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Register a new userQuantum Quasi-Monte Carlo Technique for Many-Body Perturbative Expansions
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High order perturbation theory has seen an unexpected recent revival for controlled calculations of quantum many-body systems, even at strong coupling. We adapt integration methods using low-discrepancy sequences to this problem. They greatly outperform state-of-the-art diagrammatic Monte Carlo. In practical applications, we show speed-ups of several orders of magnitude with scaling as fast as $1/N$ in sample number $N$; parametrically faster than $1/sqrt{N}$ in Monte Carlo. We illustrate our technique with a solution of the Kondo ridge in quantum dots, where it allows large parameter sweeps.
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We use the stochastic series expansion quantum Monte Carlo method, together with the eigenstate-to-Hamiltonian mapping approach, to map the localized ground states of the disordered two-dimensional Heisenberg model, to excited states of a target Hamiltonian. The localized nature of the ground state is established by studying the spin stiffness, local entanglement entropy, and local magnetization. This construction allows us to define many body localized states in an energy resolved phase diagram thereby providing concrete numerical evidence for the existence of a many-body localized phase in two dimensions.
We extend the recently developed Quantum Quasi-Monte Carlo (QQMC) approach to obtain the full frequency dependence of Green functions in a single calculation. QQMC is a general approach for calculating high-order perturbative expansions in power of the electron-electron interaction strength. In contrast to conventional Markov chain Monte Carlo sampling, QQMC uses low-discrepancy sequences for a more uniform sampling of the multi-dimensional integrals involved and can potentially outperform Monte Carlo by several orders of magnitudes. A core concept of QQMC is the a priori construction of a model function that approximates the integrand and is used to optimize the sampling distribution. In this paper, we show that the model function concept extends to a kernel approach for the computation of Green functions. We illustrate the approach on the Anderson impurity model and show that the scaling of the error with the number of integrand evaluations $N$ is $sim 1/N^{0.86}$ in the best cases, and comparable to Monte Carlo scaling $sim 1/N^{0.5}$ in the worst cases. We find a systematic improvement over Monte Carlo sampling by at least two orders of magnitude while using a basic form of model function. Finally, we compare QQMC results with calculations performed with the Fork Tensor Product State (FTPS) method, a recently developed tensor network approach for solving impurity problems. Applying a simple Pade approximant for the series resummation, we find that QQMC matches the FTPS results beyond the perturbative regime.
We propose a novel approach to nonequilibrium real-time dynamics of quantum impurities models coupled to biased non-interacting leads, such as those relevant to quantum transport in nanoscale molecular devices. The method is based on a Diagrammatic Monte Carlo sampling of the real-time perturbation theory along the Keldysh contour. We benchmark the method on a non-interacting resonant level model and, as a first non-trivial application, we study zero temperature non-equilibrium transport through a vibrating molecule.
We present a method for calculating the time-dependent many-body wavefunction that follows a local quench. We apply the method to the voltage-driven nonequilibrium Kondo model to find the exact time-evolving wavefunction following a quench where the dot is suddenly attached to the leads at $t=0$. The method, which does not use Bethe ansatz, also works in other quantum impurity models (we include results for the interacting resonant level and the Anderson impurity model) and may be of wider applicability. In the particular case of the Kondo model, we show that the long-time limit (with the system size taken to infinity first) of the time-evolving wavefunction is a current-carrying nonequilibrium steady state that satisfies the Lippmann-Schwinger equation. We show that the electric current in the time-evolving wavefunction is given by a series expression that can be expanded either in weak coupling or in strong coupling, converging to all orders in the steady-state limit in either case. The series agrees to leading order with known results in the well-studied regime of weak antiferromagnetic coupling and also reveals another universal regime of strong ferromagnetic coupling, with Kondo temperature $T_K^{(F)} = D e^{-frac{3pi^2}{8} rho |J|}$ ($J<0$, $rho|J|toinfty$). In this regime, the differential conductance $dI/dV$ reaches the unitarity limit $2e^2/h$ asymptotically at large voltage or temperature.
We extend the general formalism discussed in the previous paper [A. B. Culver and N. Andrei, Phys. Rev. B 103, 195106 (2021)] to two models with charge fluctuations: the interacting resonant level model and the Anderson impurity model. In the interacting resonant level model, we find the exact time-evolving wavefunction and calculate the steady state impurity occupancy to leading order in the interaction. In the Anderson impurity model, we find the nonequilibrium steady state for small or large Coulomb repulsion $U$, and we find that the steady state current to leading order in $U$ agrees with a Keldysh perturbation theory calculation.
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