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Thermodynamics is the study of energy and its transformation. It deals with macroscopic systems and how they respond to changes in energy. Condensed matter physics is the study of materials in which the atoms are close together, and they interact strongly with each other. This includes solids, liquids, and some gases.
Electronic Structure: The Sommerfeld and Band Theories
In the free-electron Sommerfeld theory, electrons are treated as free particles moving in a periodic potential created by the ions. This model explains some properties of metals, such as their electrical conductivity and heat capacity. However, it fails to account for the diversity of properties observed in different materials.
The band theory of solids describes the electronic structure of materials in terms of energy bands. In a periodic crystal lattice, the wave functions of the electrons overlap, forming bands of allowed energies. The electrons can occupy these bands, up to a maximum called the Fermi energy. This theory explains the diversity of electronic properties of materials, including their conductivity, magnetism, and optical properties.
Phonons and Heat Capacity
Phonons are quantized vibrations of the atoms in a crystal lattice. They contribute to the heat capacity of a solid by carrying energy and increasing the number of ways in which energy can be distributed among the atoms.
Structure, Bonding, and Properties of Solids
The properties of solids depend on their atomic structure and bonding. The types of bonding include ionic, covalent, and metallic bonding. The electronic structure and bonding determine the electrical conductivity, magnetism, and other properties of the material.
(a) Prove the finite temperature version of the fluctuation dissipation theorem
$$
\chi^{\prime \prime}(q, \omega)=\frac{1}{2}\left(e^{-\beta \omega}-1\right) S(q, \omega),
$$
and
$$
S(q, \omega)=-2\left(n_B(\omega)+1\right) \chi^{\prime \prime}(q, \omega)
$$
where $S(q, \omega)=\int d \vec{x} d t e^{-i \vec{q}-\vec{x}} e^{i \omega x}\langle\rho(\vec{x}, t) \rho(0,)\rangle_T$ and $n_B(\omega)=\left(e^{\beta \omega}-1\right)^{-1}$ is the Bose occupation factor.
The fluctuation-dissipation theorem relates the correlation function of a fluctuating variable to its response to an external perturbation. Specifically, it relates the imaginary part of the susceptibility, $\chi^{\prime \prime}(q,\omega)$, to the power spectrum of the fluctuations, $S(q,\omega)$.
Consider a system described by a Hamiltonian $H$ at temperature $T$. Let $\rho(\vec{x},t)$ be an operator that measures the local density fluctuations of the system. Then, the susceptibility $\chi(q,\omega)$ is defined as the linear response of the density fluctuations to a spatially varying potential $V(\vec{x})$ of wavevector $\vec{q}$ and frequency $\omega$:
$$\delta \langle\rho(\vec{q},\omega)\rangle_T = \chi(q,\omega) \delta \langle V(\vec{q},\omega)\rangle_T$$
where $\delta$ denotes the fluctuation about the thermal equilibrium value and $\langle \rangle_T$ denotes the thermal average.
The imaginary part of the susceptibility is related to the power spectrum of the density fluctuations as follows:
$$\chi^{\prime\prime}(q,\omega) = \frac{1}{2i} \left[\chi(q,\omega) – \chi^*(q,\omega)\right]$$
where $\chi^*(q,\omega)$ is the complex conjugate of $\chi(q,\omega)$.
Using the Kubo formula, the susceptibility can be expressed in terms of the time-dependent density correlation function:
$$\chi(q,\omega) = \frac{1}{i\omega} \int_0^\infty dt, e^{i\omega t} \langle[\rho(\vec{q},t),\rho(-\vec{q},0)]\rangle_T$$
where $[\cdot,\cdot]$ denotes the commutator.
The power spectrum of the density fluctuations is defined as:
$$S(q,\omega) = \int dt \int d\vec{x}, e^{-i\vec{q}\cdot\vec{x}} e^{i\omega t} \langle\rho(\vec{x},t)\rho(0,0)\rangle_T$$
Using these definitions, we can now derive the finite temperature version of the fluctuation-dissipation theorem.
First, we express the susceptibility in terms of the density correlation function using the Kubo formula:
$$\chi(q,\omega) = \frac{1}{i\omega} \int_0^\infty dt, e^{i\omega t} \langle[\rho(\vec{q},t),\rho(-\vec{q},0)]\rangle_T$$
Taking the imaginary part, we have:
$$\chi^{\prime\prime}(q,\omega) = -\frac{1}{2}\int_0^\infty dt, e^{i\omega t} \langle[\rho(\vec{q},t),\rho(-\vec{q},0)]\rangle_T + \text{c.c.}$$
where $\text{c.c.}$ denotes the complex conjugate of the first term.
(b) Show that $\chi^{\prime \prime}(q, \omega)=-\chi^{\prime \prime}(-q,-\omega)$ and $S(-q,-\omega)=e^{-\beta \omega} S(q, \omega)$. In terms of the scattering probability, show that this is consistent with detailed balance.
To show that $\chi^{\prime \prime}(q, \omega)=-\chi^{\prime \prime}(-q,-\omega)$, we start with the definition of the dynamic structure factor:
$\chi^{\prime \prime}(q, \omega)=-\frac{1}{\pi} \operatorname{Im} \int_{-\infty}^{\infty} d t e^{i \omega t} \int d^3 r e^{-i q \cdot r}\langle\hat{\rho}(\mathbf{r}, t) \hat{\rho}(0,0)\rangle$
where $\hat{\rho}(\mathbf{r}, t)$ is the density operator at position $\mathbf{r}$ and time $t$, and $\langle \rangle$ denotes the statistical average over an ensemble of systems in thermal equilibrium. Using the fact that the correlation function $\left\langle\hat{\rho}(\mathbf{r}, t) \hat{\rho}(0,0)\right\rangle$ is real, we can rewrite the integral as
$\chi^{\prime \prime}(q, \omega)=\frac{1}{\pi} \int_{-\infty}^{\infty} d t \sin (\omega t) \int d^3 r e^{-i q \cdot r}\langle\hat{\rho}(\mathbf{r}, t) \hat{\rho}(0,0)\rangle$
Changing variables $t\rightarrow -t$ and $r\rightarrow -r$ gives
$\begin{aligned} \chi^{\prime \prime}(-q,-\omega) & =\frac{1}{\pi} \int_{-\infty}^{\infty} d t \sin (-\omega t) \int d^3 r e^{i(-q) \cdot r}\langle\hat{\rho}(-\mathbf{r},-t) \hat{\rho}(0,0)\rangle \ & =\frac{1}{\pi} \int_{-\infty}^{\infty} d t \sin (\omega t) \int d^3 r e^{-i q \cdot r}\langle\hat{\rho}(\mathbf{r}, t) \hat{\rho}(0,0)\rangle \ & =-\chi^{\prime \prime}(q, \omega)\end{aligned}$
Thus, we have shown that $\chi^{\prime \prime}(q, \omega)=-\chi^{\prime \prime}(-q,-\omega)$.
Next, we show that $S(-q,-\omega)=e^{-\beta \omega} S(q, \omega)$. Starting with the definition of the dynamic structure factor,
$S(q, \omega)=\frac{1}{\pi} \frac{\operatorname{Im} \chi(q, \omega)}{1-e^{-\beta \omega}}$
we can rewrite $\chi(q,\omega)$ using the fluctuation-dissipation theorem as
$\chi(q, \omega)=\frac{1}{\beta}\left[\chi^{\prime \prime}(q, \omega)+i \frac{\chi^{\prime}(q, \omega)}{\omega}\right]$
(a) Using linear response theory, derive the following expression for the magnetic susceptibility $\chi_{|}=\partial M_z / \partial H_z$.
$$
\chi_{|}=\lim _{q \rightarrow 0} \int \frac{d \omega}{2 \pi}\left\langle S_z(q, \omega) S_z(-q,-\omega)\right\rangle \frac{\left(1-e^{-\frac{\hbar \omega}{k T}}\right)}{\omega}
$$
Linear response theory is a framework for understanding the response of a physical system to small perturbations. In this case, we want to calculate the magnetic susceptibility $\chi_{|}$, which is the response of the magnetization $M_z$ to a small magnetic field $H_z$. We can use the Kubo formula to relate the susceptibility to the spin-spin correlation function:
$$\chi_{|}=\frac{1}{k_BT}\lim_{\omega\rightarrow 0}\frac{1}{\omega}\int_{0}^{\infty}dt\int d\mathbf{r} e^{i\omega t}\left\langle\left[\hat{S}_z(\mathbf{r},t),\hat{S}z(\mathbf{0},0)\right]\right\rangle{H_z=0}$$
where $\hat{S}z(\mathbf{r},t)$ is the $z$-component of the spin density operator at position $\mathbf{r}$ and time $t$, and $\left\langle\cdots\right\rangle{H_z=0}$ denotes the thermal average in the absence of an external magnetic field. The Fourier transform of the spin density operator is defined by
$$\hat{S}_z(\mathbf{q},\omega)=\int dt\int d\mathbf{r} e^{i(\mathbf{q}\cdot\mathbf{r}-\omega t)}\hat{S}_z(\mathbf{r},t)$$
Using this definition, we can rewrite the susceptibility as
$$\chi_{|}=\frac{1}{k_BT}\lim_{\omega\rightarrow 0}\frac{1}{\omega}\int_{0}^{\infty}dt\int d\mathbf{r} e^{i\omega t}\int\frac{d\mathbf{q}}{(2\pi)^3}e^{-i\mathbf{q}\cdot\mathbf{r}}\left\langle\left[\hat{S}_z(\mathbf{q},\omega),\hat{S}z(-\mathbf{q},-\omega)\right]\right\rangle{H_z=0}$$
Now, we use the fact that the commutator is related to the imaginary part of the retarded correlation function:
$$\left[\hat{S}_z(\mathbf{q},\omega),\hat{S}z(-\mathbf{q},-\omega)\right]=2i\operatorname{Im}\left{\hat{\chi}{zz}(\mathbf{q},\omega)\right}$$
where $\hat{\chi}_{zz}(\mathbf{q},\omega)$ is the Fourier transform of the spin-spin correlation function $\left\langle\hat{S}z(\mathbf{r},t)\hat{S}z(\mathbf{0},0)\right\rangle{H_z=0}$. Substituting this into the expression for $\chi{|}$ and taking the limit $\omega\rightarrow 0$, we obtain
$$\chi_{|}=\frac{2}{k_BT}\int_{0}^{\infty}dt\int d\mathbf{r}\int\frac{d\mathbf{q}}{(2\pi)^3}e^{-i\mathbf{q}\cdot\mathbf{r}}\operatorname{Im}\left{\hat{\chi}_{zz}(\mathbf{q},\omega=0)\right}$$
