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arXiv:2209.02734 · Physical review. B./Physical review. B

Gross-Neveu-Heisenberg criticality from 2 + ε expansion

Konstantinos Ladovrechis, Shouryya Ray, Tobias Meng, Lukas Janssen

Published Jan 30, 2023• 20 citations
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Abstract

The Gross-Neveu-Heisenberg universality class describes a continuous quantum phase transition between a Dirac semimetal and an antiferromagnetic insulator. Such quantum critical points have originally been discussed in the context of Hubbard models on $\ensuremath{\pi}$-flux and honeycomb lattices, but more recently also in Bernal-stacked bilayer models, of potential relevance for bilayer graphene. Here, we demonstrate how the critical behavior of this fermionic universality class can be computed within an $\ensuremath{\epsilon}$ expansion around the lower critical space-time dimension of two. This approach is complementary to the previously studied expansion around the upper critical dimension of four. The crucial technical difference near the lower critical dimension is the presence of different four-fermion interaction channels at the critical point, which we take into account in a Fierz-complete way. By interpolating between the lower and upper critical dimensions, we obtain improved estimates for the critical exponents in $2+1$ space-time dimensions. For the situation relevant to single-layer graphene, we find an unusually small leading-correction-to-scaling exponent, arising from the competition between different interaction channels. This suggests that corrections to scaling may need to be taken into account when comparing analytical estimates with numerical data from finite-size extrapolations.

DOI
10.1103/physrevb.107.035151

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The Gross-Neveu-Heisenberg universality class describes a continuous quantum phase transition between a Dirac semimetal and an antiferromagnetic insulator. Such quantum critical points have originally been discussed in the context of Hubbard models on \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux and honeycomb lattices, but more recently also in Bernal-stacked bilayer models, of potential relevance for bilayer graphene. Here, we demonstrate how the critical behavior of this fermionic universality class can be computed within an \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion around the lower critical space-time dimension of two. This approach is complementary to the previously studied expansion around the upper critical dimension of four. The crucial technical difference near the lower critical dimension is the presence of different four-fermion interaction channels at the critical point, which we take into account in a Fierz-complete way. By interpolating between the lower and upper critical dimensions, we obtain improved estimates for the critical exponents in 2+12+12+1 space-time dimensions. For the situation relevant to single-layer graphene, we find an unusually small leading-correction-to-scaling exponent, arising from the competition between different interaction channels. This suggests that corrections to scaling may need to be taken into account when comparing analytical estimates with numerical data from finite-size extrapolations.
Two-dimensional van der Waals materials offer a highly tunable platform for engineering electronic band structures and interactions. By employing techniques such as twisting, gating, or applying pressure, these systems enable precise control over the electronic excitation spectrum. In moir\'e bilayer graphene, the tunability facilitates the transition from a symmetric Dirac semimetal phase through a quantum critical point into an interaction-induced long-range ordered phase with a finite band gap. At charge neutrality, the ordered state proposed to emerge from twist-angle tuning is the Kramers intervalley-coherent insulator. In this case, the transition falls into the quantum universality class of the relativistic Gross-Neveu-XY model in 2+12+12+1 dimensions. Here, we refine estimates for the critical exponents characterizing this universality class using an expansion around the lower critical space-time dimension of two. We compute the order-parameter anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\varphi}}$ and the correlation-length exponent $\ensuremath{\nu}$ at one-loop order, and the fermion anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\psi}}$ at two-loop order. Combining these results with previous findings from the expansion around the upper critical dimension, we obtain improved estimates for the universal exponents in 2+12+12+1 dimensions via Pad\'e interpolation. For Nf=4{N}_{\text{f}}=4Nf​=4 four-component Dirac fermions, relevant to moir\'e bilayer graphene, we estimate $1/\ensuremath{\nu}=0.916(5), {\ensuremath{\eta}}_{\ensuremath{\varphi}}=0.926(13)$, and ${\ensuremath{\eta}}_{\ensuremath{\psi}}=0.0404(13)$. For Nf=2{N}_{\text{f}}=2Nf​=2, potentially relevant to recent tetralayer WSe2{\mathrm{WSe}}_{2}WSe2​ experiments, the Gross-Neveu-XY fixed point may be unstable owing to a fixed-point collision at Nf,c{N}_{\text{f,c}}Nf,c​, with ${N}_{\text{f,c}}=1+\sqrt{2}+\mathcal{O}(\ensuremath{\epsilon})$ in the expansion around the lower critical dimension.
Electrons on the half-filled honeycomb lattice are expected to undergo a direct continuous transition from the semimetallic into the antiferromagnetic insulating phase with increase of onsite Hubbard repulsion. We attempt to further quantify the critical behavior at this quantum phase transition by means of functional renormalization group (RG), within an effective Gross-Neveu-Yukawa theory for an SO(3)\mathrm{SO}(3)SO(3) order parameter (``chiral Heisenberg universality class''). Our calculation yields an estimate of the critical exponents $\ensuremath{\nu}\ensuremath{\simeq}1.31$, ${\ensuremath{\eta}}_{\ensuremath{\phi}}\ensuremath{\simeq}1.01$, and ${\ensuremath{\eta}}_{\ensuremath{\Psi}}\ensuremath{\simeq}0.08$, in reasonable agreement with the second-order expansion around the upper critical dimension. To test the validity of the present method, we use the conventional Gross-Neveu-Yukawa theory with Z2{\mathbb{Z}}_{2}Z2​ order parameter (``chiral Ising universality class'') as a benchmark system. We explicitly show that our functional RG approximation in the sharp-cutoff scheme becomes one-loop exact both near the upper as well as the lower critical dimension. Directly in 2+12+12+1 dimensions, our chiral Ising results agree with the best available predictions from other methods within the single-digit percent range for $\ensuremath{\nu}$ and ${\ensuremath{\eta}}_{\ensuremath{\phi}}$ and the double-digit percent range for ${\ensuremath{\eta}}_{\ensuremath{\Psi}}$. While one would expect a similar performance of our approximation in the chiral Heisenberg universality class, discrepancies with the results of other calculations here are more significant. Discussion and summary of various approaches is presented.
Two-dimensional spin-orbital magnets with strong exchange frustration have recently been predicted to facilitate the realization of a quantum critical point in the Gross-Neveu-SO(3) universality class. In contrast to previously known Gross-Neveu-type universality classes, this quantum critical point separates a Dirac semimetal and a long-range-ordered phase, in which the fermion spectrum is only partially gapped out. Here, we characterize the quantum critical behavior of the Gross-Neveu-SO(3) universality class by employing three complementary field-theoretical techniques beyond their leading orders. We compute the correlation-length exponent $\ensuremath{\nu}$, the order-parameter anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\phi}}$, and the fermion anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\psi}}$ using a three-loop $\ensuremath{\epsilon}$ expansion around the upper critical space-time dimension of four, a second-order large-NNN expansion (with the fermion anomalous dimension obtained even at the third order), as well as a functional renormalization group approach in the improved local potential approximation. For the physically relevant case of N=3N=3N=3 flavors of two-component Dirac fermions in 2+12+12+1 space-time dimensions, we obtain the estimates $1/\ensuremath{\nu}=1.03(15), {\ensuremath{\eta}}_{\ensuremath{\phi}}=0.42(7)$, and ${\ensuremath{\eta}}_{\ensuremath{\psi}}=0.180(10)$ from averaging over the results of the different techniques, with the displayed uncertainty representing the degree of consistency among the three methods.
We perform large-scale quantum Monte Carlo simulations of the Hubbard model at half filling with a single Dirac cone close to the critical point, which separates a Dirac semimetal from an antiferromagnetically ordered phase where SU(2) spin rotational symmetry is spontaneously broken. We discuss the implementation of a single Dirac cone in the SLAC formulation for eight Dirac components and the influence of dynamically induced long-range superexchange interactions. The finite-size behavior of dimensionless ratios and the finite-size scaling properties of the Hubbard model with a single Dirac cone are shown to be superior compared to the honeycomb lattice. We extract the critical exponent believed to belong to the chiral Heisenberg Gross-Neveu-Yukawa universality class: the critical exponent ν = 1.02 ( 3 ) coincides for the two lattice types once honeycomb lattices of linear dimension L > 15 are considered. In contrast to the SLAC formulation, where the anomalous dimensions are estimated to be η ϕ = 0.73 ( 1 ) and η ψ = 0.09 ( 1 ) , they remain less stable on honeycomb lattices, but tend towards the estimates from the SLAC formulation.
Wilson–Fisher expansion near upper critical dimension has proven to be an invaluable conceptual and computational tool in our understanding of the universal critical behavior in the [Formula: see text] field theories that describe low-energy physics of the canonical models, such as Ising, XY, and Heisenberg. Here, I review its application to a class of the Gross–Neveu–Yukawa (GNY) field theories, which emerge as possible universal description of a number of quantum phase transitions in electronic two-dimensional systems such as graphene and d-wave superconductors. GNY field theories may be viewed as minimal modifications of the [Formula: see text] field theories in which the order parameter is coupled to relativistic Dirac fermions through Yukawa term and which still exhibit critical fixed points in the suitably formulated Wilson–Fisher [Formula: see text]-expansion. I discuss the unified GNY field theory for a set of different symmetry-breaking patterns, with focus on the semimetal-Néel-ordered-Mott insulator quantum phase transition in the half-filled Hubbard model on the honeycomb lattice, for which a comparison between the state-of-the-art [Formula: see text]-expansion, quantum Monte Carlo, large N, and functional renormalization-group calculations can be made.
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.