Introduction

In some condensed matter systems the electron excitation spectrum near the Fermi surface can be described by a Dirac-type matrix equation. This equation does not arise from relativistic considerations, but rather by linearizing the energy dispersion near (a finite number of) Dirac points (intersections of the energy dispersion with the Fermi level). Such systems can exhibit fermion fractionalization if the Dirac equation possesses isolated bound states in the gap between negative-energy (valence band) states and positive-energy (conduction band) states.

A familiar example is 1-dimensional polyacetylene (1). As is generally the case in one dimension, there are two Dirac points for polyactylene. Therefore a Dirac description of electron motion near each Dirac point is possible. A distortion of the underlying lattice (Peierls' instability) perturbes the electron motion in a way that couples the two Dirac points and opens a gap in the energy spectrum. In the Dirac equation description this is achieved by coupling the Dirac field $\Psi$ to a scalar field $\varphi$, which is a measure of the lattice distortion. $\varphi$ enjoys a $Z_2$ symmetry with two ground states in which it takes homogenous values $\pm \, \varphi_0$. This coupling leads to a Dirac mass $\propto \vert \varphi_0 \vert$. But $\varphi$ can also take a position-dependent kink profile (soliton) $\varphi_s$ that interpolates between the two vacua $\pm \varphi_0$. This ``twisting" of the mass parameter describes a defect in the lattice distortion. The Dirac equation with the kink profile $\varphi_s$ replacing the homogenous mass $\pm \varphi_0$possesses a single mid-gap (zero-energy) eigenstate. This gives rise to fractional fermion number for the electrons: 1/2 per spin degree of freedom (1).

Recently a similar story has been told by C.-Y. Hou, A. Chamon and C. Mudry (2) (HCM) about (monolayer) graphene. This is a 2-dimensional array of carbon atoms formed from a superposition of two triangular sublattices, A and B.

The generators of lattice A are ${\bf a} \$   and$\ {\bf a_2}$. The three vectors $\bf s_{i}$ connect any site from lattice A to its nearest neighbor sites belonging to B. They are

$${\bf s_{1}}= (0, -1) a, {\bf s_{2}}= \left({\frac{\sqrt{2}}{2}}, ... ...{1}{2}\right) a, {\bf s_{3}}= \left(-{\frac{\sqrt{3}}{2}}, \frac{1}{2}\right) a$$ (1)

where $a$ is the lattice spacing.

When no lattice distortion is considered, the tight-binding Hamiltonian, with uniform hopping constant $t$, is taken as

$$H_{0} = - t\, \sum\limits _{{\bf r} \epsilon A} \ \sum\limits _{i... ...\, b ({\bf r} + {\bf s}_{i}) + b^{\dagger} ({\bf r} + {\bf s}_{i}) a ({\bf r}))$$ (2)

where the fermion operators $a$ and $b$ act on sublattices A and B.

$H_{0}$ is diagonal in momentum space

$$\left\{\!\!\!\!\begin{array}{c} a\, ({\bf k})\\ b\, ({\bf k}) \en... ...\!\!\begin{array}{c} a\, ({\bf r})\\ b\, ({\bf k}) \end{array}\!\!\!\! \right\}$$ (3)

$$\begin{eqnarray}H_{0} = \sum_{\bf b} \ \bigg(\Phi\, ({\bf k})\, a... ...\sum\limits _{i=1,2,3} e^{i {\bf b}\cdot {\bf s}_i }\hspace{.5in}\end{eqnarray}$$ (4a)

The single particle energy spectrum $E({\bf k})= \pm \mid \Phi ({\bf k})\mid$ contains two zero-energy Dirac points at

$${\bf k}= {\bf K_{\pm}}= \pm \left(\frac{4\pi}{3\sqrt{3}\, a}, 0\right) ; \quad \Phi \, ({\bf K_{\pm}}) = 0$$

This parallels the 1-dimensional case, but is rare in two dimensions.

$H_0$ is linearized around the two Dirac points, ${\bf k}= {\bf K_{\pm}}+ {\bf p}$, and supplemented by a term arising from a (Kulé) distortion of the lattice; this couples the two Fermi points.

$$H = \sum\limits_{\bf k}\ (\varphi_+\, ({\bf k})\, a^\dagger_+\, b_+\, ({\bf p})+ \varphi^\ast_+\, ({\bf p})\, b^\dagger_+\, ({\bf k})\, a_+\, ({\bf p})$$

$$+ \varphi_- \, ({\bf p})\, a^\dagger_+\, ({\bf k})\, b_-\, ({\bf k})+ \varphi ^\ast\, ({\bf k})\, b^+_-\, ({\bf k})\, a_-\, ({\bf p})$$

$$+\sum\limits_{\bf k}\ \bigg(\triangle_0 a^\dagger_+\, ({\bf k})\, b_-\, ({\bf p})+ \triangle^\ast_0\, b^\dagger_-\, ({\bf p})\, as_+\, ({\bf p})$$

$$+ \triangle^\ast_0 \, a^\dagger_-\, ({\bf p})\, b_+ \, ({\bf p})+ \triangle_0\, b^\dagger_- \, ({\bf p})\, a_- ({\bf p})\bigg)$$ (5)

${\varphi_{\pm}}$ is the linearization of $\Phi: {\varphi_{\pm}}({\bf p})= \pm v_F\, (p_x \pm i \, p_y), v_F = 3 t a/2$, (hence forth $v_F$ is set to unity) and $a_\pm , b_\pm$ are fermion operators near the Dirac points: ${a_{\pm}}({\bf p})\equiv a\, ({\bf K_{\pm}}+ {\bf p}), b_\pm ({\bf P})\equiv b ({\bf K_{\pm}}+ {\bf p})$. In the second sum $\triangle_0$ is a homogenous, but complex order parameter, which effects the coupling between the Fermi points ${\bf K_{\pm}}$ and leads to a mass gap in the single-particle energy dispersion:

$$\epsilon\, ({\bf k})= \pm \, \sqrt{{\bf p}+\mid\sigma_0\mid^2}$$

To find zero modes for this system HCM promote the mixing parameter $\triangle_0$ to an inhomogenous complex function $\triangle$ with a profile like the scalar field in the Nielsen-Olesen-Landau-Ginsburg-Abrikosor vortex. To this end the Hamiltonian (5) is presented in coordinate spaces as

$$H =\int d^2 r \Psi^\ast\, ({\bf r})\, K \Psi ({\bf r})$$ (6)

where $\Psi\, ({\bf r})$ is a 4-component ``spinor"

$$\Psi = \left( \begin{array}{c} \psi^b_+\\ [1ex] \psi^a_+\\ [... ... d^2 k \, e^{-i {\bf p}\cdot {\bf r}}\, {b_{\pm}}\,({\bf p})$}$$

and with $K$ is the 4x4 matrix

$$K = \left( \begin{array}{cccc} 0 & -2 i \partial_{z} & \triangle ... ... [1ex] 0 & \triangle^{\ast}({\bf r})&2 i \partial_z\ast & 0 \end{array} \right)$$ (7)

with $-2 i \partial_{z} = \frac{1}{i}\ (\partial_{x} -i \, \partial_{y})$.

HCM take $\triangle ({\bf r})$ in the $n$-vortex form: $\triangle (r)\, e^{i n \theta}$ where n is an integer, $\triangle ({\bf r})$ vanishes as $r ^{\vert n\vert}$ for small $r$, and approaches the mass-generating value $\triangle_0$ at large $r$. They then establish the occurrence of $\vert n\vert$ zero modes, i.e. solutions to $K \Psi = 0$, on lattice A (B) for negative (positive) $n$, and they construct explicitly the solutions for $n= +1$. Therefore Fermion number $\alpha \int d^2 r \Psi^\ast \Psi$ is fractionalized.