Quantum impurity models describe an atom or molecule embedded in a host material with which it can exchange electrons. They are basic to nanoscience as representations of quantum dots and molecular conductors and play an increasingly important role in the theory of "correlated electron" materials as auxiliary problems whose solution gives the "dynamical mean field" approximation to the self energy and local correlation functions. These applications require a method of solution which provides access to both high and low energy scales and is effective for wide classes of physically realistic models. The continuous-time quantum Monte Carlo algorithms reviewed in this article meet this challenge. We present derivations and descriptions of the algorithms in enough detail to allow other workers to write their own implementations, discuss the strengths and weaknesses of the methods, summarize the problems to which the new methods have been successfully applied and outline prospects for future applications.

Continuous-time Monte Carlo methods for quantum impurity models

The physics of one-dimensional interacting bosonic systems is reviewed. Beginning with results from exactly solvable models and computational approaches, the concept of bosonic Tomonaga-Luttinger liquids relevant for one-dimensional Bose fluids is introduced, and compared with Bose-Einstein condensates existing in dimensions higher than one. The effects of various perturbations on the Tomonaga-Luttinger liquid state are discussed as well as extensions to multicomponent and out of equilibrium situations. Finally, the experimental systems that can be described in terms of models of interacting bosons in one dimension are discussed.

/pdf/one-dimensional-bosons-from-condensed-matter-systems-to-4vp39jxn82.pdf

One dimensional bosons: From condensed matter systems to ultracold gases

We introduce the concept of directed loops in stochastic series expansion and path-integral quantum Monte Carlo methods. Using the detailed balance rules for directed loops, we show that it is possible to smoothly connect generally applicable simulation schemes (in which it is necessary to include backtracking processes in the loop construction) to more restricted loop algorithms that can be constructed only for a limited range of Hamiltonians (where backtracking can be avoided). The ``algorithmic discontinuities'' between general and special points (or regions) in parameter space can hence be eliminated. As a specific example, we consider the anisotropic $S=1/2$ Heisenberg antiferromagnet in an external magnetic field. We show that directed-loop simulations are very efficient for the full range of magnetic fields (zero to the saturation point) and anisotropies. In particular, for weak fields and anisotropies, the autocorrelations are significantly reduced relative to those of previous approaches. The back-tracking probability vanishes continuously as the isotropic Heisenberg point is approached. For the $\mathrm{XY}$ model, we show that back tracking can be avoided for all fields extending up to the saturation field. The method is hence particularly efficient in this case. We use directed-loop simulations to study the magnetization process in the two-dimensional Heisenberg model at very low temperatures. For $L\ifmmode\times\else\texttimes\fi{}L$ lattices with L up to $64,$ we utilize the step structure in the magnetization curve to extract gaps between different spin sectors. Finite-size scaling of the gaps gives an accurate estimate of the transverse susceptibility in the thermodynamic limit: ${\ensuremath{\chi}}_{\ensuremath{\perp}}=0.0659\ifmmode\pm\else\textpm\fi{}0.0002.$

Quantum Monte Carlo with directed loops.

A cluster update (the ``operator loop'') is developed within the framework of a numerically exact quantum Monte Carlo method based on the power series expansion of $\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}H)$ (stochastic series expansion). The method is generally applicable to a wide class of lattice Hamiltonians for which the expansion is positive definite. For some important models the operator-loop algorithm is more efficient than loop updates previously developed for ``worldline'' simulations. The method is here tested on a two-dimensional anisotropic Heisenberg antiferromagnet in a magnetic field.

Stochastic series expansion method with operator-loop update

We present a numerically exact continuous-time quantum Monte Carlo algorithm for fermions with a general interaction nonlocal in space-time The new determinantal grand-canonical scheme is based on a stochastic series expansion for the partition function in the interaction representation The method is particularly applicable for multiband, time-dependent correlations since it does not invoke the Hubbard-Stratonovich transformation The test calculations for exactly solvable models, as well results for the Green function and for the time-dependent susceptibility of the multiband supersymmetric model with a spin-flip interaction are discussed

Continuous-time quantum Monte Carlo method for fermions

We propose a new method to study path integrals for discrete quantum systems in which we work directly in the Euclidean time continuum. The method is of general interest. Here it is applied to the Heisenberg quantum antiferromagnet using a continuous-time version of a loop cluster algorithm. This algorithm is exploited to confirm the predictions of chiral perturbation theory in the extreme low temperature regime, down to $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.01J$. A fit of the low-energy parameters of chiral perturbation theory gives excellent agreement with previous results and with experiments.

Simulations of Discrete Quantum Systems in Continuous Euclidean Time.

We propose a new method to study path integrals for discrete quantum systems, in which we work directly in the Euclidean time continuum. The method is applied here to the Heisenberg quantum antiferromagnet using a continuous-time version of a loop cluster algorithm. This algorithm is exploited to confirm the predictions of chiral perturbation theory in the extreme low temperature regime, down to T = 0.01 J . A fit of the low-energy parameters of chiral perturbation theory gives excellent agreement with previous results and with experiments.

Simulations of discrete quantum systems in continuous euclidean time

Path integrals are usually formulated in discrete Euclidean time using the Trotter formula. We propose a new method to study discrete quantum systems, in which we work directly in the Euclidean time continuum. The method is of general interest and can be applied to studies of quantum spin systems, lattice fermions, and in principle also lattice gauge theories. Here it is applied to the Heisenberg quantum antiferromagnet using a continuous-time version of a loop cluster algorithm. The computational advantage of this algorithm is exploited to confirm the predictions of chiral perturbation theory in the extreme low temperature regime, down to $T = 0.01 J$. A fit of the low-energy parameters of chiral perturbation theory gives excellent agreement with previous results and with experiments.

Simulations of Discrete Quantum Systems in Continuous Euclidean Time

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Simulations of Discrete Quantum Systems in Continuous Euclidean Time.

Simulations of discrete quantum systems in continuous euclidean time

Simulations of Discrete Quantum Systems in Continuous Euclidean Time