Electron interactions in bilayer graphene: Marginal Fermi liquid and zero-bias anomaly
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Citation Nandkishore, Rahul and Leonid Levitov. "Electron interactions in bilayer graphene: Marginal Fermi liquid and zero-bias anomaly." Physical Review B 82.11 (2010) : 115431. © 2010 The American Physical Society
As Published http://dx.doi.org/10.1103/PhysRevB.82.115431
Publisher American Physical Society
Version Final published version
Citable link http://hdl.handle.net/1721.1/60659
Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW B 82, 115431 ͑2010͒
Electron interactions in bilayer graphene: Marginal Fermi liquid and zero-bias anomaly
Rahul Nandkishore and Leonid Levitov Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA ͑Received 6 June 2010; published 17 September 2010͒ We analyze the many-body properties of bilayer graphene ͑BLG͒ at charge neutrality, governed by long- range interactions between electrons. Perturbation theory in a large number of flavors is used in which the interactions are described within a random phase approximation, taking account of dynamical screening effect. Crucially, the dynamically screened interaction retains some long-range character, resulting in log2 renormal- ization of key quantities. We carry out the perturbative renormalization group calculations to one loop order and find that BLG behaves to leading order as a marginal Fermi liquid. Interactions produce a log squared renormalization of the quasiparticle residue and the interaction vertex function while all other quantities renormalize only logarithmically. We solve the RG flow equations for the Green’s function with logarithmic accuracy and find that the quasiparticle residue flows to zero under RG. At the same time, the gauge-invariant quantities, such as the compressibility, remain finite to log2 order, with subleading logarithmic corrections. The key experimental signature of this marginal Fermi liquid behavior is a strong suppression of the tunneling density of states, which manifests itself as a zero bias anomaly in tunneling experiments in a regime where the compressibility is essentially unchanged from the noninteracting value.
DOI: 10.1103/PhysRevB.82.115431 PACS number͑s͒: 73.22.Pr
I. INTRODUCTION lated by a Ward identity which stems from gauge invariance symmetry. Therefore, at log2 order, BLG behaves as a mar- Bilayer graphene ͑BLG͒, due to its unique electronic ginal Fermi liquid. structure of a two-dimensional gapless semimetal with qua- We solve the RG flow equations with logarithmic accu- dratic dispersion,1 offers an entirely new setting for investi- racy, finding that the quasiparticle residue flows to zero un- gating many-body phenomena. In sharp contrast to single der RG. This behavior manifests itself in a zero bias anomaly ͑ ͒ layer graphene, the density of states in BLG does not vanish in the tunneling density of states TDOS . We conclude by ͑ ͒ at charge neutrality and thus even arbitrarily weak interac- extracting the subleading single log renormalization of the tions can trigger phase transitions. Theory predicts instabili- electron mass, as a correction to the log square RG. This ties to numerous strongly correlated gapped and gapless calculation allows us to predict the interaction renormaliza- tion of the electronic compressibility in BLG, a quantity states in BLG.2–6 These instabilities have been analyzed in which is interesting both because it is directly experimentally models with unscreened long-range interactions,2 dynami- measurable and because it allows us to contrast the slow cally screened long-range interactions,3 and in models where 4–6 single log renormalization of the compressibility with the the interactions are treated as short range. Irrespective of fast log2 renormalization of the TDOS. whether one works with short-range interactions or with The structure of the perturbative RG for BLG has strong screened long-range interactions, the instability develops similarities to the perturbative RG treatment of the one di- only logarithmically with the energy scale. However, dy- mensional Luttinger liquids.4,8–10 We recall that in the Lut- namically screened Coulomb interactions have been shown tinger liquids, the Green’s function acquires an anomalous to produce log2 renormalization of the self-energy7 and ver- scaling dimension, which manifests itself in a power law tex function.3 Such strong renormalization can result in sig- behavior of a quasiparticle residue that vanishes on shell. nificant departures from noninteracting behavior on energy In addition, the electronic compressibility in the Luttinger scales much greater than those characteristic for the onset of liquids remains finite even as the quasiparticle residue flows gapped states. However, there is as yet no systematic treat- to zero. Finally, in the Luttinger liquids, there are logarithmic ment of the log2 divergences. In this paper, we provide a divergences in Feynman diagrams describing scattering in systematic treatment of the effects of dynamically screened the particle-particle and particle-hole channels, correspond- Coulomb interactions, focusing on the renormalization of the ing to mean-field instabilities to both Cooper pairing and Green’s function, and using the framework of the perturba- charge density wave ordering. However, when both instabili- tive renormalization group ͑RG͒. ties are taken into account simultaneously within the frame- We analyze the RG flow perturbatively in the number of work of the RG, they cancel each other out, so that there flavors, given by N=4 in BLG. We use perturbation theory is no instability to any long-range ordered phase at low developed about the noninteracting fixed point and calculate energies.8 the renormalization of the fermion Green’s function and of Exactly the same behavior follows from our RG analysis the Coulomb interactions. We demonstrate that the quasipar- of BLG, including the cancellation of the vertices respon- ticle residue and the Coulomb vertex function undergo log2 sible for the pairing and charge density ordering. However, renormalization while all other quantities renormalize only the diagrams in this instance are log2 divergent and even logarithmically. The quasiparticle residue and the Coulomb after the leading log2 divergences are canceled out, there vertex function, moreover, are not independent but are re- remains a subleading single log instability. Nevertheless, this
1098-0121/2010/82͑11͒/115431͑11͒ 115431-1 ©2010 The American Physical Society RAHUL NANDKISHORE AND LEONID LEVITOV PHYSICAL REVIEW B 82, 115431 ͑2010͒ single log instability manifests itself on much lower energy while the dielectric constant incorporates the effect of po- scales than the log2 RG flow. Therefore, over a large range of larization of the substrate. Note that the single-particle energies, bilayer graphene can be viewed as a two dimen- Hamiltonian H0 takes the same form for each of the four sional analog of the one-dimensional Luttinger liquids. fermion flavors and is thus SU͑4͒ invariant under unitary Our treatment of the log2 renormalization in BLG is rotations in the flavor space. somewhat reminiscent of the situation arising in two- The Coulomb interaction sets a characteristic length scale ͑ dimensional disordered metals.11 In the latter, the log2 diver- and a characteristic energy scale Bohr radius and Rydberg ͒3 gences of the Green’s function and of the vertex function energy stem from the properties of dynamically screened Coulomb ប2 e2 1.47 a = Ϸ 10 Å, E = Ϸ eV. ͑3͒ interactions, which exhibit “unscreening” for the transferred 0 me2 0 a 2 frequencies and momenta such that /q2 is large compared 0 to the diffusion coefficient. Furthermore, the divergent cor- In Eq. ͑1͒, we have approximated by assuming that the in- rections to the Fermi-liquid parameters, as well as conduc- terlayer and intralayer interaction are equal. This approxima- tivity, compressibility and other two-particle quantities in tion may be justified by noting that the interlayer spacing d Ϸ these systems, are only logarithmic. This allows to describe 3 Å is much less than the characteristic length scale a0, the RG flow of the Green function due to the log2 di- Eq. ͑3͒. Within this approximation, Hamiltonian ͑1͒ is invari- vergences by a single RG equation12 of the form ant under SU͑4͒ flavor rotations.13 ϳ We note that for 1 the energy E0 value is comparable ϳ =− G, to the energy gap parameter W 0.4 eV of the higher BLGץ/Gץ 42g bands ͑see Ref. 15 for a discussion of four band model of BLG͒. This suggests that there is some interaction induced where g is the dimensionless conductance. The suppression mixing with the higher bands of BLG. However, since a of the quasiparticle residue, described by this equation, four-band analysis is exceedingly tedious, here we focus on manifests itself in a zero-bias anomaly in the tunneling den- the weak coupling limit E ӶW, where the two-band ap- sity of states, readily observable by transport measurements. 0 proximation, Eq. ͑1͒, is rigorously accurate. We perform all our calculations in this weak coupling regime and then ex- Ϸ −2 II. DYNAMICALLY SCREENED INTERACTION trapolate the result to E0 1.47 eV . Since the low energy properties should be independent of the higher bands, we We begin by reviewing some basic facts about BLG. BLG believe this approximation correctly captures, at least quali- consists of two AB stacked graphene sheets ͑Bernal stack- tatively, the essential physics in BLG. Meanwhile since W is ing͒. The low-energy Hamiltonian can be described in a the maximum energy scale up to which the two-band Hamil- “two-band” approximation, neglecting the higher bands that tonian, Eq. ͑1͒, is valid, we use W as the initial ultraviolet are separated from the Dirac point by an energy gap W ͑UV͒ cutoff for our RG analysis. 1 ϳ0.4 eV. There is fourfold spin/valley degeneracy. The We wish to obtain a RG flow for the problem ͑1͒ by wave function of the low-energy electron states resides on systematically integrating out the high-energy modes. How- the A sublattice of one layer and B sublattice of the other ever, the implementation of this strategy is complicated by layer. The noninteracting spectrum consists of quadratically the long-range nature of the unscreened Coulomb interac- 2 dispersing quasiparticle bands EϮ = Ϯp /2m with band mass tion. Within perturbation theory, the long-range interaction Ϸ m 0.054me. We work throughout at charge neutrality, when gives contributions which are relevant at tree level, making it the Fermi surface consists of Fermi points. The discrete difficult to come up with a meaningful perturbative RG pointlike nature of the Fermi surface is responsible for most scheme. Therefore, it is technically convenient to perform a of the similarities to the Luttinger liquids. two-step calculation, where we first take into account screen- Although the canonical Hamiltonian has opposite chirality ing within the random-phase approximation ͑RPA͒ and then in the two valleys, a suitable unitary transformation on the carry out an RG calculation with the RPA screened effective spin-valley-sublatttice space brings the Hamiltonian to a interaction. We emphasize that it is necessary to consider the form where there are four flavors of fermions, each governed full dynamic RPA screening of the Coulomb interaction since 13 by the same 2ϫ2 quadratic Dirac-type Hamiltonian. We a static screening approximation does not capture the effects introduce the Pauli matrices that act on the sublattice space we discuss below. Ϯ Ϯ i, and define Ϯ = 1 i 2, and pϮ =px ipy, and hence The dynamically screened interaction may be calculated 14 write by summing over the RPA series of bubble diagrams, to ob- e2 n͑x͒n͑xЈ͒ tain a screened interaction. The RPA approach to screening H = H + ͚ , ͑1͒ may be justified by invoking the large number N=4 of fer- 0 2 ͉x − xЈ͉ x,xЈ mion species in BLG. The screened interaction takes the form p2 p2 2 † ͩ + − ͪ ͑ ͒ 2 e H0 = ͚ + + − p,. 2 ͑ ͒ ͑ ͒ p, U ,q = 2 . 4 p, 2m 2m ͉q͉ −2e ⌸͑,q͒
Here =1,2,3,4 is a flavour index, n͑x͒=͚n͑x͒ is the Here ⌸͑,q͒ is the noninteracting polarization function, electron density, summed over spins, valleys and sublattices which can be evaluated analytically.3,16 Here we will need an
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expression for ⌸͑,q͒ in terms of Matsubara frequencies , We will employ an RG scheme which treats frequency derived in Ref. 3, where it was shown that the quantity on the same footing as p2 /2m, in order to preserve the form ⌸͑,q͒ depends on a single parameter 2m/q2, and is well of the free action Eq. ͑7͒ under RG. Thus, we integrate out described by the approximate form the shell of highest energy fermion modes q2 p2 2 ln 4 ⌳Ј Ͻ ͱ2 + ͩ ͪ Ͻ⌳ ͑9͒ Nm 2m 4ln2 4 2m ⌸͑,q͒ =− , u = , ͑5͒ 2 2 2 / 2 q →͑⌳/⌳Ј͒ p→p͑⌳/⌳Ј͒1 z ͱͩ ͪ + u2 and subsequently rescale , , 2m where z is the dynamical critical exponent,9 which takes value z=2 at tree level. Because the value z=2 is not pro- where N=4 is the number of fermion species. The depen- tected by any symmetry, it may acquire renormalization cor- ͓ ͑ ͔͒ ⌸͑ ͒ dence Eq. 5 reproduces ,q exactly in the limits rections. However, it will follow from our analysis that the Ӷ 2 / ӷ 2 / q 2m and q 2m, and interpolates accurately in be- quasiparticle spectrum does not renormalize at leading log2 ͑ ͒ ͑ ͒ tween. We discover upon substituting Eq. 5 in Eq. 4 that order, so that the exponent z does not flow at leading order. the dynamically screened interaction is retarded in time but We therefore use z=2 for the rest of the paper, which corre- crucially is only marginal at tree level. It therefore becomes sponds to scaling dimensions ͓͔=1 and ͓p2͔=1. Under such possible to develop the RG analysis perturbatively in weak an RG transformation, the Lagrangian density in momentum ӷ coupling strength, by taking the limit of N 1. space has scaling dimension ͓L͔=2, and we have tree level ⌸͑ ͒ → Since the quantity ,q vanishes when q 0, the RPA scaling dimensions ͓͔=1/2 and ͓⌫͔=͓Z͔=0, respectively. ͓ ͑ ͔͒ screened interaction Eq. 4 retains some long-range char- Given these tree level scaling dimension values, it can be acter, exhibiting unscreening for ӷq2 /2m. This will lead 2 seen that all potentially relevant terms arising as part of S2 to divergences in Feynman diagrams of a log character. must involve four fermion fields. Indeed, any term involving more than four fields will be irrelevant at tree level under III. SETTING UP THE RG RG, and may be neglected. The terms with odd numbers of To calculate the RG flow of the Hamiltonian, Eq. ͑1͒,in fields are forbidden by charge conservation while the qua- ⌬ † the weak coupling regime, we begin by writing the zero- dratic terms ij i j cannot be generated under perturbative temperature partition function ⌽ as an imaginary-time func- RG since they break the symmetries of the Hamiltonian tional field integral. We have listed above.17 Thus, the only potentially relevant terms that could arise under perturbative RG take the form of a four- ⌽ ͵ † ͑ ͓† ͔ ͓† ͔͒ ͑ ͒ point interaction which may be written as = D D exp − S0 , − S1 , , 6 1 ͵ 3 3 Ј⌼Ј† ͑ ͒ ͑ ͒† ͑ Ј͒ ͑ Ј͒ S2 = d xd x x ,j x x ,l x , ijkl ,i Ј,k Ј dd2p − i + H ͑p͒ 2 ͚ ͵ † ͩ 0 ͪ ͑ ͒ S0 = 3 ,,p ,,p, 7 ͑10͒ ͑2͒ Z where x=͑r,t͒, xЈ=͑rЈ,tЈ͒. Here ⌼ is an effective four par- 1 dd2p ticle vertex, which is marginal at tree level, the indices ,Ј ͵ ⌫2 ͑ ͒ ͑ ͒ S1 = U ,q n,qn−,−q + S2. 8 refer to the flavour ͑spin-valley͒ of the interacting particles, 2 ͑2͒3 and i, j,k,l are sublattice indices. Here the fields are Grassman valued ͑fermionic͒ fields with The symmetries of the Hamiltonian, Eq. ͑1͒, impose flavour ͑spin-valley͒ index while is a fermionic Matsub- strong constraints on the spin-valley-sublattice structure of ara frequency, ⌫ is a vertex renormalization parameter, Z is the four-point vertex ⌼. Since the Coulomb interaction does ͑ ͒ the quasiparticle residue, and n,q is the Fourier transform of not change fermion flavour spin or valley , and the electron the electron density, summed over spins, valleys and sublat- Green’s function is diagonal in flavour space, the vertex ⌼ tices. The effective interaction U͑,q͒ is given by Eq. ͑4͒. cannot change fermion flavour. Moreover, the SU͑4͒ flavour ⌼ The term S2 is included tentatively to represent more com- symmetry of the Hamiltonian implies that does not depend plicated interactions that may be generated under RG. In the on the flavour index of the interacting particles and we may ⌫ ͑ ͒ bare theory, =1, Z=1, and S2 =0. The theory is defined with therefore drop the indices , Ј in Eq. 10 . Finally, the bare ⌳ ͑ ͒ ͑ ͒ the initial UV cutoff 0. Since the two-band model, Eq. 1 , Hamiltonian 1 is invariant under combined pseudospin/ is only justified on energy scales less than the gap W spatial rotations through ei 3R͑/2͒. This symmetry further Ϸ0.4 eV to the higher bands in BLG, we conservatively restricts the form of four-point vertices in Eq. ͑10͒ to have ⌳ ⌳ ⌼ ⌼ 18 identify 0 =W. Our main results will be independent of 0. sublattice structure iijj or ijji only. That is, the allowed As we shall see, the RG flow will inherit the symmetries scattering processes are restricted to ͑AA͒→͑AA͒, ͑AB͒ of the Hamiltonian, Eq. ͑1͒, strongly constraining the pos- →͑AB͒ and ͑AB͒→͑BA͒. We note that the processes ͑AB͒ →͑ ͒ ͑ ͒→͑ ͒ sible terms S2. The relevant symmetries are particle-hole AB and AB BA are distinct since the particles have symmetry, time reversal symmetry, SU͑4͒ flavour flavour, and the interaction ͓Eq. ͑4͔͒ is not short range. symmetry,13 and the symmetry of the Hamiltonian under the Below we obtain the RG flow for bilayer graphene, work- transformation ei 3R͑/2͒, where R͑͒ generates spatial ro- ing in the manner of Ref. 9. We consider the partition func- Ϯi tations, R͑͒pϮ =e pϮ. tion, Eq. ͑6͒, where the interaction is given by Eq. ͑4͒. Start-
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ץ/ ⌺ץ It was shown in Ref. 7 that i AA and 2 ͒ / 2 ͑ץ/ ⌺ץ AB q+ 2m are both log divergent, and are equal to leading order ͑see below and Sec. VIII for alternative deri- vation͒. Thus the self-energy can be written, with logarithmic accuracy, as ⌳ ⌺ץ (a) (b) ⌺͑,q͒ =−iZ G−1͑,q͒ + Oͩln ͪ. ͑16͒ 0 ⌳Јץ 0 FIG. 1. Diagrammatic representation of ͑a͒ self-energy and ͑b͒ is due to theץ/⌺ץ vertex correction ͓Eqs. ͑13͒ and ͑27͔͒. Straight lines with arrows Here, it is understood that nonvanishing represent fermion propagator, Eq. ͑12͒, wavy lines represent dy- modes that have been integrated out, Eq. ͑9͒. Within the namically screened long-range interaction, Eq. ͑4͒. leading log approximation, the electron Green’s function, Eq. ͑12͒, retains its noninteracting form, whereby the self-energy, ⌳ ͑ ͒ ing from this action, supplied with UV cutoff 0,we upon substitution into Eq. 11 , can be absorbed entirely into systematically integrate out the shell of highest energy fer- a redefinition of the quasiparticle residue, as mion modes, Eq. ͑9͒. We perform the integrals perturbatively ⌺ץ ͑ ͒ 1 AA in the interaction, Eq. 4 . This corresponds to a perturbation ␦G͑,q͒ = ␦Z, ␦Z =−i Z2. ͑17͒ 0ץ ͒ ͑ 2 2 theory in small ⌫ Z /N. We carry out our calculations to one i − H0 q loop order and examine the renormalization, in turn, of the ͑ ͒ ⌫ We emphasize that the lack of renormalization of the mass electron Green’s function Sec. IV , the vertex function 2 ͑Sec. V͒ and the four-point vertex ⌼ ͑Sec. VI͒. only holds at log order. The subleading single log renormal- ization of the mass will be analyzed in Sec. VIII. IV. SELF-CONSISTENT RENORMALIZATION The renormalization of the quasiparticle residue, Eq. ͑17͒, . Taking ⌺ץ/⌺ץOF THE ELECTRON GREEN’S FUNCTION can be evaluated explicitly by calculating i from Eq. ͑13͒, we write At first order in the interaction, the fermion Green’s func- tion acquires a self-energy ⌺, represented diagrammatically p2 2 ͑to leading order in the interaction͒ by Fig. 1͑a͒. A self- ͩ ͪ − 2 ⌺ dd2p 2m 2⌫2Z e2ץ consistent expression for the change in the fermion propaga- ͯ ͯ ͵ 0 0 i =− 3 2 2 2 . ͑2͒ p 2mץ tor G is =0 ͫͩ ͪ + 2ͬ p −2e2⌸ͩ ͪ 2m p2 ␦ ͑ ͒ ͑ ͒⌺͑ ͒ ͑ ͒ ͑ ͒ G ,q = G0 ,q ,q G0 ,q , 11 ͑18͒ Z ␣ G ͑,q͒ = 0 , ͑12͒ We express the momenta in polar coordinates px =p cos , 0 ͑ ͒ ␣ Ͻ␣Ͻ i − H0 q py =p sin , and straightaway integrate over − .We further change to pseudopolar coordinates in the frequency- dd2p momentum space, =r cos , p2 /2m=r sin , with the “po- ⌺͑ ͒ ͵ ⌫2 ͑ ͒ ͑ ͒ ,q =− 0U,pG0 + ,p + q , 13 lar angle” 0ϽϽ. Using the Rydberg energy E , Eq. ͑3͒, ͑2͒3 0 as units for r, we have where ⌺ isa2ϫ2 matrix in sublattice space. ⌺ ⌳ ͑ 2 2 ͒⌫2ץ A number of general properties of the self-energy can be dr d sin − cos Z0 i =−͵ ͵ 0 , ͑19͒ ⌳Ј r 2 2ץ established based on symmetry considerations. It follows ͑ ͒ ⌺͑ ͒ ͑ ͒ 0 ͱ2r sin − ⌸͑͒ from Eq. 13 that 0,0 vanishes, since the part of G ,p m which is invariant under rotations of p is an odd function of frequency . Likewise, the expressions for diagonal entries where ⌸͑͒ is the dimensionless polarization function, given ⌺ ͑ ͒ ⌺ ͑ ͒ AA 0,q and BB 0,q , which involve an integral of an odd by Eq. ͑5͒ with quasiparticle mass m suppressed and function of , vanish on integration over . For the same 2m /p2 =cot . We note that ⌸͑͒ goes to zero when ⌺ ͑ ͒ reason, the expressions for off-diagonal entries AB ,0 →0,, and these zeros of the polarization function dominate ⌺ ͑ ͒ 2 and BA ,0 vanish upon integrating the momentum p over the integral and lead to the log divergence. Since ⌸͑͒ is angles. Hence, nonvanishing contributions arise at lowest or- even about =/2, the log2 contribution can be evaluated by der when the right hand side of Eq. ͑13͒ is expanded to replacing ⌸͑͒ in Eq. ͑19͒ by its asymptotic Ӷ form, leading order in small and q. We obtain Nm ⌺ ͑0,0͒ ⌸͑͒Ϸ tan . ͑20͒ ץ i ⌺ ͑,q͒ =−i AA + O͑2,q2,q4͒, ͑14͒ 4 ץ AA In the region Ӷ, we may approximate sin Ϸ, tan ⌺ ͑ ͒ Ϸ Ϸ ץ 2 q AB 0,0 , and cos 1. Including a factor of 2 for the region ⌺ ͑,q͒ = + + O͑2,q2,q4͒, ͑15͒ Ϸ q2/2m͒ , which gives a contribution identical to that of the region͑ץ AB 2m + Ϸ0, we can express the integral Eq. ͑19͒ with logarithmic ء⌺ ⌺ ⌺ ⌺ where AA= BB and AB= BA by symmetry. accuracy as
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⌳ / ⌺ dr 2 d ⌫2Zץ i =2͵ ͵ 0 0 . ͑21͒ ⌳Ј r 2 Nץ 0 ͱ2r + 2
Performing the integral over and assuming rӶN2, yields
⌳ ⌺ 2⌫2Z dr N22ץ 0 0 ͵ ͑ ͒ (a) (b) i = 2 ln . 22 N ⌳Ј r 8rץ FIG. 2. The renormalization of the four-point vertex ⌼ proceeds ͑ ͒ Integrating over ⌳ЈϽrϽ⌳ ͓see Eq. ͑9͔͒, we obtain through repeated scattering in a the particle-particle channel and in ͑b͒ the particle-hole channel, known as the BCS loop and the ZSЈ loop in the Luttinger liquid literature ͑Ref. 9͒. The RPA bubble ⌳ ⌳ ⌺ ⌫2 22ץ 2 Z0 N E0 1 ͑ ͒ i = 0 ͩln ln − ln2 ͪ. ͑23͒ diagrams ZS loop in the language of Ref. 9 , which arise in the N2 8⌳Ј ⌳Ј 2 ⌳Ј same order of perturbation theory, have already been taken intoץ account in the screened interaction, Eq. ͑4͒. We now consider an infinitesimal RG transformation. Defin- ing an RG time p2 2 ͩ ͪ − 2 2 ⌫3 2 2 ⌳ ⌳ d d p 2m 2 0Z0e 0 ␦⌫ =−͵ . =ln , ␦ =ln , ͑24͒ ͑2͒3 p2 2 2 2m ⌳ ⌳Ј ͫͩ ͪ + 2ͬ p −2e2⌸ͩ ͪ 2m p2 we rewrite the recursion relation, Eq. ͑23͒,as ͑27͒
-⌺ 2⌫2Z N22E This is the same expression as for the residue renormalizaץ i = 0 0 ͑ + c͒d, c =ln 0 . ͑25͒ tion ͓Eqs. ͑17͒ and ͑18͔͒, with ⌫ replacing Z, and a sign ⌳ 2ץ N 8 0 change. Hence, we obtain 3͑͒Z2͑͒⌫2 ⌫ץ ,The constant term c describes corrections subleading in log2 = ͑ + c͒͑28͒ N2ץ and thus may seem to be irrelevant. However, we shall retain it in the RG equation since it will determine the form of renormalization near the UV cutoff ͑see discussion of TDOS which is identical to the flow equation for Z, albeit with a in Sec. VII͒. reversed sign. Therefore, the product ⌫Z does not renormal- In our derivation of Eq. ͑22͒ it was assumed that our ize at log square order, and we can write ⌳ ϽN22E / ⌳ initial UV cutoff 0 0 8. Such choice of 0 is cer- ⌫͑͒Z͑͒ =1. ͑29͒ tainly justified when N is large, which is the limit we worked on thus far. Better still, this condition remains en- This result is not a coincidence since the residue Z and the tirely reasonable for the physical value N=4, leading to vertex function ⌫ are not independent quantities. The Hamil- 22 / −2 ͑ ͒ N E0 8=24 eV , which is much bigger than the band- tonian, Eq. 1 , is invariant under a gauge transformation width for BLG. of electron wave function Ј=ei, accompanied by energy ٌ ץ Substituting Eq. ͑25͒ into Eq. ͑17͒, we obtain a differen- and momentum shifts Ј= − t , pЈ=p+ . This gauge in- tial equation for the flow of the quasiparticle residue, variance symmetry can be shown to lead to Eq. ͑29͒ through a Ward identity that relates the self-energy to the vertex Z 2⌫2͑͒Z3͑͒ function.19,20ץ =− ͑ + c͒. ͑26͒ N2ץ VI. RENORMALIZATION OF THE FOUR-POINT This equation encapsulates a one loop RG flow for the resi- VERTEX ⌼ Z 2 due , describing its renormalization within a log accuracy. The four-point vertex ⌼, introduced in Eq. ͑10͒, renormal- izes through the diagrams presented in Figs. 2͑a͒ and 2͑b͒, V. SELF-CONSISTENT RENORMALIZATION which represent the repeated scattering of two particles in the OF THE VERTEX FUNCTION ⌫ electron-electron and electron-hole channels, respectively. We follow the naming conventions used in Ref. 9 in the The screened Coulomb interaction renormalizes through context of the Luttinger liquid, and name these two dia- the vertex correction, pictured in Fig. 1͑b͒. The RPA bubble grams, the BCS loop and the ZS’ loop, pictured in Figs. 2͑a͒ diagrams, which have already been taken into account in and 2͑b͒, respectively. In the one dimensional Luttinger liq- moving from an unscreened to a screened interaction, Eq. uids, the two processes famously cancel,8 so that the four- ͑4͒, do not contribute to renormalization. It may be verified point vertex does not renormalize. In higher dimensions, by an explicit calculation that the vertex correction in Fig. such a cancellation is rare. However, the discrete nature of 1͑b͒ is given by the Fermi surface in BLG results in a Luttinger liquid-
115431-5 RAHUL NANDKISHORE AND LEONID LEVITOV PHYSICAL REVIEW B 82, 115431 ͑2010͒ like cancellation of the processes Figs. 2͑a͒ and 2͑b͒, as will dd2p ⌼BCS ⌫4͵ ͑ ͒ be discussed below. AAAA = U,pU−,p−qGAA E1 + ,k1 + p ͑2͒3 We argued in Sec. III that the RG-relevant scattering pro- ϫ ͑ ͒ ͑ ͒ cesses allowed by symmetry must have sublattice structure GAA E2 − ,k2 − p . 33 ͑A,A͒→͑A,A͒, ͑A,B͒→͑A,B͒,or͑A,B͒→͑B,A͒. To see Here, the interaction U͑,p͒ is defined by Eq. ͑4͒, the the mathematical origin of such selection, it is instructive to Green’s functions are defined by Eq. ͑30͒, and the integral explicitly write out the form of the electron Green’s function. goes over the shell defined by Eq. ͑9͒. We have As always in a RG analysis, we assume that the external frequencies and momenta are small compared to the internal − Zi frequencies and momenta G ͑,p͒ = = G ͑,p͒, ͑30͒ AA 2 ͑ 2/ ͒2 BB + p 2m q2 qЈ2 p2 2 maxͩ,Ј, , ͪ Ӷ⌳Ј Ͻ ͱ2 + ͩ ͪ Ͻ⌳. 2m 2m 2m − Zp2/2m ͑34͒ ͒ ͑ ͒ ͑ ء + ͒ ͑ GAB ,p = 2 2 2 = GBA ,p . 31 + ͑p /2m͒ In such a case, the standard approach to handling the integrals over and p involves setting the external frequency When the diagrams Figs. 2͑a͒ and 2͑b͒ are evaluated in any and momenta to zero at first, and restoring their finite ͑ ͒ channel other than these three channels, they vanish upon values later to regulate the infrared IR divergences. How- integration over inner momentum variables, due to the chiral ever, a straightforward application of this recipe to the inte- ͑ ͒ ͑ ͒ structure of the sublattice changing Green’s functions, Eq. grals in Eqs. 32 and 33 proves impossible, because these ͑31͒. integrals are power law divergent when all external momenta Similar reasoning leads to a conclusion that the ͑A,B͒ are set to zero. The divergence arises from the region Ϸ ͓ ͑ ͔͒ →͑B,A͒ vertex cannot exhibit a log2 divergence. As we saw near p 0 which lies within the shell defined by Eq. 9 , above, the log2 divergences arise because the effective inter- where the interaction is nearly unscreened. In this region, we have action U,p has a pole at p=0 and finite . However, the ͑ ͒ sublattice changing Green’s functions, Eq. 31 , have zeros at 1 small p, which cancel the contribution of the pole in the U U ϳ ͑35͒ ,p − ,p−q ͉͑p͉ + ␣͉p͉2͉͒͑p − q͉ + ␣͉p − q͉2͒ interaction. Thus, the diagrams in Fig. 2 can only be log2 divergent if all internal Green’s functions are sublattice pre- with ␣=Ne2 /2. At finite q, the poles in this expression are serving, given by Eq. ͑30͒. Since the process ͑AB͒→͑BA͒ split apart, and thus the singular contribution of each pole, involves two sublattice changing Green’s functions, it fol- p=0 and p=q, is regularized by the integration measure d2p lows that the integrals associated with this processes cannot so that the integrals in Eqs. ͑32͒ and ͑33͒ remain well de- be log2 divergent, and hence this process does not contribute fined. However, when all external momenta are zero, the at leading log2 order. poles from the two interaction lines coincide, and the expres- Thus, at leading order, we need to consider only the pro- sions ͑32͒ and ͑33͒ acquire a second-order pole at p=0. cesses ͑AA͒→͑AA͒ and ͑AB͒→͑AB͒. Moreover since the When we integrate over this second-order pole, we pick up a interaction ͓Eq. ͑4͔͒ does not distinguish between sublattices, power law divergence. the ZSЈ and BCS contributions from Figs. 2͑a͒ and 2͑b͒ in Hence, if either of the ZSЈ or BCS diagrams existed in these channels are the same. Therefore, to demonstrate that isolation, this power law divergence would indicate a strong ⌼ does not renormalize at leading order, it is sufficient to ͑power law͒ instability, which would drive ⌼ into the strong demonstrate that there are no log2 divergences in the ͑AA͒ coupling regime, where our log2 RG would cease to apply. →͑AA͒ channel. However, as we will now show, the divergences in the con- In evaluating the ZSЈ and BCS diagrams ͑Fig. 2͒, it will tributions to ⌼ from the expressions ͑32͒ and ͑33͒ in fact prove important to keep track of external momenta. The ver- cancel out, so that ⌼ does not flow to log2 order. To analyze ⌼͑ ͒ tex E1 ,E2 , ,k1 ,k2 ,q then represents the amplitude for the cancellation between the ZSЈ and BCS diagrams, it is the scattering process convenient to add the integrands of Eqs. ͑32͒ and ͑33͒ to- gether before doing the integral while keeping external mo- → menta finite. Preserving finite external momenta ensures that ,A,E ,k Ј,A,E ,k ,A,E +,k +q Ј,A,E −,k −q. 1 1 2 2 1 1 2 2 the integrals Eqs. ͑32͒ and ͑33͒ are well defined. After com- ⌼ZSЈ ⌼BCS ⌼˜ bining the integrands and denoting AAAA+ AAAA= ,we Translating the ZSЈ and BCS diagrams in Fig. 2 into inte- obtain grals, we find the contributions dd2p ⌼˜ ⌫4͵ ͑ ͒ = U U G E + ,k + p ͑ ͒3 ,p − ,p−q AA 1 1 2 2 ZSЈ 4 d d p ⌼ = ⌫ ͵ U U G ͑E + ,k + p͒ ϫ͓ ͑ ͒ ͑ ͔͒ AAAA ͑2͒3 ,p − ,p−q AA 1 1 GAA E2 + − ,k2 + p − q + GAA E2 − ,k2 − p . ͑ ͒ ϫ ͑ ͒ ͑ ͒ 36 GAA E2 + − ,k2 + p − q , 32
115431-6 ELECTRON INTERACTIONS IN BILAYER GRAPHENE:… PHYSICAL REVIEW B 82, 115431 ͑2010͒
To simplify this expression we note that momentum q enters To extract the leading contribution at small q, we approxi- very differently in Eq. ͑36͒ as compared to other external mate the effective interaction as frequencies and momenta E , E , , k , and k . The momen- 1 2 1 2 ⌸−1͑,p͒ 1 tum q is needed to split the poles coming from the two U͑,p͒ =− Ϸ − . ͑41͒ interaction terms—if we take q to zero, the integral will ac- ͉p͉ ⌸͑,p͒ 1− quire a second-order pole at p=0, leading to a divergence. 2e2⌸͑,p͒ This divergence arises from within the shell that we are in- tegrating out ͓Eq. ͑9͔͒, and thus the RG will be ill defined. In From the definition of the polarization function, Eq. ͑5͒, we see that the approximation UϷ−1/⌸ holds everywhere contrast, sending the frequencies and momenta E1, E2, , k1, ͑ ͒ in the shell Eq. ͑9͒ except at pϷ0 since ⌸͑p=0͒=0. and k2 to zero by applying Eq. 34 does not cause any concern. We thus have However, in the limit p→0, the expression in brackets in Eq. ͑40͒ tends to zero because of the expansion dd2p z2 −z2 =͑p2 /m͒͑p·q/2m͒+O͑p4͒, which ensures validity of ⌼˜ ⌫4͵ ͑ ͒ + − = U U G ,p ͑2͒3 ,p ,p−q AA the approximation ͓Eq. ͑41͔͒. Hence, using Eq. ͑5͒, we obtain ϫ͓ ͑ ͒ ͑ ͔͒ ͑ ͒ GAA ,p − q + GAA − ,− p . 37 ͱ͑z2 + u2͒͑z2 + u2͒ Interestingly, the expression in square brackets vanishes ⌼˜ = ⌫4Z2͵ dd2p + − ͑ ͒ ͑ ͒ 4͑Nm ln 4͒2 identically when q=0 since GAA ,p =−GAA − ,−p . How- → ever, taking the limit q 0 is potentially problematic be- 2 z2 − z2 2 ϫ ͫ + − ͬ ͑ ͒ cause of the pole structure of U U discussed above. . 42 ,p − ,p−q z z ͑2 2͒͑2 2͒ Instead, we proceed with caution and evaluate Eq. ͑37͒ at + − + z+ + z− finite q, using the conditions ͓Eq. ͑34͔͒ to simplify the analy- Simple power counting shows that this integral is UV con- sis. vergent, IR convergent, and is completely independent of q, Given what we just said, it is now easy to see why there is which can be scaled out by defining new variables pЈ=p/q no log2 divergence in ⌼˜ . First, we note that the interaction and Ј=2m/q2. It follows that the diagrams representing Eq. ͑4͒ carries a soft UV cutoff, so the integral in Eq. ͑37͒ is repeated scattering in the particle-particle and particle-hole ⌼ 2 UV convergent ͑this property of dynamically screened inter- channels do indeed cancel, so that AAAAZ does not renor- action in BLG is discussed, e.g., in Ref. 3͒. Hence, we can malize. shift variables to pϮ =pϮq/2 and rewrite the expression Combining this with our argument demonstrating that ⌼ 2 2 ͓ ͑37͒ as ABBAZ does not renormalize at log order see discussion ͑ ͔͒ ⌼ ⌼ below Eq. 31 , and recalling that AAAA= AABB, we con- dd2p ⌼ 2 ⌼˜ ⌫4 2͵ 2 ͑ ͓͒ ͑ ͒ clude that we can set =0 with log accuracy. =− Z U U D ,p D ,p ͑2͒3 ,p+ ,p− + − VII. SOLUTION OF RG FLOW EQUATIONS. − D͑,p ͔͒ + ZERO BIAS ANOMALY IN BILAYER GRAPHENE dd2p D͑,p ͒ + D͑,p ͒ ⌫4 2͵ 2ͫ + − 2 =− Z U U Since the only quantities which renormalize at log order ͑ ͒3 ,p+ ,p− 2 2 in a one loop RG are the quasiparticle residue Z and the D͑,p ͒ − D͑,p ͒ interaction vertex function ⌫, the problem of finding the RG + − ͓ͬ ͑ ͒ ͑ ͔͒ ͑ ͒ + D ,p− − D ,p+ , 38 flow of these quantities reduces to solving Eqs. ͑26͒ and 2 ͑28͒. All other quantities do not renormalize at log square where we factored the Green’s functions as order, and may thus be treated as constants with logarithmic accuracy. 1 Additional simplification arises due to the Ward identity G ͑,p͒ = iZD͑,p͒, D͑,p͒ = . AA 2 + ͑p2/2m͒2 ⌫Z=1, Eq. ͑29͒. Using it to decouple the RG equations for Z and ⌫, we write the equation for Z as ͑39͒ Z 2ץ We note that because ⌼ should be even under q→−q the ͑ ͒ ͑ ͒ =− 2 + c Z, 43 Nץ .first term in the brackets gives zero upon integration over p ⌼˜ ͑ ͒ N22E Hence, we can rewrite the result for , Eq. 38 ,as where we retained a constant c=ln 0 corresponding to 8⌳0 ⌫4 2 2 the first term in the self-energy renormalization, Eq. ͑23͒. ˜ Z d d p 2 2 ⌼ = ͵ U U ͓D͑,p ͒ − D͑,p ͔͒ Integrating the RG equation, and taking into account the 2 ͑2͒3 ,p+ ,p− − + boundary conditions Z͑0͒=⌫͑0͒=1, we obtain ⌫4Z2 dd2p z2 − z2 2 ͵ 2ͫ + − ͬ 2c + 2 ⌳ = U U , 0 2 ͑2͒3 ,p+ ,p− ͑2 + z2͒͑2 + z2͒ Z͑͒ = expͩ− ͪ = ⌫−1͑͒, =ln . ͑44͒ + − N2 ⌳ ͑40͒ We note that in the limit of small 2 /N, we reproduce the 2 where zϮ =͉pϮ͉ /2m. perturbative result7 for the residue, Eq. ͑23͒. However, our
115431-7 RAHUL NANDKISHORE AND LEONID LEVITOV PHYSICAL REVIEW B 82, 115431 ͑2010͒
result ͓Eq. ͑44͔͒ applies for all , both small and large. The fermion propagator at arbitrary energies and momenta is then given by
i + H ͑k͒ G͑,k͒ =−Z͑͒ 0 . ͑45͒ k2 2 2 + ͩ ͪ 2m
At zero temperature, the infrared cutoff is supplied by the ⌳ 0 external frequency and momentum, such that =ln ⌳ and FIG. 3. ͑Color online͒ TDOS of BLG at charge neutrality, Eq. ⌳ ͱ2 ͑ 2 / ͒2 = + k 2m ͑51͒, is shown as a function of external bias =eV. Predicted Thus, the quasiparticle residue in undoped BLG is sup- TDOS is shown for two different values of the dielectric constant in ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ pressed to zero by electron-electron interactions, Eq. 45 . E0, Eq. 3 : =1 solid curve and =2.5 dashed curve , describ- This is reminiscent of the situation in disordered metals, ing free-standing BLG and BLG on SiO substrate, respectively. Plot where enhancement of interactions by disorder produces a is normalized so that =1 at an external bias of 100 meV. renormalization of electron self-energy of a log2 form,11 and analysis of an RG flow12 yields a suppression of the quasi- ϱ N d particle residue similar in form to our Eq. ͑45͒. The suppres- 0 ͵ ͑ ͒ ͑ ͒ Tr G = i Z − ln cosh , 48 sion of the quasiparticle spectral weight at low energies, gov- 0 cosh ͑͒ erned by the Z dependence, will manifest itself directly in ͑⌳ /͒ where =ln 0 . Noting that this integral is dominated the behavior of the tunneling density of states of BLG, simi- ϳ lar to disordered metals. by 1, we obtain an estimate of the spectral weight We note parenthetically that while keeping the constant 2 +2c ͑ ͒ 2 ͑͒Ϸ ͑ ͒ ͩ ͪ ͑ ͒ term c in the RG Eq. 43 is formally beyond the log accu- N Z = N exp − . 49 0 0 2 racy generally adopted in our analysis, it can be justified on N the same grounds as in the discussion of the zero bias The form of this expression remains unchanged, to leading 21,22 anomaly in disordered metals. Because of its fairly large log2 order, upon analytic continuation to real frequencies. 2 Ϸ value for N=4, given by c=ln2 2.98, this term may The expression in Eq. ͑49͒ can be rearranged by using Eq. significantly alter predictions for the behavior of Z at inter- ͑25͒ as Շ⌳ mediate energies 0. 22 22 To analyze the suppression of TDOS, we use its relation 2 N E0 2 N E0 11 ln −ln to the retarded Green’s function, 8 8⌳ ͑͒ 0 ͑ ͒ = N exp − . 50 0 N2 1 ͑͒ =− Im͓Tr G ͑,k͔͒, ͑46͒ ⌳ R Thus, we see that the only effect of the UV cutoff 0 is to rescale the prefactor for the TDOS without affecting the fre- quency dependence. Absorbing the dependence on ⌳ in the ͑ ͒ 0 where GR ,k is obtained from the Matsubara Green’s prefactor, function analyzed above, Eq. ͑45͒, by the analytic continua- tion of frequency from imaginary to real values, i→ 1 N22E ͑͒ = N˜ expͩ− ln2 0 ͪ. ͑51͒ +i . 0 N2 8 It is convenient to take the trace before performing the analytic continuation. The trace may be most easily taken in Tunneling measurements yield ͑=eV͒, where V is the bias a basis of free particle eigenstates ͑plane waves with appro- voltage. The interaction suppression of the density of states, priate spinor structure͒, which amounts to integrating Eq. Eq. ͑49͒, will therefore manifest itself as a zero bias anomaly ͑45͒ over all k values, Tr G=͐G͑,k͒d2k. Noting that the in tunneling experiments. The predicted behavior of the ͑ ͒ term containing H0 k vanishes upon integration due to the TDOS is shown in Fig. 3. Because of the exponential depen- angular dependence, we write dence in Eq. ͑51͒, the suppression rapidly becomes more pronounced at lower energies. ϱ Closing our discussion of the zero bias anomaly in BLG, N0 i Tr G =2 ͵ Z͑͒ dz, ͑47͒ we note that the results described above apply only to the 2 z2 0 + system at charge neutrality. Away from neutrality, with the Fermi surface size becoming finite, the effects of screening 2 / where z=k 2m and N0 is the density of electronic states in will grow stronger, resulting in a weaker effective interac- BLG in the absence of interactions. tion. Yet, even in this case, the tunneling density of states ͑ ͒/ It can be seen that the integral over z is determined by will be described by the suppression factor =eV N0 zϳ. It is therefore convenient to introduce a variable given by Eq. ͑49͒, provided that the bias voltage eV exceeds =sinh−1͑z/͒ and write the Fermi energy measured from the neutrality point.
115431-8 ELECTRON INTERACTIONS IN BILAYER GRAPHENE:… PHYSICAL REVIEW B 82, 115431 ͑2010͒
VIII. SINGLE LOG RENORMALIZATION The integral over is now fully convergent and the resulting OF ELECTRON MASS expression is only single log divergent. Integrating analyti- cally over r and then integrating numerically over ,wefind Thus far we have concentrated on log2 flows. However, the analysis may be extended to obtain the subleading single ␦m 0.56 ⌳ log flows of the action. We illustrate this procedure by cal- ⌫2 2 ͑ ͒ = 0Z0 ln . 55 culating the renormalization of the mass ͑which did not m 2N ln 4 ⌳Ј renormalize at log2 order in the RG͒. This calculation is in- teresting because it allows us to investigate the interaction Converting this recursion relation into a differential equation, renormalization of the compressibility—a directly measur- we obtain able quantity and also because it allows us to illustrate how much slower the single log flows are than the log2 flows. d ln m 0.56 = ⌫2Z2. ͑56͒ In this section, we first analyze mass renormalization by d 2N ln 4 extracting it directly from the self-energy. After that, in Sec. IX we consider electron compressibility of BLG and show This equation cannot be solved for general by applying the that the log divergent correction to the compressibility Ward identity Eq. ͑29͒ since the Ward identity only holds at matches exactly our prediction for mass renormalization ob- leading log2 order while the mass flows at subleading ͑single tained from the self-energy. log͒ order in Eq. ͑56͒. In the perturbative limit 1 Ӷ1, when ϫ N In BLG, the self-energy is a 2 2 matrix, given by Eq. ZϷ1 and ⌫Ϸ1, from Eq. ͑56͒ we obtain a logarithmic cor- ͑ ͒ 13 , which is related to the renormalized Green’s function rection to the mass by the Dyson equation, ⌺ ͑ ͒ ⌺ ͑ ͒ 0.56 AA ,q AB ,q ͑͒ ͑ ͒ ͑ ͒ G−1͑,q͒ = G−1͑,q͒ − ͫ ͬ. ͑52͒ m = m 0 ͩ1+ ͪ. 57 0 ⌺ ͑ ͒ ⌺ ͑ ͒ 2N ln 4 BA ,q BB ,q As discussed in Sec. IV, the leading log2 contribution to the We may relate this mass renormalization to a measurable ͒ / 2 ͑ץ/ ⌺ץ −1 self-energy is proportional to G0 since AB q+ 2m quantity, by noting that the electronic compressibility K is ץ/ ⌺ץ =i AA . This means that all renormalization can be at- proportional to the density of states which is proportional to tributed to the residue Z with mass remaining unchanged. the mass. Thus, the logarithmic renormalization of the mass However, as we now show, this equality is only true to lead- in Eq. ͑57͒ should manifest itself in a logarithmic enhance- ing logarithmic order. ment of the electronic compressibility. The relation between Comparison of Eq. ͑52͒ with Eqs. ͑14͒ and ͑15͒ indicates mass renormalization and compressibility will be further dis- that the mass renormalization is given by cussed in Sec. IX. ⌺ץ ⌺ץ m␦ = Z ͫi AA − AB ͬ. ͑53͒ IX. INTERACTION CORRECTION TO COMPRESSIBILITY ͒ /2 ͑ץ ץ 0 m q+ 2m ͒ ͑ ץ/ ⌺ץ Here, i AA is defined by Eq. 18 . For the second term, Here we explicitly calculate the renormalization of the we obtain the expression compressibility. By doing this we shall confirm that the com- pressibility does not renormalize at leading ͑log square͒ or- p2 2 5ͩ ͪ der and also extract the single log renormalization of the ⌺ dd2p 1 2mץ AB = ͵ − compressibility. The interaction correction to the compress- ͒3 2 2 2 2 2 ͑ ͒ /2 ͑ץ q+ 2m 2 Ά p p ibility K is given by 2 + ͩ ͪ ͫ2 + ͩ ͪ ͬ 2m 2m 2Fץ p2 4 ␦K =− , ͑58͒ 2ץ ͪ 4ͩ 2m + ⌫2ZU͑,p͒, ͑54͒ p2 2 3 · where is the chemical potential, and F is the interaction ͫ2 + ͩ ͪ ͬ 2m energy. Within the RPA framework, the interaction energy is expressed as where U͑,p͒ is given by Eq. ͑4͒. To evaluate the difference ͑ ͒ in Eq. 53 , it is convenient to subtract the integrands of Eqs. dd2p ͑18͒ and ͑54͒ before doing the integrals. Once again, we use F͑͒ = ͵ ln͓1−V͑q͒⌸͑,,q͔͒. ͑59͒ ͑2͒3 the “polar” representation of the frequency and momentum variables, =r cos , p2 /2m=r sin , and obtain Here, ⌸͑,,q͒ is the noninteracting polarization function ␦ ⌳ ⌫2 2͑ 2 4 ͒ ͑ ͒ m dr d 0Z0 3 sin − 4 sin evaluated at a chemical potential and V q is the un- = ͵ ͵ ͱ , screened Coulomb interaction V͑q͒=2e2 /q. m ⌳Ј r 2 2r sin −2⌸͑͒ 0 To evaluate the second derivative in ͓Eq. ͑58͔͒, we con- where ⌸͑͒ is the dimensionless polarization function intro- sider the difference ⌬F=F͑͒−F͑0͒. After rearranging logs ͑ ͒ duced in Eq. 19 and r is measured in units of E0 as before. under the integral, we rewrite this expression as
115431-9 RAHUL NANDKISHORE AND LEONID LEVITOV PHYSICAL REVIEW B 82, 115431 ͑2010͒
dd2q lytically over r, we find that the fractional change in the ⌬ ͵ ͓ ͑⌸͑ ͒ ⌸͑ ͔͒͒ F =− ln 1−U,q , ,q − 0, ,q , compressibility is ͑2͒3 ͑60͒ ␦K͑͒ 0.56 = ͑66͒ K͑0͒ 2N ln 4 where now U,q is the dynamically screened Coulomb inter- action, Eq. ͑4͒. Since the compressibility is obtained from a result that agrees exactly with Eq. ͑57͒. We note that an 2, the problem of calcu- enhancement of the compressibility due to interactions wasץ/2Fץ−=the free energy through K lating the interaction renormalization of the compressibility also predicted in Ref. 23. However, the effect described by is reduced to that of calculating the polarization function at Eq. ͑57͒ is much weaker than that predicted in Ref. 23 be- finite . This may be calculated through methods similar to cause we have worked with a screened interaction whereas in those developed in Ref. 3. We define Ϯ =Ϯ/2, pϮ Ref. 23 screening was not taken into account. 2 =pϮq/2, and zϮ =͉pϮ͉ /2m. The noninteracting polariza- In summary, the compressibility does not renormalize at tion function at finite is given by leading ͑log square͒ order, just as in the Luttinger liquids and ⌸͑,,q͒ =TrG͑, ,p ͒G͑, ,p ͒ while there is a subleading logarithmic correction, the pref- + + − − actor is quite small ͓0.56/͑2N ln 4͒Ϸ0.016͔. Thus, in con- dd2p 1 trast to the zero-bias anomaly in TDOS, experimental detec- =Tr͵ ͑ ͒3 ͓ ͑ ͔͓͒ ͑ ͔͒ 2 i + − − H0 p+ i − − − H0 p− tion of the interaction correction to the compressibility is likely to be challenging. The difference arises because the dd2p 2 =2N͵ single log flows are much weaker than the log flows, retro- ͑ ͒3͓ ͑ ͔͓͒ ͑ ͔͒ 2 + + i + z+ + + i − z+ spectively justifying our earlier neglect of the single log flows in the RG. Hence, strong suppression of the tunneling ͑i − ͒͑i − ͒ + z z cos 2 ϫ + − + − pq , ͑61͒ density of states at energy scales where the compressibility is ͓ ͑ ͔͓͒ ͑ ͔͒ − + i + z− − + i − z− not significantly renormalized is a key signature of the mar- ginal Fermi liquid physics in bilayer graphene. where pq is the angle between p+ and p−. We now perform the integral over by residues to obtain d2p ͑z + i + z cos 2 ͒⌰͑z − ͒ ⌸͑,,q͒ = N͵ + − pq + X. DISCUSSION AND CONCLUSIONS ͑2͒2 z2 − z2 − 2 +2iz + − + Here we briefly discuss the range of validity of our re- + ͑,q → − ,− q͒ sults. Our analysis was organized as a perturbation theory in 2 2 ⌫ Z /N. Since ⌫Z=1 at leading ͑log square͒ order, the per- z+= d2p 1 2z sin2 = N͵ + ͫ − − pq ͬ turbation theory remains well defined under the log square ͑2͒2 z + i − z ͑z + i͒2 − z2 z+=0 + − + − flows. However, our analysis neglected subleading single log flows. For ϷN2, the subleading single log flows become + ͑,q → − ,− q͒. ͑62͒ important, and the analysis leading to the expression Eq. ͑44͒ In the limit →0, this reproduces the noninteracting polar- no longer applies. A mean field theory of subleading single ization function from Ref. 3. Now we expand Eq. ͑60͒ to log effects3 indicates that a gapped state develops at 3 2 leading order in small to obtain = 13N , the scale which we tentatively identify as the limit of validity of our analysis. 2⌸͑,,q͒ץ dd2q 1 ⌬ 2͵ ͑ ͒ ͑ ͒ How can the marginal Fermi liquid physics be distin- F =− 3 U ,q 2 . 63 guished from the formation of a gapped state? We note thatץ 2͒͑ 2 The term linear in must vanish, by particle hole symmetry. at very low energies, once the gapped state has developed, Taking derivatives of Eq. ͑62͒ greatly simplifies the calcula- both the tunneling density of states and the compressibility tions since it turns the two-dimensional integral over mo- will vanish. What we have shown, however, is that there is a menta into a one dimensional integral over momentum large range of energies greater than the energy scale for gap angles, which is fully convergent, and may be evaluated nu- formation, where the tunneling density of states vanishes merically. We find while the compressibility remains essentially unchanged. Such behavior represents the key signature of the marginal 2 4 2⌸ m 32z − z q2 Fermi liquid physics discussed above, which is analogous toץ = N q q , z = , ͑64͒ .2 ͑2 2͒2 q the Luttinger liquid physicsץ 2 + zq 2m In our analysis, we neglected the short range interactions -2⌸ which are characterized by lattice scale, such as the interץ 2 dd2q ⌬F =− ͵ U͑,q͒ . ͑65͒ layer density difference interaction V = 1 ͑V −V ͒=e2d 2 − 2 AA ABץ 2͒3͑ 2 and the Hubbard-type on-site repulsion. Short range interac- We again change to the coordinates =r cos , zq =r sin , tions are nondispersive, do not renormalize the Green’s func- and measure r in units of E0. Note that even though the tion in the weak coupling limit, and hence do not alter our interaction has a pole at →0,, this pole is canceled by results. Short range interactions also produce only single log 2 having a zero at →0,. As a result, the integral renormalization4,5 and therefore do not need to be includedץ/2⌸ץ is fully convergent. Integrating numerically over and ana- in our log square RG. Similarly, we justify our neglect of the
115431-10 ELECTRON INTERACTIONS IN BILAYER GRAPHENE:… PHYSICAL REVIEW B 82, 115431 ͑2010͒ trigonal warping effect24 by noting that trigonal warping is cle residue Z and the Coulomb vertex function ⌫ both flow significant only on energy scales smaller than the character- as 2, where is the RG time. All other quantities flow only istic energy scale for onset of gapped states.3 as . The structure of the RG for Coulomb interacting BLG Finally, we note that our analysis made use of the fact that has strong similarities to the RG for the one dimensional there were no uncanceled log square divergences at one loop Luttinger liquids. In particular, we predict a strong interac- order in the RG, except for the renormalization of the quasi- tion suppression of the tunneling density of states for un- particle residue and the Coulomb vertex function, which doped BLG, even at energy scales where the electronic com- were related by a Ward identity, Eq. ͑29͒. Technically, in pressibility is essentially unchanged from its noninteracting order for our neglect of higher loop corrections to be justi- value. These predictions may be readily tested by experi- fied, we also require that there are no uncanceled log square ments. divergences beyond one loop order in the RG, except those that are constrained by Ward identities. We believe this to be ACKNOWLEDGMENTS the case, however, the proof requires a nonperturbative ap- proach, which lies beyond the scope of the present work. We acknowledge useful conversations with A. Potter and To conclude, we have examined the one-loop RG flow for P. A. Lee. This work was supported by Office of Naval Re- bilayer graphene. We have demonstrated that the quasiparti- search under Grant No. N00014-09-1-0724.
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