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arXiv:1807.00361 · Physical review. B./Physical review. B

State and superstate sampling in hybridization-expansion continuous-time quantum Monte Carlo

Alexander Kowalski, Andreas Hausoel, Markus Wallerberger, Patrik Gunacker, Giorgio Sangiovanni

Published Apr 8, 2019• 24 citations
arXivHTMLPDF

Abstract

Due to the intrinsic complexity of the quantum many-body problem, quantum Monte Carlo algorithms and their corresponding Monte Carlo configurations can be defined in various ways. Configurations corresponding to few Feynman diagrams often lead to severe sign problems. On the other hand, computing the configuration weight becomes numerically expensive in the opposite limit in which many diagrams are grouped together. Here we show that for continuous-time quantum Monte Carlo in the hybridization expansion the efficiency can be substantially improved by dividing the local impurity trace into fragments, which are then sampled individually. For this technique, which also turns out to preserve the fermionic sign, a modified update strategy is introduced in order to ensure ergodicity. Our (super)state sampling is particularly beneficial to calculations with many $d$ orbitals and general local interactions, such as full Coulomb interaction. For illustration, we reconsider the simple albeit well-known case of a degenerate three-orbital model at low temperatures. This allows us to quantify the coherence properties of the ``spin-freezing'' crossover, even close to the Mott transition.

DOI
10.1103/physrevb.99.155112

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Due to the intrinsic complexity of the quantum many-body problem, quantum Monte Carlo algorithms and their corresponding Monte Carlo configurations can be defined in various ways. Configurations corresponding to few Feynman diagrams often lead to severe sign problems. On the other hand, computing the configuration weight becomes numerically expensive in the opposite limit in which many diagrams are grouped together. Here we show that for continuous-time quantum Monte Carlo in the hybridization expansion the efficiency can be substantially improved by dividing the local impurity trace into fragments, which are then sampled individually. For this technique, which also turns out to preserve the fermionic sign, a modified update strategy is introduced in order to ensure ergodicity. Our (super)state sampling is particularly beneficial to calculations with many ddd orbitals and general local interactions, such as full Coulomb interaction. For illustration, we reconsider the simple albeit well-known case of a degenerate three-orbital model at low temperatures. This allows us to quantify the coherence properties of the ``spin-freezing'' crossover, even close to the Mott transition.
The sign problem remains a major obstacle to extending the success of Monte Carlo methods to generic quantum many-body systems. A powerful alternative relies on cluster expansions, which provide convergent representations of free energies and related quantities, with applications ranging from tensor network contractions to quantum dynamics and circuits. Within their convergence regime, these expansions often yield deterministic polynomial-time algorithms. However, such methods require exhaustive enumeration of connected clusters, leading to a computational overhead of (Nε−1)O(log⁡Δ)(N\varepsilon^{-1})^{\mathcal{O}(\logΔ)}(Nε−1)O(logΔ), where NNN is the system size, ε\varepsilonε the desired precision, and ΔΔΔ the interaction-graph degree. This becomes particularly severe in long-range systems, where Δ≡NΔ\equiv NΔ≡N, resulting in quasi-polynomial runtimes. Here, we introduce a Markov chain algorithm that samples cluster expansions directly in the space of connected clusters, replacing deterministic enumeration by Monte Carlo sampling. Because sampling occurs in an abstract cluster space rather than a physical configuration space, the conventional exponential sign problem is absent within the convergence regime. For general polymer models, we establish an FPRAS with runtime O~(N4ε−3)\tilde{\mathcal{O}}(N^4\varepsilon^{-3})O~(N4ε−3). For quantum partition functions at temperatures β≤βsampling≡O(1)β\leqβ_{\mathrm{sampling}}\equiv\mathcal{O}(1)β≤βsampling​≡O(1), this improves to O~(N3+1/Dε−3)\tilde{\mathcal{O}}(N^{3+1/D}\varepsilon^{-3})O~(N3+1/Dε−3), where DDD is the lattice dimension. The framework directly applies to long-range quantum systems with interactions ∝1/rα\propto1/r^α∝1/rα, including 222-local models with α>2Dα>2Dα>2D.
With the success of dynamical mean-field theories, solvers for quantum-impurity problems have become an important tool for the numerical study of strongly correlated systems. Continuous-time quantum Monte Carlo sampling of the expansion in powers of the hybridization between the ``impurity'' and the bath provides a powerful solver when interactions are strong. Here we show that the usual updates that add or remove a pair of creation-annihilation operators are rigorously not ergodic for several classes of broken symmetries that involve spatial components. We show that updates with larger numbers of simultaneous updates of pairs of creation-annihilation operators remedy this problem. As an example, we apply the four-operator updates that are necessary for ergodicity to the case of ddd-wave superconductivity in plaquette dynamical mean-field theory for the one-band Hubbard model. While the results are qualitatively similar to those previously published, they are quantitatively better than previous ones, being closer to those obtained by other approaches.
We explore two complementary modifications of the hybridization-expansion continuous-time Monte Carlo method, aiming at large multi-orbital quantum impurity problems. One idea is to compute the imaginary-time propagation using a matrix product state representation. We show that bond dimensions considerably smaller than the dimension of the Hilbert space are sufficient to obtain accurate results and that this approach scales polynomially, rather than exponentially with the number of orbitals. Based on scaling analyses, we conclude that a matrix product state implementation will outperform the exact-diagonalization based method for quantum impurity problems with more than 12 orbitals. The second idea is an improved Monte Carlo sampling scheme which is applicable to all variants of the hybridization expansion method. We show that this so-called sliding window sampling scheme speeds up the simulation by at least an order of magnitude for a broad range of model parameters, with the largest improvements at low temperature.
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 interaction-expansion variants of continuous-time QMC simulations 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 nontrivial single-particle basis. Such bases can be generated by diagonalizing a subset of the intracluster hoppings.
Continuous-time quantum Monte Carlo refers to a class of algorithms designed to sample the thermal distribution of a quantum Hamiltonian through exact expansions of the Boltzmann exponential in terms of stochastic trajectories which are periodic in imaginary time. Here, we show that for (sign-problem-free) quantum many-body systems with discrete degrees of freedom—such as spins on a lattice—this sampling can be done in a rejection-free manner using transition path sampling (TPS). The key idea is to converge the trajectory ensemble through updates where one individual degree of freedom is modified across all time while the remaining unaltered ones provide a time-dependent background. The ensuing single-body dynamics provides a way to generate trajectory updates exactly, allowing one to obtain the target ensemble efficiently via rejection-free TPS. We demonstrate our method on the transverse field Ising model in one and two dimensions, and on the quantum triangular plaquette (or Newman-Moore) model. We show that despite large autocorrelation times, our method is able to efficiently recover the respective quantum phase transition of each model. We also discuss the connection to rare event sampling in continuous-time Markov dynamics. Published by the American Physical Society 2024
In the context of realistic calculations for strongly correlated materials with ddd or fff electrons the efficient computation of multi-orbital models is of paramount importance. Here we introduce a set of invariants for the SU(2)SU(2)SU(2)-symmetric Kanamori Hamiltonian, which allows us to massively speed up the calculation of the fermionic trace in hybridization-expansion continuous-time quantum Monte Carlo algorithms. We show that by exploiting this set of good quantum numbers the study of the orbital-selective Mott transition in systems with up to seven correlated orbitals becomes feasible.