Non-Fermi liquid behaviour of electrons in the half-filled ...cds.cern.ch/record/258853/files/P00021014.pdfMoreover, this superconductivity can be considered a form of high—Tc superconductivity,

Ze-mail: [email protected]: [email protected]

PEEQEMM

I|II||UII||l|IIIIIIIININIIJHIIIIIIIIUINIIIJlllQERN L.IBRFIRIE5» GENEVFJ OCR Output

resembles in the infrared the one-dimensional Luttinger liquid.results are robust as the Fermi level is not changed by the interaction. The systemquasiparticle pole, and anomalous screening of the Coulomb interaction. These

characteristics. There are anomalous dimensions in the fermionic observables, noinfrared behavior of the system that flows to a fixed point with non—Fermi liquidequal to QED3. Renormalization group techniques are used to investigate theinteractions is described by a renormalizable quantum field theory similar but not

A system of electrons in the two—dimensional honeycomb lattice with Coulomb

Abstract

Avda. Mediterrdneo s/n, 28.913 Leganés {Madrid), Spain.IDepartamento de Ingenieria, Universidad Carlos III de Madrid

Cantoblanco, 28049 Madrid, Spain.

'[Instituto de Ciencia de Materiales, CSIC

Serrano 128, 28006 Madrid, Spain

§Instituto de Estructura de la Materia, CSIC

J. Gonzalez§*, F. Guineaiz and M.A.H. Vozmedianoi

(A I`€I1OI`II1H.liZ&tlOIl gI'OU.p 3.ppI`O8.Cl'l)

in the Half-filled Honeycomb Lattice

Non-Fermi Liquid Behaviour of Electrons

October 1993

hep—th/9311105

S <-U 9) U G U

/, ‘·f‘¢5f’ r o —>" '° ~ °

superconductivity. OCR Outputcovered by Anderson that is attracting a major interest in relation with the study of the high Tc

3One of the clearest examples of the mentioned difficulties is the “iuf`rared catastrophe” [3] dis

the general theorems concerning renormalizability. The more severe difference lies in thewhose typical value is much smaller than the speed of light. This prevents the use oftivistic nature of the many body physics where there is a parameter, the Fermi velocity,

quantum field theory, we encounter two main differences. The first one is the nonrela

When comparing the effective field theories that arise in this context with standard

towards a BCS state when a weak attraction is introduced between the quasiparticles.the Fermi liquid as a fixed point of this RG transformations (it is gapless) that flowsrenormalization group transformations are rescalings of p. In this way one can visualizewe end up with an effective field theory whose couplings are cutoff dependent. TheBy integrating out in the path integral all the momenta above a certain cutoff p f ,00,

scale much lower than it, namely, a very narrow strip above and below the Fermi surface.ultraviolet cutoff po (the width of the band), we are only interested in the physics at aof this approach is that, although we are dealing with problems that naturally have an[4, 5, 6] to which renormalization group techniques can be applied The main ideacharge density wave instabilities have recently been rephrased as effective field theories

The theory of the Fermi liquid and the perturbations of it that lead to the BCS andcorresponding difficulties of doing perturbations around the Fermi surface.

origin of the trouble lies on the very complex nature of the many—body vacuum and thesolution is known at the present and the situation remains controversial? The ultimateand under a theoretical point of view. Despite the great effort devoted to it, no exactmaterials are organized in two dimensional layers and experimental data are available)

The case d = 2 is physically very interesting both in practice (most of the high-Tcthe behavior of the system is always of the Luttinger type[2].quasiparticles. Things are different in d = 1 where an exact solution can be found andd where it leads to the famous theory of Landau’s Fermi liquid [1] and the concept ofThis program has been successfully addressed by Landau in three spatial dimensionsis and what is the nature of the low energy excitations of the full interacting theory.an interaction among the electrons and ask the question of what the new ground stateexcitations are electrons and holes obeying a free dispersion relation. Then switch oni. e., a free Fermi gas whose ground state is known and whose low energy elementaryreview the highlights of the problem. Start with a free theory of, say, electrons in a metal,

particular, on the nature of their normal state. To fix ideas and some notation let us

the maximum interest is centered at the present on the high—Tc superconductors, incrystal. This problem uncovers all the macroscopic properties of the materials of whichThis paper addresses the problem of interacting electrons in a band of a two-dimensional

OCR Output1 Introduction

that, despite its nonrelativistic nature and due to its gauge invariance, it is renormalOCR OutputSection 3 is devoted to the study of the renormalizability of the model. It is shown

instability which is clarified in section 4 with the help of the renormalization group.the computation of the density of states in both cases. The results announce an infraredof the relativistic versus the instantaneous electromagnetic field propagator illustrated byquantum field theory. A discussion is done on the differences to be expected by the useinteraction considered in the many—body approach. We end up with a standard localfield whose propagator reproduces, in the nonrelativistic limit, the usual four—fermionof the Coulomb interaction. In the field theory, this is mediated by an electromagneticsingle, well-defined, vacuum state. Next the free theory is completed with the additionbody physics, our effective action constitutes a genuine quantum field theory with apoint of our analysis. We emphasize the point that, unlike what usually occurs in manyon the honeycomb lattice that lead to the low energy effective action used as a starting

The paper is organized as follows. In section 2 we review results on the free electronsstructures at half—filling [13].to the physics of the fullerenes that, as mentioned above, are two-dimensional graphiticstate of high-Tc materials is not of Fermi liquid type. Besides, it can be directly appliedinvestigation relies on the fact, made clear by now from the experiments, that the normalhave shown very ellusive in more conventional approaches. The major interest of thisideal laboratory for investigating departures from Fermi liquid behavior in d : 2 whichmentioned above in analogy with the d = 1 case. ln particular it makes of this lattice anThis peculiarity opens the possibility of performing a complete analysis of the problemspresent context is that its Fermi level at half-filling consists of exactly two Fermi points.

The main feature that makes the honeycomb lattice the object of our attention in theas the number of electrons available to form Cooper pairs is unusually low.Moreover, this superconductivity can be considered a form of high—Tc superconductivity,striking of them being the superconductivity of doped crystals of C60 molecules [11, 12].honeycomb lattice. These compounds show a variety of unexpected features, the mostcompounds [9, 10] can be thought of as closed, curved surfaces derived from the planarthe case of graphite, built of carbon atoms. The rapidly growing family of fullereneperiodic systems made of identical atoms with planar, threefold coordination. That istwo—dimensional honeycomb lattice. This lattice constitutes the basic building block of

In ref. [8], we developed a model for the electronic structure of free electrons in abunch of coupling constants.as in quantum field theory, with a true vacuum around which one can expand and get aFermi surface which is also modified by the interaction. Only in the d = 1 case we deal,points in d : 1. We are then faced with couplings that are functions of the shape of theare to be described around a Fermi surface in d = 3, a Fermi line in d : 2, and Fermitem as a perturbation of the "free” system constituted by the fixed point, perturbationsabove mentioned nature of the many body vacuum. When treating the interacting sys

(2) OCR Output{aww} = {ah af} = 0 {at G3'} = 6ei

canonically anticommuting operatorswhere the sum is over pairs of nearest neighbors atoms i, j on the lattice and ai , af are

<•,J>

(1)H=—*}/Iifaj

the diagonalization of the one-particle hamiltonian

orbitals for the two atoms. In the tight—binding approximation the problem reduces to

in a variational computation of the energy eigenstates, corresponding to the respectivelattice has two different atoms per primitive cell, so that two degrees of freedom arise[8], regarding the conduction band of a two—dimensional graphite layer. The honeycombThe present one—particle description has been worked out with complete detail in ref.

honeycomb lattice, which serve afterwards as a starting point for the many-body problem.In this section we first review the one—particle electronic states in the two—dimensional

2 Many-body theory of the 2D layer

in section 6.

We set our main conclusions and a summary of the more relevant points of the article

fixed point.renormalization of the electric charge and establish the non—Fermi liquid nature of our

is the anomalous dimension found for the electron Green function. We discuss the non

both from the technical and from the physical point of view. The most significant onewith standard RG analysis of the Fermi liquid theory and find significant differencesobtained when the Fermi velocity equals the speed of light. Next we compare our modelwhich is the relevant regime in the infrared. We find a fixed point of the RG flow

The previous analysis is completed in section 5 where we study the relativistic regime

Fermi liquid behavior of the system.

dimension for the electron wave function at two—loop order that points towards a nonmake some comments on the system away from half—filling. Finally we find an anomalous

We then investigate the possibility of having a natural length scale in the theory andto zero in the infrared although there is no fixed point in the nonrelativistic regime.of the electric charge and the Fermi velocity. The RG shows that this coupling decreasesthat the perturbative expansion has an effective coupling given by the ratio of the squareimplications. The first section is devoted to the nonrelativistic limit of the theory. We see

Sections 4 and 5 contain the analysis of the RG flow of the model and its physical

electric charge does not.

izable. The Fermi velocity and the electron wave function get renormalized while the

(7) OCR OutputH : gm mak + o ((a 6k)2)

for the operator 'H in equation (5)of the two independent Fermi points by taking the limit a —> O. In either case we obtainconsidering an arbitrarily large sample, it is appropriate to amplify the region about anyto encode the low-lying excitations about the Fermi level into a simple field theory. When

more than one spatial dimension. It has important consequences since it makes possible

independent states. The existence of a finite number of Fermi points is quite unusual in

Brillouin zone and, due to the periodicity of the dual lattice, they correspond to only twoisolated points in momentum space (see figure 2). These are the six corners of the firsthas the amazing property that the Fermi level at E(k) = O is only reached by sixand, in particular, graphite layers and undoped fullerenes. The dispersion relation (6)this band at half—filling, which is the pertinent instance for the carbon-based materials

the parameter a being the lattice spacing. We will be interested in the consideration of

(6), ., E(k) : :bt(/1 + 4cos2Tak, + 4cos%—ak,, c0s§aky3 3 _

band of levels

natural energy scale in the one—particle description. The diagonalization of (5) gives theThe parameter t measures the hopping between nearest neighbors and provides the

= Em <¤>C, ( C ) 0`k·u· 0 4 E. e' J C, ,1..,. J ( ) _; Ei Q J () C0problem

eigenvector of H provided that the coefficients c, and Co are solutions of the eigenvaluethe triad made of their respective opposites (see figure l). Therefore, the state (3) is anwhere {uj} is a triad of vectors connecting a • atom with its nearest neighbors, and {V,}

_ i + + _ -1e J ac ai |0) - te J ac Ja, |O) (4)iZ2Zik·v· ik·r·ik·u· ik-r

1 0 <1,_1>1 •<z,]>

HW : 4c.,eJa$|0) - rc,eJa§*|O)x Z; Zlk"`ik`”

viceversa. We have, indeed,

1. Obviously, under the action of (1) black points are mapped into blank points andwith coefficients c, and co for black and blank points, respectively, as depicted in figure

1 0t •

(3)‘I’ = > iC•€i¤P` l0} + > lcoeial IO)ikrzk`r

generators have to be of the formThe states which are simultaneously eigenvectors of the hamiltonian and of the lattice

5From now on we replace 6k by the wavevector k with origin at the Fermi point. OCR Outputderived from the 3D diamond structure with one state per site[14].

4Similar dispersion relations can be found in higher dimensions. The best known of them is the one

{70,70} = -2 {7r,7j} = 26rj {70.7r} = 0 (12)

{70, 7} is a standard set of 7—matrices satisfying

_ n —w2+v2k2—zc U ( I_ — + v 7 · k $(0) k Z ..2*Em(W I I

transform of the propagator

Taking the statistical average with the system at half-filling, we get for the Fourier

(IO){ak_t,a;,‘t} = 6(k — kl)relations

where v : 3ta /2 is the Fermi velocity and the operators Clkvt satisfy anticommutation

` ` *Pr(¢.r) = ) ,€6 wk,+¤k,+ + ) , €€¢k,-¤k,- (9)ik- —' kt ' I'"I I Ik- 'ki “ r"’I I

we havesW} 03. In terms of the respective modes 1,bk,+, ¢»k,_ for unoccupied and occupied levels

We denote by \I/I the electron operator \Il in the interaction representation while \I/I =

(8)GI°I(t, rg t', r') = ——i(T\I/[(t, r)@I(t’, r’))

two Dirac spinors and work out the many-body electron propagator of the free theoryhole excitations of the Dirac sea. We may consider separately, for instance, each of thevacuum in the sense of quantum field theory. Its excitations are similar to the particlesecond quantized theory built from the dispersion relation (7) has, however, a definitethe elementary excitations (quasiparticles) are only stable right at the Fermi level. Thea condensate with an extended Fermi surface (in more than one spatial dimension) andsignificant. In quantum statistical theory the ground state of the system is, in general,field theory than into that of quantum statistical theory. The difference is subtle butstates, we end up with a formalism which falls more into the framework of quantumquantization. We remark that, proceeding in this way with the above set of one-particle

We now turn to introduce the electronic interaction by applying the method of secondspatial dimensions? This is all about the free theory.theory of two massless Dirac spinors (one for each independent Fermi point) in two(energy) eigenstates of the electrons in the honeycomb lattice are given by the fieldmatrices. The conclusion is that, in the long-wavelength limit a, —> O, the one—particlewhere 6k is the displacement about the Fermi point and 01,02 stand for the two Pauli

6Unless otherwise stated, we work henceforth in units fz : c : 1.

Vu/1** = O (16) OCR Output

possible by specializing to the Feynman gauge[15], which enforces the constrainttwo-dimensional layer. Although this may not be achieved in general, it turns out to be

dimensional space while we want the dynamics of the electrons to be confined to theThis poses some technical problems since the electromagnetic field propagates in three

(15)j, ~ (z@~,O¤1¤, www)

in the standard fashion, by coupling to the conserved currentThe interaction of the electromagnetic field Au and the electrons in the layer is described

(14)H = gta / dzr \I!(r)·y · V\I!(r) — e [ dzr j,,A“

tized hamiltonian6We propose, therefore, a quantum field theory description based on the second quan

gives only one marginal operator dictating the long—distance behaviour of the free theory.accidental in the description of the two-dimensional layer, as long as the expansion (7)unconventional behaviour is the massless character of the spinor YI/. This property is not

sensitive to retardation effects of the electromagnetic propagation. The reason for suchthat we support in what follows is that, actually, the model of interacting electrons isthe interaction with the electromagnetic field Au is more desirable. One of the pointsof the quantum theory as well as for computational purposes, a complete description ofinvestigation from a formal point of view. Both for the sake of studying the propertiesinteraction is, however, a highly nonlocal field theory, what makes very awkward itsneous interaction between electric charges. The model given in terms of the Coulomb

We attach to this hamiltonian the name of "Coulomb” since it describes the instanta

+_/d,,,,1`/dT_22 41r lr] — rg€22\II(r1)U3‘I!(rl)\p(r2)U3\I](r2)

HC0uI0mb Z Eta I d2r .

leads, in first instance, to the second quantized hamiltonianIn the nonrelativistic approximation, the introduction of the electronic interaction

is a feature of the many—body theory.presence of the Fermi velocity v in front of the spatial part of the scalar product, whichis equivalent to the Dirac sea in quantum field theory. We must notice however thespinor in 2 + 1 dimensions, which confirms our assertion that the system at half—iilling

The above expression coincides precisely with the Feynman propagator for a masslessand related to the above quoted Pauli matrices by 03 = —i70, al = iayyl, 02 = iayyg.

of section 4). However, as mentioned above, the propagator (19) cannot be safely used OCR Outputof the propagator G, to the first perturbative order (see also the discussion at the endThen, it becomes obvious that this kind of interaction cannot modify the w dependence

,g (19)· 1 Z -i6,,6(i · wg / élgiezk Ir rlM ° V ` W (27r)3 21r k2 + kf — ic

» I » /

this limit we have, computing always at points r, r' in the two-dimensional layerobtained from (17), for instance, by considering formally its expression for c —> oo. Inmodified to first order in perturbation theory. The instantaneous interaction can be

In the theory with the instantaneous Coulomb interaction, the density of states is not

(18)n(w) = Im / d2k Tr [G'(w,k)a3]

the density of states n(w) near the Fermi level (w : 0). This observable is given byLet us illustrate the above fact with the computation of the quantum corrections to

theory.

resort to more sophisticated methods in order to obtain information from the quantum

looses predictive power because of the mentioned infrared instabilities, and one has todifferent in the infrared regime. From a technical point of view, perturbation theoryv/c ——> O, the response of nonlocal quantities to the interaction turns out to be quitequantities) in the nonrelativistic model (13) and in the model given by (14) in the limitnot expect any discrepancy in the computation of local quantities (like cutoff dependentthe appearance of infrared divergences in perturbation theory. Thus, although we shouldality of the system is low enough so that the massless condition of the spinor reflects incomputation of a given observable taking (13) or (14) as starting point. The dimension

It is important to stress that we have to expect, in general, different answers in thelimit of all quantities by expanding in powers of v/ c.would need in a nonrelativistic theory, but we can recover at any time the nonrelativistic

plane of the two-dimensional layer. With this description we certainly get more than wetation of electronic properties we have just to take care of placing the points r, r' on theand the coupling to the 2 + 1 dimensional current is perfectly consistent. In the compu

(17)(TA,i(t,r)A,,(t’, r')) : -—i6,,,,!4 ik-(r—r’) —iw(t—t’ )

current. In this gauge we have the propagatorand has also the property of placing the A,,_ field in the same direction as the electronic

model. Though the many-body theory of the 2D layer has a natural ultraviolet cutoff, we OCR Outputgroup methods. We study first in this section the renormalization properties of the

the possibility of analyzing the infrared behaviour by application of renormalization

Henceforth we will pay attention to the local quantum field theory (14), which offers

3 Renormalmatiori

described points at new physics near the Fermi level.

theory is clearly not reliable in the infrared regime, but the kind of infrared instabilitythat higher order terms can produce increasingly large powers of log w. Perturbationfirst perturbative correction dominates over the zeroth order term, and one can presume

At the point near which we want to measure the density of levels, i.e. near w : 0, thein some way this infrared divergence, we still would be left with a more serious problem.

since the last integral in (22) is clearly divergent. Even if we could manage to regularizeIt is already appreciated that the perturbative expansion of n(w) is not well-defined,

(22). --€2 / dx;——E-——— + O(e‘*) 16vr2 0 (1- sc + v2x)_2 ac2 ,/1 - 1 :

2232vrv 81rO (1-2:%-vx)|w| 1 e2 1 1 y/1 — x Z 22-——1———l2 --2 d;——;l Mw) Tr 2 +Ogw +2€ $22 Ogx

to the density of levelsWhen inserted in equation (18) the above expression gives the first quantum corrections

167r(1_x+Ux))) ( )2 1 y/1 —— x 2 ,k:‘i- ·k d l 2k2—21— O421 R(w ) z2v·y A :c220g{v x w:v( x)+ (e

finite contribution to E

theory, the relevant diagram is shown in figure 3 and gives (in the limit v/c —+ U) thewill leave the issue of renormalization to the next section. To first order in perturbation

absorbed into 1/G(°). For the moment we are only interested in finite corrections andquantum field theory we expect these divergences to be local and susceptible of beingwhich are in general ultraviolet divergent. Since our model can be treated as a genuineE(w, k) has a perturbative expansion in terms of one-particle·irreducible diagrams[1],

1 1self—energy E(w, k)

The inverse propagator may be decomposed, as usual, into a free part and the electron

follows.

density of states, whose determination in the quantum field theory given by (14) goes asin the computation of nonlocal quantities. This remark is appropriate to the case of the

bare(25) OCR OutputZ;/QAQ

Rqlbarc 2;/"¤1¤ébare Ze€R

Ubare ZUUR

tg, CR, WR,

fulfilled, the renormalized theory may be made finite in terms of renorrnalized quantitiesof the theory, which arises in the absence of relativistic invariance. Provided that it isrenormalized by the product Z,mZ,,. This is a nontrivial check of the renormalizabilityaction vertex. It turns out therefore that the spatial components have necessarily to beficient Zm, is determined from the renormalization of the time component of the interE(w, k), as well as the renormalization coefficient for the Fermi velocity Z,,. The coefThe renormalization coefficient Zkm may be determined from the electron self—energy

(24)2,,,, ie / drdzr Yr»`(-70/10 + Zvi)-y-A)¤1»

5},,,,.8 : Zkm / dtdzr @(-7030 + Z,,v·y · V)\If

as to cancel the ultraviolet divergences. We should start therefore with the bare actionln the quantum theory the coefficients of the terms in the action have to be adjusted so

(23)ie I dtd2r E(—*yO/10 + 111 • A)\I/

S T-: / dtd27" @(-*793g + 'U'Y °

the lagrangian formalism. Corresponding to the hamiltonian (14), we have the actionThe conditions which make the theory renormalizable can be stated more clearly in

perturbation theory.

with a compelling argument showing that the model is renormalizable to all orders inin the hamiltonian. We present here the analysis at the one-loop level, and concludecan be absorbed into a redefinition of the scale of the fields and of the parametersFor this reason it is still necessary to check that the ultraviolet divergences of the theorydoes not match the speed of light c that appears in the dispersion relation of photons.are missing is relativistic invariance since the Fermi velocity of the electrons v = 3ta/ 2the usual considerations for renormalizability in Held theory, the only condition that wesimpler as our low-energy effective theory is a genuine quantum field theory. Regardingmay depend on the point chosen on the Fermi surface. In our case things become muchbody theory is, in general, a nontrivial issue, since the couplings in the effective theoryinfrared[16, 17]. As we have mentioned before, the renormalizability of a quantum manysince this is a way of extracting relevant information about the scaling properties in theinsist in absorbing ultraviolet divergent contributions into bare parameters of the theory

10 OCR Output

_ —— 1 — 2 d —-Q W2 s11¤"( " l/0 x,/$(112 + (1 -112)11)2e“ 2 2 1 — :c 1

po Z ,,0 ;(1_2,,¤)/ dx_____;.._ 16vr2 0 \/$(112 + (1 — 112)a:)

the vertex is actually different than that to the spatial components. We find thatdenote them symbolically by 1`,,. The divergent contribution to the time component offirst quantum corrections to the vertex are given by the diagram in figure 4, and we

Now we come to the renormalization of the interaction term in the action (24). The

11 —> O.

We will later use this expression in the consideration ofthe nonrelativistic approximation

+4 (1 - 2112)F 1,1,,112- 4F 1,2,,112— + 0 (64) (29)1 3 1 (5) (5)} E

Z., : 1- —e7r-F —--11—— 27r111— 211F —--111 1 1 3 1 3 3 3 2 ,; ,2 (2) ,; ,161r2 11 2 2 2 2 2 2

ZU has, for instance, an infinite power series expansion in 11as long as 11 does not match the speed of light -which, in our units, means that v ;£ 1.These coefficients have a more complicated structure than those of a relativistic theory,

2 4421 —— 1 - 2 2 11 —— 4 SM 2M, 41%+1216+044)4 1/1 - 11 1

SW2 0 (1 - rc + v2x)2 62 4 1/1 - 1 2,:1i;/11....;0+ (4) 27 (2c Z:1—1—22 d———-— 4 +811% 2)/,, 41%+1216

4 1/1 - 33 1 4

coefficients

In order to ensure the Hniteness of the electron propagator we take the renormalization

gw. - - — - k zz v 7 , _ - 0 4 2 22+ fzmte terms + (e) ( 6)1 \/1 — x 1 d ;»;A $(1*- x +Ux)E

1, :-z‘- z,, -k—‘— 1-22 11-—-—~ 44 4444 44 > 4811 1<»4< 411, 41-11.1111 1/1 — x 1

G10)

From the diagram in figure 3 we haveobject, regularizing it by working in analytical continuation to dimension d = 3 — @[15].the electron self-energy E(w, We deal here with the ultraviolet divergent part of this

In the previous section we already quoted the one-loop renormalized contribution to

12 OCR Output

magnitude of the parameters pertinent to the real layer. A standard value for the hoppinglarge size is the appropriate physical instance to look at. The first question concerns thetwo-dimensional layer. A single graphite sheet or any fullerene aggregate of sufficientlyln this section we seek the connection of our model to the real physics of electrons in a

Nonrelativistic regime.4 Scaling of the electronic interaction (I) .

bation theory.remarking the absence of renormalization of the electric charge to Hrst order in pertur

(39)2},/2 Z Z;2 : 1+ 0 (62)

We conclude, therefore, that

(38),, Up»(k) = (y.~ · )|k|dx x/$(1 · x)l e2 k k§;y$A

perturbative contribution shown diagrammatically in figure 5 is a finite quantity

of the electromagnetic field Z;is determined from the polarization tensor, whose first/2

2in standard quantum electrodynamics, that zezj/: 1. The renormalization coefficientremark. The fact that Zq, has to equal Zm, to all orders in perturbation theory implies, as

The rest of the renormalization coefficients in (25) can be obtained from the followingexpansion.

theory, this turns out to guarantee the renormalizability to all orders of the perturbativeof the action. As long as one is able to preserve the gauge invariance in the quantumit also enforces the same renormalization of the Fermi velocity v in the different termsthat Zkm = Zim. It is easily seen, by taking a gauge parameter 0 independent of t, thatis manifest at the classical level. Provided that it holds for the bare action (24), it ensures

A,) —> A,, + 0,.9(t, r) (37)

,1, __, €ie0(t,r)

The gauge symmetry

which become identical after use again of the relations among hypergeometric functions.

(36)—F —--—-2 -4F 12---2 - 4 +,,,v,,v+O(¤)vr l 3 1 3 l 1, (222) 2))E

”OCR Output1 1 1 8 5 Z2c —-— 1-2 2 -F - ----· 2) -- F 2 2·-; icwf Ulf U 2’2’ 2’” sv "2I(

13 OCR Output

at 06A+,8 (v ez); + B (v @2); — (v e2) ) G(w k A·v ez) = 0 (43) "’‘*’”’’’’’(2 at

that under a change in the scale of the cutoff Arenormalization group equation, which for the electron propagator, for instance, statesto the same renormalized theory[16, 17]. The essential information is encoded in thetherefore to relate bare theories at different scales (different cutoff) which correspondmomentum cutoff A. By means of renormalization group transformations we are ableis renormalizable, we may impose the independence of the renormalized theory on thethe scale dependence of the effective interaction. Since we have shown that the modelrely on the original interpretation of renormalization adopted by Wilson[19] to determinecharge, but in which the effective strength of the interaction ez / (47rv) is renormalized. We

We have a nonrelativistic quantum theory with no renormalization of the electric

)Z}/2 = 2;* : 1+ 0 (J

ativistic limit. We recall also the former result (39)so that we do not find wavefunction renormalization at the one—loop level in the nonrel

Zq, =1+O— (41)C4 <;) v

strength of the electronic interaction. In this limit we obtain from (35)equivalently, c ——> oo) and ez / v : const. This latter quantity becomes then the effectivea consistent perturbative expansion in the nonrelativistic limit by taking v ——> 0 (or,in the nonrelativistic approximation enters with a power (ez/v)". We define, therefore,lt is not difficult to see that at each level of the perturbative expansion the leading order

7r v

(40)Z., :1- ;—l0g A + O 162(Ei) 11(i

coefficient of the Fermi velocity Z., becomes, for instance, in the nonrelativistic limitloop level this just amounts to replace 1/6 by log A in all formulas. The renormalizationcutoff A rather than in terms of the dimensional regularization poles 1/e. At the one

the renormalization coefficients in this section as functions of the ultraviolet momentum

For a clearer description of our approach to renormalization group we will quote

keep the leading order in previous formulas.quantities to the electron physics in the layer we will have to take the limit v / c —> 0 and

This is certainly much smaller than the speed of light, for which to extract relevanton c and h, the Fermi velocity in our model turns out to be v = 3ta/(2h) z 1.5 · 10"3ccan take a ¤ 0.14- 10‘9m . From these values and restoring for a moment the dependenceparameter in graphite is t M 2.2eV . For the nearest neighbor separation in the lattice we

14 OCR Output

performing the sum of leading logarithm terms of the perturbative expansion and curing,us remark that the solution (50) of the renormalization group equation is equivalent toin which the integration of the renormalization group equation becomes consistent. Let

a particular energy scale. We then have to be very precise in establishing the regimequantum theory the statement that a coupling like e2/ (47rv) is small or large applies toin fact, larger than one. Nevertheless, we know that when dealing with a renormalized

that the coupling e2/(47rv) of the nonrelativistic theory is not a small parameter but,on perturbation theory. lf we recall our original estimate of the Fermi velocity v, we see

We have to bear in mind that, when integrating equation (49), we are relying heavily

(50)E·—··—(A//\¤)= ·~— %+——Z<>y7T Ucff 47T 'UU87T UA(— R R R 0ie 182 U2 ia A *4

so that the asymptotic behaviour of the coupling is

(49)6 6 1 2 4 p?@:—¥+O $3;- p vg 161r vcjf vcff

flow in the infrared regime. Actually, from equations (40) and (44)coupling of the electronic interaction e2/(47rv) to have a nontrivial renormalization groupsee that eeff is constant at this level, while vcff is not. We expect, therefore, the effectivetheory at large distance scale. From the one—loop order results of the previous section we

Increasing the scale p of the cutoff is equivalent to measure the observables of the

(48)x>§¢e;;(/J) = —6€(vc;;, 62;;)

p§ve;;(p) = —6t(vej;,c§,,)

where the effective parameters are given by

G k A. 2 , 2 (<¤»»r> nn ¢ )= exp v G(w, ki A, vef;(p)»¢.H(p)) (47)p dp, YThe well-known solution to this equation takes the form

(46)Ai log Zq,·r(v»€”)

AMERBAM c2)3ZE

AEKURBU(v» 62)OZU

whew

15 OCR Output

ber of particles. Regarding the polarization tensor, a nonvanishing value of H00(w ==which, after multiplying by 2 (the two independent Fermi points) gives the correct num

4% U,(53)dzk drUa(U-U|k|)6(rU*U|k|) ZY';-Z C5?/$

,u = 0. The computation of (51) givesThe fermion loop diagram is now nonvanishing, opposite to what happened before atby the nonvanishing chemical potential. This may lead, however, to finite corrections.integrals and the renormalization coefficients of the quantum field theory are not modifiedObviously, the )u—dependent part cannot change the ultraviolet behaviour of the loop

- ,U a(U - U |k|)6(U; - U |1U)) (52)G(U,, k))u¢0 Z<<>>

The correct expression of the free electron propagator at finite p is[20]

V t,>;0 (51)= — h1;rTr (T\I/(t,0)\I/l(t’, 0))charge

system away from half—filling by introducing a chemical potential y for the conservedof charge, as a consequence of doping. In order to study this effect, we may place thelevel. Thus, a finite screening length is only generated by giving to the system an excess

by quantum effects. This is due to the vanishing of the density of states at the Fermi

in our model the Coulomb interaction is not screened, in the usual metallic manner,Regarding the possible existence of a natural infrared cutoff, it can be shown that

a single graphite sheet.

observation of the phenomenon of scaling in the asymptotic region should be possible incontrolled by this value of the renormalized coupling. We believe, anyhow, that thepoint is given by 11,,;) = c, but the flow in the graphite layer may not be necessarilygroup fixed point. Later we will see that in the complete quantum field theory the fixed

the fact that, in the nonrelativistic approximation, there is no signal of a renormalization

value of vg has to be. This is related to the unlimited growth of the Fermi velocity, and totheory appears to be formally incomplete, since it does not give any hint about what the

As for the practical application of the flow (50), we remark that the nonrelativisticthe ultraviolet regime in the case of the strong interactions by the infrared regime here.providing the example of a nonrelativistic theory in which this also happens, exchangingsupported by experiment) in the quantum field theory of the strong interactions. We areat which perturbation theory becomes reliable, similar to what is believed to be true (asphysical system which has strong coupling at certain energies and an asymptotic regime

perturbation theory becomes more and more accurate. This is another instance of aat large distances (or small energies with respect to the Fermi level) and in this regimeend of section 2. Taking the limit A —> oo in the bare theory corresponds to measure

therefore, the problem of the infrared instability of the density of states addressed at the

16 OCR Output

metallic properties either.perturbation theory). The system is not an insulator, though it does not exhibit usualone-loop order (as it would be if an effective mass for the spinor had been generated innot shifted by quantum corrections. Also a gap does not open at the Fermi points to themaintained in the quantum theory, at least at the one-loop level, since the Fermi level isFermi level suppresses the usual screening of the Coulomb interaction. This property isbehaviour. ln particular, we have seen that the vanishing of the density of states at the

We have therefore an electronic system which deviates slightly from Fermi liquid

cl " dp' e4 , xp -—/ ——<p> { 161r2 p' 113,,,,(47), the electron propagator transforms under a change of scale p by the factorobservables acquire anomalous dimensions in the infrared regime. According to equationwhere cl is a nonvanishing quantity, cl z 5.49 · IO`2. We see therefore that the electron

,67,,,56 ( )z Z 1 - if-iz A o 6 w 2,0g + (e)

obtains from the sum of the two—loop diagramsusing the nonrelativistic approximation (19) and after a very lengthy calculation, onethe loop integrals overlap in such a way that the previous argument does not apply. Bycontributions on w holds except for the vertex diagrams in figure 6. In these diagramsbear any dependence on it. At the two—loop level, the same independence of the localchange of variables in the loop integral, so that the divergent part of the diagram cannotThe w—dependence in the vertex diagram in figure 3 can be absorbed then by a simplethe nonrelativistic limit in the photon propagator, which leads to the expression (19).To establish the degree of divergence of the local contributions one can safely implement

E(w,k) M f 70w -{-. (55)2 e

the form

function renormalization is given by an ultraviolet divergent self—energy contribution ofwavefunction renormalization in the limit v —·-> O is just an accident. The signal of wave

lous dimensions in the nonrelativistic limit. We remark that the absence of one-loop

We now turn, finally, to the issue of whether the electron observables may get anomawhich gives a measure of the screening effects away from half-filling.

2W U2H 0 k 0 z 00( » " )

actually the estimate for the inverse of the screening length0,k —> 0) signals the appearance of a photon effective mass. At small ,11 one finds

18 OCR Output

the electron field has a simple, anomalous scaling behaviour.this sum is consistently achieved in the relativistic regime, where the fixed point lies and

in the form of an anomalous dependence on the electron frequency. We have found that

group approach succeeds in summing up all these increasingly divergent contributions,perturbative computation of the density of states. As is well·known, the renormalization

We arrive in this way to solve the problem of the proliferation of log w terms in the

(63)G(p<».pk) = n`“°2"""2)G(<¤» k)

or, equivalently, after introducing the precise value of 7

(62)G(w, k)|A E A"<I>(w, k)

scale is

now at v = 1. At the fixed point the behaviour of the Green function under a change ofdimension acquired by the electron field \I/, which may be obtained from (27) evaluatedis definitely not a Fermi liquid. The main reason for this assertion is the anomalous

At or near the fixed point, the two—dimensional electron liquid we are describingthat they do not appear to any order in perturbation theory.the two—loop order and one may conjecture, following the above dimensional argument,type accompanying the gauge invariant expression (38) for Ilw,. These are not present atof the scale of the field Au. One would have to look for divergent contributions of log A

cutoff dependence, it can be checked that this does not lead either to a renormalizationin any event. If, however, one insists on understanding the regularization in terms of alarization of the gauge theory —it certainly is not going to break the gauge invarianceinstead of poles at d ——> 3. We believe that this scheme is the best suited for the reguthis leads to the appearance of gamma functions at half·integer values of the argument,it should diverge linearly in momentum space. In the dimensional regularization schemethe one—loop polarization tensor. A naive dimensional analysis of the diagram shows thatfound any divergent contribution of order e4 to HW. The reason is the same operating inof the charge e. Computing in the framework of dimensional regularization we have nottion of the photon field Au is directly related by gauge invariance to the renormalizationtion tensor H W to second order in perturbation theory. As stated above, the renormaliza

Regarding the renormalization of the electric charge, we have analyzed the polarizarenormalization and anomalous dimensions in the infrared regime.which are irrelevant in Fermi liquid theory but give rise in our model to wavefunctiondiagram can be discarded a priori. It is also for this reason that there are diagramsof diagrams against the rest. In our quantum field theory description it seems that noat half—filling, though, there is no argument here to conclude the relevance of one setthe infrared regime. Because of the existence of Fermi points in the honeycomb latticeof the many-body theory with an extended Fermi surface these are which dominate inrenormalization group, information of the series of ladder diagrams. In the usual analysisOCR Outputthe consideration of the one-loop graph in figure 5 is enough since it encodes, via the

19 OCR Output

more interesting is the anomalous screening of the Coulomb interactions of the model.Our analysis has important consequences in condensed matter physics. One of the

dimension of the fermionic propagator.of the renormalization group that is a relativistic quantum Held theory with anomalous

We have then found that our original system flows in the infrared towards a fixed point

in section 2 as the manifestation, in perturbation theory, of the anomalous dimension.

observables. There, the RG allows to interpret the one-loop infrared instability shown

resort to the full relativistic theory when computing the infrared behavior of nonlocal

infrared when the Fermi velocity is still far from the fixed—point value. Hence, we must

tive expansion of the nonrelativistic approximation does not have to make sense in the

found at the one-loop level, the approach is consistent in the sense that the perturba

with the retarded interaction our RG analysis shows that, despite the infrared divergence

one-loop corrections were found to the free value of the density of states. In the theory

states at the end of section 2. In the theory given by the Coulomb interaction (13), nohas been exemplified with the computation of the quantum corrections to the density ofed propagator. The effect of retardation on the infrared behavior of nonlocal quantities

Let us now comment on the comparison between the instantaneous versus the retard

the nonrenormalization of the electric charge.

stance, the finiteness of the photon self-energy up to the two-loop level and, hence, ofdifferent, it goes like l/r instead of the standard Zog r. This is the origin of, for ins from the standard propagator in QED3. In particular its infrared behavior is quitedimension. The resulting expression read as a propagator on the plane certainly differs, the two points lay on the lattice plane, allows us to integrate out the perpendicular

use in our relativistic computations. The constraint that, in the electronic interaction

Expression (17) shows the usual three-dimensional photon of QED. This is what westrength now, we are not dealing with standard massless QED.where the photon is mostly absent, and from the quantum field theory since, as we willpropagator. This use is unconventional both from the condensed matter point of view

Let us Hrst comment on the use made through the paper of the electromagnetic field

there are a few points that we would like to emphasize.Before entering into the analysis of the phenomenological consequences of our work,

one—dimensional electrons than that of a two-dimensional Fermi liquid.

renormalization group techniques and have seen that, indeed, it resembles more that of

theory. We have been able to investigate the infrared behavior of the system by usingand allows the formulation of the low energy effective theory as a standard quantum field

half-filling consists of two isolated points renders the problem similar to the d : 1 caseteracting via a Coulomb potential. The fact that the Fermi surface of this lattice atWe have studied a system of electrons in the two-dimensional honeycomb lattice in

6 Summary and conclusions

20 OCR Output

This work has been partially supported by the CICYT.Acknowledgements

as a rigorous realization of the ideas settled in [22].liquid behavior discussed in the literature [21]. On this respect, this work can be seencatastrophe mentioned in the introduction as well as with other unconventional Fermito remark the interest of investigating the connections of our work with the infrared

To conclude this summary and to point out possible future work, we would likepossibility of having an experimental test of these results remains open.should show up in a variety of properties, like the specific heat, susceptibility, etc. The

Finally, the existence of nontrivial scaling laws implies that anomalous exponentsleading to incoherent one-particle excitations.

any amount of electron-electron interaction suffices to wipe out the quasiparticle pole,ent, quasiparticles is possible. The situation resembles the 1D Luttinger liquid, wherepeak at low energies. Thus, no effective description in terms of independent, and cohertem. The wavefunction renormalization leads to the disappearance of the quasiparticle

The one-particle properties are very different from those of a conventional Fermi sys

couplings to simple on—site terms.terms of on-site interactions only. Screening effects are not strong enough to reduce thethe spectrum of charge excitations. Our results rule out the validity of descriptions inthat, although the low energy density of states goes to zero, no finite gap appears inset of two points after the corrections due to the interactions). Moreover we have seenstands on the non-renormalization of the chemical potential (the Fermi level remains aenergy, there is no metallic screening in our materials [13]. The robustness of this resultof states at the Fermi level and to the absence of renormalization of the photon self(Hubbard model). What we have seen is that, due to the vanishing of the densitythe absence of long-range interactions in the ultralocal approach to interacting electronsthe photon self-energy) generates a new scale, the screening length. This is the basis OfIn a Fermi liquid, formation of virtual electron—hole pairs (i. e., quantum corrections to

21 OCR Output

Lett, B).Fullerene Molecules”, preprint cond-mat/9303007 (to be published in Mod. Phys.

[13] J. Gonzalez, F. Guinea and M. A. H. Vozmediano, “Electrostatic Screening in

Carbon 30 (1992) 1261.[12] For a review on the properties of fullerenes, see K. Holczer and R. L. Whetten,

1154.

Huang, R. B. Kaner, K.-J. Fu, R. L. Whetten and F. Diederich, Science 252 (1991)J. Rosseinsky et al., Phys. Rev. Lett. 66 (1991) 2830. K. Holczer, O. Klein, S.—M.T. T. M. Palstra, A. P. Ramirez, and A. R. Kortan, Nature 350 (1991) 600. M.

[11] A. F. Hebard, M. J. Rosseinsky, R. C. Haddon, D. W. Murphy, S. H. Glarum,

(1990) 354.W. Kratschmer, L. D. Lamb, K. Fostiropoulos and D. R. Huffman, Nature 347[10]

(1985) 162.H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl and R. E. Smalley, Nature 318[9]

[8] J. Gonzalez, F. Guinea, and M. A. H. Vozmediano, Nucl. Phys. B406 (1993) 771.

K.G. Wilson, Phys. Rev. B4 (1971) 3174, 3184.[7]

appear.

“Renormalization Group Approach to Interacting Fermions”, Rev. Mod. Phys., to[6] R. Shankar, Physica A177 (1991) 530. For an extense review see R. Shankar,

9967.

{5] G. Benfatto and G. Gallavotti, J. Stat. Phys. 59 (1990) 541; Phys. Rev. B42 (1990)

Theory and the Fermi Surface ”, hep—th-9210046.

For a theoretical physicist’s vision of the problem see J. Polchinski, “Efective Field[4]

San Sebastian, October 1993.

"The Physics and the Mathematical Physics of the Hubbard Model”, Nato A.R.W.,For a recent review on the infrared catastrophe see the talk of P. W. Anderson in[3]

(1990) 1839; 65 2306; 66 3226.F. D. M. Haldane, J. Phys C14 (1981) 2585. P. W. Anderson, Phys. Rev. Lett. 64[2]

Pitaevskii, Statistical Physics, Vol. 2 (Pergamon Press, Oxford, 1980).Theory in Statistical Physics (Dover, New York, 1975). E. M. Lifshitz and L. P.A. A. Abrikosov, L. P. Gorkov and I. E. Dzyaloshinski, Methods of Quantum Field[1]

References

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C. M. Varma et al., Phys. Rev. Lett. 63 (1989) 1996.[22}

X. G. Wen, Phys. Rev. B42 (1990) 6623.[21]

S. A. Chin, Ann. Phys. 108 (1977) 301.[20]

K. G. Wilson, Phys. Rev. Lett. 28 (1972) 548.[19]

(Academic Press, New York, 1980).1. S. Gradshteyn and 1. M. Ryzhik, Table of Integrals, Series and Products, p. 1044[18]

(McGraw—Hill, New York, 1978).D. J. Amit, Field Theory, Renormalization Group and Critical Phenomena, Chap. 8{17]

ed by M. Alexanian and A. Zepeda) (Springer Verlag, Berlin, 1975).lished). K. Symanzik, in Particles, Quantum Fields and Statistical Mechanics (edit—J. Zinn—Justin, Lectures delivered at the Cargese Summer School 1973 (unpub[16]

1981).[15} P. Ramond, Field Theory. A Modern Primer. (Benjamin/Cummings, London,

Sorella and E. Tosatti, Phys. Rev. B 47 (1993) 16216.S. Sorella and E. Tosatti, Europhys. Lett. 19 (1992) 699, G. Santoro, M. Airoldi, S.{14]

OCR OutputFigure 1: The planar honeycomb lattice. OCR Output

ZOHC. OCR Output

relation (in units t = a = 1). The cusps appear at the six corners of the first BrillouinFigure 2: Representation in (E, k) space of the lower branch of the electronic dispersion

\ sib Q

the electron propagator and the wavy line for the photon propagator. OCR OutputFigure 3: One-loop contribution to the electron self—energy. The straight line stands for

Figure 4: First quantum corrections to the interaction vertex. OCR Output

Figure 5: One—loop polarization tensor. OCR Output

Figure 6: Two-loop diagrams contributing to wavcfunction renormalization. OCR Output

( c1)

Figure 7: Plot of the effective Fermi velocity vcff (in units c = 1) versus length scale

200 300 400100 500

0.04

0.06

0.08

0.1

0.12

0.14