# Introduction

This is an informal introduction to the Determinant Quantum Monte Carlo (DQMC) algorithm. It is capable of simulating so-called "strongly correlated" quantum systems
at high temperatures. *Strongly correlated* means that the quantum particles (typically electrons) cannot be well described by independent-electron approximations
(like density functional theory or band structure calculations); a famous example are high-\(T_c\) "cuprate" superconductors.

To get off the ground, we first need a *model* of the physical system. You have probably heard of the Schrödinger equation in quantum mechanics.
The deceptive apparent simplicity of the equation \( i \hbar \partial_t \Psi = H \Psi\) stands in stark contrast to the difficulty of actually solving it numerically,
once the wavefunction \(\Psi\) describes "many" (think 10 or more) electrons simultaneously.

*tight binding models*as pictured above. Tell me more about tight binding models

Tight binding models are succinctly expressed in the languge of "second quantization". Specifically, the picture directly translates to the following Hamiltonian: \begin{equation} \label{eq:hubbard_hamiltonian} H = -{\color{blue}t} \sum_{\langle i,j \rangle, \sigma} \left(c^{\dagger}_{i\sigma} c_{j\sigma} + c^{\dagger}_{j\sigma} c_{i\sigma}\right) + {\color{red}U} \sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} - \tfrac{1}{2}) \end{equation} The \(c^{\dagger}_{i\sigma}\) and \(c_{i\sigma}\) are the so-called fermionic creation and annihilation operators for lattice site \(i\) and spin \(\sigma\), and \(n_{i\sigma} = c^{\dagger}_{i\sigma} c_{i\sigma}\). Informally, they add or remove an electron from lattice site \(i\) with spin \(\sigma\). The first term in \eqref{eq:hubbard_hamiltonian} (so-called kinetic hopping) precisely describes the hopping to neighboring lattice sites, while the second ("interaction") term penalizes double occupancy of a lattice site due to Coulomb repulsion.

# Traces in quantum Fock-space

For a Hamiltonian of *quadratic* form
$$
H = \sum_{i,j} {\color{blue}h}_{ij}\, c^{\dagger}_i c_j
$$
the following exact identity holds:
$$
\mathrm{tr}\big[\mathrm{e}^{-\beta H}\big] = \det\!\big[I + \mathrm{e}^{-\beta {\color{blue}h}}\big]
$$
where \(I\) is the identity matrix.

This is simple to check for a single "orbital", that is, for \(H = \epsilon\, c^{\dagger} c\): $$ \mathrm{tr}\big[\mathrm{e}^{-\beta H}\big] = \langle 0 \vert \mathrm{e}^{-\beta \epsilon c^{\dagger} c} \vert 0 \rangle + \langle 1 \vert \mathrm{e}^{-\beta \epsilon c^{\dagger} c} \vert 1 \rangle = 1 + \mathrm{e}^{-\beta \epsilon}. $$ In the general case, consider a basis in which \(\color{blue}h\) is diagonal.

# Hubbard-Stratonovich transformation

However, Hubbard interaction term contains four-fermion operators … \(\leadsto\) discrete Hubbard-Stratonovich transformation (using that \(n_{i\sigma} \in \{0,1\}\), after Trotter splitting \(\beta = \Delta\tau L\)) $$ \mathrm{e}^{-\Delta\tau {\color{red}U} \sum_i (n_{i\uparrow} - \frac{1}{2})(n_{i\downarrow} - \frac{1}{2})} = \mathrm{e}^{-\Delta\tau {\color{red}U}/4}\, \tfrac{1}{2} \sum_{ {\color{forestgreen}s} = \pm 1} \mathrm{e}^{-\Delta\tau {\color{forestgreen}s} \lambda (n_{i\uparrow} - n_{i\downarrow})} $$ \(\lambda\) determined by \(\cosh(\Delta\tau \lambda) = \mathrm{e}^{\Delta\tau {\color{red}U}/2}\)