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arXiv:1708.06200 · Physical review. B./Physical review. B

Critical chiral Heisenberg model with the functional renormalization group

Benjamin Knorr

Published Feb 14, 2018• 50 citations
arXivHTMLPDF

Abstract

The critical properties of the spin-density-wave/semimetal phase transition in Dirac materials are still under debate despite all the efforts from continuum and lattice methods. Using state-of-the-art functional renormalization group methods, the author provides improved estimates of the leading critical exponents. They are in line with earlier results obtained with the same method and in disagreement with recent quantum Monte Carlo estimates. The study supports the conjecture the only adequate nonperturbative treatment of the different universality classes, related to this phase transition, is via the functional renormalization group.

DOI
10.1103/physrevb.97.075129

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The critical properties of the spin-density-wave/semimetal phase transition in Dirac materials are still under debate despite all the efforts from continuum and lattice methods. Using state-of-the-art functional renormalization group methods, the author provides improved estimates of the leading critical exponents. They are in line with earlier results obtained with the same method and in disagreement with recent quantum Monte Carlo estimates. The study supports the conjecture the only adequate nonperturbative treatment of the different universality classes, related to this phase transition, is via the functional renormalization group.
Near the critical point of a phase transition, physical observables are characterized by a set of pure numbers -- the critical exponents -- which are identical for a whole class of transitions defining a ``universality class''. Novel Dirac materials exhibit quantum critical points, which are not captured by conventional universality classes and their description defines new challenges for theory. The authors present here a comprehensive renormalization group study and give quantitative predictions for universality classes of Dirac fermions, as relevant to semimetal-insulator transitions in graphene and emergent supersymmetry in topological insulators. The authors also compare to independent conformal bootstrap results and quantum Monte Carlo estimates.
We numerically investigate the quantum criticality of the chiral Heisenberg universality class with the total number of fermion components N=8N=8N=8 in terms of the Gross-Neveu theory. Auxiliary-field quantum Monte Carlo simulations are performed for the square lattice Hubbard model in the presence of a ddd-wave pairing field, inducing Dirac cones in the single-particle spectrum. This property makes the model particularly interesting because it turns out to belong to the same universality class of the Hubbard model on the honeycomb lattice, which is the canonical model for graphene, despite the unit cells being apparently different (e.g., they contain one and two sites, respectively). We indeed show that the two phase transitions, expected to occur on the square and on the honeycomb lattices, have the same quantum criticality. We also argue that details of the models, i.e., the way of counting NNN and the anisotropy of the Dirac cones, do not change the critical exponents. The present estimates of the exponents for the N=8N=8N=8 chiral Heisenberg universality class are $\ensuremath{\nu}=1.05(5)$, ${\ensuremath{\eta}}_{\ensuremath{\phi}}=0.75(4)$, and ${\ensuremath{\eta}}_{\ensuremath{\psi}}=0.23(4)$, which are compared with the previous numerical estimations.
Quantum critical phenomena may be qualitatively different when massless Dirac fermions are present at criticality. Using our recently-discovered fermion-sign-free Majorana quantum Monte Carlo method introduced by us in (Li et al 2015 Phys. Rev. B 91 241117 ), we investigate the quantum critical phenomena of spinless Dirac fermions at their charge-density-wave phase transitions on the honeycomb lattice having sites with largest L = 24. By finite-size scaling, we accurately obtain critical exponents of this so-called Gross–Neveu chiral-Ising universality class of two (two-component) Dirac fermions in 2+1D: , , and , which are qualitatively different from the mean-field results but are reasonably close to the ones obtained from renormalization group calculations.
Abstract We investigate a quantum criticality of an antiferromagnetic phase transition in the Hubbard model on a square lattice with a d-wave pairing field by large-scale auxiliary-field quantum Monte Carlo simulations. Since the d-wave pairing filed induces Dirac cones in the non-interacting single-particle spectrum, the quantum criticality should correspond to the chiral Heisenberg universality class in terms of the Gross-Neveu theory, which is the same as those expected in the Hubbard model on the honeycomb lattice, despite the unit cells being different (e.g., they contain one and two sites, respectively). We show that both the two phase transitions, expected to occur on the square and on the honeycomb lattices, indeed have the same quantum criticality. We also argue that details of the models, i.e., the way of counting the total number N of fermion components and the anisotropy of the Dirac cones, do not change the critical exponents.
Tricriticality, as a sister of criticality, is a fundamental and absorbing issue in condensed-matter physics. It has been verified that the bosonic Wilson-Fisher universality class can be changed by gapless fermionic modes at criticality. However, the counterpart phenomena at tricriticality have rarely been explored. In this Letter, we study a model in which a tricritical Ising model is coupled to massless Dirac fermions. We find that the massless Dirac fermions result in the emergence of a new tricritical point, which we refer to as the chiral tricritical point (CTP), at the phase boundary between the Dirac semimetal and the charge-density wave insulator. From functional renormalization group analysis of the effective action, we obtain the critical behaviors of the CTP, which are qualitatively distinct from both the tricritical Ising universality and the chiral Ising universality. We further extend the calculations of the chiral tricritical behaviors of Ising spins to the case of Heisenberg spins. The experimental relevance of the CTP in two-dimensional Dirac semimetals is also discussed.