Prerequisites
Time Reversal
Time-reversal for spin-\(1/2\) systems:
\[\Theta = -is_y K.\]\(\Theta^2 = -\mathbb{1}\).
Bra-ket properties:
\[\begin{split}\bra{\psi} \Theta \ket{\phi} &= -\bra{\phi} \Theta \ket{\psi}, \\ \bra{\Theta\psi} \ket{\Theta\phi} &= \bra{\phi}\ket{\psi}, \\ \bra{\Theta \psi} \Theta A \Theta^{-1} \ket{\Theta \phi} &= \bra{\phi} A^\dagger \ket{\psi}.\end{split}\]Kramer degeneracy: if the Hamiltonian is invariant under time-reversal, energy levels at \(\vb{k}\) and \(-\vb{k}\) are identical.
Kramer pair:
\[E_{n,\uparrow}(\vb{k}) = E_{n,\downarrow}(-\vb{k}).\]Caution
When spin-orbit interaction comes into play, it makes no sense to write \(\uparrow\) and \(\downarrow\) any more, since \(S_z\) may no longer commute with \(H\).
At a TRIM \(\vb{k}\),
\[E_{n,\uparrow}(\vb{k}) = E_{n,\downarrow}(\vb{k}).\]
Note
TRS guarantees two-fold degeneracy at TRIMs. The degeneracy pairs are given by Kramers pairs.
Commutation relations: [Bernivig2013]
\[\begin{split}\Theta c_{ja\uparrow} \Theta^{-1} &= c_{ja\downarrow}, \\ \Theta c_{ja\downarrow} \Theta^{-1} &= -c_{ja\uparrow}, \\ \Theta c_{ja\uparrow}^\dagger \Theta^{-1} &= c_{ja\downarrow}^\dagger, \\ \Theta c_{ja\downarrow}^\dagger \Theta^{-1} &= -c_{ja\uparrow}^\dagger, \\\end{split}\]Note
It seems weird that we have minus signs in the above equations, since it implies
\[\Theta^2 c_{ja\uparrow} \Theta^{-2} = c_{ja\uparrow}.\]However, this is exact what we want because \(\Theta^2\) and \(\Theta^{-2}\) acts on Hilbert spaces of different particle numbers, and therefore have different values.
Ferromagnetism and antiferromagnetism breaks TRS. See 反铁磁到底有没有破缺时间反演对称性，为什么很多狄拉克材料具有反铁磁性如SrMnSb2等？ - 方辰的回答 - 知乎.
Topology
Pfaffian: for an anti-symmetric matrix \(A\),
\[\operatorname{Pf}[A]^2 = \det[A].\]
Semiclassical Theory of Electron Transport
\(\tau\) denotes the relaxation time.
Scattering probability in interval \(\dd{t}\) is given by \(\dd{t}/\tau\).
Time evolution of an electron of velocity \(\vb{v}(t)\) at time \(\vb{v}(t)\):
Without scattering:
\[\langle \vb{v}(t+\dd{t}) \rangle = \vb{v}(t) + \frac{\vb{f}(t)}{m^*} \dd{t}.\]With scattering:
\[\langle \vb{v}(t+\dd{t}) \rangle = 0.\]Overall:
\[\langle \vb{v}(t+\dd{t}) \rangle = \qty(1 - \frac{\dd{t}}{\tau}) \qty( {\vb{v}(t) + \frac{\vb{f}(t)}{m^*} \dd{t}} ).\]
Equation of motion:
\[m^* \qty({\dv{\vb{v}(t)}{t} + \frac{\vb{v}(t)}{\tau}}) \approx \vb{f}(t).\]Lorentz force law:
\[\dv{\vb{v}(t)}{t} + \frac{\vb{v}(t)}{\tau} \approx -\frac{e}{m^*} \qty( \vb{E} + \frac{1}{c} \vb{v} \times \vb{B} ).\]Stationary state, with \(\vb{B}\) along \(z\)-axis:
\[\begin{split}\begin{cases} v_x + \omega_c \tau v_y = -(e\tau/m^*) E_x, \\ v_y - \omega_c \tau v_x = -(e\tau/m^*) E_y, \\ v_z = -(e\tau / m^*) E_z. \end{cases}\end{split}\]Solution:
\[\begin{split}\begin{cases} \displaystyle J_x = \frac{ne^2\tau}{m^*} \frac{E_x - \omega_c \tau E_y}{1+(\omega_c \tau)^2}, \\ \displaystyle J_y = \frac{ne^2\tau}{m^*} \frac{E_y + \omega_c \tau E_x}{1+(\omega_c \tau)^2}, \\ \displaystyle J_z = \frac{ne^2\tau}{m^*}E_z, \end{cases}\end{split}\]where
\[J = -n(e/c)\vb{v}.\]Solution on the \(xy\)-plane:
\[\begin{split}\begin{pmatrix} J_x \\ J_y \end{pmatrix} = \frac{\sigma_0}{1+(\omega_c \tau)^2} \begin{pmatrix} 1 & -\omega_c \tau \\ \omega_c \tau & 1 \end{pmatrix} \begin{pmatrix} E_x \\ E_y \end{pmatrix},\end{split}\]where
\[\sigma_0 = \frac{ne^2 \tau}{m^*}.\]Conductivity tensor:
\[\begin{split}\sigma_{xx} &= \sigma_{yy} = \frac{1}{1+(\omega_c \tau)^2} \sigma_0, \\ \sigma_{xy} &= -\sigma_{yx} = \frac{-\omega_c \tau}{1+(\omega_c \tau)^2} \sigma_0.\end{split}\]
SdH Oscillation
\(B\) very large, \(\omega_c \tau \gg 1\),
\[\begin{split}\sigma_{xx} &\approx \frac{ne^2}{m^* \omega_c^2 \tau} = \frac{nm^* c^2}{B^2 \tau}, \\ \sigma_{xy} &\approx \frac{ne^2}{m^* \omega_c} = \frac{nec}{B}.\end{split}\]Lifshitz-Kosevich theory: oscillation of \(\sigma_{xx}\) is given by
\[\Delta \sigma_{xx} \propto \cos \qty[ 2\pi\qty(\frac{F}{B} - \frac{1}{2} + \beta)],\]where \(F\) is the frequency of oscillation, and \(2\pi \beta\) is the Berry phase of every cycle of motion.
\(\beta = 1/2\) for Dirac fermions.
Item in the \(\cos\) originates from
\[A_N = \frac{2\pi e}{\hbar c} B\qty(N + \frac{1}{2} - \beta),\]i.e. the area in the \(\vb{k}\)-space encircled by the closed path of electron motion.
Conductivity and Resistivity Tensor
If \(\sigma_{xy} = 0\),
\[\rho_{xx} = \frac{1}{\rho_{xx}}.\]If \(\abs{\sigma_{xx}} \ll \abs{\sigma_{xy}}\),
\[\rho_{xx} = \frac{\sigma_{xx}}{\sigma_{xy}^2}.\]This is the case in strong magnetic field:
\[\abs{\sigma_{xx}} \ll \abs{\sigma_{xy}} \Longleftrightarrow \omega_c\tau \gg 1.\]
Glossary
- TRIM/時間反転不変運動量/时间反演不变动量
Time-reversal invariant momentum