date: 2025.04.18

Quantum Many-Body Physics

(to be uploaded)

1. Review of Quantum Mechanics

Single-particle systems

• single-particle Hamiltonian: $\hat{H}=\frac{(\vec{p}-q\vec{A})^2}{2m^*}+U(\vec{r})$ with non-relativistic particle under a potential $U(\vec{r})$ and magnetic field $\vec{B}=\nabla \times \vec{A}$
• Schrodinger’s equation: $i\hbar \frac{\partial}{\partial t} \ket{\psi(t)}=\hat{H}\ket{\psi(t)}$
  • $(\ket{\psi(t)})^\dagger=\bra{\psi(t)}$
  • $\braket{\vec{r}|\psi(t)}=\psi(\vec{r},t)$j
  • $\braket{\phi|\psi}=\int d^3r^3\phi^*(\vec{r})\psi(\vec{r})$

Basis in the Hilbert space

• ${\ket{\phi_\alpha}}$: orthonormal basis (complete set) of the single-particle Hilbert space
• $\braket{\phi_\alpha|\phi_\beta}=\delta_{\alpha\beta}$ and $\sum_\alpha \ket{\phi_\alpha} \bra{\phi_\alpha}=\mathbb{1}$
• can write $\ket{\psi(t)}=\sum_\alpha C_\alpha (t)\ket{\phi_\alpha}=\sum_\alpha \braket{\phi_\alpha | \psi(t)}\ket{\phi_\alpha}$
• operators
  • $\hat{A}\ket{\psi}=\ket{\phi}$, $\bra{\psi}\hat{A}=\bra{\phi}$
  • representation in the $\ket{\phi_\alpha}$ basis
    • $A_{\alpha \beta}=\bra{\phi_\alpha}\hat{A}\ket{\phi_\beta}=\bra{\phi_\alpha}(\hat{A}\ket{\phi_\beta})$, $A_{\beta\alpha}^*=\bra{\phi_\beta}\hat{A}^\dagger\ket{\phi_\alpha}=(\bra{\phi_\beta}\hat{A}^\dagger)\ket{\phi_\alpha}$

Observables (Hermitian operators)

• $\hat{A}=\hat{A}^\dagger$: Hermitian operator
• $\hat{A}\ket{\phi_\alpha}=a\ket{\phi_\alpha} \rightarrow a \in \mathbb{R}$: real eigenvalues (physical observable
• $P_\psi(a)\equiv|\braket{\phi_\alpha|\psi}|^2$: probability of measuring the value for the observable $\hat{A}$ if the particle is in state $\ket{\psi}$. if the spectrum is continuous, this becomes a probability density. For example, the position operator $dP_\psi(\vec{r})\equiv|\braket{\vec{r}|\psi(t)}|^2d^3\vec{r}=|\psi(\vec{r},t)|^2d^3\vec{r}$
• example: 1D Harmonic Oscillator
  • Hamiltonian: $\hat{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega^2\hat{x}^2$ with $[\hat{x},\hat{p}]=i\hbar$
  • position representation: $\braket{\vec{r}|\hat{p}|\psi(t)}=\frac{\hbar}{i}\frac{\partial}{\partial x} \psi(x,t)$
  • Define $\hat{X}=\sqrt{\frac{m\omega}{\hbar}}\hat{x}$, $\hat{P}=\frac{1}{\sqrt{m\hbar \omega}} \hat{p}$ $\rightarrow \hat{H}=\frac{\hbar \omega}{2}(\hat{P}^2+\hat{X}^2)$
  • Define $\hat{a}=\frac{1}{\sqrt{2}}(\hat{X}+i\hat{P}), \hat{a}^\dagger=\frac{1}{\sqrt{2}}(\hat{X}-i\hat{P}) \rightarrow \hat{X}=\frac{1}{\sqrt{2}}(\hat{a}^\dagger+\hat{a}), \hat{P}=\frac{i}{\sqrt{2}}(\hat{a}^\dagger-\hat{a})$
  • $[\hat{X},\hat{P}]=i$ and $[\hat{a},\hat{a}^\dagger]=1$ and also $\hat{H}=\hbar \omega (\hat{a}^\dagger \hat{a}+\frac{1}{2})$
  • Define $\hat{N}\equiv \hat{a}^\dagger\hat{a}$ and its eigenstates $\hat{N}\ket{n}=E_n\ket{n}$
  • $E_n=\hbar\omega(n+\frac{1}{2}), n=0,1,…$, $\hat{a}^\dagger\ket{n}=\sqrt{n+1}\ket{n+1}, \hat{a}\ket{n}=\sqrt{n}\ket{n-1}, \ket{n}=\frac{(\hat{a}^\dagger)^n}{\sqrt{n!}}\ket{0}$

Single-particle spectrum

• ${\ket{n}}$: orthonormal basis of the (single-particle) Hilbert’s space
• $\braket{n|n’}=\delta_{nn’}$ and $\sum_n \ket{n}\bra{n}=\mathbb{1}$
• single-particle state: $\ket{\psi(t)}=\sum_n C_n (t)\ket{n}=\sum_n \braket{n | \psi(t)}\ket{n}$
• Probability of measuring the value $E_n$ for the energy if the particle is in state $\ket{\psi}$: $P_\psi(n)=|\braket{n|\psi}|^2$

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2. Basics of quantum N-body systems

N-particle systems (First quantization)

• Example of an N-particle Hamiltonian:

\[\underset{\text{Single-particle operators}}{\hat{H}=\sum_{i=1}^N(\frac{\|\hat{p}\|^2}{2m_i}+U(\vec{r}_i))} \quad + \underset{\text{Two-particle operators (e.g., interactions)}} {\frac{1}{2}\sum_{i,j(i\neq j)=1}^N V(\vec{r_i},\vec{r_j})}\]

• Schrodinger’s equation (Dirac’s notation):
  • $i\hbar\frac{\partial}{\partial t}\ket{\Psi(t)}=\hat{H}\ket{\Psi(t)}$
  • $\braket{\vec{r_1},\vec{r_2},\dots,\vec{r_N}|\Psi(t)}=\Psi(\vec{r_1},\dots,\vec{r_N})$
  • $\braket{\Phi|\Psi}=\int d^3\vec{r_1}d^3\vec{r_2}\cdots d^3\vec{r_N}\Phi^*(\vec{r_1},\dots,\vec{r_N})\Psi(\vec{r_1},\dots,\vec{r_N})$
• Basis out of single-particle states: ${\ket{\Phi_\alpha}}={\ket{\phi_{\alpha_1}}} \otimes {\ket{\phi_{\alpha_2}}}\otimes \cdots\otimes {\ket{\phi_{\alpha_N}}}$
• $\braket{\Phi_\alpha|\Phi_\beta}=\delta_{\alpha\beta}$ and $\sum_\alpha \ket{\Phi_\alpha} \bra{\Phi_\alpha}=\mathbb{1}$
• can write $\ket{\Psi(t)}=\sum_\alpha C_\alpha (t)\ket{\Phi_\alpha}=\sum_\alpha \braket{\Phi_\alpha | \Psi(t)}\ket{\Phi_\alpha}$
• can also use the full spectrum from a N-particle operator as a basis: ${\ket{\Psi_n}}=\hat{H}\ket{\Psi_n}=E_n\ket{\Psi_n}$

Systems of identical particles

• System of N distinguishable particles can be differentiated by performing measurements.
  • examples: classical particles, quantum particles with different mass (electrons and muons) or charge (electrons and protons)
• In a system of N indistinguishable particles, it can’t if they have the same quantum numbers (mass, charge, spin, etc.)
  • examples: system of N electrons with the same spin (spin-polarized or spinless)
  • thus the exchange of two identical particles cannot be experimentally detected; it should have the same probability density:
    • $|\Psi(\vec{r_1},\dots,\vec{r_k},\dots,\vec{r_m},\dots,\vec{r_N})|^2=|\Psi(\vec{r_1},\dots,\vec{r_m},\dots,\vec{r_k},\dots,\vec{r_N})|^2$
    • states are the same up to a global phase: $\Psi(\vec{r_1},\dots,\vec{r_k},\dots,\vec{r_m},\dots,\vec{r_N})=e^{i\theta}\Psi(\vec{r_1},\dots,\vec{r_m},\dots,\vec{r_k},\dots,\vec{r_N})$
    • experimental fact: second exchange of the same particles brings the state back to the initial one: $e^{2i\theta}=1$
      • bosons: $\theta=0$, $\Psi(\dots,\vec{r_k},\dots,\vec{r_m},\dots)=+\Psi(\dots,\vec{r_m},\dots,\vec{r_k},\dots)$
      • fermions: $\theta=\pi$, $\Psi(\dots,\vec{r_k},\dots,\vec{r_m},\dots)=-\Psi(\dots,\vec{r_m},\dots,\vec{r_k},\dots)$

An example with N=2

• Two (indistinguishable) particles in a harmonic oscillator:
  • $\hat{H}=(\frac{|\hat{p_1}|^2}{2m_i}+\frac{1}{2}m\omega\hat{x}_1^2)+\hat{H}=(\frac{|\hat{p_2}|^2}{2m_i}+\frac{1}{2}m\omega\hat{x}_2^2)=\hat{H}^{(1)}+\hat{H}^{(2)}$
  • $\hat{H}^{(1)}\ket{n_1}1=E{n_1}\ket{n_1}1$, $\hat{H}^{(2)}\ket{n_2}_2=E{n_2}\ket{n_2}_2$
  • position representation: $\braket{x|n}=\phi_n(x)$
  • $\ket{\Phi_\alpha}=\ket{n_1}\otimes\ket{n_2} \rightarrow \braket{x_1x_2|\Phi_\alpha}=\braket{x_1|n_1}1\braket{x_2|n_2}_2=\phi{n_1}(x_1)\phi_{n_2}(x_2)=\Phi_\alpha(x_1,x_2)$
  • Is this basis appropriate for indistinguishable particles? No, for example: $\phi_1(x_1)\phi_0(x_2)\neq\phi_1(x_2)\phi_0(x_1)$ Then how can we fix this?
    • Symmetrizing/Antisymmetrizing Boson and Fermions:
      • symmetrized basis (bosons): $\Phi_{2n+1}^S(x_1,x_2) = \phi_{n}(x_1)\phi_{n}(x_0)$
      • antisymmetrized basis (fermions): $\Phi_{2n}^S(x_1,x_2)=\frac{1}{\sqrt{2}}(\phi_0(x_1)\phi_n(x_2)+\phi_n(x_1)\phi_0(x_2)$
    • $\Phi_\alpha^{S,A}(x_1,x_2)=\pm \Phi_\alpha^{S,A}(x_2,x_1)$
    • In general:
    \[\Phi_\alpha^{(S,A)}(\vec{r_1},\dots,\vec{r_N})=A_\alpha\hat{S}_\pm\prod_{j=1}^N\phi_j(\vec{r_j})=\left\| \begin{array}{ccccc} \phi_1(\vec{r_1}) & \phi_1(\vec{r_2}) & \dots & \phi_1(\vec{r_N}) \\ \phi_2(\vec{r_1}) & \phi_2(\vec{r_2}) & \dots & \phi_2(\vec{r_N}) \\ \vdots & \ddots & \dots & \vdots \\ \phi_N(\vec{r_1}) & \phi_N(\vec{r_2}) & \dots & \phi_N(\vec{r_N}) \end{array} \right\|_\pm\]
  • How about spin, etc.?
    • there are other quantum numbers labeling the single-particle states
      1. count each single-particle state with a given set of quantum numbers as one single particle state: $\ket{\phi_{nlm\sigma}} \rightarrow \ket{\phi_\alpha}$
      2. we can also build many-particle states directly from sum of angular momenta (multiplets) , tricky but useful: $\ket{\phi^{(1)}{n_1l_1m_1\sigma_1}}\otimes\ket{\phi^{(2)}{n_2l_2m_2\sigma_2}}\rightarrow\ket{\Phi_{JM_J}}$

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3. Second Quantization

Symmetrizing/Antisymmetrizing Bosons and Fermions

\[\Phi_\alpha^{(S,A)}(\vec{r_1},\dots,\vec{r_N})=A_\alpha\hat{S}_\pm\prod_{j=1}^N\phi_j(\vec{r_j})=\left\| \begin{array}{ccccc} \phi_1(\vec{r_1}) & \phi_1(\vec{r_2}) & \dots & \phi_1(\vec{r_N}) \\ \phi_2(\vec{r_1}) & \phi_2(\vec{r_2}) & \dots & \phi_2(\vec{r_N}) \\ \vdots & \ddots & \dots & \vdots \\ \phi_N(\vec{r_1}) & \phi_N(\vec{r_2}) & \dots & \phi_N(\vec{r_N}) \end{array} \right\|_\pm\]

• $A_\alpha$: normalization
• $\hat{S}_\pm$: symmetrization operator: $+$ is permanant, $-$ is determinant

Number occupation representation

• alternative basis: number occupation: $\ket{n_1, \dots, n_N}$
• $n_i \rightarrow$ number of particles in state $\ket{\phi_i}$
• $\sum_j n_j=N$ (constraints), for fermions $n_j=0,1$, for bosons $n_j=0,1,2, …$
• number operator $\hat{n}_k$:
  • defined such that $\hat{n}_k\ket{n_1m\dots,n_N}=n_k\ket{n_1m\dots,n_N}$
  • vacuum state: $\ket{0}\equiv\ket{n_1=0, \dots, n_N=0}$
• examples: $n=2$, single-particle basis: ${\ket{\phi_0},\ket{\phi_2},\ket{\phi_3}}$
  • bosons:
    • $\ket{\phi_0}\otimes\ket{\phi_0} \rightarrow \ket{n_0=2, n_1=0,n_2=0}_+$
    • $\frac{1}{\sqrt{2}}(\ket{\phi_1}\otimes\ket{\phi_2}+\ket{\phi_2}\otimes\ket{\phi_1}) \rightarrow \ket{n_0=0, n_1=1,n_2=1}_+$
  • fermions:
    • $\frac{1}{\sqrt{2}}(\ket{\phi_0}\otimes\ket{\phi_2}-\ket{\phi_2}\otimes\ket{\phi_0}) \rightarrow \ket{n_0=1, n_1=0,n_2=1}_-$
    • $\frac{1}{\sqrt{2}}(\ket{\phi_1}\otimes\ket{\phi_2}-\ket{\phi_2}\otimes\ket{\phi_1}) \rightarrow \ket{n_0=1, n_1=1,n_2=1}_-$

  (with the order matters that

  • $\ket{n_0=1, n_1=0,n_2=1}+=+\ket{n_0=1, n_2=0,n_1=1}+$
  • $\ket{n_0=1, n_1=0,n_2=1}-=-\ket{n_0=1, n_2=0,n_1=1}-$)

Bosonic creation and destruction operators

• bosons: define operators $\hat{b}_i$ and $\hat{b}_i^\dagger$:
  • $\hat{b}j^\dagger\ket{n_1,\dots,n_N}=B+(n_j)\ket{n_1,\dots,n_{j}+1,\dots,n_N}$
  • $\hat{b}\ket{n_1,\dots,n_N}=B_-(n_j)\ket{n_1,\dots,n_{j}-1,\dots,n_N}$ ($n_j$ : normalization)

  • in this basis, the only non-zero matrix elements are

    $\braket{n_1,\dots,n_j+1,\dots,n_N|\hat{b}_j^\dagger|n_1,d\dots,n_N}$ = $(\braket{n_1,\dots,n_j,\dots,n_N|\hat{b}_j|n_1,d\dots,n_j+1,n_N})^*$

• question: $\hat{b}_k^\dagger\hat{b}_j^\dagger\ket{n_1,\dots,n_N}=\hat{b}_j^\dagger\hat{b}_k^\dagger\ket{n_1,\dots,n_N}$? yes!
  • thus $[\hat{b}_k^\dagger,\hat{b}_j^\dagger]=0$ and by the same reasoning $[\hat{b}_k,\hat{b}_j]=0$
• question: if $k\neq j$, $\hat{b}_k\hat{b}_j^\dagger\ket{n_1,\dots,n_N}=\hat{b}_j^\dagger\hat{b}_k\ket{n_1,\dots,n_N}$? yes!
  • thus $[\hat{b}_k,\hat{b}_j^\dagger]=0$ ($k\neq j$)
• what if $k=j$? $\hat{b}_j\hat{b}_j^\dagger\ket{n_1,\dots,n_N}=\hat{b}_j^\dagger\hat{b}_j\ket{n_1,\dots,n_N}$
  • vacuum state: $\ket{0}\equiv\ket{n_1=0, \dots, n_N=0}$
  • clearly, we must impose: $\hat{b}_j\ket{0}=0$ $\forall j$
  • thus $\hat{b}_j^\dagger\hat{b}_j\ket{0}=0$ however $\hat{b}_j\hat{b}_j^\dagger\ket{0} \propto \ket{0}$
  • we choose the norm. constant such that $\hat{b}_j\hat{b}_j^\dagger\ket{0}=\ket{0} \Rightarrow [\hat{b}_j,\hat{b}_j^\dagger]\ket{0}=0$
  • In general, let’s assume the following operator identities hold: $[\hat{b}k\hat{b}_j^]=[\hat{b}_k^\dagger\hat{b}_j^\dagger]=0$, $[\hat{b}_k\hat{b}_j^\dagger]=\delta{kj}$

Bosonic number operator

• show that: $[\hat{b}_k^\dagger\hat{b}_k,\hat{b}_k]=-\hat{b}_k$, $[\hat{b}_k^\dagger\hat{b}_k,\hat{b}_k^\dagger]=+\hat{b}_k^\dagger$
• then show that: $\hat{b}_k\ket{\dots,n_k,\dots}=\sqrt{n_k}\ket{\dots,n_k-1,\dots}$, $\hat{b}_k^\dagger\ket{\dots,n_k,\dots}=\sqrt{n_k}\ket{\dots,n_k+1,\dots}$
• finally, show that: $\hat{n}_k^\equiv\hat{b}_k^\dagger\hat{b}_k$ such that $\hat{n}_k\ket{n_1,\dots,n_N}=n_k\ket{n_1,\dots,n_N}$

Fermionic creation and destruction operators

• fermions: define operators $\hat{c}_i$ and $\hat{c}_i^\dagger$:
  • $\hat{c}j^\dagger\ket{n_1,\dots,n_N}=C+(n_j)\ket{n_1,\dots,n_{j}+1,\dots,n_N}$
  • $\hat{c}j\ket{n_1,\dots,n_N}=C-(n_j)\ket{n_1,\dots,n_{j}-1,\dots,n_N}$ $n_j=0,1$
  • the order of the occupations is: $\ket{\dots,n_k,\dots,n_j,\dots}=-\ket{\dots,n_j,\dots,n_k,\dots}$, $\hat{c}_k^\dagger\hat{c}_j^\dagger\ket{\dots,0,\dots,0,\dots}=-\hat{c}_j^\dagger\hat{c}_k^\dagger\ket{\dots,0,\dots,0,\dots}$
  • let’s assume those holds as an operator identity: $\hat{c}_k^\dagger\hat{c}_j^\dagger=-\hat{c}_j^\dagger\hat{c}_k^\dagger \Rightarrow {c_k^\dagger,c_j^\dagger}$ by the same reasoning, ${\hat{c}_k,\hat{c}_j}=0, {\hat{c}_k,\hat{c}_j^\dagger}=0$ $(k\neq j)$
  • if $k=j$, ${\hat{c}k,\hat{c}_k^\dagger}=1$, thus ${\hat{c}_k,\hat{c}_j^\dagger}=\delta{kj}$, ${\hat{c}_k,\hat{c}_j}=0={\hat{c}_k^\dagger,\hat{c}_j^\dagger}$ $\rightarrow (\hat{c}_k^\dagger)^2\ket{\dots,n_k,\dots}=0, (\hat{c}_k)^2\ket{\dots,n_k,\dots}=0$

Fermionic number operator

• show that: $[\hat{c}_k^\dagger\hat{c}_k,\hat{c}_k]=-\hat{c}_k$, $[\hat{c}_k^\dagger\hat{c}_k,\hat{c}_k^\dagger]=+\hat{c}_k$
• then, show that $(\hat{c}_k^\dagger\hat{c}_k)^2\ket{\dots,n_k,\dots}=\hat{c}_k^\dagger\hat{c}_k\ket{\dots,n_k,\dots}$
• thus, if $\hat{c}_k^\dagger\hat{c}_k\ket{\dots,n_k,\dots}=n_k\ket{\dots,n_k,\dots}\Rightarrow n_k=0,1$
• from, this, we can associate $\hat{n}_k\equiv\hat{c}_k^\dagger\hat{c}_k$ such that $\hat{n}_k\ket{n_1,\dots,n_N}=n_k\ket{n_1,\dots,n_N}$

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4. Operators in Second Quantization

One-body operators

• one-body operator written in a single-particle basis: $T_{kj}=\braket{\phi_k|\hat{T}|\phi_j}\Leftrightarrow \hat{T}=\sum_{kj}T_{kj}\ket{\phi_k}\bra{\phi_j}$
• matrix element: $T_{kj}=\int d^3\vec{r}\phi_k^*(\vec{r})T(\vec{r})\phi_j(\vec{r})$
• example: kinetic energy $K_{kj}=\left(-\frac{\hbar^2}{2m}\right)\int d^3\vec{r}\phi_k^*(\vec{r})\nabla^2_{\vec{r}}\phi_j(\vec{r})$

Two-body operators

• two-body operator written in a two-particle basis:
  • $\ket{\Phi_\alpha}=S_\pm\ket{\phi_i}\otimes \ket{\phi_j}=\ket{\phi_i}\ket{\phi_j}$
  • $V_{\alpha\beta}=\braket{\Phi_\alpha|\hat{V}|\Phi_\beta}\Leftrightarrow\hat{V}=\sum_{\alpha\beta}V_{\alpha\beta}\ket{\Phi_\alpha}\bra{\Phi_\beta}$
• two-body operator written in a single-particle basis:
  • $\Rightarrow \hat{V}=\sum_{ijkm}V_{ijkm}\ket{\phi_i}\ket{\phi_j}\bra{\phi_k}\bra{\phi_m}+\dots$
  • matrix elements: $V_{ijkm}=\int d^3\vec{r}_1d^3\vec{r}_2\phi_i^(\vec{r}_1)\phi_j^(\vec{r}_2)V(\vec{r}_1,\vec{r}_2)\phi_k(\vec{r}_1)\phi_m(\vec{r}_2)$

Operators in the N-body basis

• one-body operator written in an N-particle basis:
  • $\hat{T}^{(n)}\ket{\phi_{k_1}}\dots\ket{\phi_{k_n}}\dots\ket{\phi_{k_N}}=\sum_{ij}T_{kj}^{(n)}\delta_{jk_n}\ket{\phi_{k_1}}\dots\ket{\phi_k}\dots\ket{\phi_{k_N}}$
  • sum of N one-body operators written in a N-particle basis: $\hat{T}{tot}=\sum{n-1}^N\hat{T}^{(n)}$, $\hat{T}{tot}\ket{\phi{k_1}}\dots\ket{\phi_{k_N}}=\sum_{n=1}^N\sum_{kj}T_{kj}^{(n)}\delta_{jk_n}\ket{\phi_{k_1}}\dots\ket{\phi_{k_k}}\dots\ket{\phi_{k_N}}$

Operators in the N-body bosonic basis

• orbitals: $\ket{\phi_k}=\hat{b}_k^\dagger\ket{0}, \bra{\phi_k}=\bra{0}\hat{b}_k$
• one-body operator: $\hat{T}=\sum_{kj}T_{kj}\ket{\phi_k}\bra{\phi_j}$
• two-body operator: $\hat{V}=\sum_{ijkm}V_{ijkm}\ket{\phi_i}\ket{\phi_j}\bra{\phi_k}\bra{\phi_m}$
• N-body operators (sum):
  • $\hat{T}{tot}=\sum{n=1}^N\hat{T}^{(n)}$
  • $\hat{V}=\frac{1}{2}\sum_{n,n’=1(n\neq n’)}^N \hat{V}^{(n,n’)}$
• number occupation representation: $\sum_j n_j=N$, $\ket{n_1,\dots,n_k,\dots,n_N} \propto (\hat{b}_1^\dagger)^{n_1}\dots(\hat{b}_k^\dagger)^{n_k}\dots(\hat{b}_N^\dagger)^{n_N}\ket{0}$
• we can show that $\hat{T}{tot}\ket{n_1,\dots,n_k,\dots,n_N}=\sum{kj}T_{kj}\hat{b}_k^\dagger\hat{b}_j\ket{n_1,\dots,n_k,\dots,n_j,\dots n_N}$
• also show that $\hat{V}{tot}\ket{n_1,\dots,n_N}=\sum{ijkm}V_{ijkm}\hat{b}_i^\dagger \hat{b}_j^\dagger\hat{b}_m\hat{b}_k\ket{n_1,\dots,n_N}$

Operators in second quantized form

• we can write:
  • $\hat{T}{tot}=\sum{kj}T_{kj}\hat{b}_k^\dagger\hat{b}_j^\dagger$
  • $\hat{V}{tot}=\sum{ijkm}V_{ijkm}\hat{b}_i^\dagger\hat{b}_j^\dagger\hat{b}_m\hat{b}_k$
• it turns out that, for fermions, they take the same form:
  • $\hat{T}{tot}=\sum{kj}T_{kj}\hat{c}_k^\dagger\hat{c}_j^\dagger$
  • $\hat{V}{tot}=\sum{ijkm}V_{ijkm}\hat{c}_i^\dagger\hat{c}_j^\dagger\hat{c}_m\hat{c}_k$

Example: non-interacting system

• Hamiltonian: $\hat{H}=\sum_{n=1}^N\hat{h}^{(n)}$, in this basis : $\hat{H}_{tot}=\sum_k\epsilon_k\hat{n}_k$
• single-particle basis: $\hat{h}\ket{\phi_i}=\epsilon_i\ket{\phi_i}$, $\hat{n}_k=\hat{c}_k^\dagger\hat{c}_k$ (fermions) , $\hat{n}_k=\hat{b}_k^\dagger\hat{b}_k$ (bosons)
• many-body spectrum:
  • $\hat{H}_{tot}\ket{n_1,\dots,n_N}=(\sum_k \epsilon_kn_k)\ket{n_1,\dots,n_N}$
  • $E_{n_1,\dots,n_N}=\sum_{n=1}^N\epsilon_kn_k$

Density operator

• canonical: $\hat{\rho}\equiv e^{-\beta \hat{H}_{tot}}$, $\beta=\frac{1}{kT}$
• grand-canonical: $\hat{\rho}G\equiv e^{-\beta(\hat{H}{tot}-\mu\hat{N})}$, $\hat{N}=\sum_{k=1}^N\hat{n}_k$
• many-particle spectrum: $\hat{H}{tot}\ket{\alpha}=E\alpha\ket{\alpha}$
• partition function: $\mathcal{Z}=\sum_\alpha e^{-\beta E_\alpha}=\text{Tr}(\hat{\rho})$
• thermal average: $\braket{\hat{A}}=\frac{\text{Tr}(\hat{\rho}\hat{A})}{\text{Tr}(\hat{\rho})}=\frac{1}{\mathcal{Z}}\sum_\alpha A_\alpha e^{-\beta E_\alpha}$

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Non-interacting electron gas

Mean-field theory

Time evolution and representations in quantum mechanics

Equations of motion and time-ordered correlation functions

Retarded and and advanced Green’s functions

The Anderson impurity model

Schrieffer-Wolff transformation (Kondo model)

Linear response theory and Kubo formula

Kubo formula for the conductivity and the conductance

Electronic transports in the Anderson model

Kondo effect and numerical renormalization group

Time-ordered Green’s functions and WIck’s theorem

Feynman diagrams

Dyson’s equations and self-energy

Random Phase Approximation (RPA)

Matsubara formalism

Phonons

Phonon Green’s functions

Cooper instability

Introduction to BCS theory

TBA