Principles of continuous-time quantum Monte Carlo algorithm

Now we already have an impurity model Hamiltonian $H_{\text{imp}}$, the question is how to solve it using the Monte Carlo algorithm?

We first split the impurity Hamiltonian $H_{\text{imp}}$ into two separate parts,

$$H_{\text{imp}} = H_1 + H_2,$$

then treat $H_2$ as a perturbation term, and expand the partition function $\mathcal{Z}$ in powers of $H_2$, $$\begin{equation} \mathcal{Z} = \text{Tr} e^{-\beta H} = \sum_{n=0}^{\infty} \int_{0}^{\beta} \cdots \int_{\tau_{n-1}}^\beta \omega(\mathcal{C}_n), \end{equation}$$ with $$\begin{equation} \omega(\mathcal{C}_n)=d\tau_1 \cdots d\tau_n \text{Tr}\left\{ e^{-\beta H_1}[-H_2(\tau_n)]\cdots [-H_2(\tau_1)]\right\}, \end{equation}$$ where $H_2(\tau)$ is defined in the interaction picture with

$$H_2(\tau) = e^{\tau H_1} H_2 e^{-\tau H_1}.$$

Each term in the right side of the second equation can be regarded as a diagram or configuration (labeled by $\mathcal{C}$), and $\omega(\mathcal{C}_n)$ is the diagrammatic weight of a specific order-$n$ configuration. Next we use a stochastic Monte Carlo algorithm to sample the terms of this series. This is the core spirit of the continuous-time quantum Monte Carlo impurity solver. The idea is very simple, but the realization is not.

Depending on the different choices of $H_{2}$ term, there are multiple variations for the continuous-time quantum Monte Carlo impurity solver. According to our knowledge, the variations at least include

• CT-INT
• CT-HYB
• CT-J
• CT-AUX

In the CT-INT and CT-AUX quantum impurity solvers^1,2, the interaction term is the perturbation term, namely, $H_2 = H_{\text{int}}$, while $H_2 = H_{\text{hyb}}$ is chosen for the CT-HYB quantum impurity solver³. The CT-J quantum impurity solver is designed for the Kondo lattice model only⁴. We won't discuss it at here. In the intermediate and strong interaction region, CT-HYB is much more efficient than CT-INT and CT-AUX. We could even say that it is the most powerful and efficient quantum impurity solver so far. This is also the main reason that we only implemented the CT-HYB quantum impurity solvers in the $i$QIST software package.

Reference

¹ A. N. Rubtsov, V. V. Savkin, and A. I. Lichtenstein, Phys. Rev. B 72, 035122 (2005)

² Emanuel Gull, Philipp Werner, Olivier Parcollet, Matthias Troyer, EPL 82, 57003 (2008)

³ Philipp Werner, Armin Comanac, Luca de’ Medici, Matthias Troyer, and Andrew J. Millis, Phys. Rev. Lett. 97, 076405 (2006)

⁴ Junya Otsuki, Hiroaki Kusunose, Philipp Werner, and Yoshio Kuramoto, J. Phys. Soc. Jpn. 76, 114707 (2007)