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:1407.0450

Comment on "Polar and antiferromagnetic order in f=1 boson systems"

Bao Cheng-guang

Published Jul 2, 2014• 0 citations
arXivHTMLPDF

Abstract

An inequality for the lower bound of the average number of hyperfine component $μ=0$ particles in the ground state of spin-1 condensates under a magnetic field has been derived in ref.\cite{tasa13}. It is shown in this comment that, in a broad domain of parameters usually accessed in experiments, the lower bound appears to be negative. Thus the applicability of the inequality is very limited.

DOI
10.48550/arxiv.1407.0450

Related papers

Lower bound for the population of hyperfine component $μ=0$ particles in the ground state of spin-1 condensates

C. G. Bao

An analytical expression for the lower bound of the average number of hyperfine component $μ=0$ particles in the ground state of spin-1 condensates (denoted as $\overset{\_\_}{ρ_{0}}$) under a magnetic field has been derived. In the derivation the total magnetization $M$ is kept rigorously conserved. Numerical examples are given to show the applicability of the analytical expression. It was found that, in a broad domain of parameters specified in the paper, the lower bound is very close to the actual $\overset{\_\_}{ρ_{0}}$. Thereby, in this domain, $\overset{\_\_}{% ρ_{0}}$ can be directly evaluated simply by using the analytical expression.

arXiv:1407.1351•Jul 5, 2014•0 citations

Lower bound for the hyperfine populations of spin-2 condensates against a magnetic field under the single-spatial-mode approximation

Y. M. Liu, Y. Z. He, Bao Cheng-guang

The variation of the hyperfine populations $N{\ensuremath{\rho}}_{\ensuremath{\mu}}$ of the ground state (g.s.) of spin-2 condensates against a magnetic field $B$ is studied. A $N$-body theory which is rigorous in dealing with the spin degrees of freedom has been adopted. The conservation of the total magnetization $M$ is assumed. A combined hyperfine population $N{\ensuremath{\rho}}_{\text{comb}}=N({\ensuremath{\rho}}_{0}\ensuremath{-}3{\ensuremath{\rho}}_{2}\ensuremath{-}3{\ensuremath{\rho}}_{\ensuremath{-}2})$, which is an observable, is defined. The lower bound for this population has been derived analytically. It turns out that the lower bound is identical to the actual population when $B=0$ and $\ensuremath{\infty}$, and the population is monotonically uprising in between. Thus, via the analytical form of the lower bound, the variation of the population against $B$ can be roughly known in advance. Numerical examples are given to demonstrate the applicability of the bound. When $M=0$, the g.s. in the polar phase was found to be highly unstable against $B$ because the neighboring level density is extremely high. When $M=N$, the g.s. in the cyclic phase was also found to be highly unstable. This is because the appearance of $B$, even if it is very weak, will break the high degeneracy of the g.s. and result in having numerous levels emerging in the neighborhood of the g.s.

Physical Review A•Dec 28, 2015•0 citations

Polar and Antiferromagnetic Order in f=1 Boson Systems

Hal Tasaki

In a system of interacting f=1 bosons (in the subspace where the total spin in the z direction is vanishing), we prove inequalities for the ground state expectation value of the density of spin-0 bosons. The inequalities imply that the ground state possesses "polar" or "antiferromagnetic" order when the quadratic Zeeman term q is large enough. In the low density limit, the inequalities establish the existence of a sharp transition at q=0 when q is varied.

arXiv:1302.6349•Physical Review Letters•Jun 4, 2013•10 citations

Asymptotic expressions for the hyperfine populations in the ground state of spin-1 condensates against a magnetic field

Y. M. Liu, Y. Z. He, Bao Cheng-guang

Based on perturbation theory up to second order, analytical asymptotic expressions for the variation of the population of hyperfine component $\ensuremath{\mu}=0$ particles in the ground state of spin-1 condensates against a magnetic field $B$ have been derived. The ranges of $B$ in which the asymptotic expressions are applicable have been clarified via a comparison of the numerical results from the analytical expressions and from a diagonalization of the Hamiltonian in a complete spin space. It was found that, for $^{87}\mathrm{Rb}$, the two analytical expressions, one for a weak field and the other one for a strong field, together cover the whole range of $B$ from zero to infinity. For Na, the analytical expressions are valid only if $B$ is very weak or sufficiently strong.

arXiv:1509.05687•Physical Review A•Oct 20, 2015•2 citations

Lower bound for the variation of the hyperfine populations in the ground state of spin-1 condensates against a magnetic field

Wenfang Xie, Y.Z. He, Bao Cheng-guang

Physics Letters A•Jun 13, 2015•1 citation

Extension to the case of a magnetic field of Feynman’s path-integral upper bound on the ground-state energy: Application to the Fröhlich polaron

J. T. Devreese, F. Brosens

The Feynman inequality ${\mathit{E}}_{\mathit{G}}$\ensuremath{\le}${\mathit{E}}_{\mathrm{trial}}$+${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S-${\mathit{S}}_{\mathrm{trial}}$〉/\ensuremath{\beta} for path integrals provides a powerful upper bound on the ground-state energy ${\mathit{E}}_{\mathit{G}}$ of a large variety of systems. ${\mathit{E}}_{\mathrm{trial}}$ is the ground-state energy of some trial system with action ${\mathit{S}}_{\mathrm{trial}}$ for imaginary values of the time variable, and S is the action (also expressed in imaginary time variables) of the system under study. \ensuremath{\beta}=1/${\mathit{k}}_{\mathit{B}}$T, where ${\mathit{k}}_{\mathit{B}}$ is the Boltzmann constant and T the temperature. However, the Feynman inequality is not a priori justified for a system in a magnetic field, because imaginary terms subsist in the action also after transforming to imaginary time variables. Replacing or extending this inequality when magnetic fields are present has therefore been a long-standing problem. In the present paper we solve this problem. We first derive an inequality, providing an upper bound for the ground-state energy, that is valid even in the case of a nonzero magnetic field, ${\mathrm{E}}_{\mathrm{G}}$\ensuremath{\le}${\mathrm{E}}_{\mathrm{trial}}$+〈\ensuremath{\infty}\ensuremath{\Vert}scrT{${\mathit{markU}}_{\mathrm{trial}}$ (\ensuremath{\infty},-\ensuremath{\infty})[V(0)-${\mathit{V}}_{\mathrm{trial}}$(0)]}-\ensuremath{\infty}〉,for a system with Hamiltonian ${\mathrm{H}}_{0}$+V. T time-ordering operator, and ${\mathit{U}}_{\mathrm{trial}}$ is the time evolution operator of a trial system with Hamiltonian ${\mathit{H}}_{0}$+${\mathit{V}}_{\mathrm{trial}}$ in the interaction representation, with the interactions V(t) and ${\mathit{V}}_{\mathrm{trial}}$(t) switched on adiabatically. Because of the time ordering, retardation effects are also properly taken into account. The contribution of the magnetic field is included in the unperturbed Hamiltonian ${\mathit{H}}_{0}$. If the time-dependent integrands occurring in the matrix element in the right-hand side of our generalized inequality satisfy certain analyticity conditions in the complex-time plane, this inequality reduces to the Feynman inequality for path integrals. If these analyticity conditions are not satisfied, our generalized inequality may introduce supplementary terms ${\mathit{E}}^{\mathrm{DB}}$ in the right-hand side of the Feynman upper bound, ${\mathrm{E}}_{\mathrm{G}}$\ensuremath{\le}${\mathrm{E}}_{\mathrm{t}\mathbf{i}\mathrm{al}}$+${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S-${\mathit{S}}_{\mathrm{trial}}$〉/\ensuremath{\beta} +${\mathit{E}}^{\mathrm{DB}}$,because different branch lines or singularities have to be taken into account in the transformation to imaginary time variables. As an important illustration, our generalized inequality is applied to the problem of the Fr\"ohlich polaron in a magnetic field. From the generalization of the Feynman an inequality derived in the present paper, we determine the conditions to be imposed on the variational parameters in the trial action, such that the Feynman upper bound in its original form remains valid for a polaron in a magnetic field. Some limiting cases are studied analytically to illustrate the relevance of our additional constraints on the variational parameters of the trial system. In the free-particle limit and for a particular value of one of the variational parameters, we explicitly derive the contributions from the branch lines in the complex-time plane which arise if these additional constraints are not satisfied.

Physical review. B, Condensed matter•Mar 15, 1992•23 citations
An inequality for the lower bound of the average number of hyperfine component μ=0μ=0μ=0 particles in the ground state of spin-1 condensates under a magnetic field has been derived in ref.\cite{tasa13}. It is shown in this comment that, in a broad domain of parameters usually accessed in experiments, the lower bound appears to be negative. Thus the applicability of the inequality is very limited.
Lower bound for the population of hyperfine component μ=0μ=0μ=0 particles in the ground state of spin-1 condensates
An analytical expression for the lower bound of the average number of hyperfine component μ=0μ=0μ=0 particles in the ground state of spin-1 condensates (denoted as ρ0__\overset{\_\_}{ρ_{0}}ρ0​__​) under a magnetic field has been derived. In the derivation the total magnetization MMM is kept rigorously conserved. Numerical examples are given to show the applicability of the analytical expression. It was found that, in a broad domain of parameters specified in the paper, the lower bound is very close to the actual ρ0__\overset{\_\_}{ρ_{0}}ρ0​__​. Thereby, in this domain, $\overset{\_\_}{% ρ_{0}}$ can be directly evaluated simply by using the analytical expression.
The variation of the hyperfine populations $N{\ensuremath{\rho}}_{\ensuremath{\mu}}$ of the ground state (g.s.) of spin-2 condensates against a magnetic field BBB is studied. A NNN-body theory which is rigorous in dealing with the spin degrees of freedom has been adopted. The conservation of the total magnetization MMM is assumed. A combined hyperfine population $N{\ensuremath{\rho}}_{\text{comb}}=N({\ensuremath{\rho}}_{0}\ensuremath{-}3{\ensuremath{\rho}}_{2}\ensuremath{-}3{\ensuremath{\rho}}_{\ensuremath{-}2})$, which is an observable, is defined. The lower bound for this population has been derived analytically. It turns out that the lower bound is identical to the actual population when B=0B=0B=0 and $\ensuremath{\infty}$, and the population is monotonically uprising in between. Thus, via the analytical form of the lower bound, the variation of the population against BBB can be roughly known in advance. Numerical examples are given to demonstrate the applicability of the bound. When M=0M=0M=0, the g.s. in the polar phase was found to be highly unstable against BBB because the neighboring level density is extremely high. When M=NM=NM=N, the g.s. in the cyclic phase was also found to be highly unstable. This is because the appearance of BBB, even if it is very weak, will break the high degeneracy of the g.s. and result in having numerous levels emerging in the neighborhood of the g.s.
Based on perturbation theory up to second order, analytical asymptotic expressions for the variation of the population of hyperfine component $\ensuremath{\mu}=0$ particles in the ground state of spin-1 condensates against a magnetic field BBB have been derived. The ranges of BBB in which the asymptotic expressions are applicable have been clarified via a comparison of the numerical results from the analytical expressions and from a diagonalization of the Hamiltonian in a complete spin space. It was found that, for 87Rb^{87}\mathrm{Rb}87Rb, the two analytical expressions, one for a weak field and the other one for a strong field, together cover the whole range of BBB from zero to infinity. For Na, the analytical expressions are valid only if BBB is very weak or sufficiently strong.
The Feynman inequality EG{\mathit{E}}_{\mathit{G}}EG​\ensuremath{\le}Etrial{\mathit{E}}_{\mathrm{trial}}Etrial​+${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S-Strial{\mathit{S}}_{\mathrm{trial}}Strial​〉/\ensuremath{\beta} for path integrals provides a powerful upper bound on the ground-state energy EG{\mathit{E}}_{\mathit{G}}EG​ of a large variety of systems. Etrial{\mathit{E}}_{\mathrm{trial}}Etrial​ is the ground-state energy of some trial system with action Strial{\mathit{S}}_{\mathrm{trial}}Strial​ for imaginary values of the time variable, and S is the action (also expressed in imaginary time variables) of the system under study. \ensuremath{\beta}=1/kB{\mathit{k}}_{\mathit{B}}kB​T, where kB{\mathit{k}}_{\mathit{B}}kB​ is the Boltzmann constant and T the temperature. However, the Feynman inequality is not a priori justified for a system in a magnetic field, because imaginary terms subsist in the action also after transforming to imaginary time variables. Replacing or extending this inequality when magnetic fields are present has therefore been a long-standing problem. In the present paper we solve this problem. We first derive an inequality, providing an upper bound for the ground-state energy, that is valid even in the case of a nonzero magnetic field, EG{\mathrm{E}}_{\mathrm{G}}EG​\ensuremath{\le}Etrial{\mathrm{E}}_{\mathrm{trial}}Etrial​+〈\ensuremath{\infty}\ensuremath{\Vert}scrT{markUtrial{\mathit{markU}}_{\mathrm{trial}}markUtrial​ (\ensuremath{\infty},-\ensuremath{\infty})[V(0)-Vtrial{\mathit{V}}_{\mathrm{trial}}Vtrial​(0)]}-\ensuremath{\infty}〉,for a system with Hamiltonian H0{\mathrm{H}}_{0}H0​+V. T time-ordering operator, and Utrial{\mathit{U}}_{\mathrm{trial}}Utrial​ is the time evolution operator of a trial system with Hamiltonian H0{\mathit{H}}_{0}H0​+Vtrial{\mathit{V}}_{\mathrm{trial}}Vtrial​ in the interaction representation, with the interactions V(t) and Vtrial{\mathit{V}}_{\mathrm{trial}}Vtrial​(t) switched on adiabatically. Because of the time ordering, retardation effects are also properly taken into account. The contribution of the magnetic field is included in the unperturbed Hamiltonian H0{\mathit{H}}_{0}H0​. If the time-dependent integrands occurring in the matrix element in the right-hand side of our generalized inequality satisfy certain analyticity conditions in the complex-time plane, this inequality reduces to the Feynman inequality for path integrals. If these analyticity conditions are not satisfied, our generalized inequality may introduce supplementary terms EDB{\mathit{E}}^{\mathrm{DB}}EDB in the right-hand side of the Feynman upper bound, EG{\mathrm{E}}_{\mathrm{G}}EG​\ensuremath{\le}Etial{\mathrm{E}}_{\mathrm{t}\mathbf{i}\mathrm{al}}Etial​+${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S-Strial{\mathit{S}}_{\mathrm{trial}}Strial​〉/\ensuremath{\beta} +EDB{\mathit{E}}^{\mathrm{DB}}EDB,because different branch lines or singularities have to be taken into account in the transformation to imaginary time variables. As an important illustration, our generalized inequality is applied to the problem of the Fr\"ohlich polaron in a magnetic field. From the generalization of the Feynman an inequality derived in the present paper, we determine the conditions to be imposed on the variational parameters in the trial action, such that the Feynman upper bound in its original form remains valid for a polaron in a magnetic field. Some limiting cases are studied analytically to illustrate the relevance of our additional constraints on the variational parameters of the trial system. In the free-particle limit and for a particular value of one of the variational parameters, we explicitly derive the contributions from the branch lines in the complex-time plane which arise if these additional constraints are not satisfied.