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Physical review. B, Condensed matter

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

Eugene B. Kolomeisky

Published Dec 1, 1992• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.46.13956

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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.
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 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 (nad{\mathit{na}}^{\mathit{d}}nad${)}^{(2\mathrm{\ensuremath{-}}\mathit{d})/\mathit{d}}$. For some special cases the leading universal term of the expansion E=Cd{\mathit{C}}_{\mathit{d}}Cd​${\mathrm{\ensuremath{\Elzxh}}}^{2}$n2/d{\mathit{n}}^{2/\mathit{d}}n2/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(nfd{\mathit{nf}}^{\mathit{d}}nfd${)}^{(\mathit{d}\mathrm{\ensuremath{-}}2)/2}$], 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 present a nonperturbative renormalization-group approach to the Bose-Hubbard model. By taking as initial condition of the renormalization-group flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with quantum Monte Carlo simulations, and reproduces the two universality classes of the superfluid-Mott-insulator transition. The critical behavior near the multicritical points, where the transition takes place at constant density, agrees with the original predictions of Fisher et al. [Phys. Rev. B 40, 546 (1989)] based on simple scaling arguments. At a generic transition point, the critical behavior is mean-field like with logarithmic corrections in two dimensions. In the weakly correlated superfluid phase (far away from the Mott insulating phase), the renormalization-group flow is controlled by the Bogoliubov fixed point down to a characteristic (Ginzburg) momentum scale kG{k}_{G}kG​, which is much smaller than the inverse healing length kh{k}_{h}kh​. In the vicinity of the multicritical points, when the density is commensurate, we identify a sharp crossover from a weakly to a strongly correlated superfluid phase where the condensate density and the superfluid stiffness are strongly suppressed and both kG{k}_{G}kG​ and kh{k}_{h}kh​ are of the order of the inverse lattice spacing.
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.
A dilute homogeneous two-dimensional (2D) Bose gas at zero temperature is studied with the method developed earlier by the authors. This method allows for considering renormalization of an arbitrary pairwise potential in a self-consistent manner, without the pseudopotential delta-function representation. Low-density expansions are derived for the chemical potential, ground-state energy, pair distribution function, kinetic and interaction energies. The expansion parameter is found to be a dimensionless in-medium scattering amplitude u obeying the equation 1/u+ln u=-ln(na(2)pi)-2gamma, where na(2) and gamma are the gas parameter and the Euler constant, respectively. It is shown that the ground-state energy is mostly kinetic in the low-density limit. This result does not depend on a specific form of the pairwise interaction potential, contrary to the 3D case. A new form of the 2D Gross-Pitaevskii equation is proposed within our scheme.
We consider systems of nonrelativistic, interacting electrons at finite density and zero temperature in d=2,3,ldots dimensions. Our main concern is to characterize those systems that, under the renormalization flow, are driven away from the Landau Fermi-liquid (LFL) renormalization-group fixed point. We are especially interested in understanding under what circumstances such a system is a marginal Fermi-liquid (MFL) when the dimension of space is d\ensuremath{\geqslant}2. The interacting electron system is analyzed by combining renormalization-group (RG) methods with so called 'Luther-Haldane' bosonization techniques. The RG calculations are organized as a double expansion in the inverse scale parameter ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ , which is proportional to the width of the effective momentum space around the Fermi surface and in the running coupling constant ${\mathrm{g}}_{\ensuremath{\lambda}}$ , which measures the strength of electron interactions at energy scales \ensuremath{\sim}vF{\mathrm{v}}_{\mathrm{F}}vF​ kF{\mathrm{k}}_{\mathrm{F}}kF​ /\ensuremath{\lambda}. For systems with a strictly convex Fermi surface, superconductivity is the only symmetry-breaking instability. Excluding such an instability, the system can be analyzed by means of bosonization. The RG and the underlying perturbation expansion in powers of ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ serve to characterize the approximations involved by bosonizing the system. We argue that systems with short-range interactions flow to the LFL fixed point. Within the approximations involved by bosonization, the same holds for systems with long-range, longitudinal, density-density interactions. For electron systems interacting via long-range, transverse, current-current interactions, a deviation from LFL behavior is possible: if the exponent \ensuremath{\alpha} parametrizing the singularity of the interaction potential in momentum space by V-hat(|p|)\ensuremath{\sim}1/|p${\mathrm{|}}^{\mathrm{\ensuremath{\alpha}}}$ is greater than or equal to d-1, the results of the bosonization calculation are consistent with a MFL.