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:cond-mat/9508101 · Physical review. B, Condensed matter

Strong-coupling expansions for the pure and disordered Bose-Hubbard model

James K. Freericks, H. Monien

Published Feb 1, 1996• 353 citations
arXivHTMLPDF

Abstract

A strong-coupling expansion for the phase boundary of the (incompressible) Mott insulator is presented for the Bose-Hubbard model. Both the pure case and the disordered case are examined. Extrapolations of the series expansions provide results that are as accurate as the Monte Carlo simulations and agree with the exact solutions. The shape difference between Kosterlitz-Thouless critical behavior in one-dimension and power-law singularities in higher dimensions arises naturally in this strong-coupling expansion. Bounded disorder distributions produce a ``first-order'' kink to the Mott phase boundary in the thermodynamic limit because of the presence of Lifshitz's rare regions. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.2691

Related papers

Dynamics and thermodynamics of the Bose-Hubbard model

N. Elstner, H. Monien

We report results from a systematic analytic strong-coupling expansion of the Bose-Hubbard model in one- and two-spatial dimensions. We obtain numerically exact results for the dispersion of single-particle and single-hole excitations in the Mott insulator. The boundary of the Mott phase can be determined with previously unattainable accuracy in one and two dimensions. In one dimension, we observe the occurrence of reentrant behavior from the compressible to the insulating phase in a region close to the critical point, which was conjectured in earlier work. Our calculation can be used as a benchmark for the development of numerical techniques for strongly correlated systems.

arXiv:cond-mat/9807033•Physical review. B, Condensed matter•May 15, 1999•191 citations

Strong-coupling expansion for the momentum distribution of the Bose-Hubbard model with benchmarking against exact numerical results

James K. Freericks, H. R. Krishnamurthy, Yasuyuki Kato +2 more

A strong-coupling expansion for the Green's functions, self-energies, and correlation functions of the Bose-Hubbard model is developed. We illustrate the general formalism, which includes all possible (normal-phase) inhomogeneous effects in the formalism, such as disorder or a trap potential, as well as effects of thermal excitations. The expansion is then employed to calculate the momentum distribution of the bosons in the Mott phase for an infinite homogeneous periodic system at zero temperature through third order in the hopping. By using scaling theory for the critical behavior at zero momentum and at the critical value of the hopping for the Mott insulator--to--superfluid transition along with a generalization of the random-phase-approximation-like form for the momentum distribution, we are able to extrapolate the series to infinite order and produce very accurate quantitative results for the momentum distribution in a simple functional form for one, two, and three dimensions. The accuracy is better in higher dimensions and is on the order of a few percent relative error everywhere except close to the critical value of the hopping divided by the on-site repulsion. In addition, we find simple phenomenological expressions for the Mott-phase lobes in two and three dimensions which are much more accurate than the truncated strong-coupling expansions and any other analytic approximation we are aware of. The strong-coupling expansions and scaling-theory results are benchmarked against numerically exact quantum Monte Carlo simulations in two and three dimensions and against density-matrix renormalization-group calculations in one dimension. These analytic expressions will be useful for quick comparison of experimental results to theory and in many cases can bypass the need for expensive numerical simulations.

arXiv:0902.3435•Physical Review A•May 20, 2009•84 citations

A numerical exact solution of the Bose-Hubbard model

N. Elstner, H. Monien

In this paper we report results from a systematic strong-coupling expansion of the Bose-Hubbard model in one and two spatial dimensions. We obtain numerically exact results for the structure factor and the spectrum of single particle and single hole excitations in the Mott insulator. This enables the determination of the zero-temperature phase diagram and the location of the critical endpoints of the Mott lobes. In one dimension we confirm the occurrence of reentrance behavior from the compressible to the insulating phase in a region close to the critical point.

arXiv:cond-mat/9905367•May 25, 1999•2 citations

Phase diagram of the Bose-Hubbard Model

J. K Freericks, H Monien

The first reliable analytic calculation of the phase diagram of the Bose gas on a d -dimensional lattice with on-site repulsion is presented. In one dimension, the analytic calculation is in excellent agreement with the numerical Monte Carlo results. In higher dimensions, the deviations from the Monte Carlo calculations are larger, but the correct shape of the Mott-insulator lobes is still obtained. Explicit expressions for the energy of the Mott and the "defect" phase are given in a strong-coupling expansion.

arXiv:cond-mat/9307011•Europhysics Letters (EPL)•Jun 1, 1994•206 citations

Strong-coupling expansion for the spin-1 Bose-Hubbard model

T. Kimura

In this study, we perform a strong-coupling expansion up to third order of the hopping parameter $t$ for the spin-1 Bose-Hubbard model with antiferromagnetic interaction. As expected from previous studies, the Mott-insulator phase is considerably more stable against the superfluid phase when filling with an even number of bosons than when filling with an odd number of bosons. The phase-boundary curves are consistent with the perturbative mean-field theory in the limit of infinite dimensions. The critical value of the hopping parameter ${t}_{\mathrm{C}}$ at the peak of the Mott lobe depends on the antiferromagnetic interaction. This result indicates the reliability of the strong-coupling expansion when ${U}_{2}$ possesses large (intermediate) values for Mott lobe with an even (odd) number of bosons. Moreover, in order to improve our results, we apply a few extrapolation methods up to infinite order in $t$. The fitting results of the phase-boundary curves agree better with those of the perturbative mean-field approximation. In addition, the linear fit error of ${t}_{\mathrm{C}}$ is very small for the strong antiferromagnetic interaction.

arXiv:1210.2322•Physical Review A•Apr 18, 2013•12 citations

High-order strong-coupling expansion for the Bose-Hubbard model

Tao Wang, Xuefeng Zhang, Chunfeng Hou +2 more

Based on the process-chain method, we work out in detail a high-order strong-coupling expansion (HSCE) algorithm for determining the quantum phase boundary between the Mott insulator and the superfluid phase of the Bose-Hubbard model in different dimensions. A comparison with quantum Monte-Carlo results reveals the high efficiency and accuracy of the HSCE. With a symbolized evaluation of the algorithm we obtain an analytic expression for the strong-coupling boundary for different fillings up to eighth order. A subsequent Pad\'e approximation then yields a scaling function, which rescales lobes with different fillings to one lobe. As a further crosscheck, we perform a hopping expansion of the phase boundaries obtained from the effective potential Landau theory (EPLT) and find that the respective expansion coefficients coincide with those of HSCE.

•Jan 5, 2018•2 citations
A strong-coupling expansion for the phase boundary of the (incompressible) Mott insulator is presented for the Bose-Hubbard model. Both the pure case and the disordered case are examined. Extrapolations of the series expansions provide results that are as accurate as the Monte Carlo simulations and agree with the exact solutions. The shape difference between Kosterlitz-Thouless critical behavior in one-dimension and power-law singularities in higher dimensions arises naturally in this strong-coupling expansion. Bounded disorder distributions produce a ``first-order'' kink to the Mott phase boundary in the thermodynamic limit because of the presence of Lifshitz's rare regions. \textcopyright{} 1996 The American Physical Society.
We report results from a systematic analytic strong-coupling expansion of the Bose-Hubbard model in one- and two-spatial dimensions. We obtain numerically exact results for the dispersion of single-particle and single-hole excitations in the Mott insulator. The boundary of the Mott phase can be determined with previously unattainable accuracy in one and two dimensions. In one dimension, we observe the occurrence of reentrant behavior from the compressible to the insulating phase in a region close to the critical point, which was conjectured in earlier work. Our calculation can be used as a benchmark for the development of numerical techniques for strongly correlated systems.
A strong-coupling expansion for the Green's functions, self-energies, and correlation functions of the Bose-Hubbard model is developed. We illustrate the general formalism, which includes all possible (normal-phase) inhomogeneous effects in the formalism, such as disorder or a trap potential, as well as effects of thermal excitations. The expansion is then employed to calculate the momentum distribution of the bosons in the Mott phase for an infinite homogeneous periodic system at zero temperature through third order in the hopping. By using scaling theory for the critical behavior at zero momentum and at the critical value of the hopping for the Mott insulator--to--superfluid transition along with a generalization of the random-phase-approximation-like form for the momentum distribution, we are able to extrapolate the series to infinite order and produce very accurate quantitative results for the momentum distribution in a simple functional form for one, two, and three dimensions. The accuracy is better in higher dimensions and is on the order of a few percent relative error everywhere except close to the critical value of the hopping divided by the on-site repulsion. In addition, we find simple phenomenological expressions for the Mott-phase lobes in two and three dimensions which are much more accurate than the truncated strong-coupling expansions and any other analytic approximation we are aware of. The strong-coupling expansions and scaling-theory results are benchmarked against numerically exact quantum Monte Carlo simulations in two and three dimensions and against density-matrix renormalization-group calculations in one dimension. These analytic expressions will be useful for quick comparison of experimental results to theory and in many cases can bypass the need for expensive numerical simulations.
In this paper we report results from a systematic strong-coupling expansion of the Bose-Hubbard model in one and two spatial dimensions. We obtain numerically exact results for the structure factor and the spectrum of single particle and single hole excitations in the Mott insulator. This enables the determination of the zero-temperature phase diagram and the location of the critical endpoints of the Mott lobes. In one dimension we confirm the occurrence of reentrance behavior from the compressible to the insulating phase in a region close to the critical point.
The first reliable analytic calculation of the phase diagram of the Bose gas on a d -dimensional lattice with on-site repulsion is presented. In one dimension, the analytic calculation is in excellent agreement with the numerical Monte Carlo results. In higher dimensions, the deviations from the Monte Carlo calculations are larger, but the correct shape of the Mott-insulator lobes is still obtained. Explicit expressions for the energy of the Mott and the "defect" phase are given in a strong-coupling expansion.
In this study, we perform a strong-coupling expansion up to third order of the hopping parameter ttt for the spin-1 Bose-Hubbard model with antiferromagnetic interaction. As expected from previous studies, the Mott-insulator phase is considerably more stable against the superfluid phase when filling with an even number of bosons than when filling with an odd number of bosons. The phase-boundary curves are consistent with the perturbative mean-field theory in the limit of infinite dimensions. The critical value of the hopping parameter tC{t}_{\mathrm{C}}tC​ at the peak of the Mott lobe depends on the antiferromagnetic interaction. This result indicates the reliability of the strong-coupling expansion when U2{U}_{2}U2​ possesses large (intermediate) values for Mott lobe with an even (odd) number of bosons. Moreover, in order to improve our results, we apply a few extrapolation methods up to infinite order in ttt. The fitting results of the phase-boundary curves agree better with those of the perturbative mean-field approximation. In addition, the linear fit error of tC{t}_{\mathrm{C}}tC​ is very small for the strong antiferromagnetic interaction.
Based on the process-chain method, we work out in detail a high-order strong-coupling expansion (HSCE) algorithm for determining the quantum phase boundary between the Mott insulator and the superfluid phase of the Bose-Hubbard model in different dimensions. A comparison with quantum Monte-Carlo results reveals the high efficiency and accuracy of the HSCE. With a symbolized evaluation of the algorithm we obtain an analytic expression for the strong-coupling boundary for different fillings up to eighth order. A subsequent Pad\'e approximation then yields a scaling function, which rescales lobes with different fillings to one lobe. As a further crosscheck, we perform a hopping expansion of the phase boundaries obtained from the effective potential Landau theory (EPLT) and find that the respective expansion coefficients coincide with those of HSCE.