Lecture 1

02/20/23, M.

Reading: GY 15.1 -- 15.3.

Condensed matter physics: physical properties of matter in condensed form
Matter: atoms and everything made up of them (solids, liquids, gases, plasma, condensates, quark-gluon plasma, ... )
$\Rightarrow$ solids, liquids and condensates
Properties: macroscopic (thermodynamics, electrodynamics) and microscopic (statistical mechanics, quantum mechanics) theories.
Microscopic degrees of freedom: atom, electron, and emergent (quasiparticles, exciton, Cooper pairs ...)
$\hbar=1$ $\hbar =0$ side of the story (classical phase transition, soft condensed matter) can be found in the the text by Chaikin and Lubensky.

Why classify? group with common properties, compare and contrast
Based on atomic arrangement:
-- liquids (translational and rotational invariance, zero shear rigidity),
-- crystals (discrete translational symmetry of a lattice)
-- amorphous solids
-- quasicrystals ( discrete rotational symmetry without translational symmetry)
Electronic structure and responses
$\sigma=e^2D \partial n/\partial \mu$ )
-- Band insulator vs Mott/Anderson insulators
-- Zoo of magnetisms: para-, ferro-, antiferro-, ferri-, dia-magnets
Symmetries
-- A symmetry leaves the Hamiltonian (action) unchanged. translation, rotation, inversion, time reversal
-- Spontaneous symmetry breaking: a crystal breaks translational invariance; a magnet breaks spin rotational invariance and time reversal
-- Order parameter characterizes the pattern of symmetry breaking: a measurable quantity that transforms nontrivially under a symmetry of the Hamiltonian. e.g.
- $SU(2)$ breaking
- density (or its Fourier components) for a crystal, translation breaking
- $U(1)$ breaking
Beyond symmetries: quantum Hall system, topological and geometric effects.

First principles VS emergence: Fermi liquid, BCS theory
More is different: renormalization group
Symmetry: Landau, Ginsburg-Landau, spontaneous symmetry breaking and Goldstone modes
Topology and geometry

\begin{matrix} (1) & \begin{aligned} H & = \hat{T} + {\hat{V}}_{ext} + \hat{U} \\ = \sum_{i} [\frac{p_{i}^{2}}{2 m} + v_{ext} (r)] + \frac{1}{2} \sum_{i \neq j} u (r_{i}, r_{j}) \end{aligned} \end{matrix}

Central to the understanding of condensed matter system, this problem is generally hard to solve.

\begin{matrix} (2) & \begin{matrix} H_{0} = \frac{p^{2}}{2 m} \\ ψ_{k σ} (r) = \frac{e^{i k \cdot r}}{(2 π)^{3 / 2}}, ε_{k σ} = \frac{ℏ^{2} k^{2}}{2 m} \\ ⟨ ψ_{k σ} | ψ_{k^{'} σ^{'}} ⟩ = δ_{σ σ^{'}} δ (k - k^{'}) \end{matrix} \end{matrix}

In second quantization

\begin{matrix} (3) & H_{0} = \sum_{k σ} ε_{k σ} a_{k σ}^{†} a_{k σ} \end{matrix}

The ground state (g.s. 基态) energy is given by

\begin{matrix} (4) & E_{0} = ⟨ H_{0} ⟩ = \sum_{k σ} ε_{k σ} f_{0} (ε_{k σ}) \end{matrix}

where the Fermi-Dirac distribution function is defined as

\begin{matrix} (5) & f_{0} (ε) = \frac{1}{e^{β (ε - μ)} + 1} \end{matrix}

$T=0$ $f(\varepsilon)= \theta(\mu-\varepsilon)$ $\mu(T=0)=\varepsilon_{\text F}$

\begin{matrix} (6) & E_{0} = \sum_{k σ} ε_{k σ} θ (ε_{F} - ε) = 2 V \int_{k < k_{F}} \frac{d^{3} k}{(2 π)^{3}} ε_{k σ} = \frac{3}{5} N ε_{F} \end{matrix}

that is, average energy per electron is

\begin{matrix} (7) & \overset{―}{ε} = \frac{3}{5} ε_{F} \end{matrix}

$r_s$ parameter characterizes the linear scale of a electron gas

\begin{matrix} (8) & \frac{4}{3} π (r_{s} a_{0})^{3} = \frac{V}{N} \end{matrix}

$a_0 = 4\pi\epsilon_0\hbar^2/me^2\approx 0.529 \text{ \AA}$ $1 \text{ Ry} = e^2/4\pi\epsilon_0(2a_0)\approx 13.6$ eV. From the relation

\begin{matrix} (9) & \sum_{k σ}^{k < k_{F}} = N \Rightarrow 3 π^{2} n = k_{F}^{3} \end{matrix}

we have the kinetic energy of a uniform electron gas (with or without interaction)

\begin{matrix} (10) & \frac{E_{0}}{N} = \frac{2.21}{r_{s}^{2}} Ry . \end{matrix}

\begin{matrix} (11) & \hat{U} = \frac{1}{2} \sum_{k_{1} σ_{1}} \sum_{k_{2} σ_{2}} \sum_{k_{3} σ_{3}} \sum_{k_{4} σ_{4}} U_{k_{1} σ_{1}, k_{2} σ_{2}, k_{3} σ_{3}, k_{4} σ_{4}} a_{k_{1} σ_{1}}^{†} a_{k_{2} σ_{2}}^{†} a_{k_{3} σ_{3}} a_{k_{4} σ_{4}} \end{matrix}

The matrix element is

\begin{matrix} (12) & \begin{aligned} U_{k_{1} σ_{1}, k_{2} σ_{2}, k_{3} σ_{3}, k_{4} σ_{4}} & = ⟨ k_{1} σ_{1}, k_{2} σ_{2} | U (r^{(1)} - r^{(2)}) | k_{3} σ_{3}, k_{4} σ_{4} ⟩ \\ = ⟨ k_{1} k_{2} | U (r^{(1)} - r^{(2)}) | k_{3} k_{4} ⟩ δ_{σ_{1} σ_{3}} δ_{σ_{2} σ_{4}} \end{aligned} \end{matrix}

for the spin-independent two-body interaction.

\begin{matrix} (13) & \begin{aligned} ⟨ k_{1} k_{2} | U (r^{(1)} - r^{(2)}) | k_{3} k_{4} ⟩ \\ = \int d r^{(1)} \int d r^{(2)} ⟨ k_{1} k_{2} | r^{(1)} r^{(2)} ⟩ U (r^{(1)} - r^{(2)}) ⟨ r^{(1)} r^{(2)} | k_{3} k_{4} ⟩ \\ = \frac{1}{V^{2}} \int d r^{(1)} \int d r^{(2)} e^{- i (k_{1} - k_{3}) \cdot r^{(1)}} e^{- i (k_{2} - k_{4}) \cdot r^{(2)}} U (r^{(1)} - r^{(2)}) \end{aligned} \end{matrix}

$\boldsymbol r^{(1)} = \frac{1}{2}\boldsymbol r+\boldsymbol R$ $\boldsymbol r^{(2)} = -\frac{1}{2}\boldsymbol r+\boldsymbol R$ ,

\begin{matrix} (14) & \begin{aligned} ⟨ k_{1} k_{2} | U (r^{(1)} - r^{(2)}) | k_{3} k_{4} ⟩ \\ = \frac{1}{V^{2}} \int d r^{(1)} \int d r^{(2)} e^{- i (k_{1} + k_{2} - k_{3} - k_{4}) \cdot R} e^{- i (k_{1} - k_{2} - k_{3} + k_{4}) \cdot r / 2} U (r) \\ = \frac{1}{V} δ_{k_{1} + k_{2}, k_{3} + k_{4}} U (k_{1} - k_{3}) \end{aligned} \end{matrix}

So the interaction energy is

\begin{matrix} (15) & \hat{U} = \frac{1}{2 V} \sum_{q \neq 0, k, p, σ, σ^{'}} U (q) a_{k + q, σ}^{†} a_{p - q, σ^{'}}^{†} a_{p, σ^{'}} a_{k, σ} \end{matrix}

$q=0$ component is the electrostatic energy cancelled out with the direct Hartree energies from the background.

Then the first-order energy is

\begin{matrix} (16) & \frac{E^{(1)}}{N} = \frac{1}{2 V N} \sum_{q \neq 0, k, p, σ, σ^{'}} U (q) ⟨ a_{k + q, σ}^{†} a_{p - q, σ^{'}}^{†} a_{p, σ^{'}} a_{k, σ} ⟩ \end{matrix}

\begin{matrix} (17) & \begin{aligned} \frac{E^{(1)}}{N} & = - \frac{1}{2 V N} \sum_{q \neq 0, k, σ} U (q) ⟨ n_{k + q σ} n_{k σ} ⟩ \\ \approx - \frac{2}{2 V N} \sum_{q \neq 0, k} U (q) θ (k_{F} - | k + q |) θ (k_{F} - k) \end{aligned} \end{matrix}

where we have used the notation

\begin{matrix} (18) & \int [d k] = \int \frac{d^{d} k}{(2 π)^{d}} . \end{matrix}

Using the Coulomb interaction

\begin{matrix} (19) & U (q) = \frac{e^{2}}{ϵ_{0} q^{2}} \end{matrix}

we find

\begin{matrix} (20) & \frac{E^{(1)}}{N} = - \frac{0.916}{r_{s}} Ry . \end{matrix}

Then we have for the interacting electron gas

\begin{matrix} (21) & \frac{E}{N} = \underset{Hartree-Fock approximation}{\underset{⏟}{\underset{kinetic}{\underset{⏟}{\frac{2.21}{r_{s}^{2}}}} + \overset{Hartree}{\overset{⏞}{0}} \underset{exchange}{\underset{⏟}{- \frac{0.916}{r_{s}}}}}} + \overset{correlation}{\overset{⏞}{\dots}} \end{matrix}