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AIP conference proceedings

Large Spins, High Order Interaction, and Bosonic Problems

Naoki Kawashima

Published Jan 1, 2003• 0 citations
Publisher

Abstract

The directed loop algorithm proposed recently by Syljuåsen and Sandvik [Phys. Rev. E 66 (2002) 046701] provides a new framework for the Monte Carlo simulation that possesses the advantages of the loop update and the worm update. Starting from another formulation of the algorithm based on the path‐integral representation, we discuss the coarse‐grained algorithm for quantum spin models and its extension to the boson models from the new viewpoint.

DOI
10.1063/1.1632132

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Monte Carlo Simulations of Quantum Systems with Global Updates

Alejandro Muramatsu

We start this series of lectures on quantum Monte Carlo methods with the simplest model for a strongly correlated system, namely an antiferromagnetic Heisenberg S‐1/2 chain. We will review methods for its simulation starting with the world‐line algorithm and then introducing the loop‐algorithm with global updates. We discuss next a model for doped antiferromagnets, with emphasis on the strong correlation limit that is central for high temperature superconductors and related materials, namely the so‐called t‐J model. An exact canonical transformation for this model that leads to a formulation with separated charge and spin degrees of freedom will be discussed. Results for a single hole in antiferromagnetic chains and planes will be summarized, giving an initial picture of charge dynamics in a quantum antiferromagnet. An additional algorithmic element, namely the determinantal method, will be introduced next, in order to deal with a finite number of charge degrees of freedom. Merging the loop‐ and the determinantal methods leads to a new one, the hybrid‐loop algorithm, that combines the efficiency of the loop‐algorithm for spins with the one of the determinantal method for fermions. Finally we discuss a first application of the hybrid‐loop algorithm, where the spectral function of the t‐J model in one dimension with finite doping is determined. There we can see how charge spin separation shows up in the one‐particle spectral function, and a detailed description can be achieved by comparison with an analitically soluble model, namely the t‐J model with 1/r2 interaction. The results show that in addition to spinons and holons expected in one‐dimensional metals, antiholons with charge Q = 2e, spin S = 0, and twice the mass of the holons are necessary to describe the inverse photoemission spectra at finite doping.

AIP conference proceedings•Jan 1, 2004•0 citations
The directed loop algorithm proposed recently by Syljuåsen and Sandvik [Phys. Rev. E 66 (2002) 046701] provides a new framework for the Monte Carlo simulation that possesses the advantages of the loop update and the worm update. Starting from another formulation of the algorithm based on the path‐integral representation, we discuss the coarse‐grained algorithm for quantum spin models and its extension to the boson models from the new viewpoint.
Recently, Syljuåsen and Sandvik [Phys. Rev. E. (to be published)] proposed a new framework for constructing algorithms of quantum Monte Carlo simulation. While it includes new classes of powerful algorithms, it is not straightforward to find an efficient algorithm for a given model. Based on their framework, we propose an algorithm that is a natural extension of the conventional loop algorithm with the split-spin representation. A complete table of the vertex density and the worm-scattering probability is presented for the general XXZ model of an arbitrary S with a uniform magnetic field.
The directed-loop quantum Monte Carlo method is generalized to the case of retarded interactions. Using the path integral, fermion-boson or spin-boson models are mapped to actions with retarded interactions by analytically integrating out the bosons. This yields an exact algorithm that combines the highly efficient loop updates available in the stochastic series expansion representation with the advantages of avoiding a direct sampling of the bosons. The application to electron-phonon models reveals that the method overcomes the previously detrimental issues of long autocorrelation times and exponentially decreasing acceptance rates. For example, the resulting dramatic speedup allows us to investigate the Peierls quantum phase transition on chains of up to 1282 sites.
Efficient schemes called directed loops for cluster updates in Quantum Monte Carlo simulations have recently been proposed, which improve both the efficiency and precision for simulations of quantum models. In this work we show that local detailed balance is not necessary in order to fulfill global detailed balance during the construction of these loops. We therefore propose to insert additional degrees of freedom into the directed loop equations, resulting in even more efficient algorithms. Our approach works directly in the natural representation of an extended space where the matrix elements of the operators defining the worm (broken worldline segments) are taken into account.
We start this series of lectures on quantum Monte Carlo methods with the simplest model for a strongly correlated system, namely an antiferromagnetic Heisenberg S‐1/2 chain. We will review methods for its simulation starting with the world‐line algorithm and then introducing the loop‐algorithm with global updates. We discuss next a model for doped antiferromagnets, with emphasis on the strong correlation limit that is central for high temperature superconductors and related materials, namely the so‐called t‐J model. An exact canonical transformation for this model that leads to a formulation with separated charge and spin degrees of freedom will be discussed. Results for a single hole in antiferromagnetic chains and planes will be summarized, giving an initial picture of charge dynamics in a quantum antiferromagnet. An additional algorithmic element, namely the determinantal method, will be introduced next, in order to deal with a finite number of charge degrees of freedom. Merging the loop‐ and the determinantal methods leads to a new one, the hybrid‐loop algorithm, that combines the efficiency of the loop‐algorithm for spins with the one of the determinantal method for fermions. Finally we discuss a first application of the hybrid‐loop algorithm, where the spectral function of the t‐J model in one dimension with finite doping is determined. There we can see how charge spin separation shows up in the one‐particle spectral function, and a detailed description can be achieved by comparison with an analitically soluble model, namely the t‐J model with 1/r2 interaction. The results show that in addition to spinons and holons expected in one‐dimensional metals, antiholons with charge Q = 2e, spin S = 0, and twice the mass of the holons are necessary to describe the inverse photoemission spectra at finite doping.