Pis’ma v ZhETF, vol. 111, iss. 4, pp. 242 - 243

February 25

Bilayer, hydrogenated and fluorinated graphene: QED vs SU(2) QCD

theory

V. Yu. Irkhin^+∗1), Yu. N. Skryabin⁺

⁺M. N. Mikheev Institute of Metal Physics, 620108 Ekaterinburg, Russia

^∗Ural Federal University, 620002 Ekaterinburg, Russia

Submitted 3 January 2020

Resubmitted

3 January 2020

Accepted 4

January 2020

DOI: 10.31857/S0370274X20040074

The electron spectrum of standard graphene with

quantum phase transition at the quantum critical point

weakly correlated sp-orbitals is described in terms

[7]. Such an approach enables us to trace the hierarchy

of Dirac fermions corresponding to one-electron band

of symmetries - from SU(2) to U(1) and Z₂ spin liquids,

cones with a gap which occurs owing to spin-orbit inter-

the latter being the most stable one.

action. Thus the system has properties of a topological

In the low energy infrared (IR) limit, the La-

insulator. In some cases graphene systems demonstrate

grangian of Quantum Electrodynamics in 2+1 dimen-

strong electron correlations, including twisted magic-

sions, QED₃, reads [8]:

angle bilayer system [1] and monolayer graphene inter-

∑

calated by gadolinium [2].

L=i

ψ_j

(1)

/a^ψj^,

In the strongly correlated regime the excitation spec-

j=1

trum may change drastically. At the same time, the

model still includes Dirac fermions at the nodal points.

where

/a=γµDa,µ^{isthegaugecovariantDiracop-}

Such a spectrum occurs in the mean-field approxima-

erator, ψ_j is a two-component Dirac fermion with four

tion corresponding to the deconfinement spinon picture

flavors (N_f = 4) labeled by j,

ψ = ψ^†γ⁰, a_µ is a dynam-

[3]. The corresponding non-magnetic Dirac spin liquid

ical U(1) gauge field. The theory assumes that the U(1)

(DSL) [4] is characterized by a quantum topological or-

gauge flux, i.e., the total flux of the magnetic field, is

der. However, the stability of DSL should be further ex-

conserved. This noncompact N_f = 4 QED₃ theory flows

amined and is more probable in frustrated systems. In

to a stable critical fixed point in the IR limit.

[5], the spinon picture was applied to bilayer graphene;

For bipartite lattices, in the mean field approxima-

here we investigate the corresponding models in more

tion one can continuously tune the Hamiltonian, with-

detail.

out breaking any symmetry or changing the low-energy

In [6], a frustrated ground state for single-side hy-

Dirac dispersion, to reach a point with particle-hole

drogenated (C₂H) and fluorinated (C₂F) graphene was

symmetry [8]. This theory will then have a larger gauge

predicted, which sheds light on the absence of a conven-

symmetry of SU(2)_g,

tional magnetic ordering in defective graphene demon-

∑

ψ_vγ^µ(∂_µ - ia_µ)ψ_v,

(2)

strated in experiments despite presence of magnetic mo-

v=1,2

ments. This suggests a highly correlated magnetic be-

havior at low temperatures offering the possibility of a

where a is an SU(2) gauge field, and ψ_1,2 are two SU(2)-

quantum spin-liquid state.

fundamental fermions. This theory has an SO(5) sym-

In the present work, we apply to this problem

metry.

the gauge-field formalism of quantum electrodynamics

In an alternative theory, the SU(2) gauge symmetry

(QED) and chromodynamics (QCD) [7] and treat the

in QCD₃ is lowered to U(1) owing to the Higgs phe-

spin-liquid state in terms of U(1) QED and parent SU(2)

nomenon [7].

QCD theories. The former theory describes deconfine-

ment situation and Dirac spin liquid. The latter theory

∑

includes a monopole operator which carries trivial quan-

ψ_iγ^µ(∂_µ - ia_µ)ψ_i + (λM_a + h.c.),

(3)

i=1

tum numbers and the Neel to valence bond solid (VBS)

where a_µ is now a U(1) gauge field, and the term M_a

¹⁾e-mail: valentin.irkhin@imp.uran.ru

represents instanton tunneling.

242

Письма в ЖЭТФ том 111 вып. 3 - 4

2020

Bilayer, hydrogenated and fluorinated graphene: QED vs SU(2) QCD theory

243

The flavor symmetry of QCD₃ at N_f = 2 is SO(5). In

Triangular versus honeycomb lattice problem for bi-

both the theories (2) and (3), the Dirac fermions trans-

layer graphene was considered in [10]. Although the

form in the spinor representation of the SO(5) group,

charge density is concentrated on the triangular lattice

the SO(5)-vector operators being time-reversal invari-

sites of the moire pattern, the Wannier states of the

ant mass operators. None of the duality field theories

tight-binding model must be centered on different sites

possesses the full SO(5) symmetry (combining antifer-

which form a honeycomb lattice.

romagnetic and VBS order parameters) explicitly, the

Generalized triangular lattice Hubbard models have

symmetry being at best emergent in the IR limit. One

been proposed to describe flat moire bands in twisted

assumes that at least one of the theories (2) and (3) will

van der Waals transition metal dichalcogenide heterobi-

flow to the deconfined critical point in the IR limit [7].

layers [11]. Recently a heterostructure of ABC-stacked

The most probable scenario for the QCD₃ theory

trilayer graphene and boron nitride, which also forms a

(tuned to an SO(5) symmetric point) describes the de-

triangular moire superlattice even at zero twist angle,

confined critical point, and perturbing it drives it into

was studied [12].

either the VBS phase or the Neel phase [7].

An effective Heisenberg model was built in [6] for the

The Dirac spin liquid can be unstable with respect to

C₂H and and C₂F systems, which includes competing

proliferation of monopoles, and different ordered states

exchange interactions on different p-orbitals and com-

can be reached from DSL, the symmetry properties

bines features of honeycomb and triangle lattices. The

of the magnetic monopoles being different on different

case of C₂H turns out to be more complicated due to the

lattices [4]. For bipartite lattices, there is always one

presence of the two nonequivalent magnetic sublattices

monopole operator which transforms trivially under all

comprising the honeycomb lattice. Thus frustration can

microscopic symmetries owing to the existence of a par-

lead to a DSL state provided that monopoles are irrel-

ent SU(2) gauge theory. This is a spin singlet which

evant.

carries no non-trivial quantum numbers and therefore

Full text of the paper is published in JETP Letters

provides an allowed perturbation to the Hamiltonian,

journal. DOI: 10.1134/S0021364020040025

destabilizing DSL. On the non-bipartite lattices such a

destabilization does not occur.

1. Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken,

Thus the situation for bipartite (honeycomb) and

J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe,

non-bipartite (triangle) lattices is different. For bipar-

T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-

tite situation, there is no additional topological symme-

Herrero, Nature 556, 80 (2018).

try since the flux of SU(2) gauge field (unlike that of

2. S. Link, S. Forti, A. Stoehr, K. Kuester, M. Roes-

U(1) field) is not conserved [8]. For the non-bipartite

ner, D. Hirschmeier, C. Chen, J. Avila, M. C. Asen-

lattice, monopoles do not prevent stability of spin liq-

sio, A. A. Zakharov, T. O. Wehling, A. I. Lichtenstein,

uid (DSL is transformed to Z₂ spin liquid by inclusion

M. I. Katsnelson, and U. Starke, Phys. Rev. B 100,

121407(R) (2019).

of the Higgs field). For frustrated bipartite lattices, spin

3. P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys.

liquid is expected to exist at the quantum critical point

78, 17 (2006).

only, but the quantum critical behavior can be observed

4. X.-Y. Song, Ch. Wang, A. Vishwanath, and Y.-Ch. He,

at finite temperatures.

Nat. Commun. 10, 4254 (2019).

As discussed in [9], experimental data for twisted bi-

5. V. Yu. Irkhin and Yu. N. Skryabin, JETP Lett. 107, 651

layer graphene indicate that the electron charge density

(2018).

is concentrated on a moire triangular lattice, so that

6. A. N. Rudenko, F. J. Keil, M. I. Katsnelson, and

the consequences of local correlations should be similar

A. I. Lichtenstein, Phys. Rev. B 88, 081405(R) (2013).

to those on the triangular lattice. On the other hand,

7. C. Wang, A. Nahum, M. A. Metlitski, C. Xu, and

symmetry and topological aspects of the band struc-

T. Senthil, Phys. Rev. X 7, 031051 (2017).

ture require that the model should be formulated using

8. X.-Y. Song, Y.-Ch. He, A. Vishwanath, and Ch. Wang,

the Wannier orbitals of a honeycomb lattice. Besides

arXiv:1811.11182.

the minimal phenomenological model of antiferromag-

9. A. Thomson, S. Chatterjee, S. Sachdev, and

M. S. Scheurer, Phys. Rev. B 98, 075109 (2018).

netism on the triangular lattice, the authors of [9] con-

10. H. C. Po, L. Zou, A. Vishwanath, and T. Senthil, Phys.

sidered the model where the spin density is centered on

Rev. X 8, 031089 (2018).

the bonds of the dual bipartite honeycomb lattice. The

11. F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald,

half-filled triangular lattice model and the quarter-filled

Phys. Rev. Lett. 121, 026402 (2018).

honeycomb-lattice model can be consistent with exper-

12. G. Chen, L. Jiang, S. Wu, B. Lyu, H. Li, L. Chit-

imental observations. The half-filled honeycomb-lattice

tari, K. Watanabe, T. Taniguchi, Z. Shi, Y. Zhang, and

model requires the additional Kekule VBS order which

F. Wang, Nature Phys. 15, 237 (2019).

is in agreement with the Monte Carlo calculations.

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2020

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