Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1401.7592 · Physical Review E

Valence-bond quantum Monte Carlo algorithms defined on trees

Andreas Deschner, Erik Schwartz Sørensen

Published Sep 10, 2014• 0 citations
arXivHTMLPDF

Abstract

We present a class of algorithms for performing valence-bond quantum Monte Carlo of quantum spin models. Valence-bond quantum Monte Carlo is a projective T=0 Monte Carlo method based on sampling of a set of operator strings that can be viewed as forming a treelike structure. The algorithms presented here utilize the notion of a worm that moves up and down this tree and changes the associated operator string. In quite general terms, we derive a set of equations whose solutions correspond to a whole class of algorithms. As specific examples of this class of algorithms, we focus on two cases. The bouncing worm algorithm, for which updates are always accepted by allowing the worm to bounce up and down the tree, and the driven worm algorithm, where a single parameter controls how far up the tree the worm reaches before turning around. The latter algorithm involves only a single bounce where the worm turns from going up the tree to going down. The presence of the control parameter necessitates the introduction of an acceptance probability for the update.

DOI
10.1103/physreve.90.033304

Related papers

Valence Bond Calculations for Quantum Spin Chains: From Impurity Entanglement and Incommensurate Behaviour to Quantum Monte Carlo

Andreas Deschner

In this thesis I present three publications about the use of valence bonds to gain information about quantum spin systems. Valence bonds are an essential ingredient of low energy states present in many compounds. The first part of this thesis is dedicated to two studies of the antiferromagnetic J1-J2 chain with S=1/2. We show how automated variational calculations based on valence bond states can be performed close to the Majumdar-Ghosh point (MG-point). At this point, the groundstate is a product state of dimers (valence bonds between nearest neighbours). In the dimerized region surrounding the MG-point, we find such variational computations to be reliable. The first publication is about the entanglement properties of an impurity attached to the chain. We show how to use the variational method to calculate the negativity, an entanglement measure between the impurity and a distant part of the chain. We find that increasing the impurity coupling and a minute explicit dimerization, suppress the long-ranged entanglement present in the system for small impurity coupling at the MG-point. The second publication is about a transition from commensurate to incommensurate behaviour and how its characteristics depend on the parity of the length of the chain. The variational technique is used in a parameter regime inaccessible to DMRG. We find that in odd chains, unlike in even chains, a very intricate and interesting pattern of level crossings can be observed. The publication of the second part is about novel worm algorithms for a popular quantum Monte Carlo method called valence bond quantum Monte Carlo (VBQMC). The algorithms are based on the notion of a worm moving through a decision tree. VBQMC is entirely formulated in terms of valence bonds. In this thesis, I explain how the approach of VBQMC can be translated to the Sz-basis. The algorithms explained in the publication can be applied to this Sz-method.

•Mar 30, 2014•0 citations

Directed loop updates for quantum lattice models

Olav F. Syljuåsen

This article outlines how the quantum Monte Carlo directed loop update recently introduced can be applied to a wide class of quantum lattice models. Several models are considered: spin-s XXZ models with longitudinal and transverse magnetic fields, boson models with two-body interactions, and one-dimensional spinful fermion models. Expressions are given for the parameter regimes where very efficient "no-bounce" quantum Monte Carlo algorithms can be found.

arXiv:cond-mat/0211513•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Apr 3, 2003•82 citations

Ground State Projection of Quantum Spin Systems in the Valence-Bond Basis

Anders W. Sandvik

A Monte Carlo method for quantum spin systems is formulated in the valence-bond basis. The nonorthogonality allows for an efficient importance-sampled projection of the ground state out of an arbitrary state. The method provides access to resonating valence-bond physics, enables a direct estimator for the singlet-triplet gap, and extends the class of models that can be studied without negative-sign problems. As a demonstration, the valence-bond distribution in the ground state of the 2D Heisenberg antiferromagnet is calculated. Generalizations of the method to fermion systems are also discussed.

arXiv:cond-mat/0509558•Physical Review Letters•Nov 8, 2005•106 citations

Directed geometrical worm algorithm applied to the quantum rotor model

Fabien Alet, Erik Schwartz Sørensen

We discuss the implementation of a directed geometrical worm algorithm for the study of quantum link-current models. In this algorithm the Monte Carlo updates are made through the biased reptation of a worm through the lattice. A directed algorithm is an algorithm where, during the construction of the worm, the probability for erasing the immediately preceding part of the worm, when adding a new part, is minimal. We introduce a simple numerical procedure for minimizing this probability. The procedure only depends on appropriately defined local probabilities and should be generally applicable. Furthermore, we show how correlation functions C(r,tau) can be straightforwardly obtained from the probability of a worm to reach a site (r,tau) away from its starting point independent of whether or not a directed version of the algorithm is used. Detailed analytical proofs of the validity of the Monte Carlo algorithms are presented for both the directed and undirected geometrical worm algorithms. Results for autocorrelation times and Green's functions are presented for the quantum rotor model.

arXiv:cond-mat/0303080•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 12, 2003•42 citations

World-Line Monte Carlo Methods

Naoki Kawashima, Kenji Harada

World-line quantum Monte Carlo methods are reviewed with an emphasis on breakthroughs made in recent years. In particular, three algorithms -- the loop algorithm, the worm algorithm, and the directed-loop algorithm -- for updating world-line configurations are presented in a unified perspective. Detailed descriptions of the algorithms in specific cases are also given.

•Dec 29, 2003•0 citations

Resummation-based quantum Monte Carlo for quantum paramagnetic phases

Nisheeta Desai, Sumiran Pujari

For spin rotational symmetric models with a positive-definite high-temperature expansion of the partition function, a stochastic sampling of the series expansion upon partial resummation becomes logically equivalent to sampling an uncolored closely packed loop-gas model in one higher dimension. Based on this, we devise quantum Monte Carlo updates that importance-sample loop configurations for general $SU(N)$ in fundamental and higher-symmetric representations. The algorithmic performance systematically improves with an increase in (continuous) $N$, allowing the efficient simulation of quantum paramagnets. The underlying reason for the increased efficacy is the correspondence of quantum paramagnetic phases such as valence bond solids to short-loop phases on the loop-gas side rather than the particular value of $N$. This also gives a connection between Sandvik's $JQ$ model class and classical loop-gas models in the deconfined universality class.

arXiv:2104.06792•Physical review. B./Physical review. B•Aug 9, 2021•10 citations
We present a class of algorithms for performing valence-bond quantum Monte Carlo of quantum spin models. Valence-bond quantum Monte Carlo is a projective T=0 Monte Carlo method based on sampling of a set of operator strings that can be viewed as forming a treelike structure. The algorithms presented here utilize the notion of a worm that moves up and down this tree and changes the associated operator string. In quite general terms, we derive a set of equations whose solutions correspond to a whole class of algorithms. As specific examples of this class of algorithms, we focus on two cases. The bouncing worm algorithm, for which updates are always accepted by allowing the worm to bounce up and down the tree, and the driven worm algorithm, where a single parameter controls how far up the tree the worm reaches before turning around. The latter algorithm involves only a single bounce where the worm turns from going up the tree to going down. The presence of the control parameter necessitates the introduction of an acceptance probability for the update.
In this thesis I present three publications about the use of valence bonds to gain information about quantum spin systems. Valence bonds are an essential ingredient of low energy states present in many compounds. The first part of this thesis is dedicated to two studies of the antiferromagnetic J1-J2 chain with S=1/2. We show how automated variational calculations based on valence bond states can be performed close to the Majumdar-Ghosh point (MG-point). At this point, the groundstate is a product state of dimers (valence bonds between nearest neighbours). In the dimerized region surrounding the MG-point, we find such variational computations to be reliable. The first publication is about the entanglement properties of an impurity attached to the chain. We show how to use the variational method to calculate the negativity, an entanglement measure between the impurity and a distant part of the chain. We find that increasing the impurity coupling and a minute explicit dimerization, suppress the long-ranged entanglement present in the system for small impurity coupling at the MG-point. The second publication is about a transition from commensurate to incommensurate behaviour and how its characteristics depend on the parity of the length of the chain. The variational technique is used in a parameter regime inaccessible to DMRG. We find that in odd chains, unlike in even chains, a very intricate and interesting pattern of level crossings can be observed. The publication of the second part is about novel worm algorithms for a popular quantum Monte Carlo method called valence bond quantum Monte Carlo (VBQMC). The algorithms are based on the notion of a worm moving through a decision tree. VBQMC is entirely formulated in terms of valence bonds. In this thesis, I explain how the approach of VBQMC can be translated to the Sz-basis. The algorithms explained in the publication can be applied to this Sz-method.
A Monte Carlo method for quantum spin systems is formulated in the valence-bond basis. The nonorthogonality allows for an efficient importance-sampled projection of the ground state out of an arbitrary state. The method provides access to resonating valence-bond physics, enables a direct estimator for the singlet-triplet gap, and extends the class of models that can be studied without negative-sign problems. As a demonstration, the valence-bond distribution in the ground state of the 2D Heisenberg antiferromagnet is calculated. Generalizations of the method to fermion systems are also discussed.
We discuss the implementation of a directed geometrical worm algorithm for the study of quantum link-current models. In this algorithm the Monte Carlo updates are made through the biased reptation of a worm through the lattice. A directed algorithm is an algorithm where, during the construction of the worm, the probability for erasing the immediately preceding part of the worm, when adding a new part, is minimal. We introduce a simple numerical procedure for minimizing this probability. The procedure only depends on appropriately defined local probabilities and should be generally applicable. Furthermore, we show how correlation functions C(r,tau) can be straightforwardly obtained from the probability of a worm to reach a site (r,tau) away from its starting point independent of whether or not a directed version of the algorithm is used. Detailed analytical proofs of the validity of the Monte Carlo algorithms are presented for both the directed and undirected geometrical worm algorithms. Results for autocorrelation times and Green's functions are presented for the quantum rotor model.
For spin rotational symmetric models with a positive-definite high-temperature expansion of the partition function, a stochastic sampling of the series expansion upon partial resummation becomes logically equivalent to sampling an uncolored closely packed loop-gas model in one higher dimension. Based on this, we devise quantum Monte Carlo updates that importance-sample loop configurations for general SU(N)SU(N)SU(N) in fundamental and higher-symmetric representations. The algorithmic performance systematically improves with an increase in (continuous) NNN, allowing the efficient simulation of quantum paramagnets. The underlying reason for the increased efficacy is the correspondence of quantum paramagnetic phases such as valence bond solids to short-loop phases on the loop-gas side rather than the particular value of NNN. This also gives a connection between Sandvik's JQJQJQ model class and classical loop-gas models in the deconfined universality class.