on approximate solutions to Schwinger-Dyson equations, consistent with a putative UV-stable renormalisation group fixed point. To address the non-perturbative issue of the value $N_c$ below which such a fixed point exists we present preliminary results obtained with $N=2$.