Abstract
We review some recent developments about strongly interacting relativistic Fermi theories in three spacetime dimensions. These models realize the asymptotic safety scenario and are used to describe the low-energy properties of Dirac materials in condensed matter physics. We begin with a general discussion of the symmetries of multi-flavor Fermi systems in arbitrary dimensions. Then we review known results about the critical flavor number Ncrit of Thirring models in three dimensions. Only models with a flavor number below Ncrit show a phase transition from a symmetry-broken strong-coupling phase to a symmetric weak-coupling phase. Recent simulations with chiral fermions show that Ncrit is smaller than previously extracted with various non-perturbative methods. Our simulations with chiral SLAC fermions reveal that for four-component flavors Ncrit=0.80(4). This means that all reducible Thirring models with Nr=1,2,3,… show no phase transition with order parameter. Instead, we discover footprints of phase transitions without order parameter. These new transitions are probably smooth and could be used to relate the lattice Thirring models to Thirring models in the continuum. For a single irreducible flavor, we provide previously unpublished values for the critical couplings and critical exponents.
Highlights
The Thirring model is a relativistic field theory for interacting fermions ψ, ψ
Since parity is broken for Nf = 1 and unbroken for Nf → ∞ we must conclude that there exists a critical flavor number Nfcrit separating the systems with symmetry breaking from those without symmetry breaking
We reviewed our current knowledge about spontaneous symmetry breaking in 1+2D Thirring models
Summary
The Thirring model is a relativistic field theory for interacting fermions ψ, ψ. 2 dimensions, the model is not soluble and not renormalizable in perturbation theory. We shall discretize the models on a (euclidean) spacetime lattice with chiral fermions keeping all continuum symmetries besides Poincare invariance. We shall focus on the lattice models in 3 dimensions and discuss the possible breaking of symmetries, depending on the number of flavors and the interaction strength. In even dimensions there exists one irreducible representation of the Clifford algebra, whereas in odd dimensions there are two. In even dimensions it anticommutes with the γμ and in odd dimensions it commutes with the γμ and is 1 in one irreducible representation and −1 in the other irreducible representation. One should note that S and P are not independent in odd dimensions because γ∗ ∝ 1 is trivial
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