Torus Spectroscopy of the Gross-Neveu-Yukawa Quantum Field Theory: Free Dirac versus Chiral Ising Fixed Point

Abstract

We establish the universal torus low-energy spectra at the free Dirac fixed point and at the strongly coupled 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 = 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 orderparameter 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.

Designer Monte Carlo simulation for the Gross-Neveu-Yukawa transition

“…In addition, it can be shown that this Dirac mass carry a nontrivial topological index, and results in a dynamically generated quantum spin Hall insulator [21]. Furthermore, the interplay between bosons and fermions also -0.2 -0.17 -0. changes the universality of the quantum critical point, from Ising to the GN chiral Ising, which is believed to be among the simplest fermionic QCPs where possibly analytical and numerical approaches might be able to give consistent critical exponents to build a concrete CFT description [19,24,31,32].…”

“…For theoretical treatment, due to the absence of Fermi surfaces and the emergent Lorentz and conformal symmetry, some of the GN QCPs are believed to be among the "simplest" fermionic QCPs, and thus hold the highest expectation for achieving agreement between controlled analytical calculations and numerical simulations. In recent years, such efforts have been attempted from both numerical and theoretical sides, e.g., the chiral-Ising GN transitions have been studied via high-order expansions [19,20] and quantum Monte Carlo simulations [21][22][23][24].…”

Fractionalized fermionic quantum criticality in spin-orbital Mott insulators

“…Our findings call for more detailed theoretical investigations of the Gross-Neveu* criticalities, in particular in the SO(3) case, for which currently only leading-order estimates are available. It would also be interesting to study the universal finite-size spectra, e.g., on the torus [17,18,62,78].…”

Fermion enhanced first-order phase transition and chiral Gross-Neveu tricritical point

“…The fermion fluctuations not only change the type of phase transition but also influence the critical properties. A prevalent example is the chiral universality class, in which the Dirac fermions drive the Wilson-Fisher fixed point for the pure boson model into the Gross-Neveu fixed point 25,[42][43][44] been devoted to unveil the critical properties in these systems 35,[42][43][44][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62] . As a generalization of the chiral critical point, the CTP manifest itself when the usual bosonic tricritical point is coupled to the Dirac fermions.…”

“…The critical properties of the interacting Dirac fermion systems have attracted attentions from condensed matter to high-energy physics communities, not only in the discussion of quantum electrodynamics with fermionic matter [36][37][38][39][40][41] , but also in that the Dirac fermion drives the Wilson-Fisher fixed point into the chiral Gross-Neveu fixed point 25,[42][43][44][45] . Although enormous investigations have been devoted to this issue 35,[42][43][44][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62] , the numerical verification for type-II FIQCP is still lacking. Remarkably, recent studies based on the fieldtheoretical effective model propose that type-II FIQCP can also feature new tricritical behaviors, controlled by the chiral tricritical point (CTP) 63 .…”