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Register a new userContinuous-time auxiliary field Monte Carlo for quantum impurity models
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We present a continuous-time Monte Carlo method for quantum impurity models, which combines a weak-coupling expansion with an auxiliary-field decomposition. The method is considerably more efficient than Hirsch-Fye and free of time discretization errors, and is particularly useful as impurity solver in large cluster dynamical mean field theory (DMFT) calculations.
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We derive equations of motion for Greens functions of the multi-orbital Anderson impurity model by differentiating symmetrically with respect to all time arguments. The resulting equations relate the one- and two-particle Greens function to correlators of up to six particles at four times. As an application we consider continuous-time quantum Monte Carlo simulations in the hybridization expansion, which hitherto suffered from notoriously high noise levels at large Matsubara frequencies. Employing the derived symmetric improved estimators overcomes this problem.
In the present paper, we present an efficient continuous-time quantum Monte Carlo impurity solver with high acceptance rate at low temperature for multi-orbital quantum impurity models with general interaction. In this hybridization expansion impurity solver, the imaginary time evolution operator for the high energy multiplets, which decays very rapidly with the imaginary time, is approximated by a probability normalized $delta$-function. As the result, the virtual charge fluctuations of $f^{n}rightarrow f^{npm1}$ are well included on the same footing without applying Schrieffer-Wolff transformation explicitly. As benchmarks, our algorithm perfectly reproduces the results for both Coqblin-Schriffeer and Kondo lattice models obtained by CT-J method developed by Otsuki {it et al}. Furthermore, it allows capturing low energy physics of heavy-fermion materials directly without fitting the exchange coupling $J$ in the Kondo model.
We present an extension of constrained-path auxiliary-field quantum Monte Carlo (CP-AFQMC) for the treatment of correlated electronic systems coupled to phonons. The algorithm follows the standard CP-AFQMC approach for description of the electronic degrees of freedom while phonons are described in first quantization and propagated via a diffusion Monte Carlo approach. Our method is tested on the one- and two-dimensional Holstein and Hubbard-Holstein models. With a simple semiclassical trial wavefunction, our approach is remarkably accurate for $omega/(2text{d}tlambda) < 1$ for all parameters in the Holstein model considered in this study. In addition, we empirically show that the autocorrelation time scales as $1/omega$ for $omega/t lesssim 1$, which is an improvement over the $1/omega^2$ scaling of the conventional determinant quantum Monte Carlo algorithm. In the Hubbard-Holstein model, the accuracy of our algorithm is found to be consistent with that of standard CP-AFQMC for the Hubbard model when the Hubbard $U$ term dominates the physics of the model, and is nearly exact when the ground state is dominated by the electron-phonon coupling scale $lambda$. The approach developed in this work should be valuable for understanding the complex physics arising from the interplay between electrons and phonons in both model lattice problems and ab-initio systems.
The negative sign problem in quantum Monte Carlo (QMC) simulations of cluster impurity problems is the major bottleneck in cluster dynamical mean field calculations. In this paper we systematically investigate the dependence of the sign problem on the single-particle basis. We explore both the hybridization-expansion and the interaction-expansion variants of continuous-time QMC for three-site and four-site impurity models with baths that are diagonal in the orbital degrees of freedom. We find that the sign problem in these models can be substantially reduced by using a non-trivial single-particle basis. Such bases can be generated by diagonalizing a subset of the intracluster hoppings.
We propose a novel technique for speeding up the self-learning Monte Carlo method applied to the single-site impurity model. For the case where the effective Hamiltonian is expressed by polynomial functions of differences of imaginary-time coordinate between vertices, we can remove the dependence of CPU time on the number of vertices, $n$, by saving and updating some coefficients for each insertion and deletion process. As a result, the total cost for a single-step update is drastically reduced from $O(nm)$ to $O(m^2)$ with $m$ being the order of polynomials in the effective Hamiltonian. Even for the existing algorithms, in which the absolute value is used instead of the difference as the variable of polynomial functions, we can limit the CPU time for a single step of Monte Carlo update to $O(m^2 + m log n)$ with the help of balanced binary search trees. We demonstrate that our proposed algorithm with only logarithmic $n$-dependence achieves an exponential speedup from the existing methods, which suffer from severe performance issues at low temperatures.
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