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. B, Condensed matter

Renormalization-group analysis of the ground-state properties of dilute Bose systems indspatial dimensions

Eugene B. Kolomeisky, Joseph P. Straley

Published Nov 1, 1992• 83 citations
Publisher

Abstract

A low-density system of Bose particles of mass m and density n interacting through a short-range potential with range a is considered in d dimensions using a renormalization approach. The expansion parameter of the ground-state energy E per particle for d2 is (${\mathit{na}}^{\mathit{d}}$${)}^{(2\mathrm{\ensuremath{-}}\mathit{d})/\mathit{d}}$. For some special cases the leading universal term of the expansion E=${\mathit{C}}_{\mathit{d}}$${\mathrm{\ensuremath{\Elzxh}}}^{2}$${\mathit{n}}^{2/\mathit{d}}$/m+..., where ${\mathit{C}}_{\mathit{d}}$ is a universal number, gives an exact expression for the ground-state energy. For d>2 the energy expansion scales as E=(1/2)nt[1+const(${\mathit{nf}}^{\mathit{d}}$${)}^{(\mathit{d}\mathrm{\ensuremath{-}}2)/2}$], where t is the pseudopotential connected to the bare potential and f is the corresponding scattering length. This is a generalization of the generally accepted results (for d=3) of Lee and Yang [Phys. Rev. 105, 1119 (1957)], but a stronger condition for validity is found: ${\mathit{nf}}^{\mathit{d}}$\ensuremath{\ll}(f/a${)}^{2}$\ensuremath{\le}1. The possibility of the formation of quantum crystal phases is considered. We argue that for d2, crystal phases and melting transitions in the traditional sense are impossible. For d>2 the liquid phase is stable relative to the solid one. The corresponding energy difference diminishes for d\ensuremath{\rightarrow}2+0, and for d=2 it is much less than was suggested by Nelson [J. Stat. Phys. 57, 511 (1989)]; here the liquid phase is stable only in the extremely dilute regime. For d>2, attractive Bose systems exhibit a second-order phase transition connected to the formation of the bound state. For d\ensuremath{\le}2 only the bound state can occur.

DOI
10.1103/physrevb.46.11749

Related papers

Ground-state properties of Bose liquids with long-range interactions indspatial dimensions

Eugene B. Kolomeisky, Joseph P. Straley

We consider a low-density system of Bose particles in general dimensionality interacting through a potential having a short-range part and a long-range tail that falls off like 1/${\mathit{r}}^{\mathit{s}}$, focusing mainly on the special case s=2. We use a renormalization-group approach for the computation of the ground-state properties, which we argue is asymptotically exact in the dilute limit. The strength of the long-range part determines fixed-point values for the amplitude of the short-range part; several phases can occur, depending on the relative strengths of the short-range part and the fixed-point amplitudes. For the case in which the physics is determined by the flow away from the unstable fixed point (or when there are no fixed points at all), there results a collapsed state. When the flow is towards the stable fixed point, the outcome depends on whether the fixed-point value is negative or positive: In the former case we predict a kind of bound state, which is characterized by nontrivial correlations, even though the particles are not pairwise bound; in the latter case, a density expansion for the ground-state properties is given. The leading term of this expansion gives an exact result (a generalization of the one-dimensional result previously given by Sutherland) whenever the renormalization flow ends at the stable fixed point (including in particular the case that the flow starts at the fixed point). For the case that the renormalization flow stops before reaching the fixed point, the expansion parameter is proportional to the density raised to a nonuniversal power. The case s>2 can be reduced to the treatment of purely short-range interactions, whereas the repulsive case of s2 cannot be handled without additionally introducing a background of opposite charge. The case of attraction for s2 always leads to a collapse.

Physical review. B, Condensed matter•Dec 1, 1992•19 citations

Ground-state properties for ‘‘charged’’ bosons inddimensions from a renormalization-group treatment

Eugene B. Kolomeisky

A low-density Bose system comprised of two sets of particles with different intraset and interset short-ranged interactions are studied in arbitrary space dimension d with use of a renormalization-group approach. For the range of parameters where the ground-state is nonbinding, a density expansion for the chemical potential will be given. For d2 the character of this expansion is governed by the renormalization flow to a nontrivial stable fixed point of the renormalization equations. For d>2 the expansion is governed by the flow to a trivial noninteracting fixed point and the results are expressed in the terms of two pseudopotentials. The case d=2 is marginal and is characterized by the presence of logarithmic corrections. For the range of parameters where the ground-state is binding, several different phases and corresponding phase boundaries will be located. For d\ensuremath{\le}2 we find a strong-coupling molecular phase, the constituents of which are formed from the opposite charges of the original system, and a condensed phase. For d>2 we find in addition a weak-coupling binding phase, and a strong-coupling phase with correlations characterizing a tendency to interset pairing.

Physical review. B, Condensed matter•Dec 1, 1992•3 citations

Dilute Bose gas in two dimensions

Daniel S. Fisher, Pierre C. Hohenberg

The weakly interacting Bose gas in two dimensions is considered in the dilute limit ${n}^{1/2}$a\ensuremath{\ll}1, where n is the particle density and a is the range of the potential. The standard many-body perturbation theory for this system has two separate divergences: the first, associated with classical phase fluctuations, is responsible for the vanishing of the long-range order; the second is quantum mechanical and is connected with the vanishing of the scattering t matrix at long wavelengths and low energies. An earlier diagrammatic theory of Popov, which provides a consistent description of the system in the dilute limit, is rederived heuristically from a quasiparticle picture, and also using the renormalization group. It is shown that the superfluid transition temperature is ${T}_{c}$\ensuremath{\approxeq}4\ensuremath{\pi}(${\ensuremath{\Elzxh}}^{2}$/2m)n/[ln ln(1/${\mathrm{na}}^{2}$)], and the condition of validity of the dilute limit is ln ln(1/${\mathrm{na}}^{2}$)\ensuremath{\gg}1. The connection to the dilute Bose gas in dimensions d>2 and the universal behavior beyond the extreme asymptotic low-density domain are also discussed.

Physical review. B, Condensed matter•Apr 1, 1988•388 citations

Mathematical physics of dilute Bose gases

Jan Philip Solovej

We discuss recent progress in the mathematical analysis of dilute Bose gases. We review results in one to three dimensions, but the focus will be on three dimensions. In all dimensions we have a two term asymptotic expansion of the ground state energy density by an expression that depends only on the scattering length of the potential. In dimension three this is the celebrated Lee–Huang–Yang formula. In dimensions two and three the dilute limit is a weakly interacting regime whereas in dimension one it is rather strongly interacting. We sketch briefly the mathematical difficulties and review some remaining open problems in the field.

arXiv:2504.03314•Comptes Rendus Physique•Apr 8, 2025•1 citation

Ground-State Energy and Condensate Density of a Dilute Bose Gas Revisited

Kazumasa Tsutsui, Takafumi Kita

The ground-state energy per particle $E/N$ and condensate density $n_0$ of a dilute Bose gas are studied with a self-consistent perturbation expansion satisfying the Hugenholtz-Pines theorem and conservation laws simultaneously. A class of Feynman diagrams for the self-energy, which has escaped consideration so far, is shown to add an extra constant $c_{ip}\sim O(1)$ to the expressions reported by Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957) ] as $E/N=(2π\hbar^2 an/m)[1+(128/15\sqrt{π}+16c_{ip}/5)\sqrt{a^3n}]$ and $n_0/n=1-(8/3\sqrt{π}+c_{ip})\sqrt{a^3n}$, where $a$, $n$, and $m$ are are the s-wave scattering length, particle density, and particle mass, respectively. We present a couple of estimates for $c_{ip}$; the third-order perturbation expansion yields $c_{ip}=0.412$.

arXiv:1305.0657•Journal of the Physical Society of Japan•May 20, 2013•3 citations

Short-range particle correlations in a dilute Bose gas

Alexander Yu. Cherny, Arkady A Shanenko

The thermodynamics of a homogeneous dilute Bose gas with an arbitrarily strong repulsion between particles is investigated on the basis of the exact relation connecting the pair correlation function with the in-medium pair wave functions and occupation numbers. It is shown that the effective-interaction scheme, which is reduced to the Bogoliubov model with the effective pairwise potential, is not acceptable for investigating the short-range particle correlations in a dilute strongly interacting Bose gas. In contrast to this scheme, our model is thermodynamically consistent and free of the ultraviolet divergences due to accurate treatment of the short-range boson correlations. An equation for the in-medium scattering amplitude is derived that makes it possible to find the in-medium renormalization for the pair wave functions at short boson separations. Low-density expansions for the main thermodynamic quantities are reinvestigated on the basis of this equation. In addition, the expansions are found for the interaction and kinetic energies per particle. It is demonstrated that for a many-boson system of hard spheres the interaction energy is equal to zero for any boson density. The exact relationship between the chemical potential and in-medium pair wave functions is also established.

arXiv:cond-mat/9909199•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 1, 2000•21 citations
A low-density system of Bose particles of mass m and density n interacting through a short-range potential with range a is considered in d dimensions using a renormalization approach. The expansion parameter of the ground-state energy E per particle for d2 is ({\mathit{na}}^{\mathit{d}}$${)}^{(2\mathrm{\ensuremath{-}}\mathit{d})/\mathit{d}}. For some special cases the leading universal term of the expansion E={\mathit{C}}_{\mathit{d}}$${\mathrm{\ensuremath{\Elzxh}}}^{2}$${\mathit{n}}^{2/\mathit{d}}/m+..., where Cd{\mathit{C}}_{\mathit{d}}Cd​ is a universal number, gives an exact expression for the ground-state energy. For d>2 the energy expansion scales as E=(1/2)nt[1+const({\mathit{nf}}^{\mathit{d}}$${)}^{(\mathit{d}\mathrm{\ensuremath{-}}2)/2}], where t is the pseudopotential connected to the bare potential and f is the corresponding scattering length. This is a generalization of the generally accepted results (for d=3) of Lee and Yang [Phys. Rev. 105, 1119 (1957)], but a stronger condition for validity is found: nfd{\mathit{nf}}^{\mathit{d}}nfd\ensuremath{\ll}(f/a)2{)}^{2})2\ensuremath{\le}1. The possibility of the formation of quantum crystal phases is considered. We argue that for d2, crystal phases and melting transitions in the traditional sense are impossible. For d>2 the liquid phase is stable relative to the solid one. The corresponding energy difference diminishes for d\ensuremath{\rightarrow}2+0, and for d=2 it is much less than was suggested by Nelson [J. Stat. Phys. 57, 511 (1989)]; here the liquid phase is stable only in the extremely dilute regime. For d>2, attractive Bose systems exhibit a second-order phase transition connected to the formation of the bound state. For d\ensuremath{\le}2 only the bound state can occur.
We consider a low-density system of Bose particles in general dimensionality interacting through a potential having a short-range part and a long-range tail that falls off like 1/rs{\mathit{r}}^{\mathit{s}}rs, focusing mainly on the special case s=2. We use a renormalization-group approach for the computation of the ground-state properties, which we argue is asymptotically exact in the dilute limit. The strength of the long-range part determines fixed-point values for the amplitude of the short-range part; several phases can occur, depending on the relative strengths of the short-range part and the fixed-point amplitudes. For the case in which the physics is determined by the flow away from the unstable fixed point (or when there are no fixed points at all), there results a collapsed state. When the flow is towards the stable fixed point, the outcome depends on whether the fixed-point value is negative or positive: In the former case we predict a kind of bound state, which is characterized by nontrivial correlations, even though the particles are not pairwise bound; in the latter case, a density expansion for the ground-state properties is given. The leading term of this expansion gives an exact result (a generalization of the one-dimensional result previously given by Sutherland) whenever the renormalization flow ends at the stable fixed point (including in particular the case that the flow starts at the fixed point). For the case that the renormalization flow stops before reaching the fixed point, the expansion parameter is proportional to the density raised to a nonuniversal power. The case s>2 can be reduced to the treatment of purely short-range interactions, whereas the repulsive case of s2 cannot be handled without additionally introducing a background of opposite charge. The case of attraction for s2 always leads to a collapse.
A low-density Bose system comprised of two sets of particles with different intraset and interset short-ranged interactions are studied in arbitrary space dimension d with use of a renormalization-group approach. For the range of parameters where the ground-state is nonbinding, a density expansion for the chemical potential will be given. For d2 the character of this expansion is governed by the renormalization flow to a nontrivial stable fixed point of the renormalization equations. For d>2 the expansion is governed by the flow to a trivial noninteracting fixed point and the results are expressed in the terms of two pseudopotentials. The case d=2 is marginal and is characterized by the presence of logarithmic corrections. For the range of parameters where the ground-state is binding, several different phases and corresponding phase boundaries will be located. For d\ensuremath{\le}2 we find a strong-coupling molecular phase, the constituents of which are formed from the opposite charges of the original system, and a condensed phase. For d>2 we find in addition a weak-coupling binding phase, and a strong-coupling phase with correlations characterizing a tendency to interset pairing.
The weakly interacting Bose gas in two dimensions is considered in the dilute limit n1/2{n}^{1/2}n1/2a\ensuremath{\ll}1, where n is the particle density and a is the range of the potential. The standard many-body perturbation theory for this system has two separate divergences: the first, associated with classical phase fluctuations, is responsible for the vanishing of the long-range order; the second is quantum mechanical and is connected with the vanishing of the scattering t matrix at long wavelengths and low energies. An earlier diagrammatic theory of Popov, which provides a consistent description of the system in the dilute limit, is rederived heuristically from a quasiparticle picture, and also using the renormalization group. It is shown that the superfluid transition temperature is Tc{T}_{c}Tc​\ensuremath{\approxeq}4\ensuremath{\pi}(${\ensuremath{\Elzxh}}^{2}$/2m)n/[ln ln(1/na2{\mathrm{na}}^{2}na2)], and the condition of validity of the dilute limit is ln ln(1/na2{\mathrm{na}}^{2}na2)\ensuremath{\gg}1. The connection to the dilute Bose gas in dimensions d>2 and the universal behavior beyond the extreme asymptotic low-density domain are also discussed.
We discuss recent progress in the mathematical analysis of dilute Bose gases. We review results in one to three dimensions, but the focus will be on three dimensions. In all dimensions we have a two term asymptotic expansion of the ground state energy density by an expression that depends only on the scattering length of the potential. In dimension three this is the celebrated Lee–Huang–Yang formula. In dimensions two and three the dilute limit is a weakly interacting regime whereas in dimension one it is rather strongly interacting. We sketch briefly the mathematical difficulties and review some remaining open problems in the field.
The ground-state energy per particle E/NE/NE/N and condensate density n0n_0n0​ of a dilute Bose gas are studied with a self-consistent perturbation expansion satisfying the Hugenholtz-Pines theorem and conservation laws simultaneously. A class of Feynman diagrams for the self-energy, which has escaped consideration so far, is shown to add an extra constant cip∼O(1)c_{ip}\sim O(1)cip​∼O(1) to the expressions reported by Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957) ] as E/N=(2πℏ2an/m)[1+(128/15π+16cip/5)a3n]E/N=(2π\hbar^2 an/m)[1+(128/15\sqrt{π}+16c_{ip}/5)\sqrt{a^3n}]E/N=(2πℏ2an/m)[1+(128/15π​+16cip​/5)a3n​] and n0/n=1−(8/3π+cip)a3nn_0/n=1-(8/3\sqrt{π}+c_{ip})\sqrt{a^3n}n0​/n=1−(8/3π​+cip​)a3n​, where aaa, nnn, and mmm are are the s-wave scattering length, particle density, and particle mass, respectively. We present a couple of estimates for cipc_{ip}cip​; the third-order perturbation expansion yields cip=0.412c_{ip}=0.412cip​=0.412.
The thermodynamics of a homogeneous dilute Bose gas with an arbitrarily strong repulsion between particles is investigated on the basis of the exact relation connecting the pair correlation function with the in-medium pair wave functions and occupation numbers. It is shown that the effective-interaction scheme, which is reduced to the Bogoliubov model with the effective pairwise potential, is not acceptable for investigating the short-range particle correlations in a dilute strongly interacting Bose gas. In contrast to this scheme, our model is thermodynamically consistent and free of the ultraviolet divergences due to accurate treatment of the short-range boson correlations. An equation for the in-medium scattering amplitude is derived that makes it possible to find the in-medium renormalization for the pair wave functions at short boson separations. Low-density expansions for the main thermodynamic quantities are reinvestigated on the basis of this equation. In addition, the expansions are found for the interaction and kinetic energies per particle. It is demonstrated that for a many-boson system of hard spheres the interaction energy is equal to zero for any boson density. The exact relationship between the chemical potential and in-medium pair wave functions is also established.