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arXiv:1609.03824 · Physical review. B./Physical review. B

Ising and Gross-Neveu model in next-to-leading order

Benjamin Knorr

Published Dec 1, 2016• 60 citations
arXivHTMLPDF

Abstract

We study scalar and chiral fermionic models in next-to-leading order with the help of the functional renormalization group. Their critical behavior is of special interest in condensed-matter systems, in particular graphene. To derive the $\ensuremath{\beta}$ functions, we make extensive use of computer algebra. The resulting flow equations were solved with pseudospectral methods to guarantee high accuracy. New estimates of critical quantities for both the Ising and the Gross-Neveu model are provided. For the Ising model, the estimates agree with earlier renormalization-group studies of the same level of approximation. By contrast, the approximation for the Gross-Neveu model retains many more operators than all earlier studies. For two Dirac fermions, the results agree with both lattice and large-${N}_{f}$ calculations, but for a single flavor, different methods disagree quantitatively, and further studies are necessary.

DOI
10.1103/physrevb.94.245102

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We study scalar and chiral fermionic models in next-to-leading order with the help of the functional renormalization group. Their critical behavior is of special interest in condensed-matter systems, in particular graphene. To derive the \ensuremathβ\ensuremath{\beta}\ensuremathβ functions, we make extensive use of computer algebra. The resulting flow equations were solved with pseudospectral methods to guarantee high accuracy. New estimates of critical quantities for both the Ising and the Gross-Neveu model are provided. For the Ising model, the estimates agree with earlier renormalization-group studies of the same level of approximation. By contrast, the approximation for the Gross-Neveu model retains many more operators than all earlier studies. For two Dirac fermions, the results agree with both lattice and large-Nf{N}_{f}Nf​ calculations, but for a single flavor, different methods disagree quantitatively, and further studies are necessary.
Dirac and Weyl fermions appear as quasiparticle excitations in many different condensed-matter systems. They display various quantum transitions which represent unconventional universality classes related to the variants of the Gross-Neveu model. In this paper we study the bosonized version of the standard Gross-Neveu model---the Gross-Neveu-Yukawa theory---at three-loop order, and compute critical exponents in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions for a general number of fermion flavors. Our results fully encompass the previously known two-loop calculations, and agree with the known three-loop results in the purely bosonic limit of the theory. We also find the exponents to satisfy the emergent superscaling relations in the limit of a single-component fermion, order by order up to three loops. Finally, we apply the computed series for the exponents and their Pad\'e approximants to several phase transitions of current interest: metal-insulator transitions of spin-1/2 and spinless fermions on the honeycomb lattice, emergent supersymmetric surface field theory in topological phases, as well as the disorder-induced quantum transition in Weyl semimetals. Comparison with the results of other analytical and numerical methods is discussed.
We analyze fermionic criticality in relativistic 2+12+12+1 dimensional fermion systems using the functional renormalization group (FRG), concentrating on the Gross-Neveu (chiral Ising) and the Thirring model. While a variety of methods, including the FRG, appear to reach quantitative consensus for the critical regime of the Gross-Neveu model, the situation seems more diverse for the Thirring model with different methods yielding vastly different results. We present a first exploratory FRG study of such fermion systems including momentum-dependent couplings using pseudo-spectral methods. Our results corroborate the stability of results in Gross-Neveu-type universality classes, but indicate that momentum dependencies become more important in Thirring-type models for small flavor numbers. For larger flavor numbers, we confirm the existence of a non-Gaussian fixed point and thus a physical continuum limit. In the large-NNN limit, we obtain an analytic solution for the momentum dependence of the fixed-point vertex.
We study the chiral Ising, the chiral XY, and the chiral Heisenberg models at four-loop order with the perturbative renormalization group in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and compute critical exponents for the Gross-Neveu-Yukawa fixed points to order $\mathcal{O}({\ensuremath{\epsilon}}^{4})$. Further, we provide Pad\'e estimates for the correlation length exponent, the boson and fermion anomalous dimension, as well as the leading correction to scaling exponent in 2+12+12+1 dimensions. We also confirm the emergence of supersymmetric field theories at four loops for the chiral Ising and the chiral XY models with N=1/4N=1/4N=1/4 and N=1/2N=1/2N=1/2 fermions, respectively. Furthermore, applications of our results relevant to various quantum transitions in the context of Dirac and Weyl semimetals are discussed, including interaction-induced transitions in graphene and surface states of topological insulators.
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.
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.
It is expected that the Gross-Neveu-Yukawa (GNY) chiral Ising transition of Dirac fermions coupled with a scalar field in (2+1)(2+1)(2+1) dimensions will be the first fermionic quantum critical point that various methods, such as conformal bootstrap, perturbative renormalization group, and quantum Monte Carlo (QMC) simulations, would yield converged critical exponents---serving the same role as the Ising and O(N)O(N)O(N) models in the textbooks of statistical and quantum physics. However, such an expectation has not been fully realized from the lattice QMC simulations due to the obstacles introduced by the UV finite-size effect. In this Letter, by means of the elective-momentum ultrasize (EMUS)-QMC method, we compute the critical exponents of the $O{(N/2)}^{2}\ensuremath{\rtimes}{\mathbb{Z}}_{2}$ GNY N=8N=8N=8 chiral Ising transition on a two-dimensional $\ensuremath{\pi}$-flux fermion lattice model between Dirac semimetal and quantum spin Hall insulator phases. With the matching of fermionic and bosonic momentum transfer and collective update in momentum space, our QMC results provide fully consistent exponents with those obtained from the bootstrap and perturbative approaches. In this way, the EMUS now live happily on the N=8N=8N=8 island and could explore the chiral Gross-Neveu-Yukawa archipelago with ease.