Confinement Phase in Carbon-Nanotubes and the Extended Massive Schwinger Model
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Confinement Phase in Carbon-Nanotubes and the Extended Massive Schwinger Model
Takashi Oka and Hideo Aoki
Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan (Dated: February 27, 2018)Carbon nanotube with electric fluxes confined in one dimension is studied. We show that a Coulombinteraction ∝ | x | leads to a confinement phase with many properties similar to QCD in 4D. Low-energy physics is described by the massive Schwinger model with multi-species fermions labeledby the band and valley indices. We propose two means to detect this state. One is through anoptical measurement of the exciton spectrum, which has been calculated via the ’t Hooft-Berknoffequation with the light-front field theory. We show that the Gell-Mann − Oakes − Renner relation issatisfied by a dark exciton. The second is the nonlinear transport which is related to Coleman’s“half-asymptotic” state.
PACS numbers: 73.63.Fg ,03.70.+k ,78.67.Ch ,71.35.-y
Even after two decades from its discovery, fascinationcontinues with the carbon nanotube[1]. One line of re-search is motivated by its possible application as opto-electronic devices. Another is more academic, in whichpossibilities are explored to realize new states of matter.In low dimensions quantum fluctuations are enhanced,which makes the nanotube an ideal host for strongly cor-related and clean quantum systems. One monumentalresult along this line is the Tomonaga-Luttinger liquidin single-wall metallic nanotubes [2–4]. In the presentwork, we theoretically predict a realization of yet an-other interesting state, namely the confinement phase , innanotubes. In this state, charged particles cannot existas free asymptotic states where all the excitations areneutral bound states (i.e., excitons in the present case).A most famous realization of the confinement phaseis the hadronic system where the effective quantum fieldtheory is QCD in (3+1)D. By contrast, a carbon nan-otube is a one-dimensional object, and no dynamical non-abelian gauge field exists. Nonetheless, we propose herea confinement phase may exist; A key is the dimension-ality. In a seminal paper by Schwinger, a (1+1)D ver-sion of QED, i.e. the Schwinger model, was studied[5].In this model, Dirac particles interact through a one-dimensional Coulomb potential ( ∝ | x | ). The long-rangeinteraction mimics the confinement potential, and thegroundstate is indeed in a confinement phase. TheSchwinger model and its extensions were studied as a toymodel of (3+1)D QCD to understand non-perturbativeaspects of the confinement phase [6–12]. However, upto now, no physical realization of this model is known.Our claim here is that the low-energy effective modelfor electrons in a nanotube can be, in certain situations,modeled by an extended massive Schwinger model. Itis well known that a multi-species Dirac spectrum is re-alized in the band structure of nanotube within the ef-fective mass approximation. In order to realize the one-dimensional Coulomb potential, the electric fluxes mustbe confined along the tube. The situation with shielded1D electric flux can possibly be realized in (a) multi-wall nanotubes with metallic outer tubes, (b) a tube sur-rounded by metallic tubes, or (c) tubes embedded in a superconductor. While one may wonder if such shieldingcan really take place, there is a prominent example intwo-dimensional (2D) organic crystals studied by Yam-aguchi et al. [13]. They have experimentally shown thata layered structure with a large difference in the dielec-tric constants leads to confinement of electric fluxes inthe 2D plane where the Coulomb potential becomes log-arithmic. In this experiment, the long-range Coulombinteraction leads to a power-law current-electric field ( J - E ) characteristics with a temperature dependent power.This gives us a strong motivation to study nanotube witha 1D Coulomb potential. The properties of the confine-ment phase is reflected in the excitation spectrum. Inorder to explore this, we propose two observable proper-ties, one optical and another transport. Model —
We study nanotubes within the effective-mass formalism. There are infinitely many bands cor-responding to different modes along the circumference ofthe tube, which we label with n = 0 , ± , . . . . The wayin which the tube is wound is characterized by an index ν = 0 , ±
1, which in turn specifies whether a discrete set ofmomenta along the tube circumference intersect the twoDirac cones at α =K, K’ points in the graphene Brillouinzone ( ν = 0; the “(semi)metallic” case), or not ( ν = ± σ = ↑ , ↓ with the Zee-man effect neglected here. Each band is characterized bya mass ~ v F κ α ( n ), where κ K , K ′ ( n ) = πL ( n ± ϕ − ν/
3) and v F the Fermi velocity in graphene[14]. We take ~ v F = 1to be the unit of energy. Here we have introduced amagnetic field whose flux passing through the tube is ϕ in units of the flux quantum ϕ = ch/e , which acts toshift the discrete set of momenta. Assuming 1D electro-magnetic fields, the system is described by the extendedmassive Schwinger model with a Lagrangian L = − F µν F µν + X n,σ,α ¯ ψ n,σ,α [ i / ∂ − e / A − κ α ( n )] ψ n,σ,α , (1)where F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic fieldtensor, ψ = ( ψ R , ψ L ) T the fermion field with ψ R , ψ L the left and right moving components, e = e / √ ε r thescreened charge with a dielectric constant ε r , and weuse the relativistic convention g µν = diag(1 , −
1) with γ = ( ) , γ = ( −
11 0 ) and / A = A µ γ µ . The modelhas a spin SU(2) symmetry, which is enlarged to SU(4)if K and K’ are degenerate. However, perturbationscan break this into SU(2) × SU(2). When the electricfluxes are confined in 1D, the inter-electron potential is V ( x ) = e | x | /
2, which enforces confinement of charges.The coupling strength is e / ( πv F ) = 0 . /ε r deter-mined by the screening factor ε r (see e.g. [15]). Exciton spectrum —
Excitons are of central impor-tance in understanding the optical properties of carbon-nanotubes [16–20]. Specifically, Kishida et al. have ob-served not only the bright excitons, but also the darkexcitons[19], while Wang et al. observed excitons inmetallic nanotubes[20]. In the confinement phase, theoptical spectrum is dominated by excitons, and no two-particle continuum exists. Since there is a possibilitythat the lightest fermion mass vanishes, the conventionalweak-coupling approaches cannot be used, and strong-coupling methods are required. Two powerful meth-ods are applicable; One is the light-front quantizationscheme[9–11], and the other is bosonization [6–8]. Forstudying the crossover from metallic to semiconductingtubes, we employ the former, since bosonization for amassive model is still an open problem[8]. With the light-cone coordinates x µ = ( x + , x − ) ≡ ( x + x , x − x ) / √ ψ iR (with a shorthand i ≡ ( n, σ, α )) by means of the equation of motion. TheLagrangian reads L = Z dx − L = i √ Z dx − X i : ψ † iR ∂ + ψ iR :+ i √ Z dx − dy − X i κ ( n ) ψ † iR ( x − ) ε ( x − − y − ) ψ iR ( y − )+ e Z dx − dy − j + ( x − ) | x − − y − | j + ( y − ) (2)with the U(1) current j µ = P i : ¯ ψ i γ µ ψ i : , γ + =( √ ) , γ − = ( √
20 0 ). The index i runs over infinite num-ber of modes with the mass term depending on n , whichcontrasts with the standard SU( N ) massive Schwingermodel where the mass is common to all i ’s. The free-fieldexpansion is ψ iR ( x − ) = / R ∞ dk + π √ k + [ b i ( k + ) e − ik + x − + d † i ( k + ) e ik + x − ] , where b † creating electrons and d † holessatisfy a canonical commutation, { b i ( k + ) , b † j ( l + ) } = { d i ( k + ) , d † j ( l + ) } = 2 πk + δ ij δ ( k + − l + ) . The virtue of usingthe light-front formalism is that the groundstate, whichis a confinement phase (CP), is described by the Fockvacuum | i CP = | i Fock (with b j | i Fock = d j | i Fock = 0).Now let us look at the two-particle excitation, i.e., anexciton with a wave function | ψ i = Z P dk dk π √ k k X i =1 ψ i ( k , k ) b † i ( k ) d † i ( k ) | i CP , (3) where the integral is restricted to k + k = P . From theLorentz invariance, the exciton wave function satisfiesthe Einstein-Schr¨odinger equation, 2 P − P + | ψ i = M | ψ i ,where P − is the light-cone Hamiltonian, P + the momen-tum operator for the center of mass momentum P with P + | ψ i = P | ψ i , and M the excitation energy (“mass”)of the boundstate. With the light-cone operators, theEinstein-Schr¨odinger equation for the wave function isgiven explicitly as M ψ i (˜ k, − ˜ k ) = (cid:20) κ ( n )2 − e π (cid:21) (cid:18) k + 11 − ˜ k (cid:19) ψ i (˜ k, − ˜ k ) − e π Z d ˜ k ′ ψ i (˜ k ′ , − ˜ k ′ )(˜ k − ˜ k ′ ) + e π Z d ˜ k ′ X j ψ j (˜ k ′ , − ˜ k ′ )(4)with re-scaled momenta ˜ k = k/P, ˜ k ′ = k ′ /P [21]. Thisis an extension of the ’t Hooft-Bergknoff equation[9, 22],here possessing infinite number of modes labeled by i .The last term is the anomaly term, which physically cor-responds to a virtual process of exciton pair-annihilatedinto a photon and then regenerated as an exciton. Theprocess is intimately related to the photon-exciton cou-pling, and for bright excitons this term is nonzero. Bycontrast, the term disappears for dark excitons with P i ψ i = 0, and eqn.(4) reduces to the ’t Hooft equationfor planar QCD[22]. For the optical activity the brightexcitons must satisfy the condition that ψ n ↑ α = ψ n ↓ α tobe a SU(2) spin singlet (note d † nσl creates a hole withspin − σ ), and, when K and K’ are degenerate, an addi-tional condition ψ nσ K = ψ nσ K ′ is imposed to make it aSU(2) valley singlet. We note that in eqn. (4) the effectof vacuum polarization and self-energy corrections dueto “meson” propagators [12] are neglected for simplisity.We solve[23] the ’t Hooft-Bergknoff equation using thebasis-function method[10, 11].Let us first look at the excitation spectrum (excitonenergy M ) against the magnetic field for a metallic nan-otube with ν = 0 in Fig. 1. We immediately notice thatthe system is no longer metallic due to charge confine-ment , namely, the spectrum for the bright exciton has agap, and an excitation continuum does not exist, either.This is in sharp contrast with the case for the conven-tional weak-coupling picture with a 1 /r potential (insetof (a)), where a continuum exists down to zero energyat ϕ = 0. The spectrum has a periodicity with a pe-riod ϕ = 1, which originates from the mass structure ofthe fermion modes. We label each exciton mode with( n, l ; α ), where α =K,K’ is the dominant valley characternear ϕ = 0, and l = 0 , , . . . the exciton quantum numberthat labels the bound state in a trapping potential (seeFig. 3 (c)). Odd- l states are parity odd and one-photonallowed, while even- l states are only two-photon accessi-ble. For ν = 0 the ( n, l ; K ) and ( − n, l ; K ′ ) excitons aredegenerate due to the valley symmetry, so we can omit α from the index.What can we learn more about the spectrum? Com-parison with the meson spectrum in QCD becomes inter-esting. In fact, the bright and dark excitons with low- (a) (b) (0,0)(0,1)(1,0)(-1,0)(0,2)(-1,1) (0,0)(0,1)(1,0)(-1,0)(0,2)(-1,1) ν =0, bright ν =0, dark (c) (d) E x c it a ti on E n e r gy ( v ) F magnetic flux ( ϕ ) ν =1, bright ν =1, dark (0,0;K)(0,0;K')(1,0;K)(1,0;K')(0,1;K)(0,1;K') (0,0;K)(0,0;K')(1,0;K)(1,0;K')(0,1;K)(0,1;K') magnetic flux ( ϕ ) E x c it a ti on E n e r gy ( v ) F ϕ M cont.exciton ( n , l ) =( n , l ; α ) = ϕ α η -particle π -particle η -particle π -particle FIG. 1: (Color online) Two-body excitation spectrum againstmagnetic field for bright (a,c) or dark (b,d) excitons for metal-lic (a,b; ν = 0) or semiconducting (c,d; ν = 1) nanotubes with L = 10 , e /πv F = 0 .
2. Solid (dashed) lines correspond tostates with even (odd) l . Inset in (a) is a schematic spectrumfor a system with 1 /r potential near ϕ = 0. est energies (Fig.1) have many aspects shared by the η and π (pion) particles, respectively, in QCD[7]. Inter-estingly, the η and π boundstates behave differently inthe strong-coupling limit; As the fermion mass goes tozero, the π mass vanishes, while the η mass remains fi-nite due to a U(1) anomaly. For QCD there is the Gell-Mann − Oakes − Renner relation [24], which is a relation, M π ∝ m / , between the pion and quark masses. Whentranslated into the present problem, the relation appliesto the lightest dark exciton. For example, as shown inFig. 1 (b,d), the energy of the lightest dark exciton, i.e.,pion in the QCD context, goes to zero at ϕ = ¯ ϕ = 0 , ϕ = 1 / , / M ∝ | ϕ − ¯ ϕ | α with a power α ≃ . | ϕ − ¯ ϕ | , we can regardthis as a manifestation of the Gell-Mann − Oakes − Rennerrelation in nanotubes , which holds even in 1+1D systemswhere chiral symmetry is ‘almost’ broken [25].The situation is even more interesting for the lightestbright excitons ( ∼ η -particles), since they remain massiveeven though the fermions are massless. This can be seenin the (0 ,
0) and (0 , K ) states in Fig. 1 (a) (c), whichhave nonzero minima at ϕ = ¯ ϕ . This is due to a U(1) anomaly coming from the pair creation-annihilation pro-cess, i.e., the last term in eqn. (4). The physical pictureis the following. In an exciton, electrons and holes are (a) ( b ) (c) e / π ν =0, bright ν =1, bright (0,0)(0,2)(0,1) (-1,0)(1,0) (0,0;K)(0,0;K')(1,0;K)(1,0;K')(0,1;K) L L ν =0, bright (d) ν =1, bright (0,0)(0,1) (1,0)(-1,0)(0,2) (0,0;K)(0,0;K')(1,0;K)(1,0;K')(0,1;K) ( v ) F E x c it a ti on E n e r gy ( v ) F E x c it a ti on E n e r gy ( v ) F e / π ( v ) F ( n , l ) = ( n , l ; α ) = FIG. 2: (Color online) Two-body excitation spectrum in zeromagnetic field against circumference L (a,c; with e /πv F =0 .
2) or against coupling strength (b,d; with L = 10) for metal-lic (a,b; ν = 0) or semiconducting (c,d; ν = 1) nanotubes.Solid (dashed) lines correspond to states with even (odd) l . continuously created and annihilated when the fermionmass is small, and a cloud of photon is formed. Theelectromagnetic energy of the photon cloud is finite, andcontributes to the exciton energy. This process is not re-stricted to the 1D Coulomb potential, and similar effectmay take place in metallic single-wall nanotubes whereexcitons with finite binding energy were observed [20].In experiments it often happens that the nanotubeshave random circumference L . In Fig.2(a,c), we plot thedependence of the exciton energy on the circumference.The mixing between different modes n is small when L is small, since the energy difference between modes is ∝ /L , and in the limit L →
0, the system approachesto the pure SU( N ) massless Schwinger model [7]. In thislimit, the lightest bright exciton ( η -particle) mass is givenby M η = p N e /π , which amounts to M η = 0 . ν = 0, SU(4)) nanotubes and M η = 0 . ν = 1, SU(2)) ones for e /πv F = 0 . ,
0) mode converges to thedark mode in the SU( ∞ ) massless Schwinger model. InFig.2 (b,d), we plot the dependence of the spectrum onthe screened interaction parameter e . Starting from 2 κ (weak-coupling limit), the exciton energy increases as theinteraction becomes stronger, where the increase is largerfor larger quantum number l . M - κ ( e ) / (a) (b) J ( σ ) J (c) contiuum T =1 T =0.01 T =0.1 T =1 T =0.01 T =0.1 E/E cr E/E cr E/E = cr E/E = cr E/E = cr E/E cr (d) l =0 l =2 l =1 l= l= E=e /2 -e /2 e /2 -e /2 FIG. 3: (Color online) J - E characteristics(a) with a logarith-mic plot in the inset, and exciton levels(b) for semiconductingnanotubes (for ξ = 100) in the weak-coupling, nonrelativisticlimit. (c) Schematic trapping potential V ( x ) and eigenstatesfor various values of the electric field. (d) Coleman’s halfasymptotic state when the external field is E = E cr . Nonlinear transport and half-asymptotic state —
Non-linear transport in correlated systems is attracting con-siderable interests (see e.g. [26]), and in nanotubes itserves as a powerful probe in experimentally detectingnew states of matter [2]. Here let us show that the semi-conducting nanotubes with 1D interaction should exhibita power-law J - E characteristics in external electric fields,as in the 2D organic systems[13]. Unlike the 2D case,however, the carriers in the field-induced metallic stateare only “half-free”. We explain how this has to do withthe half-asymptotic state predicted by Coleman[7].Let us look at a semiconducting nanotube with a finitefermion mass κ in a strong electric field. Since the strong-coupling analysis in finite electric fields is still an openissue, here we focus on a weak-coupling approach wherethe U(1) anomaly does not contribute with the bright anddark excitons becoming nearly degenerate. In the nonrel- ativistic approximation, Dirac particles becomes nonrel-ativistic fermions with quadratic kinetic terms ( p / M ),and the exciton binding problem in an electric field E isreduced to solving a 1D Schr¨odinger equation (c.f. eqn.(4.1) in [7]),[ − ∂ x + V ( x )] ψ ( l ) ( x ) = ( M l − κ ) ψ ( l ) ( x ) , (5) V ( x ) = e | x | e −| x | /ξ + e Ex, (6)where M l is the exciton energy, V ( x ) the trapping po-tential, and we have introduced an exponential dampingfactor with a cutoff ξ . We plot the J - E characteristicsin Fig. 3(a) for various values of temperature, where thenonlinear current is seen to behave like J = σ exp[ − ∆( E ) / (2 k B T )] E, (7)as in ref.[13] with ∆( E ) the activation energy. The cur-rent follows a power-law until the conductivity reaches apeak at a critical field E = E cr ≡ e / ε r . The powerof the conductivity is proportional to T − as in ref. [13]but there is also a cutoff ξ dependence. At the criticalfield, one side of the trapping potential for a test chargebecomes flat as depicted in Fig. 3(c). Then a contin-uum spectrum emerges, as seen in Fig. 3(b) where weplot the eigenenergies of the Hamiltonian (eqn.(5)) ob-tained by gluing two Airy functions. There, Coleman’shalf-asymptotic state – a configuration with alternatingcharges but possibly random displacements (Fig. 3(d)) –has the lowest energy, where the external field and theforce from surrounding charges balance with each other.Strictly speaking, carriers with opposite charges mustswitch their position in order for the current to flow, andthis violates the half-asymptotic state condition, whichmay modify the simple relation eqn.(7) for the nonlinearcurrent. This is out of the scope of the present work, butwill merit further studies.TO wishes to thank Kazuhiro Kuroki for valuable dis-cussions, and was supported by a Grant-in-Aid for YoungScientists (B) from MEXT and by Grant-in-Aid for Sci-entific Research on Priority Area “New Frontier of Mate-rials Science Opened by Molecular Degrees of Freedom”. [1] S. Iijima, Nature , 56 (1991).[2] M. Bockrath, et al. , 397 Nature, 598 (1999).[3] R. Egger, and A. Gogolin, Phys. Rev. Lett. , 5082(1997).[4] C. L. Kane, L. Balents, and M. Fisher, Phys. Rev. Lett. , 5086 (1997).[5] J. Schwinger, Phys. Rev. , 2425 (1962).[6] S. Coleman, R. Jackiw, and L. Susskind, Ann. Phys. ,267 (1975).[7] S. R. Coleman, Ann. Phys. , 239 (1976).[8] J. E. Hetrick, Y. Hosotani, and S. Iso, Phys. Lett. B , 92 (1995).[9] H. Bergknoff, Nucl. Phys. B , 215 (1977).[10] Y. Mo and R. J. Perry, J. Comp. Phys. , 159 (1993).[11] K. Harada et al., Phys. Rev. D , 4226 (1994).[12] K. Harada, Phys. Rev. D , 065005 (1999).[13] T. Yamaguchi, et al. , Phys. Rev. Lett. , 136602 (2006),T. Yamaguchi, et al. , Phys. Rev. B , 235110 (2010).[14] H. Ajiki and T. Ando, J. Phys. Soc. Jpn. , 1255 (1993).[15] T. Ando, J. Phys. Soc. Jpn. , 1066 (1997).[16] M. Ichida, et al. , J. Phys. Soc. Jpn. , 3131 (1999).[17] M. J. Connell, et al. , Science , 593 (2002). [18] S. M. Bachilo, et al. , Science , 2361 (2002).[19] H. Kishida, et al. , Phys. Rev. Lett. , 097401 (2008).[20] F. Wang, et al. , Phys. Rev. Lett. , 227401 (2007).[21] The derivation of eqn.(4) is parallel to that in refs. [9–11]with a modification of a n dependent mass κ ( n ).[22] G. ’t Hooft, Nucl. Phys. B , 461 (1974).[23] We expand the wave function as ψ i (˜ k, − ˜ k ) = P N cut j =0 a nj f j ( β n , ˜ k ), where f j ( β, ˜ k ) is [˜ k (1 − ˜ k )] β + j forparity even and [˜ k (1 − ˜ k )] β + j (2˜ k −
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