Exact relations for quantum-mechanical few-body and many-body problems with short-range interactions in two and three dimensions

0.8 MB·English

by  Felix Werner

Exact relations for quantum-mechanical few-body and many-body problems
with short-range interactions in two and three dimensions
F´elix Werner
Department of Physics, University of Massachusetts, Amherst, MA 01003, USA
Yvan Castin
Laboratoire Kastler Brossel, E´cole Normale Sup´erieure,
UPMC and CNRS, 24 rue Lhomond, 75231 Paris Cedex 05, France
0
(Dated: January 6, 2010)
1
0 We derive relations between various observables for N particles with zero-range or short-range
2 interactions,incontinuousspaceoronalattice,intwoorthreedimensions,inanarbitraryexternal
potential. Some of our results generalize known relations between large-momentum behavior of
n
the momentum distribution, short-distance behavior of the pair correlation function and of the
a
J one-body density matrix, derivative of the energy with respect to the scattering length or to time,
and the norm of the regular part of the wavefunction; in the case of finite-range interactions, the
6
interactionenergyisalsorelatedtodE/da. Theexpressionrelatingtheenergytoafunctionalofthe

momentumdistributionisalsogeneralized, andisfoundtobreakdownforEfimovstateswithzero-
]
s rangeinteractions,duetoasubleadingoscillatingtailinthemomentumdistribution. Wealsoobtain
a
new expressions for the derivative of the energy of a universal state with respect to the effective
g
range, thederivativeof theenergy of an efimovian statewith respect tothethree-bodyparameter,
-
t and the second order derivative of the energy with respect to the inverse (or the logarithm in the
n
two-dimensional case) of the scattering length. The latter is negative at fixed entropy. We use
a
exact relations to computecorrections to exactly solvable three-bodyproblems and find agreement
u
with available numerics. For the unitary gas, we compare exact relations to existing fixed-node
q
Monte-Carlodata,andwetest,withexistingQuantumMonteCarloresultsondifferentfiniterange
.
t models, our prediction that the leading deviation of the critical temperature from its zero range
a
m valueis linear in the interaction effective range re with a model independentnumerical coefficient.
- PACSnumbers:
d
n
o
c I. INTRODUCTION
[

1 The experimental breakthroughs of 1995 having led to the first realization of a Bose-Einstein condensate in an
v atomic vapor [1–3] have opened the era of experimental studies of ultracold gases with non-negligible or even strong
4 interactions, in dimension lower or equal to three [4–8]. In these systems, the thermal de Broglie wavelengthand the
7 mean distance between atoms are much larger than the range of the interaction potential. This so-called zero-range
7
limit has interesting universal properties: Several quantities such as the thermodynamic functions of the gas depend
0
on the interaction potential only through the scattering length a, a length characterizing the low-energy scattering
.
1 amplitude of two atoms.
0 ThisuniversalitypropertyholdsfortheweaklyrepulsiveBosegasinthreedimensions[9]uptotheorderofexpansion
0
in (na3)1/2 corresponding to Bogoliubov theory [10], n being the gas density. It is also true for the weakly repulsive
1
Bose gas in two dimensions [11–13], even at the next order beyond Bogoliubov theory [14]. For a much larger than
:
v the range of the interaction potential, the ground state of N bosons in two dimensions is a universal N-body bound
i
X state [15–19]. In one dimension, the universality holds for any scattering length, and the Bose gas with zero-range
interaction is exactly solvable by the Bethe ansatz both in the repulsive case [20] and in the attractive case [21, 22].
r
a Forspin1/2fermions,the universalitypropertiesareexpectedtobe evenstronger. Theweaklyinteractingregimes
in3D[23–27]andin2D[28]areuniversal,andthe1DcaseisalsosolvablebyBetheansatzforanarbitraryinteraction
strength [29, 30]. Universalityis expected to hold for an arbitraryscattering length even in 3D (see however[31]), as
wasrecentlytestedbyexperimentalstudiesontheBEC-BCScrossoverusingaFeshbachresonance,seee.g.[8,32–39],
and in agreementwith unbiasedQuantum Monte Carlo calculations [40–45]. A similar universalcrossoverfrom BEC
to BCS is expected in 2D when the parameter ln(k a) varies from to + [46–50]. Universality is also expected
F
−∞ ∞
for mixtures in 2D [50–52], and in 3D for Fermi-Fermi mixtures below a critical mass ratio [51, 53, 54].
In the zero-range regime, it is intuitive that the short-range or high-momenta properties of the gas are dominated
by two-body physics. For example the pair distribution function g(2)(r ) of particles at distances r much smaller
12 12
than the de Broglie wavelength is expected to be proportional to the modulus squared of the zero-energy two-body
scattering wavefunction φ(r ), with a proportionality factor Λ depending on the many-body state of the gas.
12 g
Similarly the large momentum tail of the momentum distribution n(k), at wavevectorsmuch larger than the inverse
2
de Broglie wavelength, is expected to be proportional to the modulus squared of the Fourier component of the zero
energyscatteringstateφ˜(k), withaproportionalityfactorΛ depending onthemany-bodystateofthegas: Whereas
n
two colliding atoms in the gas have a center of mass wavevector of the order of the inverse de Broglie wavelength,
their relativewavevectorcanaccessmuchlargervalues,up to the inverseofthe interactionrange,simply because the
interactionpotentialhasawidthinthespaceofrelativemomentaofthe orderofthe inverseofitsrangeinrealspace.
For these intuitive reasons,and with the notable exception of one-dimensionalsystems, one expects that the mean
interactionenergyE ofthegas,beingsensitivetotheshapeofg(2) atdistancesoftheorderoftheinteractionrange,
int
is not universal,but diverges in the zero-rangelimit; one also expects that, apartfrom the 1D case, the mean kinetic
energy, being dominated by the large-momentum tail of the momentum distribution, is not universal and diverges
in the zero-range limit, a well known fact in the context of Bogoliubov theory for Bose gases and of BCS theory for
Fermi gases. Since the total energy of the gas is universal, and E is proportional to Λ while E is proportional
int g kin
to Λ , one expects that there exists a simple relation between Λ and Λ .
n g n
Thepreciselinkbetweenthepairdistributionfunction,thetailofthemomentumdistributionandtheenergyofthe
gaswasfirstestablishedforone-dimensionalsystems. In[20]thevalueofthepairdistributionfunctionforr =0was
12
expressed in terms of the derivative of the gas energy with respect to the one-dimensional scattering length, thanks
to the Hellmann-Feynman theorem. In [55] the largemomentum tail of n(k) wasalso related to this derivative of the
energy,byusingasimple andgeneralpropertyoftheFouriertransformofafunctionhavingdiscontinuousderivatives
in isolated points.
In three dimensions, results in these directions were first obtained for weakly interacting gases. For the weakly
interactingBosegas,Bogoliubovtheorycontainstheexpectedproperties,inparticularontheshortdistancebehavior
of the pair distribution function [56–58] and the fact that the momentum distribution has a slowly decreasing tail.
Forthe weaklyinteractingtwo-componentFermigas,itwasshownthatthe BCSanomalousaverage(orpairingfield)
ψˆ (r )ψˆ (r ) behaves at short distances as the zero-energy two-body scattering wavefunction φ(r ) [59], resulting
1 2 12
ihn↑a g(2)↓funcition indeed proportional to φ(r )2 at short distances. It was however understood later that the
12
| |
corresponding proportionality factor Λ predicted by BCS theory is incorrect [60], e.g. at zero temperature the BCS
g
prediction drops exponentially with 1/a in the non-interacting limit a 0 , whereas the correct result drops as a
−
→
power law in a.
Morerecently,ina seriesoftwoarticles[61,62], explicitexpressionsforthe proportionalityfactorsΛ andΛ were
g n
obtainedintermsofthederivativeofthegasenergywithrespecttotheinversescatteringlength,foratwo-component
interacting Fermi gas in three dimensions, for an arbitrary value of the scattering length, that is, not restricting to
the weakly interacting limit. Later on, these results were rederived in [63–65], and also in [66] with very elementary
methods building on the intuition that g(2) φ(r )2 at short distances and n(k) φ˜(k)2 at large momenta.
12
∝ | | ∝ | |
These relations were recently tested by numerical four-body calculations [67]. An explicit relation between Λ and
g
the interaction energy was derived in [65]. Another fundamental relation discovered in [61] and recently generalized
in [68] to bosons, to Fermi-Bose mixtures and to fermions in 2D, expresses the total energy as a functional of the
momentum distribution and the spatial density.
In the present work we derive generalizations of the relations of [20, 55, 61, 62, 65, 68] to two dimensional gases,
to the general case of a mixture of an arbitrary number of atomic species and spin component, and to the case of a
smallbut non-zerointeractionrange(bothonalattice andincontinuousspace). We alsofind entirelynew resultsfor
the first order derivative of the energy with respect to the effective range and, in presence of the Efimov effect, with
respect to the three-body parameter, as well as the second order derivative with respect to the scattering length.
Thearticleisorganizedasfollows. InSectionIIwetreatindetailthecaseoftwo-componentFermigases. Relations
holding for any system eigenstate for zero-range interactions are derived in Section IIB and summarized in Table II.
We then considerlattice models (Tab.III, Sec.IIC) and finite-rangemodels incontinuous space (Tab.IV, Sec.IID).
In Section IIE we derive a model-independent expression for the correction to the energy due to a finite range or a
finite effective range of the interaction. The generalization to thermodynamic equilibrium, where the system is in a
statisticalmixtureofeigenstates,isdiscussedinSectionIIF. InSectionIIIweturntothecaseofspinlessbosons. We
focus on the case of zero-rangeinteractionswhere, in 3D, the Efimoveffect leads to modifications or even breakdown
of some relations,and to the appearance of a new relation. Then we show briefly in Section IV how to treat the case
of an arbitrary mixture and present results for zero-range interactions (Tab. VI). Finally we present applications of
exact relations: For three particles we compute corrections to exactly solvable cases and compare them to numerics
(Sec. VA), and we check that exact relations are satisfied by existing fixed-node Monte-Carlo data for correlation
functions of the unitary gas. We expect from our expressionfor the leading finite-range correctionto the energy that
theleadingfinite-rangecorrectiontothecriticaltemperatureintheBEC-BCScrossoverdependsonlyontheeffective
range of the interaction, an expectation that we test against the Quantum Monte Carlo calculations of [41, 44]. We
conclude in Section VI.
3
II. TWO-COMPONENT FERMIONS
In this Section we consider spin-1/2 fermions. For a fixed number N of particles in each spin state σ = , , one
σ
↑ ↓
can consider that particles 1,...,N have a spin and particles N +1,...,N +N = N have a spin , i.e. the
wavefunctionψ(r ,...,r )changes↑signwhenone↑exchangestheposi↑tionsoftwo↑partic↓leshavingthesame↓spin[132].
1 N
A. Models
Here we introduce the three models used in this work to model interparticle interactions.
1. Zero-range model
In this well-known model (see e.g. [69–76] and refs. therein) the interaction potential is replaced by contact
conditions on the many-body wavefunction: For any pair of particles i = j, there exists a function A , hereafter
ij
6
called regular part of ψ, such that in 3D
1 1
ψ(r ,...,r ) = A (R ,(r ) )+O(r ), (1)
1 N ij ij k k=i,j ij
rij→0(cid:18)rij − a(cid:19) 6
and in 2D
ψ(r ,...,r ) = ln(r /a)A (R ,(r ) )+O(r ), (2)
1 N ij ij ij k k=i,j ij
rij→0 6
where the limit of vanishing distance r between particles i and j is taken for a fixed position of their center of mass
ij
R = (r +r )/2 and fixed positions of the remaining particles (r ) . Fermionic symmetry of course imposes
ij i j k k=i,j
A =0 if particles i and j have the same spin. When none of the r ’s c6oincide, there is no interaction potential and
ij i
Schr¨odinger’s equation reads
Hψ(r ,...,r )=Eψ(r ,...,r ) (3)
1 N 1 N
with
N ~2
H = ∆ +U(r ) ψ (4)
−2m ri i
i=1(cid:20) (cid:21)
X
where m is the atomic mass and U is an external potential. The crucial difference between the Hamiltonian H and
the non-interacting Hamiltonian is the boundary condition (1,2).
2. Lattice models
These models were used for quantum Monte-Carlo calculations [40–43, 45, 77]. They can also be convenient for
analytics, as used in [14, 78, 79] and in this work. Here particles live on a lattice, i. e. the coordinates are integer
multiples of the lattice spacing b. The Hamiltonian reads
H =H +g W (5)
0 0
where
N ~2
H = ∆ +U(r ) (6)
0 −2m ri i
i=1(cid:20) (cid:21)
X
W = δri,rjb−d (7)
i<j
X
4
Three dimensions Two dimensions
1 1
ψ(r ,...,r ) = A (R ,(r ) )+O(r ) ψ(r ,...,r ) = ln(r /a)A (R ,(r ) )+O(r )
1 N rij→0(cid:18)rij − a(cid:19) ij ij k k6=i,j ij 1 N rij→0 ij ij ij k k6=i,j ij
(A(1),A(2)) ddr ddR A(1)(R ,(r ) )∗A(2)(R ,(r ) )
≡ k ij ij ij k k6=i,j ij ij k k6=i,j
Xi<jZ (cid:16)kY6=i,j (cid:17)
TABLE I: Notation for the regular part A of themany-bodywavefunction appearing in the contact conditions (first line) and
for the scalar product between such regular parts(second line).
Three dimensions Two dimensions
dE 4π~2 dE 2π~2
1 = (A,A) = (A,A)
d( 1/a) m d(lna) m
−
4πm dE 2πm dE
2 C lim k4n (k)= C lim k4n (k)=
≡k→+∞ σ ~2 d( 1/a) ≡k→+∞ σ ~2 d(lna)
−
r r C 1 r r C
3 d3Rg(2) R+ ,R d2Rg(2) R+ ,R ln2r
↑↓ 2 − 2 r∼→0 (4π)2r2 ↑↓ 2 − 2 r→∼0 (2π)2
Z (cid:16) (cid:17) Z (cid:16) (cid:17)
~2C ~2C aΛeγ
4 E E = E E = lim ln
− trap 4πma − trap Λ→∞ −2πm 2
(cid:20) (cid:18) (cid:19)
d3k ~2k2 C d2k ~2k2
+ n (k) + n (k)
(2π)3 2m σ − k4 (2π)2 2m σ
σ Z (cid:20) (cid:21) σ Zk<Λ (cid:21)
X X
r r C r r C
5 d3Rg(1) R+ ,R = N r+O(r2) d2Rg(1) R+ ,R = N + r2lnr+O(r2)
σσ 2 − 2 r→0 σ− 8π σσ 2 − 2 r→0 σ 4π
Z (cid:16) (cid:17) Z (cid:16) (cid:17)
6 1 3 d3Rg(1) R+ rui,R rui = N 1 2 d2Rg(1) R+ rui,R rui = N
3 σσ 2 − 2 r→0 2 σσ 2 − 2 r→0
Xi=C1Xσ Zm (cid:16) ~2C (cid:17) CXi=1Xσ Zr (cid:16) m (cid:17)
−4πr− 3~2 E−Etrap− 4πma r2+o(r2) +4πr2 ln a + 3F2 − 2~2 (E−Etrap)r2+o(r2)
(cid:18) (cid:19) (cid:20) (cid:16) (cid:17) (cid:21)
1 d2E 4π~2 2 (A(n),A)2 1 d2E 2π~2 2 (A(n),A)2
7 = | | = | |
2d( 1/a)2 m E E 2d(lna)2 m E E
− (cid:18) (cid:19) n,EXn6=E − n (cid:18) (cid:19) n,EXn6=E − n
d2F d2E d2F d2E
8 <0, <0 <0, <0
d( 1/a)2 d( 1/a)2 d(lna)2 d(lna)2
(cid:18) − (cid:19)T (cid:18) − (cid:19)S (cid:18) (cid:19)T (cid:18) (cid:19)S
dE ~2C d( 1/a) N dE ~2C d(lna) N
9 dt = 4πm −dt + ∂tU(ri,t) dt = 2πm dt + ∂tU(ri,t)
DXi=1 E DXi=1 E
TABLEII:Relations fortwo-componentfermionswith zero-rangeinteractions. TheregularpartAisdefinedinTableI. Lines
1-7holdforanyeigenstate,andcanbegeneralizedtofinitetemperaturebytakingathermalaverageinthecanonicalensemble
and by taking the derivatives of E with respect to a at constant entropy S. Line 8 holds in the canonical ensemble. Line 9
holds for any time-dependence of scattering length and trapping potential and any corresponding time-dependent statistical
mixture.
5
4πm dE 2πm dE
1 C C
≡ ~2 d( 1/a) ≡ ~2 d(lna)
−
dE 4π~2 dE 2π~2
2 = (A,A) = (A,A)
d( 1/a) m d(lna) m
−
~2 2 C
3 E =
int m g
(cid:18) (cid:19) 0
~2C ~2C aqeγ
4 E E = E E = lim ln
− trap 4πma − trap q→0(− 2πm (cid:18) 2 (cid:19)
d3k ~2 2 d2k ~2 ~2
+ σ ZD (2π)3ǫk"nσ(k)−C(cid:18)2mǫk(cid:19) # + σ ZD (2π)2ǫk(cid:20)nσ(k)−C2mǫkP2m(ǫk−ǫq)(cid:21))
X X
1d2E (A(n),A)2
5 = φ(0)4 | |
2 dg02 | | n,EXn6=E E−En
d2F d2E
6 <0, <0
dg2 dg2
(cid:18) 0 (cid:19)T (cid:18) 0 (cid:19)S
C C
7 b3g(2)(R,R)= φ(0)2 b2g(2)(R,R)= φ(0)2
↑↓ (4π)2| | ↑↓ (2π)2| |
R R
X X
In thezero-range regime k b 1
typ
≪
r r C r r C
8 b3g(2) R+ ,R φ(r)2 for r k−1 b2g(2) R+ ,R φ(r)2 for r k−1
↑↓ 2 − 2 ≃ (4π)2| | ≪ typ ↑↓ 2 − 2 ≃ (2π)2| | ≪ typ
XR (cid:16) (cid:17) XR (cid:16) (cid:17)
~2 2
9 n (k) C for k k
σ ≃ (cid:18)2mǫk(cid:19) ≫ typ
TABLE III: Relations for two-component fermions in a lattice model. C is definedin line 1.
in first quantization, i.e.
ddk
H = ǫ c (k)c (k)+ bdU(r)(ψ ψ )(r) (8)
0 (2π)d k †σ σ σ† σ
σ ZD r,σ
X X
W = bd(ψ†ψ†ψ ψ )(r) (9)
↑ ↓ ↓ ↑
r
X
in second quantization. Here d is the space dimension, ǫ is the dispersion relation and c (k) is creates a particle
k †σ
in the plane wave state k defined by rk = eikr for any k belonging to the first Brillouin zone D = π,π d.
| i h | i · −b b
Accordingly the operator ∆ in (6) is the discrete representation of the Laplacian defined by ~2 r∆ k ǫ rk .
−2mh | r| i≡(cid:0) kh |(cid:3)i
~2k2
The simplest choice for the dispersion relation is ǫ = [14, 42, 45, 78, 79]. Another choice, used in [41, 77], is
k
2m
6
Three dimensions Two dimensions
4πm dE 2πm dE
1 C C
≡ ~2 d( 1/a) ≡ ~2 d(lna)
−
C C
2 E = d3rV(r)φ(r)2 E = d2rV(r)φ(r)2
int (4π)2 | | int (2π)2 | |
Z Z
~2C ~2C R
3 E E = E E = lim ln
− trap 4πma − trap R→∞(2πma (cid:18)a(cid:19)
d3k ~2k2 C d2k ~2k2 C
+ n (k) φ˜′(k)2 + n (k) φ˜′ (k)2
σ Z (2π)3 2m (cid:20) σ − (4π)2| | (cid:21) σ Z (2π)2 2m (cid:20) σ − (2π)2| R | (cid:21))
X X
In thezero-range regime k b 1
typ
≪
r r C r r C
5 d3Rg(2) R+ ,R φ(r)2 for r k−1 d2Rg(2) R+ ,R φ(r)2 for r k−1
↑↓ 2 − 2 ≃ (4π)2| | ≪ typ ↑↓ 2 − 2 ≃ (2π)2| | ≪ typ
Z (cid:16) (cid:17) Z (cid:16) (cid:17)
C C
6 n (k) φ˜(k)2 for k k n (k) φ˜(k)2 for k k
σ ≃ (4π)2| | ≫ typ σ ≃ (2π)2| | ≫ typ
TABLE IV: Relations for two-component fermions with a finite-range interaction potential V(r) in continuous space. C is
defined in line 1.
~2 d
the dispersion relation of the Hubbard model: ǫ = [1 cos(k b)]. More generally, what follows applies to
k mb2 − i
i=1
~2k2 X
any ǫ such that ǫ sufficiently rapidly and ǫ =ǫ .
k k k k
b→0 2m −
A key quantity is t→he zero-energy scattering state φ(r), defined by the two-body Schr¨odinger equation (with the
center of mass at rest)
~2 δ
∆ +g r,0 φ(r)=0 (10)
−m r 0 bd
(cid:18) (cid:19)
and by the normalization conditions
1 1
φ(r) in 3D (11)
r≃b r − a
≫
φ(r) ln(r/a) in 2D. (12)
r≃b
≫
A straightforwardtwo-body analysis,detailed in App. A, yields the relation between the scattering length and the
bare coupling constant g :
0
1 m d3k 1
= in 3D (13)
g 4π~2a − (2π)32ǫ
0 ZD k
1 m d2k 1
= lim ln(aqeγ/2)+ in 2D (14)
g0 q→0−2π~2 ZD (2π)2P2(ǫq−ǫk)
where γ =0.577216...is Euler’s constant and is the principal value. Other useful relations derived in App. A are
P
4π~2
φ(0) = in 3D (15)
−mg
0
2π~2
φ(0) = in 2D (16)
mg
0
7
Three dimensions Two dimensions
∂E 4π~2 dE 2π~2
= (A,A) = (A,A)
∂( 1/a) m d(lna) m
(cid:18) − (cid:19)Rt
8πm ∂E 4πm dE
C lim k4n(k)= C lim k4n(k)=
≡k→+∞ ~2 ∂( 1/a) ≡k→+∞ ~2 d(lna)
(cid:18) − (cid:19)Rt
r r C 1 r r C
d3Rg(2) R+ ,R d2Rg(2) R+ ,R ln2r
2 − 2 r→∼0 (4π)2r2 2 − 2 r∼→0 (2π)2
Z (cid:16) (cid:17) Z (cid:16) (cid:17)
E E if∃=lim ~2C E E = lim ~2C ln aΛeγ
− trap 8πma − trap Λ→∞ −4πm 2
d3k ~2k2 C d2k(cid:20) ~2k2 (cid:18) (cid:19)
+ lim n(k) + n(k)
Λ→+∞Zk<Λ (2π)3 2m (cid:20) − k4(cid:21) Zk<Λ (2π)2 2m (cid:21)
1 ∂2E 4π~2 2 (A(n),A)2 1 d2E 2π~2 2 (A(n),A)2
= | | = | |
2 ∂( 1/a)2 m E E 2d(lna)2 m E E
(cid:18) − (cid:19)Rt (cid:18) (cid:19) n,EXn6=E − n (cid:18) (cid:19) n,EXn6=E − n
∂E ~2√3 d2F
= s 2N(N 1)(N 2) <0
(cid:18)∂lnRt(cid:19)a m 32| 0| − − (cid:18)d(lna)2(cid:19)T
d2E
dC dr ...dr B(C,r ,...,r )2 <0
× 4 N| 4 N | d(lna)2
Z Z (cid:18) (cid:19)S
TABLEV:Mainresultsforspinlessbosonsinthelimitofazerorangeinteraction. Inthreedimensions,thederivativesaretaken
forafixedthree-bodyparameterR . Asdiscussedinthetext,inthreedimensions,therelationbetweenenergyandmomentum
t
distribution is valid if the large cut-off limit Λ + exists, which is not thecase for Efimovian states (i.e. eigenstates whose
→ ∞
energy dependson R ). In thelast relation in three-dimensions, B is thethree-bodyregular part definedin (152).
t
and
4π~2d( 1/a)
φ(0)2 = − in 3D (17)
| | m dg
0
2π~2d(lna)
φ(0)2 = in 2D. (18)
| | m dg
0
In the zero-range limit (b 0 with g adjusted in such a way that a remains constant), the spectrum of the
0
→
lattice modelis expected toconvergeto the one ofthe zero-rangemodel[41,79], andanyeigenfunctionψ(r ,...,r )
1 N
of the lattice model tends to the corresponding eigenfunction of the zero-range model, provided all interparticle
distancesremainmuchlargerthanb. Letusdenoteby1/k thetypicallength-scaleonwhichthezero-rangemodel’s
typ
wavefunction varies: e.g. for the lowest eigenstates, it is on the order of the mean interparticle distance, or on the
order of a in the regime where a is small and positive and dimers are formed. The zero-range limit is then reached if
k b 1.
typ
≪
For lattice models, it will prove convenient to define the regular part A by
ψ(r ,...,r =R ,...,r =R ,...,r )=φ(0)A (R ,(r ) ). (19)
1 i ij j ij N ij ij k k=i,j
6
8
Three dimensions Two dimensions
∂E 2π~2 ∂E π~2
∂(−1/aσσ′) = µσσ′(A,A)σσ′ ∂(lnaσσ′) = µσσ′(A,A)σσ′
Cσ ≡k→lim+∞k4nσ(k)=Xσ′ (1+δσσ′)8π~µ2σσ′ ∂(−1∂/Eaσσ′) Cσ ≡k→lim+∞k4nσ(k)=Xσ′ (1+δσσ′)4π~µ2σσ′ ∂(ln∂aEσσ′)
Z d3Rgσ(2σ)′(cid:18)R+ mσm+σm′ σ′r,R− mσm+σmσ′r(cid:19)r∼→0(1+δσσ′) Z d2Rgσ(2σ)′(cid:18)R+ mσm+σm′ σ′r,R− mσm+σmσ′r(cid:19)r∼→0(1+δσσ′)
µσσ′ ∂E 1 µσσ′ ∂E ln2r
×2π~2∂(−1/aσσ′)r2 ×π~2 ∂(lnaσσ′)
E E = 1 ∂E E E = lim ∂E ln aσσ′Λeγ
− trap σX≤σ′ aσσ′ ∂(−1/aσσ′) − trap Λ→∞−σX≤σ′ ∂(lnaσσ′) (cid:18) 2 (cid:19)

d3k ~2k2 C d2k ~2k2
+ n (k) σ + n (k)
σ Z (2π)3 2mσ (cid:20) σ − k4(cid:21) σ Zk<Λ (2π)2 2mσ σ #
X X
1 ∂2En = 2π~2 2 |(A(n′),A(n))σσ′|2 1 ∂2En = π~2 2 |(A(n′),A(n))σσ′|2
2∂(−1/aσσ′)2 (cid:18)µσσ′(cid:19) n′,EXn′6=En En−En′ 2∂(lnaσσ′)2 (cid:18)µσσ′(cid:19) n′,EXn′6=En En−En′
∂2F ∂2F
<0 <0
(cid:18)∂(−1/aσσ′)2(cid:19)T (cid:18)∂(lnaσσ′)2(cid:19)T
∂2E ∂2E
<0 <0
(cid:18)∂(−1/aσσ′)2(cid:19)S (cid:18)∂(lnaσσ′)2(cid:19)S
TABLEVI:Mainresultsforanarbitrarymixturewithzero-rangeinteractions. Inthreedimensions,iftheEfimoveffectoccurs,
thederivatives must be taken for fixed three-bodyparameter(s) and theexpression for E in line 4 breaks down.
In the zero-range regime k b 1, we expect that when the distance r between two particles of opposite spin is
typ ij
≪
1/k while allthe otherinterparticledistancesaremuchlargerthanb andthanr ,the many-bodywavefunction
typ ij
≪
is proportional to φ(r ), with a proportionality constant given by (19):
ij
ψ(r ,...,r ) φ(r r )A (R ,(r ) ) (20)
1 N j i ij ij k k=i,j
≃ − 6
where R = (r +r )/2. If moreover r b, φ can be replaced by its asymptotic form (11,12); since the contact
ij i j ij
≫
conditions (1), (2) of the zero-range model must be recovered, we see that the lattice model’s regular part tends to
the zero-range model’s regular part in the zero-range limit.
9
3. Finite-range continuous-space model
Such models are used in numerical few-body correlated Gaussian and many-body fixed-node Monte-Carlo calcula-
tions (see e. g. [5, 67, 80–83] and refs. therein). They are also relevant to neutron matter [84]. The Hamiltonian
reads
N↑ N
H =H + V(r ), (21)
0 ij
Xi=1j=XN↑+1
H being defined by (6) where ∆ now stands for the usual Laplacian, and V(r) is an interaction potential between
0 ri
particles of opposite spin, which vanishes for r > b or at least decays quickly enough for r b. The two-body
zero-energy scattering state φ(r) is again defined by the Schr¨odinger equation (~2/m)∆ φ+≫V(r)φ = 0 and the
r
−
boundary condition (11,12). The zero-range regime is again reached for k b 1 with k the typical relative
typ typ
≪
wavevector [133]. Equation (20) again holds in the zero-rangeregime, where A now simply stands for the zero-range
model’s regular part.
B. Relations in the zero-range limit
1. First order derivative of the energy with respect to the scattering length
We now derive relations for the zero-rangemodel. For some of the derivations we will use a lattice model and take
the zero-range limit in the end.
Three dimensions:
Let us consider a wavefunction ψ satisfying the contact condition (1) for a scattering length a . We denote by
1 1
A(1) the regular part of ψ appearing in the contact condition (1). Similarly, ψ satisfies the contact condition for a
ij 1 2
scattering length a and a regular part A(2). Then, as shown in Appendix B, the following lemma holds:
2 ij
4π~2 1 1
ψ ,Hψ Hψ ,ψ = (A(1),A(2)) (22)
1 2 1 2
h i−h i m a − a
(cid:18) 1 2(cid:19)
where the scalar product between regular parts is defined by
(A(1),A(2))≡ ddrk ddRijA(ij1)∗(Rij,(rk)k6=i,j)A(ij2)(Rij,(rk)k6=i,j). (23)
Xi<jZ (cid:16)kY6=i,j (cid:17)Z
We then apply (22) to the case where ψ and ψ are N-body eigenstates of energy E and E . The left hand side of
1 2 1 2
(22) then reduces to (E E ) ψ ψ . Taking the limit a a gives the final result
2 1 1 2 2 1
− h | i →
dE 4π~2
= (A,A) (24)
d( 1/a) m
−
for any eigenstate. This result is contained in the work of Tan [61, 62][134]. Note that, here and in what follows, we
have assumed that the wavefunction is normalized: ψ ψ =1.
h | i
Two dimensions:
The 2D version of the lemma (22) is
2π~2
ψ ,Hψ Hψ ,ψ = ln(a /a ) (A(1),A(2)), (25)
1 2 1 2 2 1
h i−h i m
as shown in Appendix B. As in 3D, we deduce from the lemma the final result
dE 2π~2
= (A,A). (26)
d(lna) m
10
2. Large-momentum tail of the momentum distribution
The momentum distribution is defined in second quantization by
n (k)= cˆ (k)cˆ (k) (27)
σ h †σ σ i
where cˆ (k) annihilates a particle of spin σ in the plane-wave state k defined by rk =eikr. This corresponds to
σ ·
| i h | i
the normalization
ddk
n (k)=N . (28)
(2π)d σ σ
Z
In first quantization,
2
nσ(k)= ddrl ddrie−ik·riψ(r1,...,rN) (29)
Xi:σ Z (cid:16)Yl6=i (cid:17)(cid:12)(cid:12)Z (cid:12)(cid:12)
(cid:12) (cid:12)
where the sum is takenover all particles of spin σ, i.e. i r(cid:12)uns from 1, to N for σ = (cid:12)and from N +1 to N for σ = .
Three dimensions: ↑ ↑ ↑ ↓
The key point is that in the large-k limit, the Fourier transform with respect to r is dominated by the contribution
i
of the short-distance divergence coming from the contact condition (1):
1
d3rie−ik·riψ(r1,...,rN) d3rie−ik·ri Aij(rj,(rk)k=i,j). (30)
Z k→≃∞Z j,j=irij 6
X6
From ∆(1/r)= 4πδ(r), we have the identity
−
1 4π
d3re ikr = , (31)
− ·
r k2
Z
so that
4π
d3rie−ik·riψ(r1,...,rN)k≃ k2 e−ik·rjAij(rj,(rl)l6=i,j). (32)
Z →∞ j,j=i
X6
Inserting this into (29) and expanding the modulus squared, the cross terms vanish in the large-k limit, so that
C =(4π)2(A,A) (33)
where C lim k4n (k). This can be rewritten using (24) as:
k σ
≡ →∞
4πm dE
C = , (34)
~2 d( 1/a)
−
in agreement with Tan [62].
Two dimensions:
The 2D contact condition (2) now gives
d2rie−ik·riψ(r1,...,rN) d2ri,e−ik·ri ln(rij)Aij(rj,(rl)l=i,j). (35)
k≃ 6
Z →∞Z j,j=i
X6
From ∆(lnr)=2πδ(r), we have the identity
2π
d2re ikrlnr = , (36)
− ·
−k2
Z
so that
2π
d2rie−ik·riψ(r1,...,rN)k≃ −k2 e−ik·rjAij(rj,(rl)l6=i,j). (37)
Z →∞ j,j=i
X6
As in 3D this leads to
C =(2π)2(A,A) (38)
where C lim k4n (k), and thus from (24):
k σ
≡ →∞
2πm dE
C = . (39)
~2 d(lna)