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arXiv:1801.01320 · Physical review. D/Physical review. D.

Large N critical exponents for the chiral Heisenberg Gross-Neveu universality class

J. A. Gracey

Published May 14, 2018• 59 citations
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Abstract

We compute the large $N$ critical exponents $\ensuremath{\eta}$, ${\ensuremath{\eta}}_{\ensuremath{\phi}}$ and $1/\ensuremath{\nu}$ in $d$ dimensions in the chiral Heisenberg Gross-Neveu model to several orders in powers of $1/N$. For instance, the large $N$ conformal bootstrap method is used to determine $\ensuremath{\eta}$ at $O(1/{N}^{3})$ while the other exponents are computed to $O(1/{N}^{2})$. Estimates of the exponents for a phase transition in graphene are given which are shown to be commensurate with other approaches. In particular the behavior of the exponents in $2<d<4$ is in qualitative agreement with a functional renormalization group analysis. The $\ensuremath{\epsilon}$ expansion of each exponent near four dimensions is in exact agreement with recent four loop perturbation theory.

DOI
10.1103/physrevd.97.105009

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We compute the large NNN critical exponents \ensuremathη\ensuremath{\eta}\ensuremathη, \ensuremathη\ensuremathϕ{\ensuremath{\eta}}_{\ensuremath{\phi}}\ensuremathη\ensuremathϕ​ and 1/\ensuremathν1/\ensuremath{\nu}1/\ensuremathν in ddd dimensions in the chiral Heisenberg Gross-Neveu model to several orders in powers of 1/N1/N1/N. For instance, the large NNN conformal bootstrap method is used to determine \ensuremathη\ensuremath{\eta}\ensuremathη at O(1/N3)O(1/{N}^{3})O(1/N3) while the other exponents are computed to O(1/N2)O(1/{N}^{2})O(1/N2). Estimates of the exponents for a phase transition in graphene are given which are shown to be commensurate with other approaches. In particular the behavior of the exponents in 2<d<42<d<42<d<4 is in qualitative agreement with a functional renormalization group analysis. The \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion of each exponent near four dimensions is in exact agreement with recent four loop perturbation theory.
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.
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+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.
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.
An important yet largely unsolved problem in the statistical mechanics of disordered quantum systems is to understand how quenched disorder affects quantum phase transitions in systems of itinerant fermions. In the clean limit, continuous quantum phase transitions of the symmetry-breaking type in Dirac materials such as graphene and the surfaces of topological insulators are described by relativistic (2+1)-dimensional quantum field theories of the Gross-Neveu-Yukawa (GNY) type. We study the universal critical properties of the chiral Ising, XY, and Heisenberg GNY models perturbed by quenched random-mass disorder, both uncorrelated or with long-range power-law correlations. Using the replica method combined with a controlled triple epsilon expansion below four dimensions, we find a variety of new finite-randomness critical and multicritical points with nonzero Yukawa coupling between low-energy Dirac fields and bosonic order parameter fluctuations, and compute their universal critical exponents. Analyzing bifurcations of the renormalization-group flow, we find instances of the fixed-point annihilation scenario—continuously tuned by the power-law exponent of long-range disorder correlations and associated with an exponentially large crossover length—as well as the transcritical bifurcation and the supercritical Hopf bifurcation. The latter is accompanied by the birth of a stable limit cycle on the critical hypersurface, which represents the first instance of fermionic quantum criticality with emergent discrete scale invariance.
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