{'fraction_non_alphanumeric': 0.03747222696974876, 'fraction_numerical': 0.008246453597675611, 'mean_word_length': 4.4722936637830255, 'pattern_counts': {'":': 0, '<': 14, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 10, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Coulomb Blockade Regime of a Single-Wall Nanotube

A model of carbon nanotube at half filling is studied. The Coulomb interaction is assumed to be unscreened. It is shown that this allows to develop the adiabatic approximation which leads to considerable simplifications in calculations of the excitation spectrum. We give a detailed analysis of the spectrum and the phase diagram at half filling and discuss effects of small doping. At small doping several phases develop strong superconducting fluctuations corresponding to various types of pairing.


I. INTRODUCTION
Carbon nanotubes have attracted an enormous amount of attention and generated an immense level of theoretical and experimental activity. It has been rightly pointed out that single-wall carbon nanotubes (SWCN) represent ideal one-dimensional systems and one may expect to observe here exotic phenomena characteristic for strong correlations in one dimension. Most of the attention has been concentrated on a possibility of Tomonaga-Luttinger liquid. Though such liquid is certainly a very interesting object, it is just one of many wonders strong correlations can produce.
In this paper we concentrate on the physics of an armchair SWCN at half filling. The features which dominate this physics are the unscreened Coulomb interaction and Umklapp processes. The importance of the unscreened Coulomb interaction for nanotubes away from half filling was already noticed in the previous studies [1], [2]. At half filling some additional factors come into play. It was realized in [3] that the long-range Coulomb interaction makes the operators responsible for the Umklapp scattering terms relevant. Consequently, the gaps for collective excitations generated by such operators are not exponentially small, as it would be away from half filling, but have power-law dependence on the Umklapp scattering matrix elements. Such enhancement of the gaps increases chances for their experimental observation. Unfortunately, in their further analysis the authors of [3] resorted to Renormalization Group (RG) equations, which for systems with many fields do not provide much insight into the properties at the strong-coupling fixed point. RG equations also do not take into account the drastic difference between velocities of the plasma modes and all other collective excitations. Instead of adding difficulties, however, the difference in velocities leads to considerable simplifications in actual calculations (see [4]). In this paper we shall exploit this feature to our advantage and develop an approach based on the adiabatic approximation similar to the one used for the problem of electron-phonon interaction. As a result we will be able to provide a rather detailed information about the spectrum and the phase diagram of the system. As will be shown below, the interplay of the long range Coulomb interaction and Umklapp processes at half filling gives rise to a phase diagram which includes several interesting strongly correlated states. The system has a hidden Z 2 ×Z 2 ×Z 2 symmetry and these phases are conveniently classified as different symmetry breaking patterns of this group.

II. THE PROBLEM OF SINGLE-CHANNEL WIRE AT HALF FILLING
To warm up, let us first recall the problem of the unscreened Coulomb interaction for a single chain (or a single channel quantum wire) at half-filling. The charge and spin sectors decouple; the Tomonaga-Luttinger liquid Hamiltonian for the charge sector should be supplemented by the Umklapp term which contains only the charge field Φ c : where Ua 0 is the 2k F = π Fourier component of the interaction. The full Hamiltonian density for the charge sector is H = H 0 + V, where v being the Fermi velocity. This model is very close to the sine-Gordon one. The spectrum contains solitons and their bound states (breathers or excitons). To get their spectrum one can just expand the cosine around its minimum and obtain: where the breather gap is m b = (2/π) √ Uǫ F , ǫ F = πv/a 0 . The breathers are effectively optical phonons of the one-dimensional Wigner crystal [5]. The soliton gap is larger; it was estimated in [4], a better estimate is The spectrum in the spin sector is gapless, and the spin velocity is approximately v.  Fig. 1). Under compactification the spectrum is divided into subbands corresponding to different quantized values of the transverse momentum. In the so-called armchair nanotubes k y = 0 remains an eigevalue and the spectrum in the lowest subband stays gapless; at half filling it is also doubly degenerate at low energies. This degeneracy is a vestige of the doublecone structure of the two-dimensional dispersion. We discuss the spectrum in some detail in Appendix A; more detailed information can be obtained from the book [6]. In Appendix B we derive the bosonized form of the effective low-energy Hamiltonian.
The total Hamiltonian density is where H 0 is given by Eq.(2); Φ (+) c is the symmetric charge mode. The label a takes three values, a = (c, −), (s, +), (s, −), corresponding to the antisymmetric charge field and sym-metric and antisymmetric spin fields, respectively (see Appendix A). The interaction density V contains Umklapp terms where the dots stand for all other terms that do not involve Φ where, as it turns out, g c = g 1 and, hence, there are only two independent coupling constants.
The unscreened Coulomb interaction strongly reduces the scaling dimension of the operator c ) in the long-wavelength limit, making it smaller than 1. Thus the Umklapp terms (6) become strongly relevant, with the scaling dimension almost equal to 1. This circumstance dramatically increases the values of the gaps.
One may well expect that the double degeneracy of the electron band in carbon nanotubes makes the problem similar to the problem of two interacting chains (the 'ladder' problem) much discussed in literature. This is indeed the case. To put the problem in a broader context, let us discuss two extreme types of ladders: (i) the ones where two chains are placed far apart such that there is no direct tunneling between them, and (ii) those in which the interchain tunneling is stronger than all the interactions. Hamiltonian (5,6) is formally equivalent to the first case describing two well separated chains. As a matter of fact, the Hamiltonian for case (ii) is not very different. The calculations done in [7] for the ladder with strong interchain tunneling yields the same Hamiltonian as (6), but with Φ c,− field being substituted by its dual counterpart, Θ c,− : Therefore one expects that the two models have the same excitation spectrum though the response functions are different due to the different field identification.
Hamiltonian (6) is not very convenient to analyse in its bosonic form since the effective potential contains mutually nonlocal and noncommuting fields, Φ s,− and Θ s,− . The physics becomes significantly more transparent when one uses the refermionization procedure [8], [9].
Fermionizing all the fields except of Φ (+) c we obtain Following [10], we shall handle Hamiltonian (8) using the adiabatic approximation, whose validity is guaranteed by the fact that the velocity in the symmetric charge sector is strongly enhanced by the long-distant Coulomb interaction with respect to the bare Fermi velocity.
The results have certain similarity with the SU(4) theory proposed in [11], but also contain important differences, which we shall discuss.
Thus, from the Φ where So as we see, the integration over the fast mode gives rise to the mass terms for all of the fermions. The fermionic modes acquire gaps: They correspond to neutral excitations with the quantum numbers given in the table that follows; see also where with the logarithmic accuracy M is either M c,− or M S . Thus which further supports the adiabatic approximation. However, as we have already mentioned, its validity is already assured by the difference in the velocities. Apart from the breather modes there are massive modes corresponding to half-period solitons in Φ The results for the spectrum are summarized in Table 1.; see also  by uniform translations of the dual field Θ (−) c ), emerges in the low-energy limit only. This symmetry leads to conservation of the "flavor" charge which, together with conservation of Q, implies independent conservation of the particle numbers at each Dirac point, Q 1,2 = (Q ± V )/2. sublattices.
The nomenclature "vorticity" we have chosen follows from the microscopic origin of the flavor density. Indeed, consider a lattice operator describing a current flowing around the elementary plaquette of sublattice a or b: From this construction it follows that the state with a nonzero j plaq,a(b) would represent an orbital antiferromanet, or a staggered flux phase, realized on the sublattice a (b), as shown in Fig.3. Projecting (13) onto the k y = 0 subband and passing to the continuum limit, one makes sure that the sum j plaq,a + j plaq,b does transform to the flavor charge density Previous attempts to study the problem of two coupled channels have mostly relied on the assumption of equal velocities with the subsequent use of RG analysis. The approach was pioneered by Lin et al. [11] who argued that at strong coupling the spectrum of the two-chain problem at half filling acquires a higher symmetry, such as SO (6) or even SO (8).
In fact, Gell-Mann-Low equations alone are insufficient to extract information about strong coupling regime (they have to be supplemented by Callan-Symanzik equations for the physical observables) and therefore cannot provide a legitimate ground for such conclusions. Our approach is based on a different assumption; here the long-range Coulomb interaction legitimates a clear separation of scales between single particle excitations and collective modes.
This significantly simplifies the calculations providing one with the well-controlled approximation. It is instructive to compare the results with those conjectured in [11]. Though the structure of the multiplets is the same as in the SU(4)∼SO (6)  They depend on the Coulomb interaction and the quasiparticle gap is much larger than the spin and the parity collective mode gaps. The gaps of the collective modes are not equal.
In nanotubes where there are only two independent coupling constants we expect that (recall that since we deal with Majorana fermions, the masses may be negative, the spectral gaps being their absolute values).

IV. THE ORDER PARAMETERS OF DIFFERENT PHASES
From now on we shall concentrate on the model describing carbon nanotubes. Different phases, shown in Fig. 3, correspond to different ground state phase lockings and are determined by the signs of the fermionic masses [8], Eqs. (10). These sign changes are not reflected in the thermodynamics which is sensitive only to absolute values of the masses.
The difference in the correlation functions may, however, be quite dramatic.
We use the following conventions [8]:   Below we give a brief characterization of each phase; for more details see the subsequent discussion.

below).
• Phase B: 0 < g 1 < g 3 . This is a Haldane spin-liquid phase whose excitation spectrum contains a coherent triplet magnon with the mass M S,t , associated with the correlation function of the site-diagonal spin operator S (−) ; see Eq. (24). Under doping it develops a power-law response of the pairing susceptibility of type described further in the text.
The corresponding superconducting order parameter transforms non-trivially under the lattice point group.
• Non-Haldane phase C: −g 3 < g 1 < 0. The magnetic singlet mode with mass M S,s becomes coherent in this phase. As a result a coherent peak appears in the correlation function of the Charge Density Wave order parameter ∆ CDW , Eq. (18). Under doping this phase develops superconducting correlations in the s-channel.
• Phase D: g 1 < −|g 3 |. There is a Charge Density Wave (CDW) long-range order in the ground state of this phase. Phases CDW and BDW are mutually dual.
• Phase E: g 3 < g 1 < 0. This is another Haldane phase dual to phase B. Phase E has a coherent magnon in the site-off-diagonal spin density Σ (−) , Eq.(25). Under doping it develops superconducting correlations of the same type as phase B.
• Phase F : 0 < g 1 < −g 3 . This phase is dual to phase C and has a coherent spin-singlet mode displayed by the BDW order-parameter correlation function. Under doping it develops superconducting correlations in the s-channel.
we define the CDW order parameter (OP) as follows: This OP has a non-zero average value in phase D. In phase C where sin( √ πΦ BDW. The Bond Density Wave order parameter is similar to the CDW one, but is off-diagonal in the site indices: This phase is dual to the CDW one. The BDW order parameter condenses in phase A. The dimerization ordering pattern in this phase is shown in Fig. 4. In the disordered phase F (which is dual to phase C), the spectral weight of the OP ∆ BDW contains a coherent magnetic singlet mode with the mass M S,s .
Notice that the CDW and BDW OPs do not contain any oscillatory pieces. These show up in the density distributions that are not uniform across a sublattice but otherwise are consistent with the uniaxial symmetry of the nanotube: Two incommensurate CDW OPs can then be constructed as follows:  Fig. 6. This kind of order can be induced by applying a modulated potential of sufficient strength that couples to ∆ − mCDW . The potential must be strong enough to overcome the energy gaps of the corresponding excitation branches and to drive the system into a state with an induced order of the mCDW − type.
Magnetization. The total magnetization is given by Its smooth part is equal to the sum of the chiral currents of the Majorana triplet: The oscillating part of the spin density is (Klein factors omitted) HereÑ = (µ 1 σ 2 σ 3 , σ 1 µ 2 σ 3 , σ 1 σ 2 µ 3 ) and N = (σ 1 µ 2 σ 3 , µ 1 σ 2 µ 3 , µ 1 µ 2 σ 3 ). This OP is never coherent. On the other hand, to define the staggered magnetization S (−) (r) associated with the bipartite lattice, we need to introduce locally averaged spin densities for the two sublattices (cf. Eq. (17)): Then The OP S (−) (r) is coherent in the Haldane spin-liquid phase B, the corresponding particle representing a massive triplet magnon. Notice that in the expression (24) the vector field N plays the role of the staggered magnetization of the effective antiferromagnetic spin-1 chain [8].
In full analogy with the BDW OP (19), one can construct the site-off-diagonal staggered magnetization: The structure of (25) indicates that phase E represents a Haldane spin liquid and is dual to phase B. Indeed, in phase E it is the vector fieldÑ that can be regarded as the staggered magnetization of the effective S=1 chain, and, hence, the spectral weight of the operator Σ (−) contains in this phase a coherent triplet magnon.
Pairing operators. Let us construct the s-wave superconducting (SC) order parameter ∆ sSC . To this end, we first build locally averaged, site-diagonal, singlet pairing operators: The OP ∆ sSC is then defined as The amplitude of this OP acquires a finite value in non-Haldane phases C and F . These two phases, with coherent singlet magnetic modes in their spectrum at half-filling, exhibit power-law sSC correlations under doping.
The s-wave type superconductivity is not the only possible singlet SC order one can imagine. One can introduce SC order parameter which transforms non-trivially under the point group: This pairing OP exhibits power-law correlations in doped Haldane phases B and E.
To see what kind of superconductivity this is, let us consider the lattice mean field Hamiltonian: where g(k) = −1 + 2 cos(k x /2)e i √ 3ky/2 . The spectrum is In the Haldane phase C under doping we get the following OP: The Aharonov-Bohm effect: the y-component of the vector potential is coupled to It is coherent in Haldane phase E with an emission of the magnetic singlet.
The staggered flux around a hexagonal plaquette is where the flavor fermion is associated with Φ (−) c field. Therefore there is an interesting possibility to drive a system into a peculiar critical state with a mixture of the magnetic singlet and nonmagnetic orbital mode by applying a magnetic field with the corresponding period. c field, though becoming gapless, still lies above the others in the most of the momentum space (see Fig. 2b). This means that one can still integrate over Φ (+) c and obtain Eq.(9) though with a prefactor < 1. Thus a small doping will decrease the gaps. It may also give rise to finite decay rates for the collective modes.
This is a pseudogap regime which is perhaps similar to the one existing in the underdoped state of the cuprate superconductors (see, for example, the recent paper [13]). The physics of this regime will be goverened by two energy scales: the scale of collective modes M c (see is close to one (see [14] for a more detailed analysis). Therefore there is a window of doping whereK c < 1/2 where the pairing susceptibility diverges. A more detailed description of the pseudogap regime will be given elsewhere. SWCNs exhibit two types of behavior: either band insulators or Luttinger liquids. Armchair CNs are believed to be metallic and be of the Tomonaga-Luttinger (TLL) type. This, in fact, is only true away from 1/2 filling. As shown in this paper, the picture changes dramatically at 1/2 filling when effects of strong correlations become crucial. To observe these effects in experimental conditions, the chemical potential must be fine tuned to the value µ = 0 at which the Fermi "surface" is represented by two degeneracy (Dirac) points. Then, due to the special role of Umklapp processes in the presence of unscreen Coulomb interac-tion, on lowering the temperature, one must be able to see a crossover from the metallic, TLL behavior to an insulator discussed in the paper. Such insulating state is Mott-type, with a much richer spectrum than in the Hubbard model. One important difference is that, unlike in the Hubbard model, the spin excitations have spectral gaps.
The estimates of interaction matrix elements for carbon nanotubes are provided in [3]: where N is the number of transverse bands of the nanotube and ǫ ≈ 1.4 is the dielectric constant. This estimates indicate that realistic nanotubes belong to phase B. Eq.(10) also gives the following estimates for the spectral gaps of the collective modes: Since α is the short distance cut-off of the bosonic theory and is therefore non universal, this formula contains a certain degree of ambiguity. For noninteracting electrons πα = a 0 (the lattice spacing). Taking N = 10, v/v c = 1/3 and a 0 = 0.246nm we get the estimate of order of 0.1 ev. The single particle gaps should be even greater.
Since the gaps should be quite sizeable, one may wonder whether they have not been already observed. The existing techniques produce simultaneously carbon nanotubes of different sizes and chiralities and one has to select the relevant ones using some criteria. The simplest one is to differ between metallic and insulating (semiconductor) nanotubes. Naturally, on the first glance Mott and band insulators look alike. Therefore it is possible that the Mott insulating armchair nanotubes have been overlooked being taken for semiconducting ones. To distinguish Mott insulator from a band one one has to measure transport and magnetic properties and compare the gaps. The clearest sign of 'mottness' is the difference in gap sizes in different response functions. This is the feature to look for experimentally.
ICTP where a part of the work was conducted. AAN is partly supported by MIUR, under project COFIN2003 "Field Theory, Statistical Mechanics and Electron Systems". He also acknowledges the support from the Institute for Theory of Strongly Correlated and Complex Systems at Brookhaven.

APPENDIX A
The original Hamiltonian describing noninteracting electrons on a honeycomb lattice can be compactly written in the two-sublattice (Nambu) representation: where and the sum in (A1) goes over the Brillouin zone. The spectrum has two degeneracy points (nodes) at Q 1,2 = (±4π/3, 0). Linearizing the noninteracting Hamiltonian near these points yields two cones associated with (2+1)-dimensional massless Dirac fermions: When a two-dimensional sheet of graphite is wrapped to produce an armchair nanotube, k y gets quantized, and the lowest-energy subband correspoding to k y = 0 stays gapless.
The resulting problem is one-dimensional because the wave function does not depend on y.
Projecting the fermionic annihilation operators of the a and b sublattices onto the k y = 0 subspace, we get: The effective 1D Hamiltonian is brought to its canonical diagonal form by linear transformations: The structure of these transformations reflects the fact that different Dirac points, k x = Q and k x = −Q, are not associated with fermions of different chiralities, as it is the case for standard chains and ladders; instead each of these points is characterized by a pair of right (R j ) and left (L j ) fields. This is because the gapless spectrum of the armchair nanotube keeps the memory of the two-cone Dirac structure of the dispersion law in the 2D graphite (in fact, the 1D spectrum is obtained from the 2D one by cutting the two cones by the plane k y = 0). For this reason, in case of nanotubes, smooth components of the physical fields are contributed not only by the diagonal "currents", R † j R j and L † j L j , but also by off-diagonal "mass bilinears", R † j L j and L † j R j . The chiral fermionic fields can be bosonized in terms of chiral bosonic fields Φ R,L jσ : Here α is the short-distance cutoff in the bosonic theory, and κ jσ are Klein factors obeying the algebra {κ jσ , κ j ′ σ ′ } = δ jj ′ δ σσ ′ . The product of the four Klein factors, Γ = κ 1↑ κ 1↓ κ 2↑ κ 2↓ , satisfies Γ 2 = 1. Since Γ is not a dynamical variable, we can conveniently choose Γ = 1.
In the bulk of this paper, we adopted the description in terms of four scalar fields, Φ (±) c , Φ (±) s , and their dual counterparts, Θ (±) c , Θ (±) s , known from earlier studies of the twochannel Kondo problem [15]. These fields describe the symmetric and antisymmetric charge excitations (equivalently, the "charge" and "flavor" modes), as well as the symmetric and antisymmetric spin excitations(or the "spin" and "spin-flavor" modes), Here Φ jσ = Φ R jσ + Φ L jσ . The corresponding dual fields, Θ (±) c and Θ (±) s are obtained from the above expressions by replacing Φ jσ by Θ jσ = −Φ R jσ + Φ L jσ .
In the UV model, which apart from the Hubbard (on-site) interaction U also includes the interaction between electrons on nearest-neighbor sites, V , the couplings g i are given by The bare Hamiltonian for R, L is standard: As already explained in section I and II, in the presence of long-range Coulomb potential, Umklapp processes with the structure R † 1 R † 2 L 3 L 4 +h.c. represent the strongly relevant part of interaction (B3). In the bosonic language, these are the processes containing the symmetric charge field Φ + c . Using bosonization rules (A8) -(A10), one straightforwardly derives Eq.
If the interaction is the Coulomb one, then the couplings g 1 , g 3 are positive. It also looks likely that g 3 > g 1 and |g 3 − g 1 | << g 1,3 . At these circumstances we have to consider two possibilities: (i) g 1 < g 3 (Haldane SL) and (ii) g 1 > g 3 (CDW) .