# Quantum Transport and Mesoscopic Physics


## 1 Introduction


**1.1 **  **Recommended Textbooks**
- J.M. Ziman *Principles of the theory of solids*, 2nd Edition
- A. Atland and B. Simons, *Condensed Matter Field Theory*
- S.M. Girvin and Kun Kang, *Modern Condensed Matter Physics*

**1.2 **  **Presentation Topics**

- Coulomb blockade.
- Kondo effect
- Orthogonality catastrophe and quantum speed limit
- Mott transition
- Spin Hall effect
- Composite fermions and the fractional quantum Hall effect
- Charge density waves
- Strongly coupled large polarons

## 2 Brownian Motion


### 2.1 Stationary 1D Brownian Motion


**Definition 2.1.1 **  A quantum particle exhibits **coherent motion** iff its lifetime $t_\ell$ is much greater than $\frac{\hbar}{E}$.
\[t_\ell >> \frac{\hbar}{E}\]

**Definition 2.1.2 **  A quantum particle exhibits **incoherent motion** iff its lifetime $t_\ell$ less than $\frac{\hbar}{E}$.
\[t_\ell \leq \frac{\hbar}{E}\]

**Definition 2.1.3 **  A **stationary 1D Brownian particle** is a system of a stationary 1D particle experiencing many fast collisions with momentum $p_i$ integrated over a small time step $\Delta t$. Defined the force experienced by the particle $f(t)$ as a function of time.
\[f(t) = \frac{2}{\Delta t}\sum_i{p_i}\]

**Definition 2.1.4 **  The **ensemble average** denoted $\langle x \rangle$ of a random variable $x$ is the integral of the variable weighted by the probability density function $P_{x}(x)$ across all possible values.
\[\langle x \rangle = \int_{-\infty}^{\infty}{x P_{x} dx}\]

**Definition 2.1.5 **  The **momentum diffusion coefficient** is defined $D=2\nu\langle p_i^2\rangle = 2\nu m_{mol} k_B T$ where $\nu$ is the average frequency of collisions and $m_{mol}$ is the mass of the molecules.

**Result 2.1.6 **  $\langle f\rangle = 0$

**Result 2.1.7 **  $\langle f^2\rangle = \frac{4}{(\Delta t)^2} \sum_i{\langle p_i^2 \rangle}=\frac{4}{\Delta t}\nu\langle p_i^2\rangle = \frac{2D}{\Delta t}$

**Result 2.1.8 **  $\langle f^{2n+1} \rangle = 0$

**Result 2.1.9 **  $\langle f^{2n} \rangle = \langle f^2 \rangle^n (2n-1)!! =\left(\frac{4}{\Delta t}\nu\langle p_i^2\rangle\right)^n (2n-1)!! = \left(\frac{2D}{\Delta t}\right)^n (2n-1)!!$

**Definition 2.1.10 **  A **Gaussian Distribution** with variance $\sigma=\langle f^2 \rangle$ is a distribution of the form:
\[P_f(x) = \frac{1}{\sqrt{2\pi\sigma}}e^{-x^2/2\sigma}\]

**Definition 2.1.11 **  A **probability density function** of a variable is a function that when integrated over a region of possible values represents the probability of the variable being in that region.

**Result 2.1.12 **  The probability density function of $f(t)$ at a particular time is Gaussian with $\sigma = \langle f^2 \rangle = \frac{2D}{\Delta t}$.

**Definition 2.1.13 **  A **functional probability density function** is a probability density function that is integrated over different functions.

**Result 2.1.14 **  The functional probability density of $f(t)$ is \[\mathcal{P}(f(t)) = C \text{exp}\left[{-\frac{1}{4D}\int_{-\infty}^{\infty}{f^2(t)dt}}\right]\]

### 2.2 1D Brownian Motion with Friction


**Definition 2.2.1 **  The **friction coefficient** is a constant that describes the amount of friction experienced by a 1D Brownian particle.
\[\Gamma = \frac{m_{mol}\nu}{M}\]

**Definition 2.2.2 **  A **1D Brownian Particle** is a system of a 1D particle of mass $m_p$ moving at velocity $v_{p}$ which is experiencing many fast collisions. Define the force experienced by the particle $f_p(t)$ in terms of the force experienced by a stationary 1D Brownian particle $f(t)$ and the friction coefficient $\Gamma$.
\[f_v(t) = f(t) - 2\Gamma m_p v_p\]

**Definition 2.2.3 **  The **position diffusion coefficient** denoted $\mathscr{D}$ is defined in terms of the noise intensity $D$, the friction coefficient $\Gamma$ and the mass of the particle $m_p$.
\[\mathscr{D} = \frac{D}{(2\Gamma m_p)^2} = \frac{k_B T}{2\Gamma m_p}\]

**Result 2.2.4 **  The motion $q(t)$ and the momentum $p(t)$ of a 1D Brownian particle is described by the following equations.
\[p(t) = \int_0^t{e^{-2\Gamma(t-\tau)f(\tau)d\tau}}\]
\[q(t) = \frac{1}{m_p}\int_0^t{p(\tau)d\tau} = \frac{1}{m_p}\int_0^\tau{f(\tau)\frac{1-e^{-2\Gamma(t-\tau)}}{2\Gamma}d\tau}\]

**Result 2.2.5 **  $\langle q\rangle = 0$

**Result 2.2.6 **  For $t>>\frac{1}{\Gamma}$, $\langle q^2\rangle = \langle (q(t)-q(0))^2 \rangle = 2\mathscr{D}\left[t - \frac{2}{2\Gamma}(1-e^{-2\Gamma t}) + \frac{1}{4\Gamma}(1-e^{-4\Gamma t})\right] \approx 2\mathscr{D}t$

### 2.3 Lagevin and Fokker-Plank Equations


**Definition 2.3.1 **  A **1D Brownian Particle in a Potential** is a system of a 1D Brownian particle experiencing a potential $U(p)$. The system can be written in terms of position $q$ and momentum $p$ or as a time dependent force function $f_U(t)$
\[\dot{q} = \frac{p}{m_p},\quad \dot{p} = f(t) - 2\Gamma p - U'(q)\]
\[f_U(t) = f(t) - 2\Gamma m_pv_p - U'\]

**Theorem 2.3.2 **  The **Lagevin and Fokker-Plank Equations** describe the probability density function $\omega(q,p,t,q(0),p(0),0)$ of a 1D Brownian particle in a potential $U(q)$.
\[\omega(q,p,0,q(0),p(0),0) = \delta(q-q(0))\delta(p-p(0))\]
\[\partial_t \omega(q,p,t)= -\partial_q\left( \frac{p}{m_p}\omega(q,p,t) \right) - \partial_p\left(- 2\Gamma p - U'(q)  + D\partial_p^2\right)\omega(q,p,t)\]

**Definition 2.3.3 **  The **Diffusion Equation** describes a Brownian particle where there is no potential or drift, that is $\Gamma = 0$ and $U(q)=0$.
\[\frac{\partial n}{\partial t} - \mathscr{D}\nabla_\mathbf{x}^2n = 0\]

**Result 2.3.4 **  The solution to the diffusion equation is of the form
\[\partial_t\omega = \mathscr{D}\partial_x^2\omega\]
\[\omega = \frac{C}{\sqrt{t}}e^{-x^2/4\mathscr{D}t}\]

**Definition 2.3.5 **  The **time average** denoted $\langle x \rangle$ of a random variable $x(t)$ is the time integral of the variable over a very long period.
\[\langle x\rangle = \lim_{T\to\infty}\frac{1}{T}\int_0^T{x(t)dt}\]

**Definition 2.3.6 **  A system is **ergodic** iff the ensemble average and the time average are equivalent.

### 2.4 Harmonic Oscillator Diffusion


**Definition 2.4.1 **  The **Brownian harmonic oscillator** is a system describing a Brownian particle in a harmonic oscillator potential $U(q)$
\[U(q) = \frac{1}{2}m\omega_0^2q^2\]
\[\partial_t \omega(q,p,t) = -\partial_q\left( \frac{p}{m_p}\omega(q,p,t) \right) - \partial_p\left(- 2\Gamma p\omega(q,p,t) - m\omega_0^2q\omega(q,p,t)  + D\partial_p^2\right)\omega(q,p,t)\]

**Result 2.4.2 **  For a Brownian harmonic oscillator, $\langle \ddot{q}\rangle = -\omega_0^2\langle q \rangle - \frac{2\Gamma}{m}\langle \ddot{q}\rangle$

**Definition 2.4.3 **  The **time correlation function** $Q_{qq}(t)$ of a variable $q(t)$ is defined $Q_{qq}(t) = \langle q(t)q(0)\rangle$. 

**Definition 2.4.4 **  The **power spectrium** $S_{qq}(\omega)$ of a variable $q(t)$ is the Fourier transform of the time correlation function $S_{qq}(\omega) = \int_{\infty}^\infty{e^{i\omega t}\langle q(t) q(0)\rangle dt}$,

**Result 2.4.5 **  For a real variable $q(t)$ the power spectrum is $S_{qq}(\omega) = 2\text{Re}\left( \int_{-\infty}^\infty{e^{i\omega t}\langle q(t) q(0)\rangle dt} \right)$

**Result 2.4.6 **  The stationary solution to the Brownian Harmonic oscillator is of the form
\[\omega(q,p,t) = C e^{-(\frac{p^2}{2m} + U(q))/\alpha}\]

**Result 2.4.7 **  The power spectrum of $q$ for the Brownian harmonic oscillator is
\[S_qq(\omega) = \frac{2\Gamma \omega_0^2}{(\omega^2-\omega_0^2)^2 + 4\Gamma^2\omega^2}\]

### 2.5 Escape and Activation via Tunneling


**Definition 2.5.1 **  The **probability current** $j(q)$ is the rate at which probability to crossing point $q_B$.
\[j(q_B) = \int{\frac{p}{m}\omega(q_B,p,t)}\]

**Definition 2.5.2 **  The **escape rate** is the reciprocal of the average lifetime $\tau$ and can be written in terms of probability current.
\[W_{esc} = \frac{j}{N} = \frac{1}{\tau}\]

**Definition 2.5.3 **  The **number of trapped particles** $N$ is defined 
\[N = \int_{-\infty}^{\infty}dp\int_{-\infty}^{q_B}dq \omega(q,p,t)\]

**Result 2.5.4 **  The time derivative of the number of particles $N$ is $\frac{\partial N}{\partial t} = - W_{esc}N$ and the decay over time is
\[N(t) = N(0)e^{-W_{esc}t}\]

### 2.6 Escape and Activation via Tunneling


**Definition 2.6.1 **  The **probability current** $j(q)$ is the rate at which probability to crossing point $q_B$.
\[j(q_B) = \int{\frac{p}{m}\omega(q_B,p,t)}\]

**Definition 2.6.2 **  The **escape rate** is the reciprocal of the average lifetime $\tau$ and can be written in terms of probability current.
\[W_{esc} = \frac{j}{N} = \frac{1}{\tau}\]

**Definition 2.6.3 **  The **number of trapped particles** $N$ is defined 
\[N = \int_{-\infty}^{\infty}dp\int_{-\infty}^{q_B}dq \omega(q,p,t)\]

**Result 2.6.4 **  The time derivative of the number of particles $N$ is $\frac{\partial N}{\partial t} = - W_{esc}N$ and the decay over time is
\[N(t) = N(0)e^{-W_{esc}t}\]

## 3 Electron Transport


### 3.1 Bloch's Theorem and Berry Phase


**Theorem 3.1.1 **  **Bloch's Theorem** states that the solution any system with a periodic potential can be represented with a periodic function $u_{\mathbf{k},n}(\mathbf{r})$ and $e^{i\mathbf{k}\cdot\mathbf{r}}$, that is
\[H\Psi_{\mathbf{k},n}(\mathbf{r}) = E_{\mathbf{k},n}\Psi_{\mathbf{k},n}(\mathbf{r}),\quad \Psi_{\mathbf{k},n}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{\mathbf{k},n}(\mathbf{r})\]

**Definition 3.1.2 **  The **Berry connection** denoted $\mathcal{A}_{n,\mathbf{k}}(\mathbf{r})$ is defined as $\mathcal{A}_{n,\mathbf{k}}(\mathbf{r}) = i\langle u_{\mathbf{k},n}(\mathbf{r}) | \nabla_{\mathbf{k}} u_{\mathbf{k},n}(\mathbf{r}) \rangle$

**Definition 3.1.3 **  The **Berry phase** denoted $\gamma_{n,\mathbf{k}}$ is defined as $\gamma_{n,\mathbf{k}} = \int \mathcal{A}_{n,\mathbf{k}}(\mathbf{r}) \cdot d\mathbf{r}$ 

**Result 3.1.4 **  For the band wavefunction $\Phi_n(\mathbf{r}) = \int{ c(\mathbf{k})\Psi_{\mathbf{k},n}(\mathbf{k})\Psi_{\mathbf{k},n}(\mathbf{r}) d\mathbf{k}}$, written in terms of eigenstates $\Psi_{\mathbf{k},n}$,
\[\mathbf{r} = i\partial_k + i\langle u_{\mathbf{k},n}(\mathbf{r}) | \nabla_{\mathbf{k}} u_{\mathbf{k},n}(\mathbf{r}) \rangle = i\partial_k + \mathcal{A}_n(\mathbf{r})\]
\[\langle \mathbf{r} \rangle = \left( \frac{(2\pi)^d}{V} \right)^2\int{c^*(\mathbf{k}) \mathbf{r} c(\mathbf{k}) d\mathbf{k}} = \left( \frac{(2\pi)^d}{V} \right)^2\int{c^*(\mathbf{k})\left(i\partial_k + \mathcal{A}_{n,\mathbf{k}}(\mathbf{r})\right) c(\mathbf{k}) d\mathbf{k}}\]
\[\langle \dot{\mathbf{r}} \rangle = \frac{\partial E_{\mathbf{k},n}}{\partial \mathbf{k}}\]

### 3.2 Anomalous Quantum Hall Effect


**Definition 3.2.1 **  The **anomalous quantum hall effect Hamiltonian** $H$ describes the behavior of many electrons in a lattice $H_0$ experiences an electric field $E$ pointing in the x direction.
\[H = H_0 - e_qEx\]

**Definition 3.2.2 **  The **Berry curvature** denoted $\mathbf{\Omega}_{n,\mathbf{k}}(\mathbf{r})$ is defined as $\mathbf{\Omega}_{n,\mathbf{k}}(\mathbf{r}) = \nabla_\mathbf{r}\times \mathcal{A}_{n,\mathbf{k}}(\mathbf{r})$

**Result 3.2.3 **  The average velocities for 2D electrons experiencing an electric field in the x direction are 
\[\langle v_x\rangle = \frac{1}{\hbar} \frac{\partial E_{n\mathbf{k}}}{\partial k_x}\]
\[\langle v_y \rangle = \frac{1}{\hbar} \frac{\partial E_{n\mathbf{k}}}{\partial k_x} + \frac{qE}{\hbar}\left(\mathbf{\Omega}_{n,\mathbf{k}}(\mathbf{r})\right)_z\]

**Result 3.2.4 **  current

**Definition 3.2.5 **  Churn number

**Result 3.2.6 **  conductivity

**3.3 **  Boltzmann Kinetic Equation

## 4 Scattering and Conductivity


**4.1 **  Impurity Scattering

**4.2 **  Electric Conductivity

**4.3 **  Thermal Conductivity

**4.4 **  Magnetoconductivity

**4.5 **  Cyclotron Resonance

## 5 Low-dimensional Systems


**5.1 **  Landauer 1D Conductivity

**5.2 **  2D Electron Systems

**5.3 **  Quantum Hall Effect

## 6 Localization


**6.1 **  Weak Localization

**6.2 **  Anderson Localization

**6.3 **  Density Matrix and the Quantum Kinetic Equation

## 7 Phonon


**7.1 **  Electron-phonon Interaction

**7.2 **  Polaronic Effect

**7.3 **  Ohmic Dissipation

**7.4 **  The Orthogonality Catastrophe

**7.5 **  Holstein Polarons

## 8 Topological Materials


**8.1 **  Variable-range Hopping

**8.2 **  The Coulomb Gap

**8.3 **  The Berry Phase

**8.4 **  Group Velocity in Topologically Nontrivial Solids

**8.5 **  The Kitaev Chain