We establish the universal torus low-energy spectra at the free Dirac fixed point and at the strongly coupled {\em chiral Ising} fixed point and their subtle crossover behaviour in the Gross-Neuveu-Yukawa field theory with nD=4 component Dirac spinors in D=(2+1) dimensions. These fixed points and the field theories are directly relevant for the long-wavelength physics of certain interacting Dirac systems, such as repulsive spinless fermions on the honeycomb lattice or π-flux square lattice. The torus spectrum has been shown previously to serve as a characteristic fingerprint of relativistic fixed points and is a powerful tool to discriminate quantum critical behaviour in numerical simulations. Here we use a combination of exact diagonalization and quantum Monte Carlo simulations of strongly interacting fermionic lattice models, to compute the critical energy spectrum on finite-size clusters with periodic boundaries and extrapolate them to the thermodynamic limit. Additionally, we compute the torus spectrum analytically using the perturbative expansion in ε=4−D, which is in good agreement with the numerical results, thereby validating the presence of the chiral Ising fixed point in the lattice models at hand. We show that the strong interaction between the spinor field and the scalar order-parameter field strongly influences the critical torus spectrum. Building on these results we are able to address the subtle crossover physics of the low-energy spectrum flowing from the chiral Ising fixed point to the Dirac fixed point, and analyze earlier flawed attempts to extract Fermi velocity renormalizations from the low-energy spectrum

1 aSchuler, Michael1 aHesselmann, Stephan1 aWhitsitt, Seth1 aLang, Thomas, C.1 aWessel, Stefan1 aLäuchli, Andreas, M. uhttps://arxiv.org/abs/1907.0537301937nas a2200157 4500008004100000245008200041210006900123260001500192300001200207490000700219520145800226100001901684700002001703700001901723856003701742 2018 eng d00aQuantum field theory for the chiral clock transition in one spatial dimension0 aQuantum field theory for the chiral clock transition in one spat c2018/11/09 a205118 0 vB 3 aWe describe the quantum phase transition in the N-state chiral clock model in spatial dimension d=1. With couplings chosen to preserve time-reversal and spatial inversion symmetries, such a model is in the universality class of recent experimental studies of the ordering of pumped Rydberg states in a one-dimensional chain of trapped ultracold alkali atoms. For such couplings and N=3, the clock model is expected to have a direct phase transition from a gapped phase with a broken global ZN symmetry, to a gapped phase with the ZN symmetry restored. The transition has dynamical critical exponent z≠1, and so cannot be described by a relativistic quantum field theory. We use a lattice duality transformation to map the transition onto that of a Bose gas in d=1, involving the onset of a single boson condensate in the background of a higher-dimensional N-boson condensate. We present a renormalization group analysis of the strongly coupled field theory for the Bose gas transition in an expansion in 2−d, with 4−N chosen to be of order 2−d. At two-loop order, we find a regime of parameters with a renormalization group fixed point which can describe a direct phase transition. We also present numerical density-matrix renormalization group studies of lattice chiral clock and Bose gas models for N=3, finding good evidence for a direct phase transition, and obtain estimates for z and the correlation length exponent ν.

1 aWhitsitt, Seth1 aSamajdar, Rhine1 aSachdev, Subir uhttps://arxiv.org/abs/1808.07056