arXiv:0704.0769v1  [cond-mat.other]  5 Apr 2007
The Fermionic Density-functional at Feshbach Resonance
Michael Seidl
Institute of Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany
Rajat K. Bhaduri
Department of Physics and Astronomy, McMaster University, Hamilton, Canada L8S 4M1
(Dated: November 17, 2018)
We consider a dilute gas of neutral unpolarized fermionic atoms at zero temperature. The atoms
interact via a short-range (tunable) attractive interaction. We demonstrate analytically a curious
property of the gas at unitarity. Namely, the correlation energy of the gas, evaluated by second
order perturbation theory, has the same density dependence as the first order exchange energy,
and the two almost exactly cancel each other at Feshbach resonance irrespective of the shape of
the potential, provided (µrs) ≫ 1. Here (µ)−1 is the range of the two-body potential, and rs is
defined through the number density, n = 3/(4πr3s). The implications of this result for universality
is discussed.
I. INTRODUCTION
Consider a dilute gas ofN ≫ 1 neutral fermionic atoms
(massM) at T = 0 interacting with a short-range attrac-
tive potential. In general, the properties of the dilute gas
are determined by the number density n, and the scatter-
ing length a. The Hamiltonian of this N -particle system
reads
Ĥ = − ~
∇2i +
|ri − rj |
. (1)
Not written explicitly here, there is also an external po-
tential vext(r) that forces the N atoms to stay within
a large box with volume Ω [where vext(r) ≡ 0]. The
attractive interaction potential is assumed to have the
2-parameter form
v(r) = −v0f(µr) (2)
where v0 > 0 is the strength of the interaction, R0 =
is its range, and f(x) is a dimensionless function.
In the true ground state of the Hamiltonian (1) the
attractive atoms may form dimers or even clusters. We
are, however, looking for a metastable state where there
is a dilute gas of separated atoms with uniform density n,
satisfying the condition (µrs) ≫ 1 where n = NΩ =
Even then, for a weak v0, there will be BCS-type pair-
ing, followed by dimer formation as the strength of the
interaction increases. This was predicted long back by
Leggett [1], and has been observed experimentally [2].
For the density functional analysis of the uniform gas
at Feshbach resonance, we shall disregard the BCS con-
densed pairs in this paper.
To study the effect of the attractive interaction v(r), we
consider the corresponding atom-atom scattering prob-
lem in the relative s-state. Separating the center-of-mass
motion, we are left with the relative Hamiltonian
Ĥrel = −
− v0f(µr) . (3)
Keeping the range of the potential small enough such
that (µrs) ≫ 1, the strength v0 is adjusted such that the
potential can support a single bound state at zero energy.
This happens when the scattering length a→ ∞, leaving
no length scale from the interaction. Such a tuning of
the interaction is possible experimentally, and gives rise
to Feshbach resonance [3]. The scattering cross section
in the given partial wave (s-wave in our case) reaches the
unitary limit, and the gas is said to be at unitarity. It is
then expected to display universal behavior [4]. Note that
at Feshbach resonance, there is no length scale left other
than the inverse of the Fermi wave number kF , where
kF = (3π
2n)1/3. The energy per particle, E/N , as a
function of the density n, should therefore scale the same
way as the noninteracting kinetic energy, 3
2k2F /2M ∝
n2/3. There has been much interest amongst theorists to
calculate the properties of the gas in the unitary regime
(kF |a| ≫ 1). In particular, at T = 0, the energy per
particle of the gas is calculated to be
, (4)
where ξ ≃ 0.44 [5]. The experimental value of ξ is about
0.5, but with large error bars [6]. Recently, there have
been two Monte Carlo (MC) finite temperature calcula-
tions [7, 8] of an untrapped gas at unitarity, where various
thermodynamic properties as a function of temperature
have been computed. It is clear that at unitarity, the
kinetic and potential energies should scale the same way.
This has been assumed a priori in a previous density
functional treatment of a unitary gas [9]. However, such
a scaling behavior is not evident from the density func-
tionals for the direct, exchange and correlation energies
[10] (see sects. II and III). The aim of the present paper
is to examine this point in some detail. In particular, we
are able to show analytically that the leading contribu-
tion of the correlation energy (calculated in second order
perturbation theory), cancels the first order exchange en-
rgy almost exactly at Feshbach resonance. This happens
irrespective of the shape of the potential as specified by
f(µr), provided the condition (µrs) ≫ 1. We show that
http://arxiv.org/abs/0704.0769v1
our general Eq.(24) (derived later in the text) that en-
sures such a cancellation is satisfied at unitarity for a
variety of 2-parameter potentials, including the square
well and the delta-shell, as well as the smoothly varying
cosh−2(µr) and Gaussian potentials. This is the main re-
sult of the present work. The implications of this result
for universality is marginal. This is because these po-
tential energy terms, in the limit of (µrs) ≫ 1, are very
small compared to the kinetic energy [4]. For a moder-
ately large value like (µrs) ≃ 3 howevr, these terms are
comparable in magnitude to the kinetic energy (sect. IV).
Even then, the cancellation of the first order exchange,
and the second order perturbative terms leave the di-
rect first order term in tact. In the electron gas, this
(repulsive) term got cancelled by the interaction of the
electrons with the positive ionic background. There is no
such mechanism of cancellation here, unless we assume,
rather arbitrarily, that the short-range interatomic repul-
sion cancels this direct (attractive) contribution. Even
without any such assumptions, however, our main result
(Table I), applicable at Feshbach reonance, is interesting
from the angle of potential theory.
II. PERTURBATION EXPANSION
Treating the interaction (2) as a weak perturbation in
the Hamiltonian (1), the unperturbed energy E(0) is the
kinetic energy of a non-interacting Fermi gas,
E(0) = Nts(rs) ≡ N
. (5)
Here, kF =
and α3 = 4
. The corresponding ground
state |Φ0〉 is a Slater determinant of plane waves.
In terms of dimensionless coordinates xi = µri, the
Hamiltonian (1) can be written as
Ĥ = −1
∇2i − λ
|xi − xj |
, λ =
This suggests that the perturbation parameter is not re-
ally small at unitarity. For example, for the square-well
potential, the zero-energy single bound state occurs when
λ = π
(see sect. III). Nevertheless the low-order terms
can point to important information, even when the ex-
pansion is divergent [11]. In our problem, there are three
parameters, µ, v0, and rs. The unitarity condition re-
lates µ and v0, so two independent parameters are left.
One of these may be taken to be the small parameter
ζ = (µrs)
−1. The remaining free parameter v0 may be
chosen independently of ζ to fulfill the unitarity condi-
tion.
A. First order
Formally, the first-order correction,
E(1) = 〈Φ0|V̂int|Φ0〉, (7)
has a direct contribution U(rs, µ) = Nu(rs, µ) with
u(rs, µ) =
d3r′v
|r− r′|
(µrs)3
f2. (8)
Here, f2 =
dxx2f(x).
The other first-order contribution is the exchange en-
ergy Ex(rs, µ) = Nex(rs, µ), [4]
ex(rs, µ) = −
drj1(kF r)
2v(r). (9)
Here, j1(z) is a spherical Bessel function. Since v(r) is
short-range and kF is small in a dilute gas, we can use
the small-z expansion j1(z) =
+O(z3) to find
ex(rs, µ) =
(µrs)3
f2 +O(µrs)
−5. (10)
B. Second order
1. General expressions
Also the second-order correction,
E(2) = −
|〈Φn|V̂int|Φ0〉|2
En − E0
dir + e
, (11)
has a direct and an exchange contribution [12],
dir(rs, µ) = −
d3q f̃
d3k1 d
q · (q+ k1 − k2)
, (12)
e(2)ex (rs, µ) = +
d3q f̃
d3k1 d
|q+ k1 − k2|
q · (q+ k1 − k2)
. (13)
While v20(2M/~
2µ2) has the dimension energy, the inte-
gration variables q, k1, and k2 are dimensionless here.
The domain of the integral over d3k1 d
3k2 depends on q,
D : |k1|, |k2| < 1; |k1 + q|, |k2 − q| > 1. (14)
Furthermore, f̃(y) is a dimensionless transform of f(x),
f̃(y) =
dxx2 f(x) j0(yx)
dxx f(x) sin(yx). (15)
To recover Eqs. (8) and (9) of Ref. [12], put M = me,
v0 = −e2µ, and f(x) = 1x or f̃(y) =
, such that v(r) =
becomes the electronic Coulomb repulsion. (Note that
Ref. [12] uses Rydberg units, mee
4/2~2 = e2/2aB = 1.)
2. The limit µrs ≫ 1
For a dilute gas (small kF ) with short-range interaction
(large µ), Eqs. (12) and (13) can be evaluated in the limit
µ/kF ≡ αµrs ≫ 1 where α3 = 49π . Following Ref. [13],
we choose a number q1 such that 1 ≪ q1 ≪ µ/kF and
split the integrals over d3q into two parts,
d3q =
d3q +
d3q. (16)
In the first part with q < q1, we have q ≪ µ/kF and
|q + k1 − k2| ≪ µ/kF (note that |k1|, |k2| < 1 ≪ q1).
Therefore, we may expand f̃(y) = f2+O(y
2) in Eqs. (12)
and (13) and keep the leading term f2 only. The sum of
the two resulting q < q1 contributions reads
q<q1(rs, µ) = −
f22 ×
d3k1 d
q · (q+ k1 − k2)
. (17)
The number q1 can be chosen independently of µ/kF ≫
1, despite the condition 1 ≪ q1 ≪ µ/kF . Then, the
integral in Eq. (17) is a finite constant and we conclude
q<q1(rs, µ) = O(µrs)
−4. (18)
In the second part q > q1 ≫ 1 of the integral (16), we can
put q + k1 − k2 ≈ q, since |k1|, |k2| < 1. The resulting
contributions to Eqs. (12) and (13) add up to
q>q1(rs, µ) = −
where
d3k1 d
3k2 = (
)2 has been used. Now,
dy f̃(y)2 (20)
where y1 = kF q1/µ ≪ 1. If
dy f̃(y)2 in Eq. (20) did
not depend on y1, expression (19) did rigorously have the
order O(µrs)
−3. However, using the small-y expansion
f̃(y) = f2 +O(y
2), we have
dyf̃(y)2 = f22 y1 +O(y
Consequently, shifting the lower limit y1 of the integral
(20) to zero does not affect the leading-order contribution
to expression (19),
q>q1(rs, µ) = O(µrs)
−3. (21)
Therefore, the quantity (18) does not contribute to the
leading order of e
c = e
dir + e
ex which is purely due to
expression (19),
e(2)c (rs, µ) = −
(µrs)3
F +O(µrs)
−4 (22)
where F =
dyf̃(y)2.
III. DENSITY SCALING AT UNITARITY
If the perturbation expansion is convergent [12], the
total energy E(rs, µ) = Ne(rs, µ) of the gas can be ex-
pressed in the form
e(rs, µ) = ts(rs) + ex(rs, µ) +
e(n)c (rs, µ). (23)
At unitarity, when the relative Hamiltonian (3) has a
single bound state at zero energy, the exchange plus cor-
relation energy ex +
n=2 e
c should display the same
density scaling as the kinetic energy, ts(rs) ∝ r−2s ∝ ρ2/3.
This is obviously not the case with any one of the present
(leading-order) results (10) and (22). However, since the
exchange energy (10) and the second-order correlation
energy (22) have opposite signs, they can cancel each
other at some value of µ. This happens when
, (24)
where f2 =
dxx2 f(x) and F =
dyf̃(y)2. This
is the main result of our paper, and we check it by tak-
ing four different potentials. The results of this analysis,
summarized in Table I, are discussed in detail below.
Generally, we need an eigenfunction ψ(r) =
of the relative Hamiltonian (3) with eigenvalue zero.
Writing u(r) = φ(µr), the corresponding dimensionless
Schrödinger Equation reads
φ′′(x) = −λf(x)φ(x), λ ≡ Mv0
. (25)
Precisely, we wish to determine that particular value λuty
of λ for which this zero-energy solution is the only bound
state. Then, φ(x) must obey φ(0) = 0, φ′(x) < 0 for
x ≥ 0, and φ(x) → const. for x → ∞. In the following
examples (A-D), the solution φ(x) can be found analyti-
cally or numerically.
(A) Square-well potential of radius R0 = 1/µ:
v(r) = −v0Θ(R0 − r) , (26)
where Θ(z) denotes the Heavyside step function, Θ(z) =
1 for z > 0 and Θ(z) = 0 for z ≤ 0. By setting the
dimensionless variable µr = x, we see that f(x) = Θ(1−
x). The square-well potential (26) supports a single zero
energy bound state when the LHS of Eq.(24) is λuty =
π2/4. It may be easily checked analytically that for the
square-well potential (26), f2 =
and F = π
so that the
RHS of Eq.(24) is 5
, very close to its LHS, π2/4 = 2.47.
(B) Rosen-Morse hyperbolic potential [5]. This poten-
tial is given by
v(r) = −v0 sech2(µr) , (27)
which suppotrs a single zero energy bound state when
the LHS of Eq.(24) is λuty = 2 instead of π
2/4. For this
potential, it is easy to check that f2 = π
2/12. The quan-
tity F , however, has to be calculated numerically, and
is given by F = 0.596. Again, Eq.(24) is approximately
satisfied, since its RHS for this potential is 2.17.
(C) Delta-shell potential [14]. Consider the potential
v(r) = −η ~
δ(r −R0) ,
= −η ~
= −v0f(µr) . (28)
Thus, we have v0 = η
, µ = 1
, and f(x) = δ(x− 1).
So we get f2 = 1, f̃(y) =
sin y
, and F = π
. Hence the
RHS of Eq. (24) is unity. The LHS is (ηR0), which is
exactly unity when the s-state scattering length goes to
infinity [14]. Thus Eq.(24) is exactly obeyed in this case.
(D) Gaussian Potential.
v(r) = −v0 exp(−µ2r2) (29)
For this example, f(x) = exp(−x2) in Eq. (2). We find
π and F = 1
)3/2 so that the RHS of Eq. (24)
becomes πf2/2F = 2
3/2. Solving Eq. (25) numerically
for this f(x), we obtain a single bound state at zero
energy when the LHS of Eq. (24) is λuty = 0.949× 23/2,
close to 23/2.
Note, however, that contributions O(µrs)
−3 may also
come from higher order terms of the perturbation expan-
sion in section II, since that expansion is carried out with
respect to the parameter λ =Mv0/~
2µ2, but not 1/µrs.
IV. DISCUSSION
The dimensionless Hamiltonian ĥ from Eq. (6) depends
on the dimensionless paramaters
TABLE I: The moments f2 and F of four different profiles
f(x) for the potential (2). λuty is the value at unitarity of the
parameter λ in Eq. (25). At unitarity, the ratio Q of the LHS
of Eq. (24) to the RHS is always close to 1.
f(x) f2 F λuty Q
Θ(1− x) 1
0.987
sech(x)2 π
0.596 2 0.922
δ(1− x) 1 π
1 1.000
exp(−x2) 1
)3/2 2.684 0.949
and, not written explicitly, xs = µrs. The perturbation
expansion of the ground-state energy of ĥ reads
ε(xs, λ) =
εn(xs)λ
n. (31)
The ground-state energy of the original Hamiltonian Ĥ ,
with three independent parameters, is then given by
E(rs, µ, λ) =
ε(µrs, λ)
εn(µrs)λ
n. (32)
For µrs ≫ 1, we may expand
εn(µrs) =
(µrs)m
. (33)
From Eq. (5), we have ε02 = N
α−2 while ε0m = 0 for
m 6= 2. Eqs. (8) and (10) imply that ε1m = 0 for m < 3
and ε13 = N(− 32 +
)f2. Eventually, due to Eq. (22),
ε2m = 0 for m < 3 and ε23 = N(− 34π )2F .
So far as the unitary point is concerned, we are inter-
ested in a situation where kF |a| ≫ 1 ≫ kFR0 ∼ (µrs)−1.
In view of the fact that the perturbation series above does
not converge at unitarity, how significant is our low order
perturbation calculation in this situation ? Note that our
first order direct and exchange (potential) energy terms
given by Eqs. (8,10) are the same as those obtained in
the Hartree-Fock calculation (see, for example, Eq.(10)
of Heiselberg [4]). How big are these terms at unitarity
compared to the kinetic energy per particle ? Taking the
example of the square-well potential discussed earlier, it
is straight forward to show that our exchange term (10)
at Feshbach resonance is
ex(rs, µ) =
. (34)
For the square-well example,
(kF a) =
(µrs)
1− tan
. (35)
At unitarity, the RHS diverges for any finite value of
(µrs), how ever large. Even in the neighbourhood of
unitarity, it is possible to have (kF |a|) ≫ 1 for (µrs) ≫ 1.
From Eq.(34), we note that too large a choice for (µrs)
would make ex negligible against
EF . Instead, taking
a modestly large value, µrs = 3, we obtain the ration
of ex to kinetic energy per particle to be about 0.56.
Noting that ex has a different density-dependence than
the kinetic energy per particle, its cancellation with the
second order perturbative correlation term helps towards
scale invariance, but only if there is a mechanism for the
direct first order term to be cancelled.
We conclude by emphasizing that the new result in this
paper is displayed in Table 1, and should be of interest
from the point of view of potential theory.
The authors would like to thank Brandon van Zyl for
discussions. This research was financed by NSERC of
Canada.
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  We consider a dilute gas of neutral unpolarized fermionic atoms at zero
temperature.The atoms interact via a short range (tunable) attractive
interaction. We demonstrate analytically a curious property of the gas at
unitarity. Namely, the correlation energy of the gas, evaluated by second order
perturbation theory, has the same density dependence as the first order
exchange energy, and the two almost exactly cancel each other at Feshbach
resonance irrespective of the shape of the potential, provided $(\mu r_s) >>
1$. Here $(\mu)^{-1}$ is the range of the two-body potential, and $r_s$ is
defined through the number density $n=3/(4\pi r_s^3)$. The implications of this
result for universality is discussed.

arXiv:0704.0769v1  [cond-mat.other]  5 Apr 2007
The Fermionic Density-functional at Feshbach Resonance
Michael Seidl
Institute of Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany
Rajat K. Bhaduri
Department of Physics and Astronomy, McMaster University, Hamilton, Canada L8S 4M1
(Dated: November 17, 2018)
We consider a dilute gas of neutral unpolarized fermionic atoms at zero temperature. The atoms
interact via a short-range (tunable) attractive interaction. We demonstrate analytically a curious
property of the gas at unitarity. Namely, the correlation energy of the gas, evaluated by second
order perturbation theory, has the same density dependence as the first order exchange energy,
and the two almost exactly cancel each other at Feshbach resonance irrespective of the shape of
the potential, provided (µrs) ≫ 1. Here (µ)−1 is the range of the two-body potential, and rs is
defined through the number density, n = 3/(4πr3s). The implications of this result for universality
is discussed.
I. INTRODUCTION
Consider a dilute gas ofN ≫ 1 neutral fermionic atoms
(massM) at T = 0 interacting with a short-range attrac-
tive potential. In general, the properties of the dilute gas
are determined by the number density n, and the scatter-
ing length a. The Hamiltonian of this N -particle system
reads
Ĥ = − ~
∇2i +
|ri − rj |
. (1)
Not written explicitly here, there is also an external po-
tential vext(r) that forces the N atoms to stay within
a large box with volume Ω [where vext(r) ≡ 0]. The
attractive interaction potential is assumed to have the
2-parameter form
v(r) = −v0f(µr) (2)
where v0 > 0 is the strength of the interaction, R0 =
is its range, and f(x) is a dimensionless function.
In the true ground state of the Hamiltonian (1) the
attractive atoms may form dimers or even clusters. We
are, however, looking for a metastable state where there
is a dilute gas of separated atoms with uniform density n,
satisfying the condition (µrs) ≫ 1 where n = NΩ =
Even then, for a weak v0, there will be BCS-type pair-
ing, followed by dimer formation as the strength of the
interaction increases. This was predicted long back by
Leggett [1], and has been observed experimentally [2].
For the density functional analysis of the uniform gas
at Feshbach resonance, we shall disregard the BCS con-
densed pairs in this paper.
To study the effect of the attractive interaction v(r), we
consider the corresponding atom-atom scattering prob-
lem in the relative s-state. Separating the center-of-mass
motion, we are left with the relative Hamiltonian
Ĥrel = −
− v0f(µr) . (3)
Keeping the range of the potential small enough such
that (µrs) ≫ 1, the strength v0 is adjusted such that the
potential can support a single bound state at zero energy.
This happens when the scattering length a→ ∞, leaving
no length scale from the interaction. Such a tuning of
the interaction is possible experimentally, and gives rise
to Feshbach resonance [3]. The scattering cross section
in the given partial wave (s-wave in our case) reaches the
unitary limit, and the gas is said to be at unitarity. It is
then expected to display universal behavior [4]. Note that
at Feshbach resonance, there is no length scale left other
than the inverse of the Fermi wave number kF , where
kF = (3π
2n)1/3. The energy per particle, E/N , as a
function of the density n, should therefore scale the same
way as the noninteracting kinetic energy, 3
2k2F /2M ∝
n2/3. There has been much interest amongst theorists to
calculate the properties of the gas in the unitary regime
(kF |a| ≫ 1). In particular, at T = 0, the energy per
particle of the gas is calculated to be
, (4)
where ξ ≃ 0.44 [5]. The experimental value of ξ is about
0.5, but with large error bars [6]. Recently, there have
been two Monte Carlo (MC) finite temperature calcula-
tions [7, 8] of an untrapped gas at unitarity, where various
thermodynamic properties as a function of temperature
have been computed. It is clear that at unitarity, the
kinetic and potential energies should scale the same way.
This has been assumed a priori in a previous density
functional treatment of a unitary gas [9]. However, such
a scaling behavior is not evident from the density func-
tionals for the direct, exchange and correlation energies
[10] (see sects. II and III). The aim of the present paper
is to examine this point in some detail. In particular, we
are able to show analytically that the leading contribu-
tion of the correlation energy (calculated in second order
perturbation theory), cancels the first order exchange en-
rgy almost exactly at Feshbach resonance. This happens
irrespective of the shape of the potential as specified by
f(µr), provided the condition (µrs) ≫ 1. We show that
http://arxiv.org/abs/0704.0769v1
our general Eq.(24) (derived later in the text) that en-
sures such a cancellation is satisfied at unitarity for a
variety of 2-parameter potentials, including the square
well and the delta-shell, as well as the smoothly varying
cosh−2(µr) and Gaussian potentials. This is the main re-
sult of the present work. The implications of this result
for universality is marginal. This is because these po-
tential energy terms, in the limit of (µrs) ≫ 1, are very
small compared to the kinetic energy [4]. For a moder-
ately large value like (µrs) ≃ 3 howevr, these terms are
comparable in magnitude to the kinetic energy (sect. IV).
Even then, the cancellation of the first order exchange,
and the second order perturbative terms leave the di-
rect first order term in tact. In the electron gas, this
(repulsive) term got cancelled by the interaction of the
electrons with the positive ionic background. There is no
such mechanism of cancellation here, unless we assume,
rather arbitrarily, that the short-range interatomic repul-
sion cancels this direct (attractive) contribution. Even
without any such assumptions, however, our main result
(Table I), applicable at Feshbach reonance, is interesting
from the angle of potential theory.
II. PERTURBATION EXPANSION
Treating the interaction (2) as a weak perturbation in
the Hamiltonian (1), the unperturbed energy E(0) is the
kinetic energy of a non-interacting Fermi gas,
E(0) = Nts(rs) ≡ N
. (5)
Here, kF =
and α3 = 4
. The corresponding ground
state |Φ0〉 is a Slater determinant of plane waves.
In terms of dimensionless coordinates xi = µri, the
Hamiltonian (1) can be written as
Ĥ = −1
∇2i − λ
|xi − xj |
, λ =
This suggests that the perturbation parameter is not re-
ally small at unitarity. For example, for the square-well
potential, the zero-energy single bound state occurs when
λ = π
(see sect. III). Nevertheless the low-order terms
can point to important information, even when the ex-
pansion is divergent [11]. In our problem, there are three
parameters, µ, v0, and rs. The unitarity condition re-
lates µ and v0, so two independent parameters are left.
One of these may be taken to be the small parameter
ζ = (µrs)
−1. The remaining free parameter v0 may be
chosen independently of ζ to fulfill the unitarity condi-
tion.
A. First order
Formally, the first-order correction,
E(1) = 〈Φ0|V̂int|Φ0〉, (7)
has a direct contribution U(rs, µ) = Nu(rs, µ) with
u(rs, µ) =
d3r′v
|r− r′|
(µrs)3
f2. (8)
Here, f2 =
dxx2f(x).
The other first-order contribution is the exchange en-
ergy Ex(rs, µ) = Nex(rs, µ), [4]
ex(rs, µ) = −
drj1(kF r)
2v(r). (9)
Here, j1(z) is a spherical Bessel function. Since v(r) is
short-range and kF is small in a dilute gas, we can use
the small-z expansion j1(z) =
+O(z3) to find
ex(rs, µ) =
(µrs)3
f2 +O(µrs)
−5. (10)
B. Second order
1. General expressions
Also the second-order correction,
E(2) = −
|〈Φn|V̂int|Φ0〉|2
En − E0
dir + e
, (11)
has a direct and an exchange contribution [12],
dir(rs, µ) = −
d3q f̃
d3k1 d
q · (q+ k1 − k2)
, (12)
e(2)ex (rs, µ) = +
d3q f̃
d3k1 d
|q+ k1 − k2|
q · (q+ k1 − k2)
. (13)
While v20(2M/~
2µ2) has the dimension energy, the inte-
gration variables q, k1, and k2 are dimensionless here.
The domain of the integral over d3k1 d
3k2 depends on q,
D : |k1|, |k2| < 1; |k1 + q|, |k2 − q| > 1. (14)
Furthermore, f̃(y) is a dimensionless transform of f(x),
f̃(y) =
dxx2 f(x) j0(yx)
dxx f(x) sin(yx). (15)
To recover Eqs. (8) and (9) of Ref. [12], put M = me,
v0 = −e2µ, and f(x) = 1x or f̃(y) =
, such that v(r) =
becomes the electronic Coulomb repulsion. (Note that
Ref. [12] uses Rydberg units, mee
4/2~2 = e2/2aB = 1.)
2. The limit µrs ≫ 1
For a dilute gas (small kF ) with short-range interaction
(large µ), Eqs. (12) and (13) can be evaluated in the limit
µ/kF ≡ αµrs ≫ 1 where α3 = 49π . Following Ref. [13],
we choose a number q1 such that 1 ≪ q1 ≪ µ/kF and
split the integrals over d3q into two parts,
d3q =
d3q +
d3q. (16)
In the first part with q < q1, we have q ≪ µ/kF and
|q + k1 − k2| ≪ µ/kF (note that |k1|, |k2| < 1 ≪ q1).
Therefore, we may expand f̃(y) = f2+O(y
2) in Eqs. (12)
and (13) and keep the leading term f2 only. The sum of
the two resulting q < q1 contributions reads
q<q1(rs, µ) = −
f22 ×
d3k1 d
q · (q+ k1 − k2)
. (17)
The number q1 can be chosen independently of µ/kF ≫
1, despite the condition 1 ≪ q1 ≪ µ/kF . Then, the
integral in Eq. (17) is a finite constant and we conclude
q<q1(rs, µ) = O(µrs)
−4. (18)
In the second part q > q1 ≫ 1 of the integral (16), we can
put q + k1 − k2 ≈ q, since |k1|, |k2| < 1. The resulting
contributions to Eqs. (12) and (13) add up to
q>q1(rs, µ) = −
where
d3k1 d
3k2 = (
)2 has been used. Now,
dy f̃(y)2 (20)
where y1 = kF q1/µ ≪ 1. If
dy f̃(y)2 in Eq. (20) did
not depend on y1, expression (19) did rigorously have the
order O(µrs)
−3. However, using the small-y expansion
f̃(y) = f2 +O(y
2), we have
dyf̃(y)2 = f22 y1 +O(y
Consequently, shifting the lower limit y1 of the integral
(20) to zero does not affect the leading-order contribution
to expression (19),
q>q1(rs, µ) = O(µrs)
−3. (21)
Therefore, the quantity (18) does not contribute to the
leading order of e
c = e
dir + e
ex which is purely due to
expression (19),
e(2)c (rs, µ) = −
(µrs)3
F +O(µrs)
−4 (22)
where F =
dyf̃(y)2.
III. DENSITY SCALING AT UNITARITY
If the perturbation expansion is convergent [12], the
total energy E(rs, µ) = Ne(rs, µ) of the gas can be ex-
pressed in the form
e(rs, µ) = ts(rs) + ex(rs, µ) +
e(n)c (rs, µ). (23)
At unitarity, when the relative Hamiltonian (3) has a
single bound state at zero energy, the exchange plus cor-
relation energy ex +
n=2 e
c should display the same
density scaling as the kinetic energy, ts(rs) ∝ r−2s ∝ ρ2/3.
This is obviously not the case with any one of the present
(leading-order) results (10) and (22). However, since the
exchange energy (10) and the second-order correlation
energy (22) have opposite signs, they can cancel each
other at some value of µ. This happens when
, (24)
where f2 =
dxx2 f(x) and F =
dyf̃(y)2. This
is the main result of our paper, and we check it by tak-
ing four different potentials. The results of this analysis,
summarized in Table I, are discussed in detail below.
Generally, we need an eigenfunction ψ(r) =
of the relative Hamiltonian (3) with eigenvalue zero.
Writing u(r) = φ(µr), the corresponding dimensionless
Schrödinger Equation reads
φ′′(x) = −λf(x)φ(x), λ ≡ Mv0
. (25)
Precisely, we wish to determine that particular value λuty
of λ for which this zero-energy solution is the only bound
state. Then, φ(x) must obey φ(0) = 0, φ′(x) < 0 for
x ≥ 0, and φ(x) → const. for x → ∞. In the following
examples (A-D), the solution φ(x) can be found analyti-
cally or numerically.
(A) Square-well potential of radius R0 = 1/µ:
v(r) = −v0Θ(R0 − r) , (26)
where Θ(z) denotes the Heavyside step function, Θ(z) =
1 for z > 0 and Θ(z) = 0 for z ≤ 0. By setting the
dimensionless variable µr = x, we see that f(x) = Θ(1−
x). The square-well potential (26) supports a single zero
energy bound state when the LHS of Eq.(24) is λuty =
π2/4. It may be easily checked analytically that for the
square-well potential (26), f2 =
and F = π
so that the
RHS of Eq.(24) is 5
, very close to its LHS, π2/4 = 2.47.
(B) Rosen-Morse hyperbolic potential [5]. This poten-
tial is given by
v(r) = −v0 sech2(µr) , (27)
which suppotrs a single zero energy bound state when
the LHS of Eq.(24) is λuty = 2 instead of π
2/4. For this
potential, it is easy to check that f2 = π
2/12. The quan-
tity F , however, has to be calculated numerically, and
is given by F = 0.596. Again, Eq.(24) is approximately
satisfied, since its RHS for this potential is 2.17.
(C) Delta-shell potential [14]. Consider the potential
v(r) = −η ~
δ(r −R0) ,
= −η ~
= −v0f(µr) . (28)
Thus, we have v0 = η
, µ = 1
, and f(x) = δ(x− 1).
So we get f2 = 1, f̃(y) =
sin y
, and F = π
. Hence the
RHS of Eq. (24) is unity. The LHS is (ηR0), which is
exactly unity when the s-state scattering length goes to
infinity [14]. Thus Eq.(24) is exactly obeyed in this case.
(D) Gaussian Potential.
v(r) = −v0 exp(−µ2r2) (29)
For this example, f(x) = exp(−x2) in Eq. (2). We find
π and F = 1
)3/2 so that the RHS of Eq. (24)
becomes πf2/2F = 2
3/2. Solving Eq. (25) numerically
for this f(x), we obtain a single bound state at zero
energy when the LHS of Eq. (24) is λuty = 0.949× 23/2,
close to 23/2.
Note, however, that contributions O(µrs)
−3 may also
come from higher order terms of the perturbation expan-
sion in section II, since that expansion is carried out with
respect to the parameter λ =Mv0/~
2µ2, but not 1/µrs.
IV. DISCUSSION
The dimensionless Hamiltonian ĥ from Eq. (6) depends
on the dimensionless paramaters
TABLE I: The moments f2 and F of four different profiles
f(x) for the potential (2). λuty is the value at unitarity of the
parameter λ in Eq. (25). At unitarity, the ratio Q of the LHS
of Eq. (24) to the RHS is always close to 1.
f(x) f2 F λuty Q
Θ(1− x) 1
0.987
sech(x)2 π
0.596 2 0.922
δ(1− x) 1 π
1 1.000
exp(−x2) 1
)3/2 2.684 0.949
and, not written explicitly, xs = µrs. The perturbation
expansion of the ground-state energy of ĥ reads
ε(xs, λ) =
εn(xs)λ
n. (31)
The ground-state energy of the original Hamiltonian Ĥ ,
with three independent parameters, is then given by
E(rs, µ, λ) =
ε(µrs, λ)
εn(µrs)λ
n. (32)
For µrs ≫ 1, we may expand
εn(µrs) =
(µrs)m
. (33)
From Eq. (5), we have ε02 = N
α−2 while ε0m = 0 for
m 6= 2. Eqs. (8) and (10) imply that ε1m = 0 for m < 3
and ε13 = N(− 32 +
)f2. Eventually, due to Eq. (22),
ε2m = 0 for m < 3 and ε23 = N(− 34π )2F .
So far as the unitary point is concerned, we are inter-
ested in a situation where kF |a| ≫ 1 ≫ kFR0 ∼ (µrs)−1.
In view of the fact that the perturbation series above does
not converge at unitarity, how significant is our low order
perturbation calculation in this situation ? Note that our
first order direct and exchange (potential) energy terms
given by Eqs. (8,10) are the same as those obtained in
the Hartree-Fock calculation (see, for example, Eq.(10)
of Heiselberg [4]). How big are these terms at unitarity
compared to the kinetic energy per particle ? Taking the
example of the square-well potential discussed earlier, it
is straight forward to show that our exchange term (10)
at Feshbach resonance is
ex(rs, µ) =
. (34)
For the square-well example,
(kF a) =
(µrs)
1− tan
. (35)
At unitarity, the RHS diverges for any finite value of
(µrs), how ever large. Even in the neighbourhood of
unitarity, it is possible to have (kF |a|) ≫ 1 for (µrs) ≫ 1.
From Eq.(34), we note that too large a choice for (µrs)
would make ex negligible against
EF . Instead, taking
a modestly large value, µrs = 3, we obtain the ration
of ex to kinetic energy per particle to be about 0.56.
Noting that ex has a different density-dependence than
the kinetic energy per particle, its cancellation with the
second order perturbative correlation term helps towards
scale invariance, but only if there is a mechanism for the
direct first order term to be cancelled.
We conclude by emphasizing that the new result in this
paper is displayed in Table 1, and should be of interest
from the point of view of potential theory.
The authors would like to thank Brandon van Zyl for
discussions. This research was financed by NSERC of
Canada.
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