Lattice Field Theory Analysis of the Chiral Heisenberg Model

Abstract

Motivated by ongoing interest in the universal behaviour of the Hubbard model of spinning electrons on honeycomb and $\pi$-flux lattices at the semi-metal -- Mott insulator phase transition, we formulate the 3D chiral Heisenberg model, a theory of relativistic fermions in three spacetime dimensions, as a lattice field theory using domain wall fermions. The contact interaction term preserves an SU(2) global symmetry. We perform numerical simulations using the Rational Hybrid Monte Carlo algorithm on system sizes $L^3\times L_s$ with $L\in \{8,\dots ,24\}$ and domain wall separation $L_s\in \{8,16,24\}$. We locate the phase transition corresponding to spontaneous SU(2)$\to$U(1) breaking, yielding critical exponent estimates ${\nu }^{-1}=0.63(3)$, ${\eta }_{\Phi }=1.42(8)$. These values are considerably removed from estimates obtained from simulations performed in (2+1)D, ie. with the time and spatial directions treated differently, but align more closely with analytic estimates obtained using 3D covariant field theory. We also present first results for the fermion correlator, ultimately needed for the determination of the exponent ${\eta }_{\Psi }$, highlighting the need to rotate the fermion source to a common reference direction in isospace in order to obtain a signal.

28 pages, 9 figures, 3 tables

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