Basic Theory of Topological Insulators
とあるトポロジカルインデックス
Berry Phase
For \(H\) depending on parameters \(\vb{R}(t)\), the Schrödinger equation becomes
\[H[\vb{R}(t)]\ket{n,\vb{R}(t)} = E_n[\vb{R}(t)]\ket{n,\vb{R}(t)}.\]Time evolution of ket as \(\vb{R}(t)\) varies with time.
\[\ket{n,t} = \exp\qty(i \gamma_n) \times \exp\qty({-\frac{i}{\hbar} \int \dd{t'} E_n[\vb{R}(t')]}) \ket{n,\vb{R}(t)}.\]The first \(\exp\) factor is known as the Berry phase term.
The second \(\exp\) factor is known as the dynamical term.
Note
Berry phase is the nontrivial phase factor caused by the time-dependence of Hamiltonian.
Berry phase for closed loop:
\[\begin{split}\gamma_n[C] &= i \oint_C \dd{\vb{R}} \cdot \bra{n,\vb{R}} \nabla_{\vb{R}} \ket{n,\vb{R}} \\ &= -\oint_C \dd{\vb{R}}\cdot \vb{A}_n(\vb{R}) \\ &= -\int_S \dd{\vb{S}} \cdot \vb{B}_n(\vb{R}).\end{split}\]Berry connection:
\[\vb{A}_n(\vb{R}) = -i \bra{n,\vb{R}} \nabla_{\vb{R}} \ket{n,\vb{R}}.\]Berry curvature:
\[\vb{B}_n(\vb{R}) = \nabla_{\vb{R}} \times \vb{A}_n(\vb{R}).\]
QHE: Chern Number and Conductivity
From Wikipedia: Quantum Hall Effect
Weak Magnetic Field
If the magnetic field is weak, we have the classical result
\[\begin{split}\rho_{xx} &\sim \mathrm{const}. \\ \rho_{xy} &\propto B.\end{split}\]
Strong Magnetic Field
For strong magnetic field, \(\rho_{xy}\) becomes significant and both \(\sigma_{xx}\) and \(\rho_{xx}\) tend to zero.
Warning
\(\sigma_{xx} \neq 1/\rho_{xx}\) in such case. Instead, one has \(\sigma_{xx} \propto \rho_{xx}\).
Hall Conductance: TKNN Derivation
Set-up: magnetic field along \(z\)-axis while electric field along \(y\).
Perturbation to the first order:
\[\ket{n}_E = \ket{n} + \sum_{m \neq n} \frac{\bra{m} (-eEy) \ket{n}}{E_n - E_m} \ket{m} + \cdots.\]Transversal current:
\[\langle j_x \rangle_E = \sum_n f(E_n) \bra{n}_E \qty(\frac{e v_x}{L^2}) \ket{n}_E.\]Hall conductance:
\[\begin{split}\sigma_{xy} &= \frac{\langle j_x \rangle_E}{E} \\ &= \nu \frac{e^2}{h}.\end{split}\]We have used
\[\bra{u_{m \vb{k}'}} v_\mu \ket{u_{n\vb{k}}} = \frac{1}{\hbar} (E_{n\vb{k}} - E_{m\vb{k}'}) \bra{u_{m \vb{k}'}} \pdv{}{k_\mu} \ket{u_{n\vb{k}}}\]to simplify the above expression.
The TKNN number is given by
\[\begin{split}\nu &= \sum_n \int_{\mathrm{BZ}} \frac{\dd{^2 \vb{k}}}{2\pi} \qty(\pdv{A_{n,y}}{k_x} - \pdv{A_{n,x}}{k_y}) \\ &= -\frac{1}{2\pi} \gamma_n[\partial \mathrm{BZ}],\end{split}\]i.e. the winding number of Berry phase along the boundary of BZ.
\(\nu \in \mathbb{Z}\).
Note
TKNN number may be also called Chern number.
Hall Conductance: Kubo Derivation
Hint
Under construction.
Hall Conductance: Laughlin Argument
See Laughlin 规范实验 - 泰勒猫爱丽丝的文章 - 知乎.
The following is the simplified version [Shen2012].
Two-dimensional electron gas rolled as a cylinder along the \(y\)-direction, i.e. \(y\) now becomes \(S^1\).
A magnetic flux \(\phi\) now threads through the \(x\)-direction.
Charge transport as magnetic flux increases:
\[\Delta Q = \sigma_{xy} \Delta \Phi.\]Quantization of magnetic flux:
\[\Delta \Phi = \frac{h}{e}.\]Gauge invariance: kets returns to their original states as magnetic flux is increased by a magnetic flux quantum. Therefore, a integer number of charge is transported.
\[\sigma_{xy} = \frac{n e^2}{h}.\]
Hall Conductance: Wilson Loop Derivation
See 霍尔电导率 via Wilson Loop - PhakeNews的文章 - 知乎.
Hint
Under construction.
Example: A Two-Band Model
Brillouin zone: \(\vb{k} \in T^2\).
Parameter space: \(\vb{h} \in \mathbb{R}^3\). We assume that \(\vb{h}(\vb{k}) \neq 0\) so that we have a two-band insulator.
\(\vb{h}(\vb{k})\) sweeps a closed surface in \(\mathbb{R}^3\).
Magnetic monopole analogy: we are going to calculate the magnetic flux through this closed surface.
We may forget everything about \(\vb{k}\) now and take \(\vb{h}\) as \(\vb{R}\) in the above derivation of Berry phase.
Hamiltonian: magnetic dipole,
\[H(\vb{k}) = \vb{h}(\vb{k}) \cdot \vb*{\sigma}.\]Ket space: \(\ket{u} \in \mathbb{C}^2\). More specifically, \(S^2\).
Dispersion relation:
\[\epsilon_{\pm}(\vb{k}) = \pm h(\vb{k}) = \pm \abs{\vb{h}(\vb{k})}.\]Ket solution: assuming filling only the lower band,
For \(\vb{h}\) lying on the southern hemisphere,
\[\begin{split}\ket{u_-^{\mathrm{S}}[\vb{h}(\vb{k})]} = \begin{pmatrix} \sin \frac{\theta}{2} \\ -e^{i\phi} \cos \frac{\theta}{2} \end{pmatrix}.\end{split}\]For \(\vb{h}\) lying on the northern hemisphere,
\[\begin{split}\ket{u_-^{\mathrm{S}}[\vb{h}(\vb{k})]} = \begin{pmatrix} e^{-i\phi} \sin \frac{\theta}{2} \\ -\cos \frac{\theta}{2} \end{pmatrix}.\end{split}\]The kets are unimportant by themselves. We want only the Berry connection.
Berry connection:
For \(\vb{h}\) lying on the southern hemisphere,
\[\vb{A}^{\mathrm{S}}_-(\vb{h}) = \frac{1}{2}(1+\cos\theta) \dd{\phi}.\]For \(\vb{h}\) lying on the northern hemisphere,
\[\vb{A}^{\mathrm{N}}_-(\vb{h}) = -\frac{1}{2}(1-\cos\theta) \dd{\phi}.\]Magnetic monopole analogy: vector potential of a magnetic dipole at the origin.
We can’t have an \(\vb{A}\) covering the whole space. We have to either discard the \(z\) axis or the equator. Here we chose the latter.
Gauge transformation on the equator:
\[\vb{A}^{\mathrm{N}}_-(\vb{h}) = \vb{A}^{\mathrm{S}}_-(\vb{h}) - \dd{\phi}.\]
TKNN number:
\[\nu = \frac{1}{2\pi} \int \dd{\vb{A}}\]counts how many times the closed surface encircles the origin.
Topologically trivial: \(\nu = 0\) if the surface does not encircle the origin.
Topologically nontrivial: \(\nu \neq 0\) if the surface does encircle the origin.
Pump: ℤ₂ Index and Polarization
Polarization and Time-Reversal Polarization
We define
\[w_{\alpha\beta}(\vb{k}) = \bra{u_{\alpha,-\vb{k}}} \Theta \ket{u_{\beta,\vb{k}}},\]where \(\alpha\) and \(\beta\) are indices of bands.
\(w\) relates time-reversal pairs.
\[\ket{u_{\alpha,-\vb{k}}} = \sum_{\beta} w^*_{\alpha\beta}(\vb{k}) \Theta \ket{u_{\beta,\vb{k}}}.\]\(w\) is unitary.
\[\sum_{\alpha} w^\dagger_{\gamma\alpha}(\vb{k}) w_{\alpha\beta}(\vb{k}) = \delta_{\beta\gamma}.\]\(w\) satisfies
\[w_{\beta\alpha}(-\vb{k}) = -w_{\alpha\beta}(\vb{k}).\]\(w\) is antisymmetric at TRIMs.
\[w_{\beta\alpha}(-\vb{\Lambda}) = -w_{\alpha\beta}(\vb{\Lambda}).\]
Berry connection matrix
\[\vb{a}_{\alpha\beta}(\vb{k}) = -i \bra{u_{\alpha,\vb{k}}} \nabla_{\vb{k}} \ket{u_{\beta,\vb{k}}}.\]\(\vb{a}\) at time-reversal momentum pairs:
(1)\[\begin{split}\vb{a}(-\vb{k}) &= w(\vb{k}) \vb{a}^*(\vb{k}) w^\dagger(\vb{k}) + iw(\vb{k}) \nabla_{\vb{k}} w^\dagger(\vb{k}), \\ \tr[\vb{a}(\vb{k})] &= \tr[\vb{a}(-\vb{k})] + i\tr[w^\dagger(\vb{k}) \nabla_{\vb{k}} w(\vb{k})].\end{split}\]
Wannier function: of a certain band,
\[\ket{R} = \sum_{k=-\pi}^\pi \frac{e^{ik(x-R)}}{\sqrt{L}} \ket{u_k}.\]Polarization: of a certain band,
\[P_\rho = -\bra{R=0} x \ket{R=0}.\]Polarization given by Berry connection:
\[P_\rho = -\frac{1}{L} \sum_{k=-\pi}^\pi \bra{u_k} i\pdv{}{k} \ket{u_k} = \int_{-\pi}^\pi \frac{\dd{k}}{2\pi} a(k).\]
Time-Reversal Polarization: of a time-reversal pair,
\[P_\theta = P_1 - P_2 = 2P_1 - P_\rho,\]where
\[\begin{split}P_i &= \int_{-\pi}^\pi \frac{\dd{k}}{2\pi} a_{ii(k)}, \\ P_\rho &= P_1 + P_2.\end{split}\]
Spin Pump: ℤ₂ Index and Time-Reversal Polarization
The Hamiltonian satisfies
\[\begin{split}H[t + T] &= H[T], \\ H[-t] &= \Theta H[t] \Theta^{-1}.\end{split}\]At \(t=0\) and \(t=T/2\), the Hamiltonian is time-reversal invariant. Therefore,
\[\begin{split}\Theta \ket{u_2(k)} &= e^{-i\chi(k)} \ket{u_1(-k)}, \\ \Theta \ket{u_1(k)} &= -e^{-i\chi(-k)} \ket{u_2(-k)}. \\ w(k) &= \begin{pmatrix} 0 & e^{-i\chi(k)} \\ -e^{-i\chi(-k)} & 0 \end{pmatrix}, \\ a_{11}(-k) &= a_{22}(k) - \pdv{}{k} \chi(k).\end{split}\]Note
The relation between \(a_{11}\) and \(a_{22}\) does not hold if TRS is broken.
Time-reversal polarization: with (1),
\[\begin{split}P_\theta &= \int_{0}^{\pi} \frac{\dd{k}}{2\pi} [A(k) - A(-k)] - \frac{i}{\pi} \log \frac{w_{12}(\pi)}{w_{12}(0)} \\ &= \frac{i}{\pi} \cdot \frac{1}{2} \log \frac{\det[w(\pi)]}{\det[w(0)]} - \frac{i}{\pi} \log \frac{w_{12}(\pi)}{w_{12}(0)} \\ &= \frac{1}{i\pi} \log \qty(\frac{\sqrt{w_{12}(0)^2}}{w_{12}(0)} \cdot \frac{w_{12}(\pi)}{\sqrt{w_{12}(\pi)^2}}).\end{split}\]Spin transport in a half-period:
\[\Delta = \abs{P_\theta(T/2) - P_\theta(0)}.\]\(\Delta\) as \(\mathbb{Z}_2\) index:
\[(-1)^\Delta = \prod_{i=1}^4 \frac{w_{12}(\Lambda_i)}{\sqrt{w_{12}(\Lambda_i)^2}},\]where \(\Lambda_i\) are TRIMs (\(T\) plays the role of \(k_y\) here since they are both \(S^1\))
\[\begin{split}\Lambda_1 &= (k=0, t=0), \\ \Lambda_2 &= (k=\pi, t=0), \\ \Lambda_3 &= (k=0, t=T/2), \\ \Lambda_4 &= (k=\pi, t=T/2).\end{split}\]Note
\(\Delta \neq 0\) indicates that the system is topologically nontrivial. The Kramers pair at \(t=0\) and \(t=T/2\) are given by different electron pairs.
\(\mathbb{Z}_2\) index of multi-band systems:
\[(-1)^\nu = \prod_{i=1}^4 \frac{\operatorname{Pf}[w(\Lambda_i)]}{\sqrt{\det[w(\Lambda_i)]}}.\]
ℤ₂ Index via Wilson Loop
See 凝聚态中的拓扑（四）：从TKNN到Z2拓扑绝缘体 - 中科院物理所的文章 - 知乎.
At given \(k_y\), we define
\[\hat{W} = \hat{A}_{12} \hat{A}_{23} \cdots \hat{A}_{N-1,N} \hat{A}_{N1},\]where
\[\begin{split}\qty(\hat{A}_{ij})_{nm} &= \bra{\phi_{\vb{k}_i,n}} \ket{\phi_{\vb{k}_j,m}}, \\ \vb{k}_i &= (k_{x,i}, k_y),\end{split}\]and \(k_{x,i}\) devides \((-\pi,\pi)\) into \(N\) intervals.
\(\hat{W} \in \mathrm{U}(2)\) and therefore takes the form
\[\hat{W} = z\hat{W}_0\]where \(z\in \mathrm{U}(1)\) and \(\hat{W}_0 \in \mathrm{SU}(2)\).
\(\hat{W}_0\) takes eigenvalues \(e^{\pm i\Theta}\).
At \(k_y = 0,\pi\), TRS guarantees either \(e^{\pm i \Theta} = 1\) or \(e^{\pm i \Theta} = -1\).
As we travel from \(k_y = 0\) to \(k_y = \pi\), \(e^{\pm i \Theta}\) draws a closed loop on the cylinder \((0,\pi) \times \mathrm{U}(1)\). The winding number \(\nu\) is the \(\mathbb{Z}_2\) index.
Warning
It may seem that \(\hat{W} \equiv \mathbb{1}\). However, as we demand that \(\hat{A}_{ij} \approx \mathbb{1}\), \(\hat{W}\) could be nontrivial. For example, if each \(k\in (-\pi,\pi)\) is associated with two orthonormal eigenvectors in \(\mathbb{R}^2\) given by rotating \((\hat{\vb{x}}, \hat{\vb{y}})\) by \(k/2\) radian, then \(\hat{W} = -\mathbb{1}\).
Three-Dimensional Topological Insulators With Time-Reversal Invariance: ℤ₂ Index and Edge States
Three-Dimensional ℤ₂ Index
I could tell if it is a TI with these 4 bits.
The \(\mathbb{Z}_2\) index of three-dimensional topological insulators are denoted as
\[(\nu_0; \nu_1 \nu_2 \nu_3),\]where
\[\begin{split}(-1)^{\nu_0} &= \prod_{n_j = 0,\pi} \delta(\Lambda_{n_1, n_2, n_3}), \\ (-1)^{\nu_i} &= \prod_{\substack{n_i = \pi \\ n_{j\neq i} = 0}} \delta(\Lambda_{n_1, n_2, n_3}), \\ \delta(\Lambda_i) &= \frac{\operatorname{Pf}[w(\Lambda_i)]}{\sqrt{\det[w(\Lambda_i)]}}.\end{split}\]The index \(\nu_z\) equals to the \(\mathbb{Z}_2\) index of the two-dimensional system on \(k_z = \pi\).
In this two-dimensional subsystem, we may let \(k_y\) play the role of \(t\) in the spin pump.
If \(\nu_z \neq 0\), there is a recombination of Kramers pairs as we move from \(k_y = 0\) to \(k_y = \pi\).
Such recombination guarantees the existence of surface states.
Strong TI: \(\nu_0 = 1\).
Weak TI: \(\nu_0 = 0\) while \(\nu_i = 1\) for some \(i\).
Inversion Symmetry
Keeping track of complex phases? That’s too complex!
Define
\[\vb{a}^c(\vb{k}) = \tr [\vb{a}(\vb{k})] \in \mathbb{R}^3.\]Berry curvature:
\[F(\vb{k}) = \nabla_{\vb{k}} \times \vb{a}^c(\vb{k}).\]Berry curvature with symmetry:
\[\begin{split}\mathrm{TRS} &\Rightarrow F(-\vb{k}) = -F(\vb{k}), \\ \mathrm{Inversion\ Symmetry} &\Rightarrow F(-\vb{k}) = +F(\vb{k}).\end{split}\]When both TRS and inversion symmetry are present,
\[F(\vb{k}) \equiv 0\]and therefore there exists a gauge such that
\[\vb{a}(\vb{k}) = 0.\]
We define
\[v_{\alpha\beta}(\vb{k}) = \bra{u_{\alpha,\vb{k}}} \Pi \Theta \ket{u_{\beta,\vb{k}}},\]where \(\alpha\) and \(\beta\) are indices of bands.
\(w\) is unitary.
\[\sum_{\alpha} v^\dagger_{\gamma\alpha}(\vb{k}) v_{\alpha\beta}(\vb{k}) = \delta_{\beta\gamma}.\]\(v\) is anti-symmetric:
\[v_{\alpha\beta} = -v_{\beta\alpha}.\]
\(\vb{a}^c\) given by \(v\):
\[\begin{split}\vb{a}^c(\vb{k}) &= \frac{i}{2} \tr\qty[v^\dagger \nabla_{\vb{k}} v] \\ &= i \nabla_{\vb{k}} \log \operatorname{Pf}[v(\vb{k})].\end{split}\]Since \(\vb{a}^c \equiv 0\), we have
\[\operatorname{Pf}[v(\vb{k})] \equiv 1.\]
We extract the parity of occupied states at each TRIM:
\[\Pi \ket{\psi_\alpha(\Lambda_i)} = \xi_\alpha(\Lambda_i) \ket{\psi_\alpha(\Lambda_i)}.\]\(w\) given by \(v\):
\[w_{\alpha\beta}(\Lambda_i) = \xi_\alpha(\Lambda_i) v_{\alpha\beta}(\Lambda_i).\]Therefore,
\[\delta(\Lambda_i) = \prod_{n=1}^N \xi_{2n}(\Lambda_i).\]
\(\mathbb{Z}_2\) index given by parity:
\[(-1)^\nu = \prod_{i=1}^4 \prod_{n=1}^N \xi_{2n}(\Lambda_i).\]Note
We need an odd number of points of odd parity in the \(4N\) \(\xi_{2n}\)’s to have a nontrivial topology.
Warning
SO interaction like \(\vb{k} \cdot \vb*{\sigma}\) breaks inversion symmetry (there are two distinct energy levels at each \(\vb{k}\)). Moreover, it disqualifies \(s_z\) as a good quantum number.
However, it is not true that all SO interactions break inversion symmetry. In the BHZ model, there is SO interaction term (coupling between \(\vb{k}\) and \(\vb*{\sigma}\)). However, in that case the inversion symmetry was not broken. Otherwise we could not have been able to apply the formula yielding \(\mathbb{Z}_2\) index using parities at TRIMs.
Note
SO interaction is necessary to maintain a nonzero gap everywhere [Kane2007].
Remark: Why Does TRS Matter?
Topological Crystalline Insulators
\(\mathbb{Z}_2\) index is based on TRS.
Topological index based on crystal symmeries, e.g. \(C_4\), \(C_6\), and \(m\) may be given.
Mirror Chern Number
Mirror Chern number:
\[n_{\mathcal{M}} = \nu_1 - \nu_{-1},\]where \(\nu_{\pm 1}\) are the Chern numbers of the subspaces with eigenvalue \(\pm 1\) under \(m\).
Mirror chirality: for 3D systems with \(\nu_0 = 1\), \(\nu_{\mathcal{M}} = \pm 1\) and therefore defines the mirror chirality.
Miscellaneous Effects
Spin-Orbit Coupling
Rashba Effect
The Rashba spin-orbit coupling is typical for systems with uniaxial symmetry, or due to the spatial inhomogeneity of an interface or surface.
\[H_{\mathrm{R}} = \alpha_{\mathrm{R}}(\vb*{\sigma}\times \vb{p})\cdot \hat{\vb*{z}}.\]
Dresselhaus Effect
Occurs when inversion symmetry is broken.
In 2D systems, the Dresselhaus Hamiltonian can be separated into a linear and a cubic term. The linear Dresselhaus Hamiltonian is usually written as
\[H^{(1)}_{\mathrm{D}} = \frac{\beta}{\hbar} (\sigma_x p_x - \sigma_y p_y).\]Dresselhaus and Rashba effects are often of similar strength in the band splitting of GaAs nanostructures.
Localization
Weak Localization
Occurs in disordered electronic systems at very low temperatures.
Increases resistivity.
Originates from interference of electrons propagating clockwise and counter-clockwise.
Anderson Localization
Also known as strong localization.
The absence of diffusion of waves in a disordered medium.
Glossary
- QHE/量子ホール効果/量子霍尔效应
A quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance exhibits steps that take on the quantized values.