One-dimensional theory: carbon nanotubes and strong correlations. Sam T Carr. University of Karlsruhe

One-dimensional theory: carbon nanotubes and strong correlations Sam T Carr University of Karlsruhe CFN Summer School on Nano-Electronics Bad Herrenalb, 5 th September 2009

Outline Part I - introduction to single wall carbon nanotubes Characterization of nanotubes Electronic structure - nanotubes as ladder models Part II - one dimensional physics Interactions - 1D is different - Luttinger liquid The lattice - umklapp terms and the Mott insulator Spin chains, spin ladders and spin liquids Part III - application to armchair nanotubes Long range Coulomb interaction and Luttinger liquid Umklapp terms Ground state of armchair nanotubes Part IV - what s different in zigzag nanotubes? Curvature gaps Interaction induced dimerization Quantum criticality Phase diagram of zigzag nanotubes

A short history... 1991 - Fullerene related structure consisting of tubes of graphene discovered. S Iijima 1993 Single Wall Carbon Nanotubes (SWCN) first fabricated. S. Iijima and Ichihashi 1997 Quantum conductance and nanotubes as quantum wires. S.J. Tans et el 1997 Calculations of correlation and Luttinger liquid effects. Egger et al, Kane et al 1999 Luttinger liquid effects and power law scaling observed in SWNT. Bockrath et el 2003 - Low temperature strong coupling phases of armchair nanotubes elucidated. Nersesyan and Tsvelik 2005 - Spin Gap observed in nanotubes. Singer et el 2009 - Mott insulating state observed in nanotubes. Deshpande et el They are prototypical one-dimensional experimental objects to play with exotic strong correlation effects and non-fermi liquid behaviours.

Single Wall Carbon Nanotubes A nanotube is characterized by the wrapping vector (n,m): na 1 +ma 2 a (n,n) Armchair b - (n,-n) Zig-zag c - (n,m) Chiral

The tight binding model The tight binding spectrum of graphene. The Fermi surface of graphene consists of isolated points (two independent). Fold into a nanotube: compactify space in one direction, and therefore quantize momentum. If (n-m) is divisible by 3, then the nanotube is metallic. There are two bands which will have zero energy excitations. Low energy theory = 2 leg ladder For an armchair nanotube, these are at zero-transverse momentum. For all other nanotubes, there will be a nontrivial structure of excitations around the waist.

The tight binding model (II) Lin, PRB (1998) Example: zig-zag nanotube Partial Fourier transform in wrapping direction: The label q takes n integer values, and the new effective hopping is given by If Δ(q)=0 for some q, then we have a uniform and therefore gapless chain. cos(πq/n)=0, q=2n/3 so if n is divisible by 3, we have gapless modes. In this case, there are two gapless chains. Low energy theory will consider only these two chains. They will be coupled by interactions.

Interactions Long range Coulomb interaction Isolated nanotubes are not a bulk material: Coulomb interaction unscreened This would not be true of for example ropes of nanotubes The long-range interaction makes the one-dimensionalness even more pronounced. Local umklapp and backscattering terms Kane, Balents and Fisher, PRL (1997) Egger and Gogolin PRL (1997) Balents and Fisher PRB (1997) Yoshioka and Odintsov PRL (1999) Undoped graphite is half-filled so umklapp terms are relevant. Opening of gaps, giving rise to Mott insulating behaviour. In fact, ground state will be a gapped state with no local order parameter: similar to a Haldane spin liquid.

Part II A brief introduction to the physics of one dimensional systems Luttinger liquid physics Mott insulator Spin chains and spin ladders

One dimension is different... Electrons have no way 'around' each other Even arbitrarily small interactions dramatically change the ground state properties of the system No coherent single particle excitations... instead we are left with propagating density waves plasmons Excitation velocity controlled by interaction strength g. The stronger the interaction, the stiffer the medium and the faster the plasmons. This state is a Luttinger liquid.

Chiral decomposition and Bosonization Single chain: 2 Fermi points, low energy excitations around these Fermi points, so expand in the slow fields, R and L. c(l) b [ e ik F l R(x) + e ik F l L(x) ] H 0 = iv F [ R (x) x R(x) L (x) x L(x) ] This form of the Hamiltonian has a linear spectrum around the two Fermi points - Tomonaga-Luttinger model. Notice that near Fermi points, particles and holes move in coherent pairs - expand in density waves: R(L) (2πα) 1/2 exp [ i π(φ Θ) ] Non-local commutation relation (c.f. Jordan-Wigner): [Φ(x), Θ(y)] = iθ(x y) c (l) b [ e ik F l R (x) + e ik F l L (x) ]

Chiral decomposition and Bosonization H 0 = iv F [ R (x) x R(x) L (x) x L(x) ] R(L) (2πα) 1/2 exp [ i π(φ Θ) ] [Φ(x), Θ(y)] = iθ(x y) Transformation and commutation relations preserve original anti-commutation relations amongst Fermions. Under transformation, non-interacting Hamiltonian becomes Gaussian model: H 0 v F [ ( x Θ) 2 + ( x Φ) 2] 2 Both models have identical correlations functions: R (x)r(y) = 1 π[φ(x)+θ(x)] 2πα e+i e i π[φ(y)+θ(y)] Power of bosonization: density is linear in bosonic field, so interaction remains quadratic: ρ(x) = R (x)r(x) + L (x)l(x) = 1 x Φ(x) π

Chiral decomposition and Bosonization H 0 = iv F [ R (x) x R(x) L (x) x L(x) ] R(L) (2πα) 1/2 exp [ i π(φ Θ) ] [Φ(x), Θ(y)] = iθ(x y) H 0 v F 2 [ ( x Θ) 2 + ( x Φ) 2] ρ(x) = 1 π x Φ(x) Consider interacting model (quartic in Fermions): H = H 0 + g(r R + L L)(R R + L L) = H 0 + gρ(x)ρ(x) Model remains quadratic in Bosonic language: H = v [ ( F ( x Θ) 2 + 1 + g ) ] ( x Φ) 2 = [K( v x Θ) 2 + 1K ] 2 πv F 2 ( xφ) 2 Luttinger liquid parameter renormalized velocity K = (1 + g/πv F ) 1/2 v = v F /K

Single particle tunneling density of states It's difficult to add another electron to the system the other electrons all have to move to make 'room' for the newcomer. In fact, single particle tunneling density of states is zero at the Fermi level. At higher energies, obtain a power law, where the power depends on the interaction strength. G R (x = 0, t) t (K+1/K 2)/8+1 = ν(ω) ω (K+1/K 2)/8

Spin-charge separation Electrons also have spin. Collisions may exchange the spins. However, because the electrons are ordered, this exchange interactions are independent of any charge fluctuations. Spin and Charge excitations are separated - may have different velocities, energy scales, etc... Add spin-index: R(L) σ (2πα) 1/2 exp [ i π(φ σ Θ σ ) ] Split bosonic fields into c/s: Ham. is sum of sectors: Φ c = Φ + Φ, H 0 v F 2 a=c,s Φ s = Φ Φ [ ( x Θ a ) 2 + ( x Φ a ) 2]

Remember nanotubes are a ladder... Two types of plasmons total charge: - and relative charge: Note that in the relative charge excitations, the total charge at any point along the chain does not change much the long range Coulomb interaction couples only to the total charge channel.

The Mott Insulator What about the lattice? - Umklapp interactions. Consider a one-dimensional tight binding model, the Hubbard model: At half filling, there will be one-electron per site: Ground state is an insulator for any repulsive interaction. (in one dimension) Technically: (R L)(R L) + (L R)(L R) 1 (2πα) 2 cos( 8πΦ c ) (sine-gordon model) There will be a residual anti-ferromagnetic interaction of order t 2 /U from virtual hopping processes. Low energy excitations will be in the spin sector.

Aside about the sine-gordon model H = v 2 [K( x Θ) 2 + 1K ( xφ) 2 ] g (2πα) 2 cos( 8πΦ)

Aside about the sine-gordon model H = v 2 [K( x Θ) 2 + 1K ( xφ) 2 ] g (2πα) 2 cos( 8πΦ) wants the field phi to fluctuate wants to lock the field phi Competition between two terms in Hamiltonian --> Quantum Phase Transition K>1... First term wins, cosine term irrelevant, LL K<1... Second term wins, spectral gap develops, Mott insulator Transition from one to other - U(1) Berezinski-Kosterlitz- Thouless transition Excitations in gapped phase are solitons.

Spin systems in one dimension Hesienberg antiferromagnet. S i S j H = i,j Ground state 1: Ground state 2:

Spin systems in one dimension Hesienberg antiferromagnet. S i S j H = i,j Ground state 1: Ground state 2: Kink:

Spin systems in one dimension (2)

Spin systems in one dimension (2) The spin-one excitation spilts into two spin-half kinks (also called solitons) a phenomena known as fractionalization. The two solitons are completely independent (deconfined). Single excitation will destroy anti-ferromagnetic order (Mermin-Wagner theorem) Quantum fluctuations: No long range order, even at T=0.

Spin Ladders - Confinement Shelton, Nersesyan, Tsvelik PRB (1996) Antiferromagnetic coupling, both in chain and interchain.

Spin Ladders - Confinement Shelton, Nersesyan, Tsvelik PRB (1996) Antiferromagnetic coupling, both in chain and interchain. Look what happens to the kinks of the chains:

Spin Ladders - Confinement Shelton, Nersesyan, Tsvelik PRB (1996) Antiferromagnetic coupling, both in chain and interchain. Look what happens to the kinks of the chains: Kinks are confined, excitations are massive (have a gap). In fact, two 'flavours' of excitations: spin singlet and spin triplet (like two charge channels). Quantum fluctuations still destroy AF order: Spin Liquid

Nanotubes Armchair Nersesyan and Tsvelik, PRB 68 235419 (2003) Part III - application to armchair nanotubes Long range Coulomb interaction and Luttinger liquid Umklapp terms Ground state of armchair nanotubes

Interactions long range Coulomb Kane, Balents and Fisher PRL (1997) The long range part of the Coulomb interaction couples only to the total density. H Coul = 2e2 π In fact, the density is fairly uniform so can estimate this expression by a simpler one: H int = e 2 ln(r s /R) dxρ 2 tot dx dy ρ tot(x)ρ tot (y) x y R R s L Gives the plasmon velocity, and Luttinger Liquid parameter (The Luttinger liquid parameter is simply a dimensionless measure of the interaction strength): v ρ = v F 1 + 8e2 v F ln R s R Most importantly, can express g in terms of mesoscopic parameters K ρ = (1 + 4E c / E) 1/2 K ρ = v F /v ρ Finite size effects Level spacing: E v F /L Charging energy: E c = e2 L ln R 2 R

Interactions Luttinger Liquid Physical observables Tunneling density of states: ν(e) E E F α α bulk = (K 1 + K 2)/8 α end = (K 1 1)/4 Transport: G(T ) T α di/dv V α Experimentally: K 0.3 Bockrath et al, Nature 397, 598 (1999)

Interactions short range part Now consider the undoped (half filled) case. Only consider umklapp terms (the ones responsible for Mott-ness): (a) in chain (b) inter-chain, no spin exchange (c) inter-chain, spin exchange Use UV model: g 1 =g c =(U-3V)/n, g 3 =U/n. U = onsite interaction, V=nearest neighbour interaction Notice in particular 1/n factor. Can also estimate g's in more advanced ways. V U V V

Low energy Hamiltonian There are four fields: total and relative charge, and total and relative spin. The Hamiltonian consists of the sum of three parts: Kinetic term: Long range Coulomb: umklapp: Notice that the total charge field is coupled to all of the others, but the other three aren't coupled to each other. The scaling dimension of each of the terms is 1+d, where d is the scaling dimension of the total charge field (small).

Long-range interaction: adiabatic approximation Long-range Coulomb interaction means that total charge field Φ + c much faster than other modes. From the point of view of total charge mode, all other modes are static. Integrate out Φ + c - basically this amounts to replacing the cosine with it s expectation value. Theory is now decoupled!!! Spin sector looks a bit strange: presence of both Φ s and Θ s. Reason is Abelian bosonization of something with nonabelian SU(2) symmetry - can refermionize.

Interactions in Armchair Nanotubes Can consider 4 independent fields: total and relative charge, and total and relative spin. Unscreened Coulomb interaction couples to total charge gives plasmon like excitations. Remaining interactions (backscattering and umklapp) can be treated as local. Result: all excitations are gapped. Mott Insulator-Spin Liquid Ground state Nersesyan and Tsvelik, PRB (2003) Spin gap seen with ingenious NMR technique Singer et al (2005), Dora et al (2007) Mott gap observed for first time very recently Deshpande et al (2009)

Nanotubes Zig-zag STC, A.A.Gogolin, A.A.Nersesyan, PRB 76 245121 (2007) Part IV - what s different in zigzag nanotubes? Curvature gaps Interaction induced dimerization Quantum criticality Phase diagram of zigzag nanotubes

Zig-zag nanotubes curvature gap Kane and Mele, PRL 1997 In fact, the two hoppings are not identical (except armchair) There is a small difference of order 1/n 2 between them due to curvature effects. The interaction effects are of order 1/n so usually we neglect this single particle gap. However, in zig-zag nanotubes there is another contribution to dimerization from staggered interaction which is of order 1/n.

Interactions: zig-zag nanotubes Interaction induced dimerization STC, Gogolin, Nersesyan, PRB (2007) A uniform interaction in the 2D graphene sheet can create a staggered interaction after compactification due to the way the C 3 symmetry is broken. This is not present in the armchair case. The low-energy theory of the zig-zag nanotube will contain all the terms in the armchair case plus dimerization. This can substantially change the physics.

Low energy Hamiltonian In zig-zag nanotubes, the low energy Hamiltonian is the sum of 4 terms: Kinetic: Long range Coulomb umklapp dimerization, both single-particle and interaction induced.

Solving the zigzag model The dimerization term would like to distribute the charge preferentially on the stronger bonds which is in a different way to the way the Mott term would like to distribute the charge. Note however, the dimerization comes from the nearest neighbor interaction V, whereas the Mottness comes from the much larger U. U>>V so consider the charge distributed in the same way as in the Mott insulator. Then all interesting physics occurs in the spin-sector of the theory we have to consider the physics of a dimerized spin ladder.

Dimerization in spin ladders Wang and Nersesyan NPB (2000) Martin-Delgado, Shankar and Sierra PRL (1996) Cabra and Grynberg PRL (1999) If we tune δ= J, and J'=2J, then we end up with a single snake chain: We have driven the system back to massless excitations. In practice, there is a whole critical line δ(j') where we can regain criticality.

Zero Temperature Phase Diagram Turns out singlet excitations compatible with dimerization. Quantum critical point arises from competition between triplet gap and dimerization gap. These two gaps have different scaling behavior with the radius of the nanotube, n. As n is varied, can move between two phases. Two phases are thermodynamically equivalent, difference is in correlation functions in the spin sector. In between, there is a gapless point. No dimerization Massive magnons Massless spinons Massive magnons m m t = ( U V ) 3/13 ( n V ) 2/13 SU 1 (2)

Zero Temperature Phase Diagram No dimerization Massive magnons Massless spinons Massive magnons m m t = ( U V ) 3/13 ( n V ) 2/13 SU 1 (2)

Beyond weak dimerization... We can look at the phase diagram as a function of the dimerization. We can study either weak dimerization on the Mott-insulating spin-liquid phase, or weak umklapp terms on a dimerized phase. No dimerization Massive magnons Massless spinons SU 1 (2) Massive magnons U(1)xSU(2)xZ 2 3+6+3 O(6) multiplets (Levitov and Tsvelik) Massive spinons O(6) No umklapp What are realistic parameters for nanotubes? How to vary `dimerization experimentally? What about chiral nanotubes? Is the U>>V approximation ok? What happens in middle of phase diagram?

In conclusion... Strongly correlated phases in metallic single wall carbon nanotubes well studied theoretically. Experimentally, need some good ideas of how to probe this fascinating physics. Many other things: any more QPT s? disorder? phonons? devices and applications?

STC, Int. J. Mod. Phys. 22, 5235 (2008) In conclusion... Strongly correlated phases in metallic single wall carbon nanotubes well studied theoretically. Experimentally, need some good ideas of how to probe this fascinating physics. Many other things: any more QPT s? disorder? phonons? devices and applications? Thank you for your attention.