Equation of state of atomic systems beyond s-wave determined by the
  lowest order constrained variational method: Large scattering length limit
Equation of state of atomic systems beyond s-wave determined by the lowest order
constrained variational method: Large scattering length limit
Ryan M. Kalas(1) and D. Blume(1,2)
(1)Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814
(2)INFM-BEC, Dipartimento di Fisica, Università di Trento, I-38050 Povo, Italy
Dilute Fermi systems with large s-wave scattering length as exhibit universal properties if the
interparticle spacing ro greatly exceeds the range of the underlying two-body interaction potential.
In this regime, ro is the only relevant length scale and observables such as the energy per particle
depend only on ro (or, equivalently, the energy EFG of the free Fermi gas). This paper investigates
Bose and Fermi systems with non-vanishing angular momentum l using the lowest order constrained
variational method. We focus on the regime where the generalized scattering length becomes large
and determine the relevant length scales. For Bose gases with large generalized scattering lengths,
we obtain simple expressions for the energy per particle in terms of a l-dependent length scale ξl,
which depends on the range of the underlying two-body potential and the average interparticle
spacing. We discuss possible implications for dilute two-component Fermi systems with finite l.
Furthermore, we determine the equation of state of liquid and gaseous bosonic helium.
PACS numbers:
I. INTRODUCTION
The experimental realization of dilute degenerate Bose
and Fermi gases has led to an explosion of activities in the
field of cold atom gases. A particularly intriguing feature
of atomic Bose and Fermi gases is that their interaction
strengths can be tuned experimentally through the ap-
plication of an external magnetic field in the vicinity of
a Feshbach resonance [1, 2]. This external knob allows
dilute systems with essentially any interaction strength,
including infinitely strongly attractive and repulsive in-
teractions, to be realized. Feshbach resonances have been
experimentally observed for s-, p- and d-wave interacting
gases [3, 4, 5, 6, 7] and have been predicted to exist also
for higher partial waves.
A Feshbach resonance arises due to the coupling of two
Born-Oppenheimer potential curves coupled through a
hyperfine Hamiltonian, and requires, in general, a multi-
channel description. For s-wave interacting systems, Fes-
hbach resonances can be classified as broad or narrow [8].
Whether a resonance is broad or narrow depends on
whether the energy width of the resonance is large or
small compared to the characteristic energy scale, such
as the Fermi energy or the harmonic oscillator energy,
of the system. In contrast to s-wave resonances, higher
partial wave resonances are necessarily narrow due to the
presence of the angular momentum barrier [9]. This pa-
per uses an effective single channel description to investi-
gate the behaviors of strongly-interacting Bose and Fermi
systems with different orbital angular momenta.
In dilute homogeneous Bose and Fermi gases with large
s-wave scattering length as, a regime has been identi-
fied in which the energy per particle takes on a universal
value which is set by a single length scale, the average
interparticle spacing ro [10, 11, 12]. In this so-called uni-
tary regime, the length scales of the s-wave interacting
system separate according to |as| ≫ ro ≫ R, where R
denotes the range of the two-body potential. The en-
ergy per particle EB,0/N (the subscripts “B” and “0”
stand respectively for “boson” and “s-wave interacting”)
for a homogeneous one-component gas of bosons with
mass m in the unitary regime has been calculated to be
EB,0/N ≈ 13.3 ~
B /m using the lowest order con-
strained variational (LOCV) method [12]. The energy
EB,0/N at unitarity is thus independent of as and R,
and depends on the single length scale ro through the
boson number density nB, ro = (4πnB/3)
−1/3. How-
ever, Bose gases in the large scattering length limit are
expected to be unstable due to three-body recombina-
tion [13, 14, 15, 16].
On the other hand, the Fermi pressure prevents the
collapse of two-component Fermi gases with equal masses
and equal number of “spin-up” and “spin-down” fermions
with large interspecies s-wave scattering length [10, 11,
17, 18]. At unitarity, the energy per particle is given by
EF,0/N ≈ 0.42EFG, where EFG = (3/10)(~
2k2F /m) de-
notes the energy per particle of the non-interacting Fermi
gas [19, 20, 21, 22]. The Fermi wave vector kF is related
to the number density of the Fermi gas by nF = k
F /3π
which implies that EF,0/N depends on ro but is indepen-
dent of as and R. We note that the inequality |as| ≫ ro
is equivalent to 1/(kF |as|) ≪ 1.
This paper investigates Bose and Fermi systems with
large generalized scattering lengths using the LOCV
method. For p- and d-wave interacting Bose systems,
we define the unitary regime [23] through the inequali-
ties |al(Erel)| ≫ ξl ≫ R, where ξl denotes a l-dependent
length scale given by the geometric combination of ro and
R, i.e., ξl = r
(1−l/4)
l/4, and Erel the relative scatter-
ing energy. The generalized energy-dependent scattering
length al(Erel) [24, 25, 26] characterizes the scattering
strength (see below). We find that the energy of p-wave
interacting two-component Bose gases and d-wave inter-
acting one- and two-component Bose gases at unitary is
determined by the combined length ξl. While Bose gases
with higher angular momentum in the unitary regime are
http://arxiv.org/abs/0704.1804v1
of theoretical interest, they are, like their s-wave cousin,
expected to be unstable. We comment that the energet-
ics of two-component Fermi gases with large generalized
scattering length may depend on the same length scales.
Furthermore, we consider s-wave interacting Bose sys-
tems over a wide range of densities. Motivated by two re-
cent studies by Gao [27, 28], we determine the energy per
particle EB,0/N of the Bose system characterized by two
atomic physics parameters, the s-wave scattering lengh
as and the van der Waals coefficient C6. Our results lead
to a phase diagram of liquid helium in the low-density
regime that differs from that proposed in Ref. [28].
Section II describes the systems under study and in-
troduces the LOCV method. Section III describes our
results for dilute s-wave interacting Bose and Fermi sys-
tems and for liquid helium. Section IV considers Bose
and Fermi systems interacting through l-wave (l > 0)
scattering. Finally, Section V concludes.
II. LOCV METHOD FOR BOSONS AND
FERMIONS
This section introduces the three-dimensional Bose
and Fermi systems under study and reviews the LOCV
method [29, 30, 31, 32]. The idea of the LOCV method
is to explicitly treat two-body correlations, but to ne-
glect three- and higher-body correlations. This allows
the many-body problem to be reduced to solving an
effective two-body equation with properly chosen con-
straints. Imposing these constraints makes the method
non-variational, i.e., the resulting energy does not place
an upper bound on the many-body energy. The LOCV
method is expected to capture some of the key physics of
dilute Bose and Fermi systems.
The Hamiltonian HB for a homogeneous system con-
sisting of identical mass m bosons is given by
HB = −
∇2i +
v(rij), (1)
where the spherically symmetric interaction potential v
depends on the relative distance rij , rij = |ri−rj|. Here,
ri denotes the position vector of the ith boson. The
Hamiltonian HF for a two-component Fermi system with
equal masses and identical spin population is given by
HF = −
∇2i −
∇2i′ +
v(rii′ ), (2)
where the unprimed subscripts label spin-up and the
primed subscripts spin-down fermions. Throughout, we
take like fermions to be non-interacting. Our primary
interest in this paper is in the description of systems
for which many-body observables are insensitive to the
short-range behavior of the atom-atom potential v(r).
This motivates us to consider two simple model poten-
tials: an attractive square well potential vsw with depth
Vo (Vo ≥ 0),
vsw(r) =
−Vo for r < R
0 for r > R ;
and an attractive van der Waals potential vvdw with hard-
core rc,
vvdw(r) =
∞ for r < rc
−C6/r
6 for r > rc .
In all applications, we choose the hardcore rc so that the
inequality rc ≪ β6, where β6 = (mC6/~
2)1/4, is satisfied.
The natural length scale of the square well potential is
given by the range R and that of the van der Waals po-
tential by the van der Waals length β6. The solutions to
the two-body Schrödinger equation for vsw are given in
terms of spherical Bessel and Neumann functions (impos-
ing the proper continuity conditions of the wave function
and its derivations at those r values where the poten-
tial exhibits a discontinuity), and those for vvdw in terms
of convergent infinite series of spherical Bessel and Neu-
mann functions [33].
The interaction strength of the short-range square well
potential can be characterized by the generalized energy-
dependent scattering lengths al(k),
al(k) = sgn[− tan δl(k)]
tan δl(k)
k2l+1
1/(2l+1)
, (5)
where δl(k) denotes the phase shift of the lth partial
wave calculated at the relative scattering energy Erel,
mErel/~2. This definition ensures that al(k) ap-
proaches a constant as k → 0 [34, 35]. For the van der
Waals potential vvdw, the threshold behavior changes for
higher partial waves and the definition of al(k) has to be
modified accordingly [34, 35]. In general, for a poten-
tial that falls off as −r−n at large interparticle distances,
al(k) is defined by Eq. (5) if 2l < n− 3 and by
al(k) = sgn[− tan δl(k)]
tan δl(k)
1/(n−2)
if 2l > n− 3. For our van der Waals potential, n is equal
to 6 and al(k) is given by Eq. (5) for l ≤ 1 and by Eq. (6)
for l ≥ 2. The zero-energy generalized scattering lengths
al can now be defined readily through
al = lim
al(k). (7)
We note that a new two-body l-wave bound state ap-
pears at threshold when |al| → ∞. The unitary regime
for higher partial waves discussed in Sec. IV is thus, as
in the s-wave case, closely related to the physics of ex-
tremely weakly-bound atom-pairs. To uncover the key
behaviors at unitarity, we assume in the following that
the many-body system under study is interacting through
a single partial wave l. While this may not be exactly
realized in an experiment, this situation may be approx-
imated by utilizing Feshbach resonances.
We now outline how the energy per particle EB,l/N
of a one-component Bose system with l-wave interac-
tions [36] is calculated by the LOCV method [29, 30,
31, 32]. The boson wave function ΨB is taken to be a
product of pair functions fl,
ΨB(r1, . . . , rN ) =
fl(rij), (8)
and the energy expectation value ofHB, Eq. (1), is calcu-
lated using ΨB. If terms depending on the coordinates of
three or more different particles are neglected, the result-
ing energy is given by the two-body term in the cluster
expansion,
fl(r)
∇2 + v(r)
fl(r) d
r. (9)
The idea of the LOCV method is now to introduce a heal-
ing distance d beyond which the pair correlation function
fl is constant,
fl(r > d) = 1. (10)
To ensure that the derivative of fl is continuous at r = d,
an additional constraint is introduced,
f ′l (r = d) = 0. (11)
Introducing a constant average field λl and varying with
respect to fl while using that fl is constant for r > d,
gives the Schrödinger-like two-body equation for r < d,
∇2 + v(r)
(rfl(r)) = λlrfl(r). (12)
Finally, the condition
f2l (r) d
r = 1 (13)
enforces that the average number of particles within d
equals 1. Using Eqs. (9), (10) and (12), the energy per
particle becomes,
v(r)d3r. (14)
The second term on the right hand side of Eq. (14) is
identically zero for the square well potential vsw but con-
tributes a so-called tail or mean-field energy for the van
der Waals potential vvdw [27, 28]. We determine the
three unknown nB, λl and d by simultaneously solving
Eqs. (12) and (13) subject to the boundary condition
given by Eq. (11). Note that nB and d depend, just as
fl and λl, on the angular momentum; the subscript has
been dropped, however, for notational convenience.
In addition to one-component Bose systems, Sec. IV
considers two-component Bose systems, characterized by
l-wave interspecies and vanishing intraspecies interac-
tions. The Hamiltonian for the two-component Bose sys-
tem is given by Eq. (2), with the sum of the two-body
interactions restricted to unlike bosons. Correspondingly,
the product wave function is written as a product of
pair functions, including only correlations between unlike
bosons. The LOCV equations are then given by Eqs. (10)
through (13) with nB in Eq. (13) replaced by nB/2.
Next, we discuss how to determine the energy EF,l/N
per particle for a two-component Fermi system within
the LOCV method [29, 30, 31, 32]. The wavefunction is
taken to be
ΨF (r1, . . . , r1′ , . . . ) = ΦFG
fl(rij′ ), (15)
where ΦFG denotes the ground state wavefunction of the
non-interacting Fermi gas. The product of pair functions
fl accounts for the correlations between unlike fermions.
In accord with our assumption that like fermions are non-
interacting, Eq. (15) treats like fermion pairs as uncor-
related. Neglecting exchange effects, the derivation of
the LOCV equations parallels that outlined above for the
bosons. The boundary conditions, given by Eqs. (10) and
(11), and the Schrödinger-like differential equation for λl,
Eq. (12), are unchanged. The “normalization condition,”
however, becomes
f2l (r)d
r = 1, (16)
where the left-hand side is the number of fermion pairs
within d. The fermion energy per particle is then the sum
of the one-particle contribution from the non-interacting
Fermi gas and the pair correlation energy λl [20],
EF,l/N = EFG +
. (17)
This equation excludes the contribution from the tail of
the potential, i.e., the term analogous to the second term
on the right hand side of Eq. (14), since this term is neg-
ligible for the fermion densities considered in this paper.
The LOCV solutions for fl, λl and d for the ho-
mogeneous one-component Bose system and the two-
component Fermi system are formally identical if the bo-
son density is chosen to equal half the fermion density,
i.e., if nB = nF /2. This relation can be understood by
realizing that any given fermion (e.g., a spin-up particle)
interacts with only half of the total number of fermions
(e.g., all the spin-down fermions). Consequently, the two-
component Fermi system appears twice as dense as the
one-component Bose system. The fact that the LOCV
solutions for bosons can be converted to LOCV solutions
for fermions suggests that some physics of the bosonic
system can be understood in terms of the fermionic sys-
tem and vice versa. In fact, it has been shown previ-
ously [20] that the LOCV energy for the first excited
gas-like state of s-wave interacting fermions at unitarity
can be derived from the LOCV energy of the energet-
ically lowest-lying gas-like branch of s-wave interacting
bosons [12]. Here, we extend this analysis and show that
the ground state energy of the Fermi gas at unitarity can
0 1 2 3 4
FIG. 1: Energy per particle EB,0/N as a function of the
density nB , both plotted as dimensionless quantities, for a
one-component Bose system interacting through the van der
Waals potential with s-wave scattering length as = 16.9β6.
The dotted line shows the gas branch and the dashed line the
liquid branch. The minimum of the liquid branch is discussed
in reference to liquid 4He in the text.
be derived from the energetically highest-lying liquid-like
branch of the Bose system. Furthermore, we extend this
analysis to higher angular momentum scattering.
III. s-WAVE INTERACTING BOSE AND
FERMI SYSTEMS
Figure 1 shows the energy per particle EB,0/N ,
Eq. (14), obtained by solving the LOCV equations for
a one-component Bose system interacting through the
van der Waals potential with s-wave scattering length
as = 16.9β6. The dotted line in Fig. 1 has positive energy
and increases with increasing density; it describes the
energetically lowest-lying “gas branch” for the Bose sys-
tem with as = 16.9β6 and corresponds to the metastable
gaseous condensate studied experimentally. The dashed
line in Fig. 1 has negative energy at small densities, de-
creases with increasing density, and then exhibits a mini-
mum; this dashed line describes the energetically highest-
lying “liquid branch” for a Bose system with as = 16.9β6.
Within the LOCV framework, these two branches arise
because the Schrödinger-like equation, Eq. (12), permits
for a given interaction potential solutions f0 with differ-
ing number of nodes, which in turn give rise to a host of
liquid and gas branches [27, 28]. Throughout this work
we only consider the energetically highest-lying liquid
branch with n nodes and the energetically lowest-lying
gas branch with n + 1 nodes. To obtain Fig. 1, we con-
sider a class of two-body potentials with fixed as/β6, and
decrease the value of the ratio rc/β6 till EB,0/N , Eq. (14),
no longer changes over the density range of interest, i.e.,
the number of nodes n of the energetically highest-lying
liquid branch is increased till convergence is reached.
0.0 4.0×10
8.0×10
1.2×10
-0.003
0.003
0.006
FIG. 2: Energy per particle EB,0/N as a function of the den-
sity nB for a one-component Bose system interacting through
the van der Waals potential with s-wave scattering lengths
as = 16.9β6 (open circles) and as = 169β6 (filled circles).
To guide the eye, dashed and dotted lines connect the data
points of the liquid and gas branches, respectively. The liq-
uid branches go to Edimer/2 as the density goes to zero. The
solid lines show EB,0/N at unitarity; see text for discussion.
Compared to Fig. 1, the energy and density scales are greatly
enlarged.
In Fig. 1, the two-body van der Waals potential is cho-
sen so that the scattering length of as = 16.9β6 coin-
cides with that of the 4He pair potential [37]. The liquid
branch in Fig. 1 can hence be applied to liquid 4He, and
has previously been considered in Refs. [27, 28]. The min-
imum of the liquid branch at a density of nB = 2.83β
or 1.82 × 1022cm−3, agrees quite well with the experi-
mental value of 2.18× 1022cm−3 [38]. The corresponding
energy per particle of −6.56 K deviates by 8.5 % from
the experimental value of −7.17 K [38]. This shows that
the LOCV framework provides a fair description of the
strongly interacting liquid 4He system, which is charac-
terized by interparticle spacings comparable to the range
of the potential. This is somewhat remarkable consid-
ering that the LOCV method includes only pair corre-
lations and that the van der Waals potential used here
contains only two parameters.
Open circles connected by a dashed line in Fig. 2 show
the liquid branch for as = 16.9β6 in the small density
region. As the density goes to zero, the energy per par-
ticle EB,0/N does not terminate at zero but, instead,
goes to Edimer/2, where Edimer denotes the energy of
the most weakly-bound s-wave molecule of vvdw. In this
small density limit, the liquid branch describes a gas of
weakly-bound molecules, in which the interparticle spac-
ing between the molecules greatly exceeds the size of the
molecules, and Edimer is to a very good approximation
given by −~2/(ma2s). As seen in Fig. 2, we find solu-
tions in the whole density range considered. In contrast
to our findings, Ref. [28] reports that the LOCV solu-
tions of the liquid branch disappear at densities smaller
than a scattering length dependent critical density, i.e.,
FIG. 3: Scaled interparticle spacing ro/β6 as a function of the
scaled density nBβ
6 for the gas branch of a one-component
Bose system interacting through the van der Waals poten-
tial with as = 169β6. The horizontal lines show the scaled
s-wave scattering length as = 169β6 and the range of the van
der Waals potential, which is one in scaled units (almost in-
distinguishable from the x-axis). This graph shows that the
unitary inequalities as ≫ ro ≫ β6 hold for nB larger than
about 10−5β−36 .
at a critical density of 8.68 × 10−7β−36 for as = 16.9β6.
Thus we are not able to reproduce the liquid-gas phase
diagram proposed in Fig. 2 of Ref. [28], which depends
on this termination of the liquid branch. We note that
the liquid branch is, as indicated by its imaginary speed
of sound, dynamically unstable at sufficiently small den-
sities. The liquid of weakly-bound bosonic molecules
discussed here can, as we show below, be related to
weakly-bound molecules on the BEC side of the BEC-
BCS crossover curve for two-component Fermi gases.
We now discuss the gas branch in more detail. Open
and filled circles connected by dotted lines in Fig. 2 show
the energy per particle for as = 16.9β6 and 169β6, respec-
tively. These curves can be applied, e.g., to 85Rb, whose
scattering length can be tuned by means of a Feshbach
resonance and which has a β6 value of 164abohr, where
abohr denotes the Bohr radius. For this system, a scat-
tering length of as = 16.9β6 corresponds to 2770abohr,
a comparatively large value that can be realized experi-
mentally in 85Rb gases. As a point of reference, a density
of 10−5β−36 corresponds to a density of 1.53× 10
13cm−3
for 85Rb.
The solid curve with positive energy in Fig. 2 shows
the energy per particle EB,0/N at unitarity, EB,0/N ≈
13.3~2n
B /m [12]. As seen in Fig. 2, this unitary limit
is approached by the energy per particle for the Bose
gas with as = 169β6 (filled circles connected by a dotted
line). To illustrate this point, Fig. 3 shows the scaled
average interparticle spacing ro/β6 as a function of the
scaled density nBβ
6 for as = 169β6. This plot indicates
that the unitary requirement, as ≫ ro ≫ R, is met for
values of nBβ
6 larger than about 10
−5. Similarly, we find
-1 -0.5 0 0.5 1
1/(kFas)
FIG. 4: Scaled energy per particle (EF,0/N)/EFG as a func-
tion of 1/(kF as) for a two-component s-wave Fermi gas inter-
acting through the square well potential for nF = 10
−6R−3.
The combined dashed and dash-dotted curve corresponds to
the BEC-BCS crossover curve and the dotted curve corre-
sponds to the first excited state of the Fermi gas. The dashed
and dotted linestyles are chosen to emphasize the connection
to the gas and liquid branches of the Bose system in Figs. 2
and 3 (see text for more details).
that the family of liquid curves converges to EB,0/N ≈
−2.46~2n
B /m (see Sec. IV for details), plotted as a solid
line in Fig. 2, when the inequalities as ≫ ro ≫ β6 are
fullfilled. We note that the unitarity curve with negative
energy is also approached, from above, for systems with
large negative scattering lengths (not shown in Fig. 2).
Aside from the proportionality constant, the power law
relation for the liquid and gas branches at unitarity is
the same.
In addition to a Bose system interacting through the
van der Waals potential, we consider a Bose system in-
teracting through the square well potential with range
R. For a given scattering length as and density nB, the
energy per particle EB,0/N for these two two-body po-
tentials is essentially identical for the densities shown in
Fig. 2. This agreement emphasizes that the details of
the two-body potential become negligible at low density,
and in particular, that the behavior of the Bose gas in
the unitary limit is governed by a single length scale, the
average interparticle spacing ro.
As discussed in Sec. II, the formal parallels between the
LOCV method applied to bosons and fermions allows the
energy per particle EF,0/N for a two-component Fermi
gas, Eq. (17), to be obtained straightforwardly from the
energy per particle EB,0/N of the Bose system. Fig-
ure 4 shows the dimensionless energy (EF,0/N)/EFG as
a function of the dimensionless quantity 1/(kFas) for the
square well potential for nF = 10
−6R−3. We find essen-
tially identical results for the van der Waals potential.
The crossover curve shown in Fig. 4 describes any dilute
Fermi gas for which the range R of the two-body poten-
tial is very small compared to the average interparticle
spacing ro. In converting the energies for the Bose sys-
tem to those for the Fermi system, the gas branches of the
Bose system (dotted lines in Figs. 2 and 3) “turn into”
the excited state of the Fermi gas (dotted line in Fig. 4);
the liquid branches of the Bose system with positive as
(dashed lines in Figs. 2 and 3) “turn into” the part of
the BEC-BCS crossover curve with positive as (dashed
line in Fig. 4); and the liquid branches of the Bose system
with negative as (not shown in Figs. 2 and 3) “turn into”
the part of the BEC-BCS crossover curve with negative
as (dash-dotted line in Fig. 4).
To emphasize the connection between the Bose and
Fermi systems further, let us consider the BEC side of
the crossover curve. If 1/(kFas) & 1, the fermion energy
per particle EF,0/N is approximately given by Edimer/2,
which indicates that the Fermi gas forms a molecular
Bose gas. Similarly, the liquid branch of the Bose sys-
tem with positive scattering length is made up of bosonic
molecules as the density goes to zero. The formal analogy
between the Bose and Fermi LOCV solutions also allows
the energy per particle EF,0/N at unitarity, i.e., in the
1/(kF |as|) → 0 limit, to be calculated from the energies
for large as of the gas and liquid branches of the Bose
system (solid lines in Fig. 2). For the excited state of the
Fermi gas we find EF,0/N ≈ 3.92EFG, and for the low-
est gas state we find EF,0/N ≈ 0.46EFG. These results
agree with the LOCV calculations of Ref. [20], which use
an attractive cosh-potential and a δ-function potential.
The value of 0.46EFG is in good agreement with the en-
ergy of 0.42EFG obtained by fixed-node diffusion Monte
Carlo calculations [21, 22].
IV. BOSE AND FERMI SYSTEMS BEYOND
s-WAVE AT UNITARITY
This section investigates the unitary regime of Bose
and Fermi systems interacting through higher angular
momentum resonances. These higher angular momen-
tum resonances are necessarily narrow [9], and we hence
expect the energy-dependence of the generalized scatter-
ing length al(k) to be particularly important in under-
standing the many-body physics of dilute atomic systems
beyond s-wave. In the following we focus on the strongly-
interacting limit. Figure 5 shows 1/al(k) as a function of
the relative scattering energy Erel for the square-well po-
tential with infinite zero-energy scattering length al for
three different angular momenta, l = 0 (solid line), l = 1
(dashed line), and l = 2 (dotted line). Figure 5 shows
that the energy-dependence of al(k) increases with in-
creasing l.
Our goal is to determine the energy per particle
EB,l/N for Bose systems with finite angular momentum
l in the strongly-interacting regime. For s-wave interac-
tions, the only relevant length scale at unitarity is the av-
erage interparticle spacing ro (see Sec. III). In this case,
the energy per particle at unitarity can be estimated an-
alytically by evaluating the LOCV equations subject to
-0.01 -0.005 0 0.005 0.01
rel /(h-
FIG. 5: R/al(Erel) as a function of the scaled relative scatter-
ing energy Erel/(~
2/mR2) for the square well potential vsw
with infinite zero-energy scattering length al, i.e., 1/al = 0,
for three different partial waves [l = 0 (solid line), l = 1
(dashed line), and l = 2 (dotted line)].
the boundary condition implied by the zero-range s-wave
pseudo-potential [12]. Unfortunatey, a similarly simple
analysis that uses the boundary condition implied by the
two-body zero-range pseudo-potential for higher partial
waves fails. This combined with the following arguments
suggests that EB,l/N depends additionally on the range
of the underlying two-body potential for finite l: i) The
probability distribution of the two-body l-wave bound
state, l > 0, remains finite as al approaches infinity and
depends on the interaction potential [39, 40]. ii) A de-
scription of l-wave resonances (l > 0) that uses a coupled
channel square well model depends on the range of the
square well potential [41]. iii) The calculation of struc-
tural expectation values of two-body systems with finite
l within a zero-range pseudo-potential treatment requires
a new length scale to be introduced [26].
Motivated by these two-body arguments (see also
Refs. [42, 43, 44] for a treatment of p-wave interacting
Fermi gases) we propose the following functional form for
the energy per particle EB,l/N of a l-wave Bose system
at unitarity interacting through the square-well potential
vsw with range R,
mRxl/2
2/3−xl/6
B . (18)
Here, Cl denotes a dimensionless l-dependent proportion-
ality constant. The dimensionless parameter xl deter-
mines the powers of the range R and the density nB,
and ensures the correct units of the right hand side of
Eq. (18). To test the validity of Eq. (18), we solve the
LOCV equations, Eqs. (11) through (14), for l = 0 to 2
for the one-component Bose system. Note that the one-
component p-wave system is unphysical since it does not
obey Bose symmetry; we nevertheless consider it here
since its LOCV energy determines the energy of two-
component p-wave Bose and Fermi systems (see below).
Figure 6 shows the energy per particle EB,l/N for
0 5×10
FIG. 6: Scaled energy per particle (EB,l/N)/(~
2/mR2) for a
one-component Bose system for the energetically lowest-lying
gas branch as a function of the scaled density nBR
3 obtained
by solving the LOCV equations [Eqs. (11) through (14)] for
vsw for three different angular momenta l [l = 0 (crosses),
l = 1 (asterisks) and l = 2 (pluses)]. The depth V0 of vsw is
adjusted so that 1/al(k) = 0. Solid, dotted and dashed lines
show fits of the LOCV energies at low densities to Eq. (18)
for l = 0, l = 1 and l = 2 (see text for details). Note that
the system with l = 1 is of theoretical interest but does not
describe a physical system.
l xl C
0 0.00 −2.46 13.3
1 1.00 −3.24 9.22
2 2.00 −3.30 6.98
TABLE I: Dimensionless parameters xl, C
l and C
l for l = 0
to 2 for a one-component Bose system obtained by fitting the
LOCV energies EB,l/N for small densities to the functional
form given in Eq. (18) (see text for details).
a one-component Bose system, obtained by solving the
LOCV equations for the energetically lowest-lying gas
branch, as a function of the density nB for l = 0 (crosses),
l = 1 (asterisks), and l = 2 (pluses) for the square well
potential, whose depth V0 is adjusted for each l so that
the energy-dependent generalized scattering length al(k)
diverges, i.e., 1/al(k) = 0. Setting al(k) to infinity en-
sures that the l-wave interacting Bose system is infinitely
strongly interacting over the entire density regime shown
in Fig. 6. Had we instead set the zero-energy scattering
length al to infinity, the system would, due to the strong
energy-dependence of al(k) [see Fig. 5], “effectively” in-
teract through a finite scattering length.
Table I summarizes the values for xl and C
l , which we
obtain by performing a fit of the LOCV energies EB,l/N
for the one-component Bose system for small densities to
the functional form given in Eq. (18). In particular, we
find xl = l, which implies that EB,l/N varies as n
B and n
B for l = 0, 1 and 2, respectively. Table I
uses the superscript “G” to indicate that the proportion-
ality constant is obtained for the energetically lowest-
lying gas branch. The density ranges used in the fit are
chosen so that Eq. (18) describes the low-density or uni-
versal regime accurately. Solid, dotted and dashed lines
in Fig. 6 show the results of these fits for l = 0, 1 and
2, respectively; in the low density regime, the lines agree
well with the symbols thereby validating the functional
form proposed in Eq. (18).
We repeat the LOCV calculations for the energeti-
cally highest-lying liquid branch of the one-component
Bose system. By fitting the LOCV energies of the liquid
branches for small densities to Eq. (18), we determine xl
and Cl [45]. We find the same xl as for the gas branch but
different Cl than for the gas branch (the proportionality
constants obtained for the liquid branch are denoted by
CLl ; see Table I). Our values for x0, C
0 and C
0 agree
with those reported in the literature [12, 20, 46].
Equation (18) can be rewritten in terms of the com-
bined length ξl,
)(l/6−2/3) ~
, (19)
where
ξl = r
(1−l/4)
l/4. (20)
For the s-wave case, ξl reduces to ro and the conver-
gence to the unitary regime can be seen by plotting
(EB,0/N)/(~
2/mr2o) as a function of as/ro [12]. To
investigate the convergence to the unitary regime for
higher partial waves, Fig. 7 shows the energy per parti-
cle EB,l/N for the energetically lowest-lying gas branch
as a function of al(Erel)/ξl for fixed energy-dependent
scattering lengths al(k), i.e., for as(k) = 10
10R, ap(k) =
1010R, and ad(k) = 10
6R [the different values of al(k)
are chosen for numerical reasons]. Figure 7 shows that
the inequality
|al(Erel)| ≫ ξl ≫ R (21)
is fulfilled when (EB,l/N)/(~
2/mξ2l ) is constant. Note
that this inequality is written in terms of the energy-
dependent scattering length (see above). We find sim-
ilar results for the liquid branches for l = 0 to 2. For
higher partial waves, we hence use the inequality given
by Eq. (21) to define the unitary regime. In the uni-
tary regime, the energy per particle EB,l/N of the Bose
system depends only on the combined length scale ξl.
For s-wave interacting systems, we have ξs = ro and
as(k) ≈ as, and Eq. (21) reduces to the well known s-
wave unitary condition, i.e., to |as| ≫ ro ≫ R.
We now discuss those regions of Fig. 7, where the
energy per particle (EB,l/N)/(~
2/mξ2l ) for the one-
component Bose system deviates from a constant. For
sufficiently large densities, the characteristic length ξl be-
comes of the order of the range R of the square well
potential. In this “high density” regime the system
exhibits non-universal behaviors. In Fig. 7, e.g., the
energy-dependent scattering length ad(k) equals 10
) /ξl
FIG. 7: Scaled energy per particle (EB,l/N)/(~
2/mξ2l ) for
the energetically lowest-lying gas branch of l-wave interacting
one-component Bose systems obtained by solving the LOCV
equations [Eqs. (11) through (14)] for vsw as a function of
al(Erel)/ξl. The depth V0 of vsw is adjusted so that al(Erel) =
1010R for l = 0 and l = 1, and al(Erel) = 10
6R for l = 2.
Note that the system with l = 1 is of theoretical interest but
does not describe a physical system. In the regime where the
inequality R ≪ ξl ≪ al(Erel) is fulfilled, the scaled energy
per particle is constant; this defines the unitary regime.
correspondingly, ξd equals R when ad(k)/ξd = 10
6. As
ad(k)/ξd approaches 10
6 from below, the system becomes
non-universal, as indicated in Fig. 7 by the non-constant
dependence of the scaled energy per particle on ad(k)/ξd.
On the left side of Fig. 7, where al(Erel)/ξl becomes of
order 1, the “low density” end of the unitary regime is
reached. When ap(Erel)/ξp equals 10, e.g., the inter-
particle spacing ro equals 10
2ap(Erel), i.e., the system
exhibits universal behavior even when the interparticle
spacing is 100 times larger than the scattering length
ap(Erel). This is in contrast to the s-wave case, where
the universal regime requires |as| ≫ ro. The different
behavior of the higher partial wave systems compared to
the s-wave system can be understood by realizing that ξl
is a combined length, which contains both the range R
of the two-body potential and the average interparticle
spacing ro. For a given al(Erel)/R, the first inequality
in Eq. (21) is thus satisfied for larger average interparti-
cle spacings ro/R, or smaller scaled densities nBR
3, as l
increases from 0 to 2.
In addition to investigating l-wave Bose gases interact-
ing through the square well potential vsw , we consider
the van der Waals potential vvdw. For the energetically
lowest-lying gas branch of the one-component “p-wave
Bose” system we find the same results as for the square
well potential if we replace R in Eqs. (18) to (20) by
β6. We believe that the same replacement needs to be
done for the liquid branch with l = 1 and for the liquid
and gas branches of d-wave interacting bosons, and that
the scaling at unitarity derived above for the square well
potential holds for a wide class of two-body potentials.
Within the LOCV framework, the results obtained for
l xl C
0 0.00 −1.55 8.40
1 1.00 −2.29 6.52
2 2.00 −2.62 5.54
TABLE II: Dimensionless parameters xl, C
l and C
l for l = 0
to 2 for a two-component Bose system (see text for details).
xl, C
l and C
l for the one-component Bose systems can
be applied readily to the corresponding two-component
system by scaling the Bose density appropriately (see
Sec. II). The resulting parameters xl, C
l and C
l for
the two-component Bose systems are summarized in Ta-
ble II.
The energy per particle EF,l/N for l-wave interact-
ing two-component Fermi systems can be obtained from
Eq. (17) using the LOCV solutions for the liquid and
gas branches discussed above for l-wave interacting one-
component Bose systems. In the unitary limit, we find
F +Bl
2/3−l/6
, (22)
where A = (3/10)(3π2)2/3 ≈ 2.87 and Bl =
2/3−l/6 (the C
l are given in Table I). The first
term on the right hand side of Eq. (22) equals EFG, and
the second term, which is obtained from the LOCV solu-
tions, equals λl/2. The energy per particleEF,l/N at uni-
tarity is positive for all densities for Bl = C
2/3−l/6.
For Bl = C
2/3−l/6, however, the energy per particle
EF,l/N at unitarity is negative for l > 0 for small den-
sities, and goes through a minimum for larger densities.
This implies that this branch is always mechanically un-
stable in the dilute limit for l > 0.
The LOCV treatment for fermions relies heavily on the
product representation of the many-body wave function,
Eq. (15), which in turn gives rise to the two terms on
the right hand side of Eq. (22). It is the competition of
these two energy terms that leads to the energy minimum
discussed in the previous paragraph. Future work needs
to investigate whether the dependence of EF,l/N on two
length scales as implied by Eq. (22) is correct. In contrast
to the LOCV method, mean-field treatments predict that
the energy at unitarity is proportional to EFG|kF re,l|,
where re,l denotes a range parameter that characterizes
the underlying two-body potential [42, 43, 44].
V. CONCLUSION
This paper investigates Bose and Fermi systems us-
ing the LOCV method, which assumes that three- and
higher-order correlations can be neglected and that the
behaviors of the many-body system are governed by two-
body correlations. This assumption allows the many-
body problem to be reduced to an effective two-body
problem. Besides the reduced numerical effort, this for-
malism allows certain aspects of the many-body physics
to be interpreted from a two-body point of view. Further-
more, it allows parallels between Bose and Fermi systems
to be drawn.
In agreement with previous studies, we find that the
energy per particle “corrected” by the dimer binding en-
ergy, i.e., EF,0/N − Edimer/2, of dilute two-component
s-wave Fermi gases in the whole crossover regime de-
pends only on the s-wave scattering length and not on
the details of the underlying two-body potential. Fur-
thermore, at unitarity the energy per particle is given by
EF,0/N = 0.46EFG. This LOCV result is in good agree-
ment with the energy per particle obtained from fixed-
node diffusion Monte Carlo calculations, which predict
EF,0/N = 0.42EFG [19, 20, 21]. This agreement may
be partially due to the cancellation of higher-order cor-
relations, and thus somewhat fortuitous. In contrast to
Ref. [28], we find that the liquid branch of bosonic he-
lium does not terminate at low densities but exists down
to zero density.
For higher angular momentum interactions, we deter-
mine the energy per particle of one- and two-component
Bose systems with infinitely large scattering lengths. For
these systems, we expect the LOCV formalism to pre-
dict the dimensionless exponent xl, which determines
the functional dependenc of EB,l/N on the range R of
the two-body potential and on the average interparticle
spacing ro, correctly. The values of the proportionality
constants CGl and C
l , in contrast, may be less accurate.
We use the LOCV energies to generalize the known uni-
tary condition for s-wave interacting systems to systems
with finite angular momentum. Since higher angular mo-
mentum resonances are necessarily narrow, leading to a
strong energy-dependence of the scattering strength, we
define the universal regime using the energy-dependent
scattering length al(k). In the unitary regime, the en-
ergy per particle can be written in terms of the length
ξl, which is given by a geometric combination of ro and
R. The LOCV framework also allows a prediction for
the energy per particle of two-component Fermi gases
beyond s-wave to be made [see Eq. (22)]. Although the
functional form of the many-body wave function for two-
component Fermi systems used in this work may not be
the best choice, we speculate that the energy scales de-
rived for strongly interacting Bose systems are also rele-
vant to Fermi systems.
This work was supported by the NSF through grant
PHY-0555316. RMK gratefully acknowledges hospitality
of the BEC Center, Trento.
[1] W. C. Stwalley, Phys. Rev. Lett. 37, 1628 (1976).
[2] E. Tiesinga, B. J. Verhaar, and H. T. C. Stoof, Phys.
Rev. A 47, 4114 (1993).
[3] S. Inouye, M. R. Andrews, J. Stenger, H. J. Miesner,
D. M. Stamper-Kurn, and W. Ketterle, Nature 392, 151
(1998).
[4] P. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van
Abeelen, and B. J. Verhaar, Phys. Rev. Lett. 81, 69
(1998).
[5] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys.
Rev. Lett. 90, 053201 (2003).
[6] J. Zhang, E. G. M. van Kempen, T. Bourdel, L.
Khaykovich, J. Cubizolles, F. Chevy, M. Teichmann, L.
Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon,
Phys. Rev. A 70, 030702(R) (2004).
[7] C. Chin, V. Vuletić, A. J. Kerman, and S. Chu, Phys.
Rev. Lett. 85, 2717 (2000).
[8] T. Köhler, K. Góral, and P. S. Julienne, Rev. Mod. Phys.
78, 1311 (2006).
[9] L. D. Landau and E. M. Lifshitz, Quantum Mechanics
(Non-relativistic theory), Vol. 3, 3rd Edition (Butter-
worth Heinemann, Oxford, 1977).
[10] G. A. Baker, Jr., Phys. Rev. C 60, 054311 (1999).
[11] H. Heiselberg, Phys. Rev. A 63, 043606 (2001).
[12] S. Cowell, H. Heiselberg, I. E. Mazets, J. Morales, V. R.
Pandharipande, and C. J. Pethick, Phys. Rev. Lett. 88,
210403 (2002).
[13] P. O. Fedichev, M. W. Reynolds, and G. V. Shlyapnikov,
Phys. Rev. Lett. 77, 2921 (1996).
[14] B. D. Esry, C. H. Greene, and J. P. Burke Jr., Phys. Rev.
Lett. 83, 1751 (1999).
[15] E. Nielsen and J. H. Macek, Phys. Rev. Lett. 83, 1566
(1999).
[16] P. F. Bedaque, E. Braaten, and H.-W. Hammer, Phys.
Rev. Lett. 85, 908 (2000).
[17] K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R.
Granade, and J. E. Thomas, Science 298, 2179 (2002).
[18] T. Bourdel, J. Cubizolles, L. Khaykovich, K. M. F. Ma-
galhaes, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov,
and C. Salomon, Phys. Rev. Lett. 91, 020402 (2003).
[19] J. Carlson, S. Y. Chang, V. R. Pandharipande, and K. E.
Schmidt, Phys. Rev. Lett. 91, 050401 (2003).
[20] S. Y. Chang, V. R. Pandharipande, J. Carlson, and K. E.
Schmidt, Phys. Rev. A 70, 043602 (2004).
[21] G. E. Astrakharchik, J. Boronat, J. D. Casulleras, and
S. Giorgini, Phys. Rev. Lett. 93, 200404 (2004).
[22] J. Carlson and S. Reddy, Phys. Rev. Lett. 95, 060401
(2005).
[23] For finite l, the term “unitary regime” refers to the regime
in which the observables of the many-body system de-
pend, as discussed in Sec. IV, in a simple way on two
length scales. Since the observables in this regime depend
on two length scales, the system does not, strictly speak-
ing, exhibit universality. However, because of the rather
simple dependence on these two length scales, we never-
theless use the term “universal” to refer to this regime.
[24] D. Blume and C. H. Greene, Phys. Rev. A 65, 043613
(2002).
[25] E. L. Bolda, E. Tiesinga, and P. S. Julienne, Phys. Rev.
A 66, 013403 (2002).
[26] R. Stock, A. Silberfarb, E. L. Bolda, and I. H. Deutsch,
Phys. Rev. Lett. 94, 023202 (2005).
[27] B. Gao, J. Phys. B 37, L227 (2004).
[28] B. Gao, Phys. Rev. Lett. 95, 240403 (2005).
[29] V. R. Pandharipande, Nucl. Phys. A178, 123 (1971).
[30] V. R. Pandharipande, Nucl. Phys. A174, 641 (1971).
[31] V. R. Pandharipande and H. A. Bethe, Phys. Rev. C 7,
1312 (1973).
[32] V. R. Pandharipande and K. E. Schmidt, Phys. Rev. A
15, 2486 (1977).
[33] B. Gao, Phys. Rev. A 58, 1728 (1998).
[34] N. F. Mott and H. S. W. Massey, The Theory of Atomic
Collisions, Third Edition (Clarendon, Oxford, 1965).
[35] J. R. Taylor, Scattering Theory (Wiley, New York, 1972).
[36] The total wave function of the Bose system has to be
symmetric with respect to the exchange of any two par-
ticles, which restricts the angular momentum l to even
values.
[37] A. R. Janzen and R. A. Aziz, J. Chem. Phys. 103, 9626
(1995).
[38] R. De Bruyn Ouboter and C. N. Yang, Physica B 144,
127 (1987).
[39] K. Riisager, A. S. Jensen, and P. Møller, Nuc. Phys. A
548, 393 (1992).
[40] A. S. Jensen, K. Riisager, D. V. Fedorov, and E. Garrido,
Rev. Mod. Phys. 76, 215 (2004).
[41] C. H. Greene, unpublished notes.
[42] T.-L. Ho and R. B. Diener, Phys. Rev. Lett. 94, 090402
(2005).
[43] C.-H. Cheng and S.-Y. Yip, Phys. Rev. Lett. 95, 070404
(2005).
[44] M. Iskin and C. A. R. Sá de Melo, Phys. Rev. Lett. 96,
040402 (2006).
[45] For the liquid l = 1 branch we do not find solutions that
obey the boundary condition given by Eq. (11). Instead,
we determine the healing distance d by looking for the
minimum of f ′1. We find that the value of f
1 at the min-
imum is much smaller than the value of f ′1 at small r.
[46] Within the LOCV formalism, the value of −2.46 for CL0
can, as discussed in Secs. II and III, be straightforwardly
converted to obtain the energy per particle EF,0/N of the
ground state of s-wave interacting two-component Fermi
gases at unitarity, EF,0/N = 0.46EFG [20]. Similarly, the
value of 13.3 for CG0 can be straightforwardly converted
to obtain the energy per particle EF,0/N of the first
excited gas state of s-wave interacting two-component
Fermi gases at unitarity, EF,0/N = 3.92EFG.
