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arXiv:0707.3784 · Physical Review B

Interaction induced dimerization in zigzag single wall carbon nanotubes

Sam T. Carr, Alexander O. Gogolin, A. A. Nersesyan

Published Dec 21, 2007• 12 citations
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Abstract

We derive a low-energy effective model of metallic zigzag carbon nanotubes at half filling. We show that there are three important features characterizing the low-energy properties of these systems: the long-range Coulomb interaction, umklapp scattering, and an explicit dimerization generated by interactions. The ratio of the dimerization induced gap and the Mott gap induced by the umklapp interactions is dependent on the radius of the nanotube and can drive the system upon increasing dimerization strength from a Haldane spin-liquid phase through a quantum phase transition with $\mathrm{SU}{(2)}_{1}$ quantum symmetry to a dimerized phase. We consider the physical properties of the phases on either side of this transition, which should be relevant for realistic nanotubes.

DOI
10.1103/physrevb.76.245121

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We derive a low-energy effective model of metallic zigzag carbon nanotubes at half filling. We show that there are three important features characterizing the low-energy properties of these systems: the long-range Coulomb interaction, umklapp scattering, and an explicit dimerization generated by interactions. The ratio of the dimerization induced gap and the Mott gap induced by the umklapp interactions is dependent on the radius of the nanotube and can drive the system upon increasing dimerization strength from a Haldane spin-liquid phase through a quantum phase transition with SU(2)1\mathrm{SU}{(2)}_{1}SU(2)1​ quantum symmetry to a dimerized phase. We consider the physical properties of the phases on either side of this transition, which should be relevant for realistic nanotubes.
A Hamiltonian is derived for a zigzag carbon nanotube with an arbitrary number of weak electron-electron charge and spin interactions, which become significant in ultraclean systems. The renormalization group and bosonization are used to determine the ground-state phase diagram. Our phase diagram contains some exotic phases which have not previously been predicted in carbon nanotubes with physically possible interaction profiles. Phases of the undoped case include Mott insulators and a variety of density wave states. In the doped case a Tomanaga-Luttinger liquid is possible.
Electron interactions reinforce minigaps induced in metallic nanotubes by an external field and turn the gap field dependence into a universal power law. An exactly solvable Gross-Neveau model with an SU(4) symmetry is derived for neutral excitations near half filling. Charge excitations, described by a sine-Gordon perturbation of Luttinger liquid theory, are composite solitons formed by the charged and neutral fields with two separate length scales. Charge compressibility at finite density, evaluated in terms of intersoliton interaction, exhibits a crossover from overlapping to nonoverlapping soliton state. Implications for the Coulomb blockade measurements are discussed.
We investigate a metallic zigzag carbon nanotube by means of a Hubbard model, which includes both on-site and nearest-neighbor interactions. Assuming weak interactions, a renormalization group analysis of the equivalent two-leg ladder followed by bosonization and refermionization results in a Gross-Neveu model with an enlarged symmetry relative to the original Hamiltonian. For the undoped case the symmetry of the Gross-Neveu model is SO(8), but for the doped case the particle-hole symmetry is broken and the symmetry reduces to SO(6). Four ground-state phases are found in the undoped carbon nanotube with repulsive interactions, a ddd-wave Mott insulator, an sss-wave Mott insulator, a ppp-density wave, and a charge density wave. The doped case has two ground-state phases, a ddd-wave superconductor and a phase where a ppp density wave and a charge density wave coexist. We also explore the global phase diagram with a general interaction profile and find several additional states, including a chiral current phase where current flows around the nanotube along the zigzag bonds.
The low--energy excited states of a system of interacting one--dimensional fermions in a conducting state are collective charge and spin density oscillations. The unusual physical properties of such a system (called ``Luttinger liquid'') are characterized by the velocities uρu_ρuρ​ and uσu_σuσ​ of the charge and spin excitations, as well as by a parameter KρK_ρKρ​ that determines the power law behavior of correlation functions. Umklapp scattering occuring at half--filling or other commensurate bandfilling can lead to a transition into an insulating state, characterized in particular by a gap in the charge excitations (the Mott--Hubbard gap). The properties in the vicinity of the transition are shown to depend on both the way the transition is approached (constant bandfilling and varying interaction, or constant interaction and varying bandfilling) and on the ``order'' of the commensurability. In particular, even and odd fractional fillings show quite different behavior. This behavior is illustrated in detail using lattice models like the Hubbard model and its extensions.
The one-dimensional SU(n)\mathrm{SU}(n)SU(n) Hubbard model is investigated numerically for n=2n=2n=2, 3, 4, and 5 at half-filling and 1∕n1∕n1∕n filling using the density-matrix renormalization-group method. The energy gaps and various quantum information entropies are calculated. In the half-filled case, finite spin and charge gaps are found for arbitrary positive UUU if n>2n>2n>2. Furthermore, it is shown that the transition to the gapped phase at Uc=0{U}_{c}=0Uc​=0 is of Kosterlitz-Thouless type and is accompanied by a bond dimerization both for even and odd nnn. In the 1∕n1∕n1∕n-filled case, the transition has similar features as the metal-insulator transition in the half-filled SU(2) Hubbard model. The charge gap opens exponentially slowly for U>Uc=0U>{U}_{c}=0U>Uc​=0, the spin sector remains gapless, and the ground state is nondimerized.
With the surge of research in quantum information, the issue of producing entangled states has gained prominence. Here, we show that judiciously bringing together two systems of strongly interacting electrons with vastly differing ground states-the gapped BCS superconductor and the Luttinger liquid-can result in quantum entanglement. We propose three sets of measurements involving single-walled metallic carbon nanotubes which have been shown to exhibit Luttinger liquid physics, to test our claim and as nanoscience experiments of interest in and of themselves.