Weyl semimetal

Weyl Semimetals¶

Tags: #weyl_nodes

Basic Hamiltonian and Properties¶

An analogous situation of Weyl fermions prevails in 3+1 dimensions. The gamma matrices are, as Dirac found, now 4 × 4 matrices and can be represented as $\gamma^0 = \mathcal{I}^{2 \times 2} \otimes \tau_x$, $\gamma^i = \sigma^i \otimes i\tau_y$ and $\gamma^5 = -\mathcal{I}^{2 \times 2} \otimes i\tau_z$. Again, if we identify chiral components $\gamma_5 \psi_\pm = \pm\psi_\pm$, where $\psi_\pm$ are effectively two component vectors, we have for the massless Dirac equation $\(i\partial_t\psi_\pm = H\pm\psi_\pm$\) $\(H_\pm = \mp\vec{p} \cdot \vec{\sigma}$\) Thus Weyl fermions propagate parallel (or antiparallel) to their spin, which defines their chirality. We will see that a single chirality of Weyl fermions cannot be realized in 3D, but momentum separated pairs can arise. These are the Weyl semimetals.

These are three-dimensional topological phases of matter with strong spin-orbit coupling. They are featured by pairs of linear band crossings, called Weyl nodes, in momentum space. Since the Weyl nodes are topologically protected and closely associated with the Chiral Anomaly, WSMs impose a plethora of characteristic phenomena, such as surface Fermi arcs, nonlocal transport, and anomalous magnetoconductance.

In principle, Weyl nodes of opposite Chirality do not need to have the same electron population. An electron population imbalance between Weyl nodes of different Chirality, called Chiral Chemical Potential can be achieved, e.g. by the chiral anomaly with applying parallel electric and magnetic fields, by a strain deformation or in a superlattice system with breaking both Time reversal symmetry and inversion symmetries.

Topological aspects of WSM¶

In Weyl semimetals, the conduction and valence bands coincide in energy over some region of the Brillouin zone. Furthermore, this band touching(determined by symmetry) is stable at least to small variations of parameters(interaction or disorder). - This could be because disorder is average and on average keeps the symmetry - Band structure of semimetal has very low density of states on the Fermi level

Spin rotation symmetry (ignore spin-orbit coupling) : Bands are doubly degenerate. Both Time reversal symmetry ($\mathcal{T}$) and inversion symmetry: Bands are doubly degenerate. Crystal momenta are invariant. Only $\mathcal{T}$ is present : Band is non-degenerate, as crystal momentum is reversed under its action. At $\textbf{k} = -\textbf{k}$ which is TRIM, there is a Kramer's degeneracy(Kramers Theorem) present. $\mathcal{T}$ is broken, and only inversion is present : Bands are typically non-degenerate.

$\mathcal{T}$ is broken, and only inversion is present$\mathcal{T}$ is broken and only inversion is present

$\(H = f_0 \mathcal{I}^{2 \times 2} + f_1\sigma_x + f_2\sigma_y + f_3\sigma_z$\) To bring the bands in coincidence we need to adjust all three coefficients $f_1 =f_2 =f_3 =0$ simultaneously, which requires three independent variables, i.e. we are in 3D. As we can then expect band touchings without any special fine-tuning, we can readily argue that the existence of Weyl nodes is stable to small perturbations of Hamiltonian parameters.

$f_1(\textbf{k})$ typically vanishes in the momentum space. Typically, this is a 2D surface. If we demand a simultaneous zero of $f_2(\textbf{k})$ and $f_3(\textbf{k})$, this specifies the intersection of three independent surfaces, which typically occurs at a point. Consider a perturbation that changes the functions by a small amount. This will also move the zero surfaces and their points of intersection by a small amount, but the intersection will persist, just at a different crystal momentum. The Weyl nodes cannot be removed by any small perturbation and may disappear only by annihilation with another Weyl node.

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Hamiltonian¶

$\(H = v\tau_x\mathbf(\sigma\cdot \textbf{k}) + m \tau_z + b\sigma_z +b^\prime \tau_z\sigma_x$\) $$ H= \left( {\begin{array}{cc} m\mathcal{I}^{2\times2} + b\sigma_z +b^\prime \tau_z\sigma_x & v\mathbf(\sigma\cdot \textbf{k}) \ v\mathbf(\sigma\cdot \textbf{k}) & -m\mathcal{I}^{2\times2} + b\sigma_z - b^\prime \tau_z\sigma_x \ \end{array} } \right) $$ Where, $\textbf{k} =(k_x,k_y,k_z)$ and $\tau_n$'s are Pauli Matrices for the pseudospin orbital degrees of freedom. m is a mass parameter, $b$ and $b^\prime$ are Zeeman field that physically can correspond to magnetic fields in the z and x directions, respectively

Inversion symmetry breaking WSMs¶

For these kinds of WSMs the anomalous Hall conductivity vanishes due to presence of $\mathcal{T}$ symmetry.