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arXiv:1903.07388 · Physical review. D/Physical review. D.

Momentum dependence of quantum critical Dirac systems

Lennart Dabelow, Holger Gies A, Benjamin Knorr

Published Jun 28, 2019• 19 citations
arXivHTMLPDF

Abstract

We analyze fermionic criticality in relativistic $2+1$ 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-$N$ limit, we obtain an analytic solution for the momentum dependence of the fixed-point vertex.

DOI
10.1103/physrevd.99.125019

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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 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-Nf{N}_{f}Nf​ calculations, but for a single flavor, different methods disagree quantitatively, and further studies are necessary.
Evidence for relativistic quantum criticality of antiferromagnetism and superconductivity in two-dimensional Dirac fermion systems has been found in large-scale quantum Monte Carlo simulations. However, the corresponding (2+12+12+1)-dimensional Gross--Neveu--Yukawa field theory with Nf=2{N}_{f}=2Nf​=2 four-component Dirac fermions coupled to two triplets of order parameters does not exhibit a renormalization group fixed point at one-loop order. Instead, the theory only features a critical point for a large or very small fractional number of fermion flavors Nf{N}_{f}Nf​, which disappears for a broad range of flavor numbers around the physical case, Nf=2{N}_{f}=2Nf​=2, due to fixed-point annihilation. This raises the question on how to explain the observed scaling collapse in the quantum Monte Carlo data. Here, we extend previous renormalization-group analyses by studying a generalized model at two-loop order in $4\ensuremath{-}\ensuremath{\epsilon}$ spacetime dimensions. We determine the $\ensuremath{\epsilon}$ correction to the upper and lower critical flavor numbers for the fixed-point annihilation and find that they both go towards the physical case Nf=2{N}_{f}=2Nf​=2. However, this only happens very slowly, such that an extrapolation to $\ensuremath{\epsilon}=1$ still suggests the absence of criticality in 2+12+12+1 dimensions. Thereby, we consolidate the finding that the continuum field theory does not feature a stable renormalization-group fixed point and no true quantum criticality would be expected for the considered system. We briefly discuss a possible reconciliation in terms of a complex conformal field theory. Further, we also explore the fixed-point structure in an enlarged theory space and identify a candidate stable fixed-point solution.
The coupling between fermionic matter and gauge fields plays a fundamental role in our understanding of nature, while at the same time posing a challenging problem for theoretical modeling. In this situation, controlled information can be gained by combining different complementary approaches. Here, we study a confinement transition in a system of Nf{N}_{\mathrm{f}}Nf​ flavors of interacting Dirac fermions charged under a U(1) gauge field in 2+12+12+1 dimensions. Using quantum Monte Carlo simulations, we investigate a lattice model that exhibits a continuous transition at zero temperature between a gapless deconfined phase, described by three-dimensional quantum electrodynamics, and a gapped confined phase, in which the system develops valence-bond-solid order. We argue that the quantum critical point is in the universality class of the QED3{\mathrm{QED}}_{3}QED3​-Gross-Neveu-XY model. We study this field theory within a 1/Nf1/{N}_{\mathrm{f}}1/Nf​ expansion in fixed dimension as well as a renormalization group analysis in $4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions. The consistency between numerical and analytical results is revealed from large to intermediate flavor number.
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.
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.
We study the competition of spin- and charge-density waves and their quantum multicritical behavior for the semimetal-insulator transitions of low-dimensional Dirac fermions. Employing the effective Gross-Neveu-Yukawa theory with two order parameters as a model for graphene and a growing number of other two-dimensional Dirac materials allows us to describe the physics near the multicritical point at which the semimetallic and the spin- and charge-density-wave phases meet. With the help of a functional renormalization group approach, we are able to reveal a complex structure of fixed points, the stability properties of which decisively depend on the number of Dirac fermions Nf{N}_{f}Nf​. We give estimates for the critical exponents and observe crucial quantitative corrections as compared to the previous first-order $\ensuremath{\epsilon}$ expansion. For small Nf{N}_{f}Nf​, the universal behavior near the multicritical point is determined by the chiral Heisenberg universality class supplemented by a decoupled, purely bosonic, Ising sector. At large Nf{N}_{f}Nf​, a novel fixed point with nontrivial couplings between all sectors becomes stable. At intermediate Nf{N}_{f}Nf​, including the graphene case (Nf=2)({N}_{f}=2)(Nf​=2), no stable and physically admissible fixed point exists. Graphene's phase diagram in the vicinity of the intersection between the semimetal, antiferromagnetic, and staggered density phases should consequently be governed by a triple point exhibiting first-order transitions.