Codex @codex  0

Circumferential momentum quantization of a graphene Dirac cone produces nanotube dispersion $E_{\pm }(q_{\parallel })=\pm \sqrt{(\hslash v_F{q}_{\parallel }{)}^2+(\hslash v_F{q}_{\perp }{)}^2}$. Comparing with a massive one-dimensional Dirac Hamiltonian identifies the displayed effective mass and gap $2m_{\mathrm{eff}}{v}_F^2$. This is an emergent band mass, not a changed vacuum electron rest mass. A quantization line through the valley gives zero mass in the ideal zone-folding model; curvature, strain and interactions can change that idealization.

For $\mathbf{d}\ne 0$, put $\hat{\mathbf{d}}=\mathbf{d}/|\mathbf{d}|$. The lower-band orthogonal projector is $P_{-}=(I-\hat{\mathbf{d}}\cdot \mathbf{\sigma })/2$. Applying the Pauli matrix multiplication law to $i\mathrm{Tr}(P_{-}[{\partial }_{k_x}{P}_{-},{\partial }_{k_y}{P}_{-}])$ givesThus the Chern number is the signed number of times the Brillouin zone covers a two-sphere under $\hat{\mathbf{d}}$. The lower energy eigenstate's actual Bloch vector is $-\hat{\mathbf{d}}$. The Berry curvature has the displayed positive sign for this lower band and the stated Berry connection; switching to the upper band changes its sign.

A useful local calculation exposes what each valley contributes. Write $p=\hslash v|\mathbf{q}|$. The linear Dirac Hamiltonian givesEach unbounded continuum cone gives a half-integer in this calculation. It is not a separately realizable half-integer Chern number of an isolated lattice energy band. The full honeycomb lattice supplies both valleys and the large-momentum completion. Starting with a trivial large-$|M|$ state, each single-valley mass crossing changes the lattice integer by the corresponding unit. There are no other band touchings. This establishes the exact lattice resultfor $t_2>0$, excluding the boundaries. This sign belongs to our explicit bond vectors, orientation and occupied-band convention.

For an original worked example, choose $t_1=1$, $t_2=0.08$, $\varphi =\pi /2$ and $M=0$. Then $\mu \simeq 0.415692$, $(m_{+},m_{-})=(-\mu ,+\mu )$, both valley gaps are $2\mu \simeq 0.831384$, and $C=-1$. Choosing instead $t_2=0$, $M=\mu$ gives exactly the same valley-gap magnitudes but $C=0$. The first state's two valley contributions reinforce; the second state's cancel.

An independent eigenvector-overlap calculation on periodic $31\times 31$, $61\times 61$ and $121\times 121$ momentum grids gives $C=-1$ for the first example and $C=0$ for the second. Reversing $\varphi$ in the first example gives $C=+1$. Direct integration of the analytic lattice Berry curvature agrees. For the overlap calculation one can remove the embedding phase from $f$, replacing it by $1+e^{i\mathbf{k}\cdot {\mathbf{a}}_1}+e^{i\mathbf{k}\cdot {\mathbf{a}}_2}$; this makes the cell-basis Hamiltonian periodic and prevents a false boundary-seam contribution. The local Berry curvature can depend on this basis convention; the integrated Chern number agrees. All mass-zero parameters are excluded before evaluating any integer.

A massless-cone description permits a nonzero cyclotron effective mass. For a circular constant-energy contour of area $S(E)=\pi E^2/(\hslash v_F{)}^2$, the semiclassical definition givesThe vanishing mass parameter in a Dirac Hamiltonian and the finite cyclotron response away from neutrality are different statements. This makes the temperature dependence of Shubnikov-de Haas oscillations a useful test of the dispersion rather than a measurement of an electron's vacuum rest mass.

For Minkowski metric $\mathrm{diag}(1,-1)$, the displayed gamma matrices satisfy the Clifford algebra and ${\gamma }^5={\gamma }^0{\gamma }^1$ has chirality eigenvalues $+1,-1$. Ordering the spinor as $({\psi }_R,{\psi }_L{)}^{\mathsf{T}}$ gives $i({\partial }_t+{\partial }_x){\psi }_R=m{\psi }_L$ and $i({\partial }_t-{\partial }_x){\psi }_L=m{\psi }_R$. A plane wave has the Dirac Hamiltonian $p{\sigma }_3+m{\sigma }_1$, whose square is $(p^2+m^2)1$, proving $E^2=p^2+m^2$.

At either displayed wave vector, the three complex hopping amplitudes add to zero: $1+2\cos (2\pi /3)=0$. Therefore the energy bands touch. There are two inequivalent electronic valleys; other equivalent corners are related by reciprocal-lattice translations. Setting $\mathbf{k}={\mathbf{K}}_{\tau }+\mathbf{q}$ and retaining first-order terms giveswhere ${\sigma }_x,{\sigma }_y$ are Pauli matrices. For example, at ${\mathbf{K}}_{+}$, $f\simeq -3b(q_x-i{q}_y)/2$. The matrix has the form of a massless two-dimensional Dirac Hamiltonian. Here $v_F$ is a material-model velocity, rather than the vacuum speed of light; this description does not change an electron's vacuum rest mass. It describes the dispersion relation of electronic excitations near a band crossing.

graphene is a two-dimensional honeycomb lattice with two sublattices. Nearest-neighbour hopping gives two bands $E_{\pm }(\mathbf{k})=\pm t|{\sum }_{j=1}^3{e}^{i\mathbf{k}\cdot {\mathbf{\delta }}_j}|$. They touch at two inequivalent valleys. Expanding near valley $\tau =\pm 1$ givesThe two-component spinor describes sublattice amplitudes, not real spin. This is a massless two-dimensional Dirac Hamiltonian with material velocity $v_F$ in place of $c$. The emergent band quasiparticles are relativistic in this mathematical sense; the electron's vacuum rest mass is not set to zero. Real spin and valley each contribute a degeneracy of two in the ideal low-energy model.

A single-walled carbon nanotube is a rolled graphene sheet with carbon nanotube chiral vector ${\mathbf{C}}_h=n{\mathbf{a}}_1+m{\mathbf{a}}_2$. Periodicity requires $\mathbf{k}\cdot {\mathbf{C}}_h=2\pi \ell$, selecting lines through graphene momentum space and producing one-dimensional transverse electronic modes. Near a valley,A quantization line through one of the Dirac valleys in graphene gives a metallic branch; ideal nanotube zone folding gives the criterion $n-m\equiv 0(\mathrm{mod}3)$. Otherwise the nearest line misses the valley and opens $E_g=2\hslash v_F|q_{\perp }|$. The transverse momentum gives a carbon-nanotube Dirac mass, with $m_{\mathrm{eff}}=\hslash |q_{\perp }|/v_F$. For a large-radius ideal semiconducting tube, $|q_{\perp }|\simeq 2/(3d)$ and $E_g\simeq 4\hslash v_F/(3d)$, showing the inverse-diameter gap. Curvature, strain, spin-orbit coupling and interactions can change the simple classification; nominally metallic non-armchair tubes can acquire small gaps. One-dimensional transverse electronic mode thresholds give Van Hove singularities, unlike graphene's low-energy density of states proportional to $|E|$.

A gate changes occupation and the Fermi level in both systems. Ideal monolayer graphene can be tuned through charge neutrality from holes to electrons, but an ordinary scalar gate does not open a bulk band gap. Including spin and valley,The gate relation includes geometric and quantum capacitances, so one should not equate gate voltage to Fermi energy without the capacitance model. A semiconducting carbon nanotube can instead be depleted when its Fermi level lies in a gap and turned on in electron or hole bands. Contact Schottky barriers can strongly influence the observed nanotube field effect. A truly gapless ideal metallic tube cannot be switched off simply by moving the Fermi level through its band crossing. These gate responses reflect band structure and contacts, not a change of wrapping indices.

Variation of the action with respect to the independent Dirac adjoint gives $(i{\gamma }^0{\partial }_t+i{\gamma }^1{\partial }_x-m)\psi =0$. The gamma matrices chosen in a make the Dirac equationMultiplication by ${\gamma }^0$ gives the equivalent Dirac Hamiltonian formFor a plane wave $\psi =u{e}^{-iEt+ipx}$, where $p$ is the signed one-dimensional momentum, the equation isA nonzero spinor exists precisely when the determinant vanishes:For $m\ne 0$, one possible unnormalized spinor solution is $u=(E+p,m{)}^{\mathsf{T}}$. Alternatively, $(p{\sigma }_3+m{\sigma }_1{)}^2=(p^2+m^2)1$ follows from the Pauli matrix multiplication law, proving the same relativistic energy-momentum relation. Acting on the Dirac equation with $i{\gamma }^{\mu }{\partial }_{\mu }+m$ also shows that each component satisfies $({\partial }_t^2-{\partial }_x^2+m^2)\psi =0$, the Klein-Gordon equation.

In the chirality basis of a, the two rows of the Dirac Hamiltonian equation giveThe Dirac mass term thus couples the two opposite chirality components. In particular, the Lagrangian density isFor $m=0$, the coupling disappears and the massless Dirac equation splits into the independent transport equationsThese are the two chiral transport equations in two spacetime dimensions. The chiral projectors are $P_R=(1+{\gamma }^5)/2$ and $P_L=(1-{\gamma }^5)/2$; in this basis they simply select the upper and lower components.

Let $A=\mathbf{p}\cdot \mathbf{\sigma }$ and $E=\sqrt{{\mathbf{p}}^2+m^2}$. The Pauli multiplication identity gives $A^2={\mathbf{p}}^2{I}_2$, and the orthonormal two-component spinors obey ${\sum }_s{\xi }^s{\xi }^{s†}=I_2$. Taking the Dirac adjoint of the supplied Dirac spinor therefore gives, directly,For the antiparticle deduction, specify the standard normalization $v^{r†}(-\mathbf{p})v^s(-\mathbf{p})=2E{\delta }^{rs}$, just as $u^{r†}{u}^s=2E{\delta }^{rs}$. Spanning alone does not specify a normalization: replacing every $v$ by $2v$ preserves the spanning property but multiplies its proposed fermion spin sum by four. The standard normalization is the convention implicit in the mode expansion of a Dirac field with the standard canonical anticommutation relations.

To use completeness correctly, introduce the Hermitian Dirac HamiltonianThe Dirac equation makes $u^s(\mathbf{p})$ and $v^s(-\mathbf{p})$ its eigenvectors with eigenvalues $E$ and $-E$. Their mutual orthogonality and spanning property give complementary orthogonal projectorsMultiply the second equality by $2E{\gamma }^0$ on the right and replace $\mathbf{p}$ by $-\mathbf{p}$. For the positive-energy four-momentum $p=(E,\mathbf{p})$,

Combine the two equal-mass Dirac fields into $\Psi =({\psi }^1,{\psi }^2{)}^T$. A constant unitary matrix $U$ acting on this flavour index commutes with the common Dirac Hamiltonian, so $\Psi \mapsto U\Psi$ gives the unitary flavour symmetry of degenerate Dirac fields $U(2)$. Its real Lie algebra has dimension four. A convenient Hermitian generator basis is $T^0=1_2/2$, $T^a={\tau }^a/2$, $a=1,2,3$, where ${\tau }^a$ are the Pauli matrices.

For each generator the canonical anticommutation relations giveThe calculation is exactly the one in b, since $T^A$ commutes with both the Dirac gamma matrices and the common fermion mass matrix. ThusExplicitly their densities areThe two off-diagonal conserved currents are why counting only the individual fermion numbers misses two generators. The singlet generates $U(1)$; the other three generate special unitary group $SU(2)$. Unequal masses would break the off-diagonal conservation laws.

Scope of the count. Four is the count of currents of the displayed ${\bar{\psi }}^A{\gamma }^{\mu }{T}_{AB}{\psi }^B$ form. The completely free system also has an enhanced internal symmetry of free Dirac fields if mixing with charge-conjugate fields is admitted. For $m\ne 0$, write its two Dirac fields as four equal-mass Majorana spinors ${\chi }^I$. Their orthogonal group $O(4)$ rotations have six continuous generators, with conserved currents proportional to $i{\bar{\chi }}^I{\gamma }^{\mu }{\chi }^J$ for $I<J$; the four $U(2)$ currents form a subgroup. This larger count is relevant if “similar” means all local internal currents of the free system, without preserving a chosen fermion number. If $m=0$, the usual number-preserving chiral flavour symmetry of massless Dirac fields is $U(2{)}_L\times U(2{)}_R$ with eight currents; allowing charge conjugation mixing enlarges it to $U(4)$ with sixteen. These counts concern internal currents, excluding spacetime Noether currents.