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arXiv:2206.04073 · Physical review. B./Physical review. B

SO(4) multicriticality of two-dimensional Dirac fermions

Igor F. Herbut, Michael M. Scherer

Published Sep 21, 2022• 7 citations
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Abstract

We study quantum multicritical behavior in a $(2+1)\text{\ensuremath{-}}\mathrm{dimensional}$ Gross-Neveu-Yukawa field theory with eight-component Dirac fermions coupled to two triplets of order parameters that act as Dirac masses and transform as $(1,0)+(0,1)$ representation under the $\mathrm{S}\mathrm{O}(4)\ensuremath{\simeq} \mathrm{S}\mathrm{O}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(3)$ symmetry group. This field theory is relevant to spin-1/2 fermions on honeycomb or $\ensuremath{\pi}$-flux lattices, for example, near the transition points between an $s$-wave superconductor and a charge-density wave, on one side, and N\'eel order, on the other. Two triplets of such order parameters always allow for a common pair of two other order parameters that would complete them to the maximal set of compatible (anticommuting) orders of five. We first derive a unitary transformation in the Nambu (particle-hole) space which maps any two such triplets, possibly containing some superconducting orders, onto purely insulating order parameters. This allows one to consider a universal SO(4) Gross-Neveu-Yukawa description of the multicriticality without any Nambu doubling. We then proceed to derive the renormalization-group flow of the coupling constants at one-loop order in $4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions, allowing also a more general set of order parameters transforming under $\mathrm{S}\mathrm{O}({n}_{a})\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{O}({n}_{b})$. While for ${n}_{a}={n}_{b}>2$ in the bosonic sector and with fermions decoupled there is a stable fixed point of the flow, the Yukawa coupling to fermions quickly leads to its elimination by a generic fixed-point collision in the relevant range of fermion flavor numbers ${N}_{f}$. This suggests the replacement of the critical behavior by a runaway flow in the physical case ${n}_{a}={n}_{b}=3$. The structure of the renormalization group flow at ${n}_{a}\ensuremath{\ne}{n}_{b}$ is also discussed, and some nonperturbative arguments in favor of the stability of the decoupled critical point when ${n}_{a}=3$ and ${n}_{b}=1$ in $D=2+1$ are provided.

DOI
10.1103/physrevb.106.115136

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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.
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