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

Oxford University Press eBooks

Bose–Hubbard models: methods of treatment

Maciej Lewenstein, Anna Sanpera, V. Ahufinger

Published Mar 8, 2012• 1 citation
Publisher

Abstract

This chapter reviews various theoretical methods that can be applied to investigate Bose–Hubbard models. In particular, it discusses the Bogolyubov method, strong coupling expansion, the perturbative mean field approach and the Gutzwiller method, exact diagonalization, quantum Monte Carlo methods, semi-classical phase space methods, and methods specifics for 1D (Wigner-Jordan transformation, Bethe ansatz, bosonization). The chapter end with a discussion about numerical renormalization approaches in 1D (DMRG, MPS) and 2D (PEPS; MERA).

DOI
10.1093/acprof:oso/9780199573127.003.0005

Related papers

Fermi and Fermi–Bose Hubbard models: methods of treatment

Maciej Lewenstein, Anna Sanpera, V. Ahufinger

This chapter presents various methods specific for fermionic systems and Fermi–Bose mixtures. It starts by discussing a lattice version of the BCS theory and then reviews BCS-BEC crossover theory for trapped Fermi gases close to the Feshbach resonance. The chapter analyses Fermi Hubbard models in the strongly correlated regime and t-J limit, and introduces the Gutzwiller projection method and the slave boson approach. It discusses in detail, in particular in the context of Fermi-Bose mixtures, effective low energy Hamiltonians and their derivation.

Oxford University Press eBooks•Mar 8, 2012•0 citations

Quantum Monte Carlo Methods and Extensions for the 2D Hubbard Model

Joseph Paki

This thesis will describe efforts to enhance our ability to simulate the 2D Hubbard model. Chapter 2 provides a background on the main computational techniques used throughout the work, including quantum Monte Carlo (QMC), dynamical mean field theory (DMFT), the dynamical cluster approximation (DCA), the continuous time auxiliary field algorithm (CTAUX), and the Maximum Entropy Method (MEM) for numeric analytic continuation. Chapter 3 presents new work on applying Twisted Boundary Conditions to the DCA framework. This method is applied in a effort to access thermodynamic (i.e. large system) information about the Hubbard model without the immense computational expense required to simulate large lattices directly. Chapter 4 describes efforts to study the extended 2D Hubbard model, which re-introduces non-local interactions between electrons that drive the formation of charge ordered phases. The work includes a thorough analysis of the phase diagram of the model away from half-filling, as well as analysis of the effect of non-local interactions on anti-ferromagnetic fluctuations and competition between the charge order and AFM states. Chapter 5 includes a derivation of the dual fermions diagrammatic expansion and the dual fermions ladder approximation used to compute corrections to single site DMFT calculations. The method is applied to the 2D Hubbard model in order to study the evolution of its spectral function, Fermi surface, and momentum dependent mass renormalization as the temperature, interaction strength, and doping are changed. The great advantage of this method is that we are able to access arbitrary momenta throughout the Brillouin zone, in contrast to the limitation to a few cluster momenta that characterize cluster methods like DCA.

•Jan 1, 2019•1 citation

Cluster Gutzwiller method for bosonic lattice systems

Dirk-Sören Lühmann

A versatile and numerically inexpensive method is presented allowing the accurate calculation of phase diagrams for bosonic lattice models. By treating clusters within the Gutzwiller theory, a surprisingly good description of quantum fluctuations beyond the mean-field theory is achieved approaching quantum Monte Carlo predictions for large clusters. Applying this powerful method to the Bose-Hubbard model, we demonstrate that it yields precise results for the superfluid to Mott-insulator transition in square, honeycomb, and cubic lattices. Due to the exact treatment within a cluster, the method can be effortlessly adapted to more complicated Hamiltonians in the fast progressing field of optical lattice experiments. This includes state- and site-dependent superlattices, large confined atomic systems, and disordered potentials, as well as various types of extended Hubbard models. Furthermore, the approach allows an excellent treatment of systems with arbitrary filling factors. We discuss the perspectives that allow for the computation of large, spatially varying lattices, low-lying excitations, and time evolution.

arXiv:1302.6761•Physical Review A•Apr 12, 2013•75 citations

Dynamical Mean Field Theory of the Bose-Hubbard Model

Luigi Amico A, Vittorio Penna

Quantum dynamics of the Bose-Hubbard model is investigated through a semiclassical Hamiltonian picture provided by the time-dependent variational principle method. The system is studied within a factorized slow/fast dynamics. The semiclassical requantization procedure allows one to account for the strong quantum nature of the system when $t/U\ensuremath{\ll}1$. The phase diagram is in good agreement with quantum Monte Carlo results and third order strong coupling perturbative expansion.

arXiv:cond-mat/9801074•Physical Review Letters•Mar 9, 1998•98 citations

Modelling and simulating strongly correlated fermions

Didier Poilblanc

I shall briefly review here some aspects of the Exact Diagonalization technique and put it in perspective with other numerical techniques such as Density Matrix Renormalization Group (DMRG) or Quantum Monte Carlo methods (partially) covered in this AIP Series by Reinhard Noack and Richard Scalettar respectively. I will then illustrate this lecture in discussing some simple applications, mainly for one dimensional systems.

AIP conference proceedings•Jan 1, 2007•0 citations

An exact reformulation of the Bose–Hubbard model in terms of a stochastic Gutzwiller ansatz

Iacopo Carusotto, Yvan Castin

We extend our exact reformulation of the bosonic many-body problem in terms of a stochastic Hartree ansatz to a stochastic Gutzwiller ansatz for the Bose–Hubbard model. The use of this ansatz makes the corresponding Monte Carlo scheme more efficient for strongly correlated bosonic phases like the Mott insulator phase or the Tonks phase. We present a first numerical application of this stochastic method to a system of impenetrable bosons on a 1D lattice showing the transition from the discrete Tonks gas to the Mott phase as the chemical potential is increased.

arXiv:cond-mat/0303259•New Journal of Physics•Jul 14, 2003•12 citations
This chapter reviews various theoretical methods that can be applied to investigate Bose–Hubbard models. In particular, it discusses the Bogolyubov method, strong coupling expansion, the perturbative mean field approach and the Gutzwiller method, exact diagonalization, quantum Monte Carlo methods, semi-classical phase space methods, and methods specifics for 1D (Wigner-Jordan transformation, Bethe ansatz, bosonization). The chapter end with a discussion about numerical renormalization approaches in 1D (DMRG, MPS) and 2D (PEPS; MERA).
This chapter presents various methods specific for fermionic systems and Fermi–Bose mixtures. It starts by discussing a lattice version of the BCS theory and then reviews BCS-BEC crossover theory for trapped Fermi gases close to the Feshbach resonance. The chapter analyses Fermi Hubbard models in the strongly correlated regime and t-J limit, and introduces the Gutzwiller projection method and the slave boson approach. It discusses in detail, in particular in the context of Fermi-Bose mixtures, effective low energy Hamiltonians and their derivation.
This thesis will describe efforts to enhance our ability to simulate the 2D Hubbard model. Chapter 2 provides a background on the main computational techniques used throughout the work, including quantum Monte Carlo (QMC), dynamical mean field theory (DMFT), the dynamical cluster approximation (DCA), the continuous time auxiliary field algorithm (CTAUX), and the Maximum Entropy Method (MEM) for numeric analytic continuation. Chapter 3 presents new work on applying Twisted Boundary Conditions to the DCA framework. This method is applied in a effort to access thermodynamic (i.e. large system) information about the Hubbard model without the immense computational expense required to simulate large lattices directly. Chapter 4 describes efforts to study the extended 2D Hubbard model, which re-introduces non-local interactions between electrons that drive the formation of charge ordered phases. The work includes a thorough analysis of the phase diagram of the model away from half-filling, as well as analysis of the effect of non-local interactions on anti-ferromagnetic fluctuations and competition between the charge order and AFM states. Chapter 5 includes a derivation of the dual fermions diagrammatic expansion and the dual fermions ladder approximation used to compute corrections to single site DMFT calculations. The method is applied to the 2D Hubbard model in order to study the evolution of its spectral function, Fermi surface, and momentum dependent mass renormalization as the temperature, interaction strength, and doping are changed. The great advantage of this method is that we are able to access arbitrary momenta throughout the Brillouin zone, in contrast to the limitation to a few cluster momenta that characterize cluster methods like DCA.
A versatile and numerically inexpensive method is presented allowing the accurate calculation of phase diagrams for bosonic lattice models. By treating clusters within the Gutzwiller theory, a surprisingly good description of quantum fluctuations beyond the mean-field theory is achieved approaching quantum Monte Carlo predictions for large clusters. Applying this powerful method to the Bose-Hubbard model, we demonstrate that it yields precise results for the superfluid to Mott-insulator transition in square, honeycomb, and cubic lattices. Due to the exact treatment within a cluster, the method can be effortlessly adapted to more complicated Hamiltonians in the fast progressing field of optical lattice experiments. This includes state- and site-dependent superlattices, large confined atomic systems, and disordered potentials, as well as various types of extended Hubbard models. Furthermore, the approach allows an excellent treatment of systems with arbitrary filling factors. We discuss the perspectives that allow for the computation of large, spatially varying lattices, low-lying excitations, and time evolution.
Quantum dynamics of the Bose-Hubbard model is investigated through a semiclassical Hamiltonian picture provided by the time-dependent variational principle method. The system is studied within a factorized slow/fast dynamics. The semiclassical requantization procedure allows one to account for the strong quantum nature of the system when $t/U\ensuremath{\ll}1$. The phase diagram is in good agreement with quantum Monte Carlo results and third order strong coupling perturbative expansion.
We extend our exact reformulation of the bosonic many-body problem in terms of a stochastic Hartree ansatz to a stochastic Gutzwiller ansatz for the Bose–Hubbard model. The use of this ansatz makes the corresponding Monte Carlo scheme more efficient for strongly correlated bosonic phases like the Mott insulator phase or the Tonks phase. We present a first numerical application of this stochastic method to a system of impenetrable bosons on a 1D lattice showing the transition from the discrete Tonks gas to the Mott phase as the chemical potential is increased.