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

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Low energy states of (1+1)-dimensional Φ6 field theories via the coupled cluster method

M Funke, Hermann G. Kümmel

Published Jul 15, 1994• 12 citations
Publisher

Abstract

The boson field theory with ${\mathrm{\ensuremath{\Phi}}}^{4}$ and ${\mathrm{\ensuremath{\Phi}}}^{6}$ coupling in (1+1) dimensions is investigated. Rigorous duality relations covering the whole range of coupling constants are derived. Vacuum, one-, and two-particle states are computed via the coupled cluster method in the four-particle approximations. Outside a critical region around second-order phase transitions the method works very well even in regions with first-order phase transitions and bound states.

DOI
10.1103/physrevd.50.991

Related papers

Coupled cluster description of field theories: Procedures and their application to the vacuum sector in (1+1)-dimensional Φ4 field theories

C.S. Hsue, Hermann G. Kümmel, P. Ueberholz

General procedures for the application of the nonperturbative coupled cluster method (CCM) are described for one-component boson field theories. The basis set is optimized first by a Hartree-Bogolubov transformation. Then ``correlations'' are added via the CCM. This procedure is applied to two-dimensional ${\ensuremath{\Phi}}^{4}$ field theory. The results are very stable outside the critical region.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Sep 15, 1985•36 citations

Approaching the critical region of two-dimensional φ4 quantum field theory with post-Gaussian approximations

M Funke, Ulrich B. Kaulfuss, Hermann G. Kümmel

We investigate the vacuum state of (1+1)-dimensional ${\ensuremath{\varphi}}^{4}$ quantum field theory utilizing a modification of the powerful coupled cluster method by the additional maximum-overlap condition. This permits us to construct the ground state of that field theory for nearly all values of the coupling strength. Only a small region has to be excluded where our method still fails. This is most probably due to critical behavior showing up in a change of the order parameter of the model. Our procedure predicts a behavior of the (${\ensuremath{\varphi}}^{4}$${)}_{2}$ model in complete agreement with some rigorous mathematical statements which is not possible in the case of a Gaussian approximation only. Perhaps somewhat unexpectedly, the symmetry-breaking Hamiltonian does not have any critical point.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jan 15, 1987•52 citations

Benchmarking the variational cluster approach by means of the one-dimensional Bose-Hubbard model

Michael Knap, Enrico Arrigoni, Wolfgang von der Linden

Convergence properties of the variational cluster approach with respect to the variational parameter space, cluster size, and boundary conditions of the reference system are investigated and discussed for bosonic many-body systems. Specifically, the variational cluster approach is applied to the one-dimensional Bose-Hubbard model, which exhibits a quantum phase transition from Mott to superfluid phase. In order to benchmark the variational cluster approach, results for the phase boundary delimiting the first Mott lobe are compared with essentially exact density matrix renormalization group data. Furthermore, static quantities, such as the ground state energy and the one-particle density matrix are compared with high-order strong coupling perturbation theory results. For reference systems with open boundary conditions the variational parameter space is extended by an additional variational parameter which allows for a more uniform particle density on the reference system and thus drastically improves the results. It turns out that the variational cluster approach yields accurate results with relatively low-computational effort for both the phase boundary as well as the static properties of the one-dimensional Bose-Hubbard model, even at the tip of the first Mott lobe where correlation effects are most pronounced.

arXiv:1002.2449•Physical Review B•Jun 17, 2010•15 citations

Critical behaviour of theO(n)-ϕ4model with an antisymmetric tensor order parameter

N. V. Antonov, Mikhail Vladimirovich Kompaniets, N. M. Lebedev

Critical behaviour of the O(n)-symmetric $ϕ^{4}$-model with an antisymmetric tensor order parameter is studied by means of the field-theoretic renormalization group (RG) in the leading order of the $\varepsilon=4-d$-expansion (one-loop approximation). For $n=2$ and 3 the model is equivalent to the scalar and the O(3)-symmetric vector models, for $n\ge4$ it involves two independent interaction terms and two coupling constants. It is shown that for $n>4$ the RG equations have no infrared (IR) attractive fixed points and their solutions (RG flows) leave the stability region of the model. This means that fluctuations of the order parameter change the nature of the phase transition from the second-order type (suggested by the mean-field theory) to the first-order one. For $n=4$, the IR attractive fixed point exists and the IR behaviour is non-universal: if the coupling constants belong to the basin of attraction for the IR point, the phase transition is of the second order and the IR critical scaling regime realizes. The corresponding critical exponents $ν$ and $η$ are presented in the order $\varepsilon$ and $\varepsilon^{2}$, respectively. Otherwise the RG flows pass outside the stability region and the first-order transition takes place.

arXiv:1307.1991•Journal of Physics A Mathematical and Theoretical•Sep 19, 2013•14 citations

Quantum phase transitions and the extended coupled cluster method

N.E. Ligterink, Niels R. Walet, R. F. Bishop

We discuss the application of an extended version of the coupled cluster method to systems exhibiting a quantum phase transition. We use the lattice O(4) nonlinear sigma model in (1+1) and (3+1) dimensions as an example. We show how simple predictions get modified, leading to the absence of a phase transition in (1+1) dimensions, and strong indications for a phase transition in (3+1) dimensions.

arXiv:hep-ph/0008040•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Feb 27, 2001•1 citation

Coupled-cluster theory for systems of bosons in external traps

L. S. Cederbaum, Ofir E. Alon, Alexej I. Streltsov

A coupled-cluster approach for systems of $N$ bosons in external traps is developed. In the coupled-cluster approach the exact many-body wave function is obtained by applying an exponential operator $\mathrm{exp}{T}$ to the ground configuration $\ensuremath{\mid}{\ensuremath{\phi}}_{0}⟩$. The natural ground configuration for bosons is, of course, when all reside in a single orbital. Because of this simple structure of $\ensuremath{\mid}{\ensuremath{\phi}}_{0}⟩$, the appearance of excitation operators $T={\ensuremath{\sum}}_{n=1}^{N}{T}_{n}$ for bosons is much simpler than for fermions. We can treat very large numbers of bosons with coupled-cluster expansions. In a substantial part of this work, we address the issue of size consistency for bosons and enquire whether truncated coupled-cluster expansions are size consistent. We show that, in contrast to the familiar situation for fermions for which coupled-cluster expansions are size consistent, for bosons the answer to this question depends on the choice of ground configuration. Utilizing the natural ground configuration, working equations for the truncated coupled-cluster with $T={T}_{1}+{T}_{2}$, i.e., coupled-cluster singles doubles are explicitly derived. Finally, an illustrative numerical example for a condensate with up to $N=10\phantom{\rule{0.2em}{0ex}}000$ bosons in an harmonic trap is provided and analyzed. The results are highly promising.

arXiv:cond-mat/0511430•Physical Review A•Apr 17, 2006•35 citations
The boson field theory with \ensuremathΦ4{\mathrm{\ensuremath{\Phi}}}^{4}\ensuremathΦ4 and \ensuremathΦ6{\mathrm{\ensuremath{\Phi}}}^{6}\ensuremathΦ6 coupling in (1+1) dimensions is investigated. Rigorous duality relations covering the whole range of coupling constants are derived. Vacuum, one-, and two-particle states are computed via the coupled cluster method in the four-particle approximations. Outside a critical region around second-order phase transitions the method works very well even in regions with first-order phase transitions and bound states.
We investigate the vacuum state of (1+1)-dimensional ${\ensuremath{\varphi}}^{4}$ quantum field theory utilizing a modification of the powerful coupled cluster method by the additional maximum-overlap condition. This permits us to construct the ground state of that field theory for nearly all values of the coupling strength. Only a small region has to be excluded where our method still fails. This is most probably due to critical behavior showing up in a change of the order parameter of the model. Our procedure predicts a behavior of the (${\ensuremath{\varphi}}^{4}$)2{)}_{2})2​ model in complete agreement with some rigorous mathematical statements which is not possible in the case of a Gaussian approximation only. Perhaps somewhat unexpectedly, the symmetry-breaking Hamiltonian does not have any critical point.
Convergence properties of the variational cluster approach with respect to the variational parameter space, cluster size, and boundary conditions of the reference system are investigated and discussed for bosonic many-body systems. Specifically, the variational cluster approach is applied to the one-dimensional Bose-Hubbard model, which exhibits a quantum phase transition from Mott to superfluid phase. In order to benchmark the variational cluster approach, results for the phase boundary delimiting the first Mott lobe are compared with essentially exact density matrix renormalization group data. Furthermore, static quantities, such as the ground state energy and the one-particle density matrix are compared with high-order strong coupling perturbation theory results. For reference systems with open boundary conditions the variational parameter space is extended by an additional variational parameter which allows for a more uniform particle density on the reference system and thus drastically improves the results. It turns out that the variational cluster approach yields accurate results with relatively low-computational effort for both the phase boundary as well as the static properties of the one-dimensional Bose-Hubbard model, even at the tip of the first Mott lobe where correlation effects are most pronounced.
Critical behaviour of the O(n)-symmetric ϕ4ϕ^{4}ϕ4-model with an antisymmetric tensor order parameter is studied by means of the field-theoretic renormalization group (RG) in the leading order of the ε=4−d\varepsilon=4-dε=4−d-expansion (one-loop approximation). For n=2n=2n=2 and 3 the model is equivalent to the scalar and the O(3)-symmetric vector models, for n≥4n\ge4n≥4 it involves two independent interaction terms and two coupling constants. It is shown that for n>4n>4n>4 the RG equations have no infrared (IR) attractive fixed points and their solutions (RG flows) leave the stability region of the model. This means that fluctuations of the order parameter change the nature of the phase transition from the second-order type (suggested by the mean-field theory) to the first-order one. For n=4n=4n=4, the IR attractive fixed point exists and the IR behaviour is non-universal: if the coupling constants belong to the basin of attraction for the IR point, the phase transition is of the second order and the IR critical scaling regime realizes. The corresponding critical exponents ννν and ηηη are presented in the order ε\varepsilonε and ε2\varepsilon^{2}ε2, respectively. Otherwise the RG flows pass outside the stability region and the first-order transition takes place.
A coupled-cluster approach for systems of NNN bosons in external traps is developed. In the coupled-cluster approach the exact many-body wave function is obtained by applying an exponential operator expT\mathrm{exp}{T}expT to the ground configuration $\ensuremath{\mid}{\ensuremath{\phi}}_{0}⟩$. The natural ground configuration for bosons is, of course, when all reside in a single orbital. Because of this simple structure of $\ensuremath{\mid}{\ensuremath{\phi}}_{0}⟩$, the appearance of excitation operators $T={\ensuremath{\sum}}_{n=1}^{N}{T}_{n}$ for bosons is much simpler than for fermions. We can treat very large numbers of bosons with coupled-cluster expansions. In a substantial part of this work, we address the issue of size consistency for bosons and enquire whether truncated coupled-cluster expansions are size consistent. We show that, in contrast to the familiar situation for fermions for which coupled-cluster expansions are size consistent, for bosons the answer to this question depends on the choice of ground configuration. Utilizing the natural ground configuration, working equations for the truncated coupled-cluster with T=T1+T2T={T}_{1}+{T}_{2}T=T1​+T2​, i.e., coupled-cluster singles doubles are explicitly derived. Finally, an illustrative numerical example for a condensate with up to N=10000N=10\phantom{\rule{0.2em}{0ex}}000N=10000 bosons in an harmonic trap is provided and analyzed. The results are highly promising.