Quantum Monte Carlo

Famous sign problem

Central limit Theorem

$⟨O{⟩}_P={\int }_{\Omega }dxP(x)O(x){\int }_{\Omega }dxP(x)=1$ Generate set of configurations $x_i$, distributed according to $P(x)$ and that $O(x_i)$ are statistically independent, then $⟨O{⟩}_P\approx \frac{1}{N}{\sum }_{i=1}^{N}O(x_i)=X$ $\mathcal{P}(X)=\frac{1}{2\pi }{e}^{(X-{)}^2/2\sigma }$

Markov Chains

Transition probability $T_{y,x}$ where

$$\sum _{y\in \Omega }{T}_{y,x}=1$$

$T$ has to satisfy

• Ergodicity $\forall x,y\in \Omega \exists s|(T^s{)}_{y,x}>0$
• Stationary ${\sum }_x{T}_{y,x}P(x)=P(y)$
• Then ${\lim }_{n\to \infty }\frac{1}{n}{\sum }_{t=1}^n[T^t{]}_{x,x_1}=P(X)$

For fermions

$$Z=\text{Tr}{e}^{-\beta \hat{H}}=\int D\{\Phi (i,\tau )\}{e}^{-S\{\Phi \}}$$

• Trotter decomposition
• Grassman variable