Is graphene in vacuum an insulator?

0.15 MB·

by  Joaquín E. Drut

Is graphene in vacuum an insulator?
Joaqu´ın E. Drut1 and Timo A. La¨hde2
1Department of Physics, The Ohio State University, Columbus, OH 43210–1117, USA and
2Department of Physics, University of Washington, Seattle, WA 98195–1560, USA
(Dated: January 15, 2009)
We present evidence, from Lattice Monte Carlo simulations of the phasediagram of graphene as
afunction oftheCoulomb couplingbetweenquasiparticles, thatgraphenein vacuumislikely tobe
an insulator. We find a semimetal-insulator transition at αcrit = 1.11±0.06, where α ≃ 2.16 in
g g
vacuum, and αg ≃ 0.79 on a SiO2 substrate. Our analysis uses the logarithmic derivative of the
9 order parameter, supplemented by an equation of state. The insulating phase disappears above a
0 critical number of four-component fermion flavors 4 < Nfcrit < 6. Our data are consistent with a
0 second-ordertransition.
2
PACSnumbers: 73.63.Bd,71.30.+h,05.10.Ln

n
a
J
Graphene, a carbon allotrope with a two-dimensional properties resemble QED in a very strongly coupled

5 honeycomb structure, has become an important player regime. This provides an exciting opportunity for the
1
attheforefrontofcondensedmatterphysics,drawingthe study of strongly coupled theories, within a condensed-

attentionoftheoristsandexperimentalistsalikeduetoits matteranaloguethatcanbeexperimentallyrealizedwith
]
l challenging nature as a many-body problem, its unusual modest equipment.
e
- electronic properties and possible technological applica- Notably, Eq. (1) satisfies a chiral U(2N ) symmetry
r f
t tions (see Refs. [1, 2] and references therein). Graphene whichcanbreakspontaneouslyatlargeenoughCoulomb
s
also belongs to a large class of planar condensed-matter
. coupling, generating a gap in the quasiparticle spec-
t
a systems, which includes other graphite-related materials trum. Whether such an effect occurs in real graphene
m as well as high-T superconductors.
c isanopenissuefromthe experimentalpointofview(see
- Adistinctivefeatureofgrapheneisthatitsbandstruc- however Ref. [4], where a substrate-induced gap is re-
d ture containstwo degenerate‘Dirac points’, in the vicin-
ported). On the theoretical side, dynamical gap genera-
n
ityofwhichthedispersionislinear,asinrelativisticthe-
o tion is described by a quantum phase transition due to
ories[3]. Thelow-energyexcitationsingraphenearethus
c the formation of particle-hole bound states. However,in
[ DiracquasiparticlesofFermivelocityv ≃c/300,wherec such a strongly coupled regime, even a qualitative anal-

2 is the speed of light in vacuum. These are described by ysis should be non-perturbative. This is especially true
the Euclidean action
v for graphene in vacuum, where α attains its maximum
g
4 N value, while it is partially screened in the presence of a
f
3 S = − d2xdtψ¯ D[A ]ψ substrate. Whilethesemimetallicpropertiesofgraphene
8 E a 0 a
0 Xa=1Z on a substrate are well established, the issue of whether
7. +21g2 d3xdt(∂iA0)2, (1) aoftarasnusbitsitornatteoraenmianinsuslautnisnegttplehda.se occurs in the absence
0 Z
8 where g2 = e2/ǫ for graphene in vacuum, ψ is a This problem has been studied using perturbative as
0 0 a well as non-perturbative methods [5, 6, 7, 8]. The lat-
four-component Dirac field in 2+1 dimensions, A is
: 0
v a Coulomb field in 3+1 dimensions, N = 2 for real ter, which are typically based on a gap equation, yield
f
i an infinite-order transition to an insulating phase above
X graphene, and
a critical coupling. On the other hand, the results of
ar D[A0] = γ0(∂0+iA0)+vγi∂i, i=1,2 (2) large-Nf analyses [5, 9, 10] find that Coulomb inter-
actions flow towards a non-interacting fixed point un-
where the Dirac matrices γ satisfy the Euclidean Clif-
µ
der renormalization-group (RG) transformations, being
ford algebra {γ ,γ } = 2δ . The strength of the
µ ν µν
thereforeunabletoinduceatransition. However,anum-
Coulomb interaction is controlled (as can be shown by
rescaling t and A ) by α = e2/(4πvǫ ), which is the ber of uncontrolledapproximationsare involved,such as
0 g 0
the reliance on large-N results that may break down
graphene analogue of the fine-structure constant α ≃ f
for small N , various approximate treatments of the gap
1/137 of quantum electrodynamics (QED). f
equationkernel,aswellasthe linearizationofthe result-
DespitethesimilaritieswithQED,thesmallnessofv/c
ing integral equation (see Appendix B of Ref. [8]).
ingraphenehasnon-trivialconsequences: Coulombinter-
actions between the quasiparticles are essentially instan- We set out to characterize the phase diagram of
taneous,thusbreakingrelativisticinvariancewhichisre- graphene in a controlled fashion, which entails a lattice
flectedinEq.(1). Theanalogueofthefine-structurecon- MonteCarloapproachandanalysisofthe chiralconden-
stant α ≃ 300α in graphene, such that the low-energy sate,whichistheorderparameterforatransitionintoan
g
2
insulating charge-density-wavephase. Such an approach
0.30
is non-perturbative, takes full account of quantum fluc- Nf = 2, m0 = 0.010
18 0.015
tuations, and has been extensively used [11, 12, 13, 14] 0.25 16 0.020
in the study of 2+1 (QED ) and 3+1 (QED ) dimen- 0.025
3 4 N = 4, m = 0.010
sionalQED,butnotforstudiesofgraphene(seehowever 0.2 16 f 0 0.015
Ref. [15], where a model for the strong-coupling limit is 16 16 Nf = 6, m0 = 0.010
0.015
investigated).
σ 16
Tothisend,wediscretizethepuregaugepartofEq.(1)
16
according to (for a recent overview,see Ref. [16]) 12
0.1
12 14
12
β 3 2 12 14
Sg[θ ] = θ −θ , (3) 0.05
E 0 2 Xn "Xi=1(cid:16) 0,n 0,n+ei(cid:17) # 12 12 12
0
where the (dimensionless) lattice coupling β ≡ v/g2, θ 0 0.05 β 0.15 0.2
0
is the lattice gauge potential, n≡(n ,...,n ) denotes a
0 3
site on the space-time lattice, and e is a unit vector in FIG.1: (Coloronline)ChiralcondensateσforN =2,4,6as
µ f
the direction µ. For studies of chiral phase transitions, afunction of β and m0,with lines intendedto guidetheeye.
staggered fermions [17] are a preferred choice, since chi- ThelatticesareofextentL3×Lz,suchthatthefermionslive
in a 2+1 dimensional cube of size L, while the gauge bosons
ral symmetry is then partially preserved. As N stag-
also propagate in the z-direction of length L . For each β,
gered flavors correspond to N = 2N continuum Dirac z
f thevalueofLisgiven nexttothedatapoints. Allresults are
flavors [18], it suffices (for real graphene) to set N = 1,
for L = 8, as larger values had no discernible effects. For
z
which gives eachdatapoint∼300uncorrelatedgaugeconfigurationswere
generated. The statistical uncertainties, which are compara-
SEf[χ¯,χ,U0] = − χ¯mDm,n[U0]χn, (4) bletothesizeofthesymbols,wereobtained bythejackknife
m,n method [24]. Finite volume effects are largest for small β.
X
where the χn are staggered fermion spinors, and (m,n)
are restricted to a 2+1 dimensional sublattice. The The chiralU(2N )symmetry of the continuumtheory
f
invariance of Eq. (1) under spatially uniform, time- canonlypartiallyberealizedonthelatticeifthedoubling
dependentgaugetransformationsisretainedbycoupling problemistobeavoided[19]. Inparticular,onlyaglobal
the fermions to the gauge field via U0 = exp(iθ0). The U(N)×U(N)symmetryremainsupondiscretization[18].
staggered form of D is We focus on the spontaneous breakdown of this symme-
try to a U(N) subgroup, characterized by a condensate
1
Dm,n[U0]= 2 δm+e0,nU0,m−δm−e0,nU0†,n (5) σ ≡ hχ¯χi =6 0 in the limit m0 → 0, which marks the
appearance of a gap in the quasiparticle spectrum. Our
1h i
+ η δ −δ +m δ results for σ are presented in Fig. 1 for β =0.05,...,0.5
2 i,m m+ei,n m−ei,n 0 m,n
Xi h i and m0 =0.010,...,0.025 (in lattice units).
Our data for N =2 in Fig. 1 are suggestive of a crit-
where η1,n = (−1)n0 and η2,n = (−1)n0+n1. The mass ical coupling β ∼f 0.06...0.09, below which σ survives
c
termbreakschiralsymmetryexplicitly,generatinganon-
in the limit m → 0. More significantly, the suscepti-
0
zero condensate which is otherwise not possible at finite
bility χ = ∂σ/∂m shown in the right panel of Fig. 2
volume. Extrapolationtom =0isthusrequired. Upon l 0
0 exhibitsamaximumwhichtendstowardsβ asm isde-
integrationof the fermionic degreesof freedom, the path c 0
creased. The (much more limited) data for N = 4 (not
integral is governedby the effective action f
shown)haveasimilarmaximumaroundβ ∼0.03. AsN
f
Seff[θ0]=−Nlndet(Dm,n[U0])+SEg[θ0], (6) ibsetinwcereenasNed,=σ4obavnidouNsly=be6c.omTehsissaugprpereesssweidt,hvlaanrgiseh-iNng
f f f
suchthatP[θ0]≡exp(−Seff[θ0])definestheMonteCarlo results[10]thatyieldaquantumcriticalpoint(andthere-
probability measure. It is straightforward to show that fore no condensate) in the limit β →0, and is consistent
the determinant is positive definite. We have sampled with recent Monte Carlo studies of that limit [15]. Our
P[θ0]usingtheMetropolisalgorithm,updatingθ0atran- results for small Nf establish that the strong-coupling
dom locations and evaluating the fermion determinant criticalpointdisappearsbelowNcrit,with4<Ncrit <6.
f f
exactly. Our approach has been tested against known For a quantitative determination of β , we compute
c
results for QED and QED . The data of Ref. [11] on the logarithmic derivative R of σ with respect to m ,
3 4 0
the chiral condensate of QED have been accurately re-
3
produced for multiple values of Nf, along with several R ≡ ∂lnσ = m0 ∂σ , (7)
randomly chosen datapoints from Refs. [12, 14]. ∂lnm σ ∂m
0(cid:12)β (cid:18) 0(cid:19)(cid:12)β
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3
Nf = 2, β = 0.050 0.111 χl several fits were performed from L = 8 up to the main
00..006777 00..124030 Nf = 2, m0 = 00..001150 fit with L = 16. Our analysis indicates that β is sta-
0.091 5.5 0.020 c
R 0.025 ble around ∼ 0.075, even though the datapoints at the
1 5.0 smallest β shift due to finite-volume effects. In contrast,
4.5 for Nf =6 chiral symmetry remains unbroken, and thus
0.8 finite-volume effects remain small in the limit β → 0.
4.0
The stability of β can be understood in terms of the
0.6 c
3.5 results for R in Fig. 2, since any fit to the condensate
should be consistent with the susceptibility as well, and
0.4 3.0
Rcombines bothpiecesofinformation. Thecharacterof
0.2 2.5 Rclearlychangesaroundβ =0.075,indirectcorrelation
2.0 with the fitted values of βc. If one allows for extreme
modifications,suchasnon-classicalcriticalexponents,or
0 0.01 0.02 0.03 0.05 0.1 0.15 0.2
m β forces the fit to account for all the datapoints, values of
0
β aslowas0.060maybefound,althoughatthepriceof
c
a much worsefit to the data. By inclusionand exclusion
FIG. 2: (Color online) Left panel: Logarithmic derivative
R as a function of m0 for different β. Right panel: Chiral of different sets of datapoints, a realistic (though some-
susceptibility χl as a function of β for different m0. All data what model-dependent) estimate of the critical coupling
are for Nf =2, with lattice sizes identical to those of Fig. 1. isβc =0.0755±0.0008. Possibledeviationsfromclassical
The lines are intended as a guide to theeye. mean-field behavior with f = σ and f = σ3 are below
1 3
the resolution of the present study.
The results in Fig. 3 suggest a second-order transi-
accordingtothemethodofRef.[20]. Inthelimitm →0,
0
tion with classical exponents, unlike Refs. [6, 8], where
thebehaviorofRisasfollows: R→1inthechirallysym-
an infinite-order transition was found. In this situation,
metric (unbroken) phase, where σ ∝ m . At the critical
0
further investigation is clearly called for. While the sen-
coupling β = β one finds that R → 1/δ, where δ is a
c sitivity of our analysis increases for smaller m , larger
universal critical exponent. R vanishes in the sponta- 0
latticevolumesarealsorequiredtokeepfinite-volumeef-
neously broken phase, where σ 6= 0 for m → 0. The
0
fects under control. A similar EOS analysis has recently
data in Fig. 2 (left panel) indicate that chiral symme-
beenperformedbyHands andStrouthos(Ref.[15])fora
try is spontaneously broken for β = 0.067, but remains
unbroken for β = 0.077, from which we conclude that
β = 0.072±0.005. This estimate can be refined by use
c σ σ2
of larger lattice volumes and smaller values of m0. Nf = 2, m0 = 00..001150 Nf = 2, m0 = 00..001150
A more precise determination of β requires an equa- 0.020 0.07 0.020
c 0.25 0.025 0.025
tion of state (EOS) of the form m = f(σ,β) for the included β = 0.050
0 excluded 0.06 0.067
extrapolation m → 0. We have considered the EOS extrap. 0.077
0 0.2 0.091
successfully applied [12, 13] to QED , 0.05 0.111
4
0.15 0.04
m X(β) = Y(β)f (σ)+f (σ), (8)
0 1 3
0.03
0.1
whereX(β)andY(β)areexpandedaroundβ suchthat
c 0.02
X(β)=X +X (1−β/β )andY(β)=Y (1−β/β ). The
0 1 c 1 c
dependence on σ is given by f1(σ)=σb and f3(σ)=σδ, 0.05 0.01
which allows for non-classical critical exponents δ and
β¯ [12, 13], where b≡ δ−1/β¯. A χ2 fit of Eq. (8) to the 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 0.25
β m /σ
data in Fig. 1 is givenin Fig. 3 (left panel), along with a 0
“Fisherplot”ofσ2versusm /σ(rightpanel). Deviations
0
fromclassicalmean-fieldbehaviormanifestthemselvesin FIG.3: (Coloronline) Leftpanel: χ2 fittothedataofFig. 1
the Fisher plot as curvature in the lines of constant β. and extrapolation to m0 = 0 for Nf = 2 using Eq. (8) with
The mostly straight lines in Fig. 3 (right panel) indicate X0,X1,Y1 andβc asfreeparameters. Thepointswithlargest
finite-volume effects have been excluded from the fit. The
that the data should be well described, up to finite-size
effects, by the classical values δ = 3 and b = 1, which is o0.p1t9im5±al0p.0a0r3a,mXet1e=r v−a0lu.0e8s9a±re0.β0c01=an0d.0Y7515=±−00..0009013±, 0X.0001=.
confirmed in Fig. 3 (left panel).
The uncertainties are purely statistical. Right panel: Fisher
For all values of m0, the finite-volume effects appear plotofσ2versusm0/σforthedataofFig.1withNf =2. The
to be dominated by a dynamically generated correlation lines connect datapoints with identical β, such that straight
lengthintheregionthatweidentifyasthespontaneously lines indicate mean-field behavior according to Eq. (8). At
brokenphase. To gaugethe impact ofsucheffects onβ , βc, theextrapolation crosses the origin.
c
4
graphene-like theory with a zero-range interaction. Un- age for computer time, and W. Detmold, M. M. Forbes,
fortunately, a meaningful comparison is not possible at R. J. Furnstahl, D. Gazit and D. T. Son for instructive
this time, as their parameter 1/g2 cannot be identified discussions.
withourgaugecouplingβ, exceptinthe strong-coupling
limit β →0.
A comparison with experiment necessitates a discus-
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2
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