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

Hypergeometric continuation of divergent perturbation series: I. Critical exponents of the Bose–Hubbard model

Sören Sanders, Martin Holthaus

Published Nov 1, 2017• 13 citations
OpenAlex

Abstract

We study the connection between the exponent of the order parameter of the Mott insulator-to-superfluid transition occurring in the two-dimensional Bose–Hubbard model, and the divergence exponents of its one- and two-particle correlation functions. We find that at the multicritical points all divergence exponents are related to each other, allowing us to express the critical exponent in terms of one single divergence exponent. This approach correctly reproduces the critical exponent of the three-dimensional XY universality class. Because divergence exponents can be computed in an efficient manner by hypergeometric analytic continuation, our strategy is applicable to a wide class of systems.

Related papers

Hypergeometric continuation of divergent perturbation series: I. Critical exponents of the Bose–Hubbard model

Sören Sanders, Martin Holthaus

We study the connection between the exponent of the order parameter of the Mott insulator-tosuperfluid transition occurring in the two-dimensional Bose-Hubbard model, and the divergence exponents of its one-and two-particle correlation functions.We find that at the multicritical points all divergence exponents are related to each other, allowing us to express the critical exponent in terms of one single divergence exponent.This approach correctly reproduces the critical exponent of the threedimensional XY universality class.Because divergence exponents can be computed in an efficient manner by hypergeometric analytic continuation, our strategy is applicable to a wide class of systems.

arXiv:1803.04876•New Journal of Physics•Oct 5, 2017•14 citations

Hypergeometric analytic continuation of the strong-coupling perturbation series for the 2d Bose-Hubbard model

Sören Sanders, Christoph Heinisch, Martin Holthaus

We develop a scheme for analytic continuation of the strong-coupling perturbation series of the pure Bose-Hubbard model beyond the Mott insulator-to-superfluid transition at zero temperature, based on hypergeometric functions and their generalizations. We then apply this scheme for computing the critical exponent of the order parameter of this quantum phase transition for the two-dimensional case, which falls into the universality class of the three-dimensional $XY$ model. This leads to anontrivial test of the universality hypothesis.

arXiv:1503.04090•Europhysics Letters (EPL)•Jul 1, 2015•8 citations

Quantum critical properties of Bose–Hubbard models

Sören Sanders, Martin Holthaus

Abstract The Mott insulator-to-superfluid transition exhibited by the Bose–Hubbard model on a two-dimensional square lattice occurs for any value of the chemical potential, but becomes critical at the tips of the so-called Mott lobes only. Employing a numerical approach based on a combination of high-order perturbation theory and hypergeometric analytic continuation we investigate how quantum critical properties manifest themselves in computational practice. We consider two-dimensional triangular lattices and three-dimensional cubic lattices for comparison, providing accurate parametrizations of the phase boundaries at the tips of the respective first lobes. In particular, we lend strong support to a recently suggested inequality which bounds the divergence exponent of the one-particle correlation function in terms of that of the two-particle correlation function, and which sharpens to an equality if and only if a system becomes critical.

arXiv:1912.00679•Journal of Physics A Mathematical and Theoretical•May 13, 2019•6 citations

Hypergeometric continuation of divergent perturbation series: II. Comparison with Shanks transformation and Padé approximation

Sören Sanders, Martin Holthaus

We explore in detail how analytic continuation of divergent perturbation series by generalized hypergeometric functions is achieved in practice. Using the example of strong-coupling perturbation series provided by the two-dimensional Bose–Hubbard model, we compare hypergeometric continuation to Shanks and Padé techniques, and demonstrate that the former yields a powerful, efficient and reliable alternative for computing the phase diagram of the Mott insulator-to-superfluid transition. In contrast to Shanks transformations and Padé approximations, hypergeometric continuation also allows us to determine the exponents which characterize the divergence of correlation functions at the transition points. Therefore, hypergeometric continuation constitutes a promising tool for the study of quantum phase transitions.

arXiv:1803.04882•Journal of Physics A Mathematical and Theoretical•Sep 26, 2017•15 citations

Quantum XY criticality in a two-dimensional Bose gas near the Mott transition

Adam Rançon, N. Dupuis

We derive the equation of state of a two-dimensional Bose gas in an optical lattice in the framework of the Bose-Hubbard model. We focus on the vicinity of the multicritical points where the quantum phase transition between the Mott insulator and the superfluid phase occurs at fixed density and belongs to the three-dimensional XY model universality class. Using a nonperturbative renormalization-group approach, we compute the pressure $P(μ,T)$ as a function of chemical potential and temperature. Our results compare favorably with a calculation based on the quantum O(2) model -- we find the same universal scaling function -- and allow us to determine the region of the phase diagram in the vicinity of a quantum multicritical point where the equation of state is universal. We also discuss the possible experimental observation of quantum XY criticality in a ultracold gas in an optical lattice.

arXiv:1307.3436•Europhysics Letters (EPL)•Oct 1, 2013•15 citations

Dynamical critical exponent of the Jaynes-Cummings-Hubbard model

Martin Hohenadler, Markus Aichhorn, Sebastian Schmidt +1 more

An array of high-$Q$ electromagnetic resonators coupled to qubits gives rise to the Jaynes-Cummings-Hubbard model describing a superfluid to Mott-insulator transition of lattice polaritons. From mean-field and strong-coupling expansions, the critical properties of the model are expected to be identical to the scalar Bose-Hubbard model. A recent Monte Carlo study of the superfluid density on the square lattice suggested that this does not hold for the fixed-density transition through the Mott lobe tip. Instead, mean-field behavior with a dynamical critical exponent $z=2$ was found. We perform large-scale quantum Monte Carlo simulations to investigate the critical behavior of the superfluid density and the compressibility. We find $z=1$ at the tip of the insulating lobe. Hence the transition falls in the three-dimensional $XY$ universality class, analogous to the Bose-Hubbard model.

arXiv:1106.0801•Physical Review A•Oct 24, 2011•34 citations
We study the connection between the exponent of the order parameter of the Mott insulator-to-superfluid transition occurring in the two-dimensional Bose–Hubbard model, and the divergence exponents of its one- and two-particle correlation functions. We find that at the multicritical points all divergence exponents are related to each other, allowing us to express the critical exponent in terms of one single divergence exponent. This approach correctly reproduces the critical exponent of the three-dimensional XY universality class. Because divergence exponents can be computed in an efficient manner by hypergeometric analytic continuation, our strategy is applicable to a wide class of systems.
We study the connection between the exponent of the order parameter of the Mott insulator-tosuperfluid transition occurring in the two-dimensional Bose-Hubbard model, and the divergence exponents of its one-and two-particle correlation functions.We find that at the multicritical points all divergence exponents are related to each other, allowing us to express the critical exponent in terms of one single divergence exponent.This approach correctly reproduces the critical exponent of the threedimensional XY universality class.Because divergence exponents can be computed in an efficient manner by hypergeometric analytic continuation, our strategy is applicable to a wide class of systems.
We develop a scheme for analytic continuation of the strong-coupling perturbation series of the pure Bose-Hubbard model beyond the Mott insulator-to-superfluid transition at zero temperature, based on hypergeometric functions and their generalizations. We then apply this scheme for computing the critical exponent of the order parameter of this quantum phase transition for the two-dimensional case, which falls into the universality class of the three-dimensional XYXYXY model. This leads to anontrivial test of the universality hypothesis.
Abstract The Mott insulator-to-superfluid transition exhibited by the Bose–Hubbard model on a two-dimensional square lattice occurs for any value of the chemical potential, but becomes critical at the tips of the so-called Mott lobes only. Employing a numerical approach based on a combination of high-order perturbation theory and hypergeometric analytic continuation we investigate how quantum critical properties manifest themselves in computational practice. We consider two-dimensional triangular lattices and three-dimensional cubic lattices for comparison, providing accurate parametrizations of the phase boundaries at the tips of the respective first lobes. In particular, we lend strong support to a recently suggested inequality which bounds the divergence exponent of the one-particle correlation function in terms of that of the two-particle correlation function, and which sharpens to an equality if and only if a system becomes critical.
We explore in detail how analytic continuation of divergent perturbation series by generalized hypergeometric functions is achieved in practice. Using the example of strong-coupling perturbation series provided by the two-dimensional Bose–Hubbard model, we compare hypergeometric continuation to Shanks and Padé techniques, and demonstrate that the former yields a powerful, efficient and reliable alternative for computing the phase diagram of the Mott insulator-to-superfluid transition. In contrast to Shanks transformations and Padé approximations, hypergeometric continuation also allows us to determine the exponents which characterize the divergence of correlation functions at the transition points. Therefore, hypergeometric continuation constitutes a promising tool for the study of quantum phase transitions.
We derive the equation of state of a two-dimensional Bose gas in an optical lattice in the framework of the Bose-Hubbard model. We focus on the vicinity of the multicritical points where the quantum phase transition between the Mott insulator and the superfluid phase occurs at fixed density and belongs to the three-dimensional XY model universality class. Using a nonperturbative renormalization-group approach, we compute the pressure P(μ,T)P(μ,T)P(μ,T) as a function of chemical potential and temperature. Our results compare favorably with a calculation based on the quantum O(2) model -- we find the same universal scaling function -- and allow us to determine the region of the phase diagram in the vicinity of a quantum multicritical point where the equation of state is universal. We also discuss the possible experimental observation of quantum XY criticality in a ultracold gas in an optical lattice.
An array of high-QQQ electromagnetic resonators coupled to qubits gives rise to the Jaynes-Cummings-Hubbard model describing a superfluid to Mott-insulator transition of lattice polaritons. From mean-field and strong-coupling expansions, the critical properties of the model are expected to be identical to the scalar Bose-Hubbard model. A recent Monte Carlo study of the superfluid density on the square lattice suggested that this does not hold for the fixed-density transition through the Mott lobe tip. Instead, mean-field behavior with a dynamical critical exponent z=2z=2z=2 was found. We perform large-scale quantum Monte Carlo simulations to investigate the critical behavior of the superfluid density and the compressibility. We find z=1z=1z=1 at the tip of the insulating lobe. Hence the transition falls in the three-dimensional XYXYXY universality class, analogous to the Bose-Hubbard model.