Dirac Matter: Graphene, Dirac Cones and Topological Insulators in Biquaternionic Form

Introduction

Dirac matter is the name for the solids whose low-energy electronic excitations obey, not the Schrödinger equation, but an equation of Dirac type. The prototype is monolayer graphene, where the honeycomb lattice leaves electrons near the Fermi level massless and linearly dispersing, forming the Dirac cones of the Brillouin zone. The same structure appears in the surface states of three-dimensional topological insulators, where the bulk is gapped and insulating and the boundary carries a gapless Dirac cone protected by time-reversal symmetry. In both cases a two-component spinor, a $\boldsymbol{\sigma}\cdot\mathbf{p}$ kinetic term and a mass term that opens a gap organise the physics; these are exactly the three objects the biquaternion framework names as its own.

This article asks what the framework contributes to Dirac matter and, just as importantly, what it does not. The framework transcribes the low-energy equations faithfully, and it locates the gap-opening term as the parent's chirality-mixing mass. It does not derive the honeycomb lattice, the Fermi velocity, or the topological invariant: those are material and many-body data. The one genuinely algebraic point offered here is a counting observation — the biquaternion field carries eight real components, the same count as the sublattice and valley isospins of graphene together — and a corresponding caution: the framework has two minimal left ideals, so it can host the chirality pair or the valley pair naturally, but the identification of its ideals with graphene's valley isospin is a choice, not a consequence. The distinction between the framework's chirality and graphene's pseudospin is the sharpest thing this article has to say.

The article is organised as follows. The next section explains why lattice electrons look relativistic and derives the Dirac cone. The third separates graphene's two isospins from the framework's two ideals and states the counting comparison. The fourth identifies the gap with the framework's mass. The fifth treats topological insulators, their edge states, the $\mathbb Z_2$ invariant and the surface Dirac cone, and cross-links the framework's Kramers article for the time-reversal structure. The sixth recalls Klein tunnelling in graphene $p$–$n$ junctions. The seventh separates transcription from framework-level content. The closing sections are the open questions, the summary, the notation table and the literature.

The conventions are those of the companion articles: $\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$ with units $e_0=1,e_1,e_2,e_3$, central $i$, $\mathbb{M}_\pm$ the Hermitian and anti-Hermitian sectors, the biquaternion Dirac equation the linear chiral pair, and the mass shell $\tilde k\tilde k^{\natural}=-m^2c^2/\hbar^2$.

Why Lattice Electrons Look Relativistic

The honeycomb lattice and the two-site cell

Graphene is a single layer of carbon atoms on a honeycomb lattice. The lattice is bipartite: it is a triangular Bravais lattice with a two-atom basis, so each unit cell carries two sites, conventionally labelled $A$ and $B$. The electronic states near the Fermi level are built from one $p_z$ orbital per site, and the tight-binding hopping $\gamma_0\approx 2.8$ eV between nearest neighbours gives two bands. A cell with two sites and two orbitals per cell necessarily gives a two-component Bloch spinor, and the two components are the amplitudes on the two sublattices. This two-valued label is the sublattice pseudospin; it is not the electron's spin.

The low-energy effective equation

Expanding the tight-binding bands about a zone corner gives the effective Hamiltonian

$$ H = v_F\,\boldsymbol{\sigma}\cdot\mathbf{p}, \qquad v_F \approx 10^6\ \mathrm{m/s} \approx \frac{c}{300}, $$

acting on the two-component sublattice spinor. This is the massless Dirac equation in two spatial dimensions, with $v_F$ in place of $c$. Its energies are

$$ E(\mathbf p) = \pm v_F|\mathbf p|, $$

linear in momentum and vanishing at $\mathbf p = 0$: the bands touch at the Dirac point and there is no gap. A finite mass term opens one,

$$ H = v_F\,\boldsymbol{\sigma}\cdot\mathbf{p} + \Delta\,\sigma_z , \qquad E = \pm\sqrt{v_F^2p^2 + \Delta^2}, $$

which is the Dirac equation with rest mass $m = \Delta/v_F^2$.

The Dirac cone

The two bands meeting linearly at a Dirac point form the upper and lower halves of a cone; the six zone corners give six cones, of which two, labelled $K$ and $K'$, are inequivalent and the rest are related by symmetry. The linear dispersion is the experimental signature of Dirac matter: it produces the half-integer quantum Hall sequence

$$ \sigma_{xy} = \pm 4\left(N + \tfrac12\right)\frac{e^2}{h}, $$

whose offset $\tfrac12$ is the Berry phase $\pi$ of a massless Dirac fermion winding about the Dirac point, and whose factor $4$ is the product of a spin degeneracy and a valley degeneracy $g_v = 2$ from the two inequivalent cones. The offset is a direct consequence of the two-component, linear, gapped-or-gapless structure that the framework names.

Two Isospins and the Framework's Two Ideals

The isospins

Graphene's low-energy spinor carries two internal two-valued labels that are easy to confuse and must be kept apart.

• Sublattice pseudospin. The two components of the $\boldsymbol{\sigma}$ in $H = v_F\boldsymbol{\sigma}\cdot\mathbf{p}$ are the amplitudes on sublattices $A$ and $B$. It is an orbital isospin; it exists because the cell has two sites. It is not the electron's spin, and rotations of it are not physical rotations of the electron.
• Valley isospin. The label $K$ versus $K'$ distinguishes the two inequivalent cones. It is a momentum-space label, protected at low energy by the large momentum separation of the valleys.

The physical electron additionally carries its genuine spin, so the full low-energy count is $2_{\mathrm{sublattice}}\times 2_{\mathrm{valley}}\times 2_{\mathrm{spin}} = 8$ complex components.

The framework's ideals

The biquaternion algebra is $\mathbb{B}\cong M_2(\mathbb{C})$, whose minimal left ideals are the two columns. The parent's Dirac field is a pair of Weyl spinors, one per ideal, and that pair is what the framework calls chirality: the left and right components that the mass term mixes. The framework therefore has, natively, exactly one two-fold internal structure — the chirality pair — and its mass is the term that couples the two ideals.

Counting versus identification

The arithmetic invites a comparison. A biquaternion field has eight real components, the same count as sublattice$\times$valley for one spin species; but the biquaternion field is also meant to carry the electron's spin in the parent's dictionary. The counts therefore do not match component-by-component, and no natural map from the valley label to the algebra's second ideal exists in the corpus. The honest statement is:

The framework can host the sublattice (or valley) pair in a minimal ideal, and it hosts the chirality pair in the two-ideal structure. The identification of the algebra's two ideals with the two valleys, or with the two sublattices, is an extra input that the framework does not supply.

This is the article's central caution. A construction that simply renames "left and right" as "$K$ and $K'$" is not a derivation from the framework; the valley structure is a property of the lattice, and the algebra is indifferent to which lattice sits on top of it.

The Gap and the Framework's Mass

Where the framework does speak clearly is the gap. In the parent the mass term is the linear, chirality-off-diagonal pair $\tilde\nabla\tilde\Psi_R = m\tilde\Psi_L$, $\tilde\nabla^{\natural}\tilde\Psi_L = m\tilde\Psi_R$, the one term that couples the two minimal left ideals. In graphene the same role is played by any perturbation that breaks the sublattice symmetry: a staggered on-site energy $\Delta$ on the two sublattices inserts $\Delta\sigma_z$ into the effective Hamiltonian and opens a gap $2\Delta$. Read in the framework's terms, this is the appearance of the mass: a term that mixes the two components of the ideal and lifts the cone from a point to a hyperboloid.

Two further gap-opening mechanisms have the same algebraic shape. The Haldane mass breaks time-reversal symmetry and gives a gapped state with a quantised Hall conductance but no net magnetic flux. The Kane–Mele mass is a spin-dependent, time-reversal-symmetric version, obtained in the framework by giving the two spin species opposite mass terms. In every case the algebraic statement is the same: the mass is the chirality-mixing term, and to make a topological insulator rather than a trivial one, one must introduce two such masses with opposite signs — or one must start from a material whose band ordering is already inverted. The framework names the term and does not choose its sign; the material does that.

Topological Insulators

Bulk gap, edge states, bulk–boundary correspondence

A topological insulator is a material with a gap in the bulk and gapless states on its boundary. The existence of the boundary states is not an accident of the surface chemistry: it follows from the bulk band structure. If the occupied bands of the material are "twisted" relative to those of vacuum, then any interface between the material and vacuum must close the gap, because the two sides cannot be joined by a continuous deformation that keeps the gap open. This bulk–boundary correspondence forces the surface to conduct. Because the twisting is a global property of the bands, local, symmetry-preserving perturbations cannot remove the surface states: the states are topologically protected. This also makes topological insulators an example of a phase not classified by the Landau theory of symmetry breaking: the trivial and topological insulators have the same symmetries and differ by a global invariant.

The invariant and time reversal

For the time-reversal-symmetric topological insulators the invariant takes values in $\mathbb Z_2$: a single bit, trivial or not. Time reversal is essential to the classification, and the reason is a sign. On a spin-$\tfrac12$ state the time-reversal operator is antiunitary and squares to

$$ \mathcal T^2 = -1 , $$

which forces every $\mathcal T$-invariant Hamiltonian to have its spectrum even-degenerate — Kramers degeneracy. The two partners of a Kramers pair cannot be split by any time-reversal-symmetric perturbation. The biquaternion framework treats this structure in the companion article Kramers Degeneracy and Antiunitary Symmetry in Biquaternionic Form, which constructs the module-level time-reversal operator as a rotation composed with the conjugation, shows that the relevant rotation element is the one squaring to $-e_0$ (the rotor of the $2\pi$ rotation), obtains $\mathcal T^2=-e_0$ on the spinor module, and identifies the resulting two-complex-dimensional commutant with a quaternionic line. The $\mathbb Z_2$ nature of the topological classification and the quaternionic nature of the Kramers structure are two faces of one sign, and the framework supplies the sign as an element of $\mathbb B$. The material still supplies the bands.

The helical edge and spin–momentum locking

In the two-dimensional case the boundary states are helical: a one-dimensional pair of counter-propagating modes whose spins are locked at right angles to their momenta, so that the spin points up for one direction of motion and down for the other. Because the two modes are a Kramers pair, backscattering them into each other requires a spin flip, which a time-reversal-symmetric perturbation cannot supply; conduction along the edge is therefore ballistic and robust. The two-dimensional phase was predicted for HgTe/CdTe quantum wells and observed there in 2007, and predicted independently for graphene with a Kane–Mele mass.

Three-dimensional topological insulators

In three dimensions the surface carries a two-dimensional gapless Dirac cone traversing the bulk gap, with the spin locked to the momentum in the surface plane. The band structure is the same object as in graphene, but the cone lives on the boundary rather than throughout the sample, and the material's bulk is a genuine insulator. The canonical materials are the bismuth and antimony chalcogenides $\mathrm{Bi_2Se_3}$, $\mathrm{Bi_2Te_3}$ and $\mathrm{Sb_2Te_3}$. Both cases are, algebraically, the massless Dirac equation of the framework localised to a boundary.

Klein Tunnelling in Graphene

Graphene's massless Dirac electrons show the Klein effect of the companion article. A $p$–$n$ junction — a region where the Fermi level is shifted from one side of the Dirac point to the other — is an electrostatic step for the Dirac cone, and the massless limit of the Klein calculation gives unit transmission for normal incidence regardless of the step height. The physical reading is that the electron crosses by briefly becoming a hole inside the barrier, so it never occupies a classically forbidden state; the framework's reading, from the companion article, is that the step shifts the mass shell and exchanges the two frequency branches. Normal-incidence transmission in graphene $p$–$n$ junctions is the clean experimental instance of the effect.

The Biquaternion Reading

What the framework transcribes

The low-energy equations of Dirac matter are the framework's Dirac equation with the parameters of the material, and the transcription is exact:

• the two-component spinor is the framework's minimal ideal;
• the kinetic term $v_F\boldsymbol{\sigma}\cdot\mathbf{p}$ is the framework's massless Dirac operator, with $c$ replaced by $v_F$;
• the linear dispersion $E=\pm v_F|\mathbf p|$ is the framework's mass shell in the massless limit;
• the gap-opening perturbation is the framework's chirality-mixing mass;
• the helical edge states are the framework's spinor structure with the Kramers sign of the companion article;
• Klein tunnelling is the framework's massless step problem.

What the framework adds

Three things, none of them a derivation from first principles.

• The location of the mass. The framework says exactly which term opens the gap: the term coupling the two minimal left ideals. Sublattice-symmetry breaking in graphene is that term, and saying so is a genuine identification rather than a relabelling, because both are the unique chirality-mixing bilinear.
• The sign structure of protection. The framework's Kramers article places $\mathcal T^2=-e_0$ in the algebra and identifies the commutant as a quaternionic line, which is the algebraic content of $\mathbb Z_2$ protection.
• A counting caution. The eight real components of the biquaternion field do not decompose into sublattice and valley the way graphene's spinor does; the framework's second ideal is chirality, not valley. This negative statement disciplines any attempt to derive valleytronics from the algebra.

What the framework does not supply

It does not supply the honeycomb lattice, the hopping $\gamma_0$, the Fermi velocity $v_F$, the positions of the Dirac points, the valley degeneracy $g_v=2$, the Berry phase $\pi$, the $\mathbb Z_2$ value of any material, or the band inversion that produces it. All of these are material and many-body inputs. The framework is a language in which the effective equation can be written and the gap and protection terms named; it is not a theory of which crystals are topological insulators. The corpus's standing disclaimer applies without modification.

Open Questions

1. The valley question. Is there any framework-natural operation that pairs two Dirac points and so supplies valley isospin, or is valley isospin irreducibly lattice-level? The counting argument of this article says the algebra has one internal two-fold structure, already spoken for by chirality.

2. The spin–valley coupling. In real materials spin–orbit coupling and intervalley scattering entangle the isospins. Whether the framework's dictionary can accommodate a term coupling the minimal ideals to a lattice label is open.

3. The Haldane and Kane–Mele masses. These are gap terms that are not the naive staggered potential. Do they have distinguishable framework-level signatures — different members of the chirality-mixing family, or genuinely different terms?

4. The Fermi velocity. The framework's $c$ is structural. Is $v_F\approx c/300$ a quantity the framework can only accept, or does any of its structure constrain the ratio?

5. The many-body question. Topological protection is a statement about a filled band. The framework's one-particle Dirac equation states the effective theory; the many-body filling is outside it, and the corpus has not yet treated a condensed-matter many-body system.

6. Empirical contact. As everywhere, whether the framework's transcription yields any prediction distinguishing it from the effective Dirac theory it reproduces is the standing open question.

Summary

Dirac matter is the family of solids whose low-energy electrons obey a Dirac equation: graphene with its massless Dirac cones, and the surface and edge states of topological insulators. The honeycomb lattice's two-site cell gives a two-component sublattice pseudospin and the effective equation $H=v_F\boldsymbol{\sigma}\cdot\mathbf{p}$, linear and gapless at the Dirac point; the two inequivalent cones add a valley isospin; a sublattice-symmetry-breaking term inserts a Dirac mass and opens a gap.

The biquaternion framework transcribes these equations exactly and locates two structures: the gap is the parent's chirality-mixing mass term, the unique bilinear coupling the two minimal left ideals; and the $\mathbb Z_2$ protection of the topological phase is the Kramers sign $\mathcal T^2=-e_0$ of the companion article, whose commutant is a quaternionic line. The framework does not supply the lattice, the Fermi velocity, the Dirac-point positions, the valley degeneracy or the invariant of any material, and — the article's sharpest caution — its two minimal ideals are chirality, not valley, so identifying them with graphene's valleys is an extra input rather than a consequence. What the framework offers is a precise language for the effective equation and the exact location of the mass and protection terms; the material supplies the rest.

Summary of Notation

Further Reading

• P. R. Wallace, "The band theory of graphite," Physical Review 71 (1947) 622–634, for the original honeycomb band structure.
• G. W. Semenoff, "Condensed-matter simulation of a three-dimensional anomaly," Physical Review Letters 53 (1984) 2449–2452, for the Dirac equation of graphene and the anomalous Landau level.
• K. S. Novoselov et al., "Two-dimensional gas of massless Dirac fermions in graphene," Nature 438 (2005) 197–200, and Y. Zhang et al., "Experimental observation of the quantum Hall effect and Berry's phase in graphene," Nature 438 (2005) 201–204, for the Dirac-fermion observation.
• A. H. Castro Neto et al., "The electronic properties of graphene," Reviews of Modern Physics 81 (2009) 109–162, for the standard review of the effective theory.
• C. L. Kane and E. J. Mele, "Quantum spin Hall effect in graphene," Physical Review Letters 95 (2005) 226801, and "$\mathbb Z_2$ topological order and the quantum spin Hall effect," Physical Review Letters 95 (2005) 146802, for the invariant and the protection.
• B. A. Bernevig, T. L. Hughes and S.-C. Zhang, "Quantum spin Hall effect and topological phase transition in HgTe quantum wells," Science 314 (2006) 1757–1761, and M. König et al., "Quantum spin Hall insulator state in HgTe quantum wells," Science 318 (2007) 766–770, for the prediction and observation.
• M. Z. Hasan and C. L. Kane, "Colloquium: Topological insulators," Reviews of Modern Physics 82 (2010) 3045–3067, and X.-L. Qi and S.-C. Zhang, "Topological insulators and superconductors," Reviews of Modern Physics 83 (2011) 1057–1110, for the reviews.
• M. I. Katsnelson, K. S. Novoselov and A. K. Geim, "Chiral tunnelling and the Klein paradox in graphene," Nature Physics 2 (2006) 620–625, for Klein tunnelling in graphene.
• The companion articles of this series: The Dirac Equation in Biquaternionic Form, Kramers Degeneracy and Antiunitary Symmetry in Biquaternionic Form, The Klein Paradox in Biquaternionic Form, and Fermion Doubling and the Nielsen–Ninomiya Theorem in Biquaternionic Form.