Universal composite boson formation in strongly interacting one-dimensional fermionic

systems

Francesc Sabater,1, 2 Abel Rojo-Francàs,1, 2 Grigori E. Astrakharchik,3, 1, 2 and Bruno Juliá-Díaz1, 2

1
Departament de Física Quàntica i Astrofísica, Facultat de Física, Universitat de Barcelona, E-08028 Barcelona, Spain
2
Institut de Ciències del Cosmos, Universitat de Barcelona,
ICCUB, Martí i Franquès 1, E-08028 Barcelona, Spain
3
Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain
(Dated: January 9, 2024)

Attractive p-wave one-dimensional fermions are studied in the fermionic Tonks-Girardeau regime in
arXiv:2309.03606v2 [cond-mat.quant-gas] 7 Jan 2024

which the diagonal properties are shared with those of an ideal Bose gas. We study the off-diagonal prop-
erties and present analytical expressions for the eigenvalues of the one-body density matrix. One striking
aspect is the universality of the occupation numbers which are independent of the specific shape of the
external potential. We show that the occupation of natural orbitals occurs in pairs, indicating the formation
of composite bosons, each consisting of two attractive fermions. The formation of composite bosons sheds
light on the pairing mechanism of the system orbitals, yielding a total density equal to that of a Bose-Einstein
condensate.

Introduction. One of the most fascinating quantum phe- exhibits partial pair condensation as has been discussed in
nomena is Bose-Einstein condensation (BEC), characterized the case of harmonic trapping [8]. Recently, Kośik and Sow-
by a macroscopic occupation of a single quantum state [1]. iński have proven that in FTG the eigenvalues of the OBDM
If BEC occurs, the largest eigenvalue of the one-body den- are independent of the shape of the trapping potential [15].
sity matrix (OBDM) ρ1 (r, r′ ) = 〈Ψ̂ † (r)Ψ̂(r′ )〉 is proportional This fact enables us to conduct a completely general and
to the total number of particles N where Ψ̂ † (r) (Ψ̂(r)) are universal study of the FTG gas without the need to con-
creation (annihilation) operators of a particle at position strain the study to a specific system or confining potential,
r, respectively [2, 3]. For a homogeneous gas, the corre- as has been done in previous works [9].
sponding eigenvector of the OBDM is the zero-momentum In this Letter we analytically study the coherence in a
state which gives rise to the presence of Off-Diagonal Long- finite-size FTG state in an arbitrary external field. We
Range Order (ODLRO), lim|r−r′ |→∞ ρ1 (r, r′ ) → n0 where n0 demonstrate that in FTG gas, fermions pair into compos-
is the condensate density. In stark contrast, due to the Pauli ite bosons. As a consequence, each fermionic pair exhibits
principle, no two fermions are allowed to have the same a density profile equivalent to that of an ideal boson. To
quantum numbers, and atomic condensation is prohibited. demonstrate that, we perform natural orbital analysis di-
Instead, in this Letter we discuss and explore the possibil- agonalizing the one-body density matrix and deduce ana-
ity that strongly attractive fermions form composite bosons, lytical expressions for its occupation numbers and natural
whose density is equal to that of a Bose-Einstein conden- orbitals. We verify the correctness of the obtained expres-
sate. sions by comparison with numerical diagonalization. For a
One-dimensional geometry is very special as due to Gi- large number of particles, we find that the occupation of the
rardeau’s mapping, bosonic and fermionic systems might OBDM asymptotically approaches a Lorentzian shape, the
possess exactly the same diagonal properties [4, 5]. In same as the one found in Ref. [16] for the momentum dis-
that way, the wave function of impenetrable bosons can be tribution of a homogeneous FTG gas in the thermodynamic
mapped to the wave function of ideal fermions, known as limit. It is remarkable that eigenvalues are universal and re-
Tonks-Girardeau (TG) gas [4, 6]. As well, single-component main exactly the same for any shape of the external poten-
fermions with a strong p-wave attraction can exhibit iden- tial [15]. In particular, for an untrapped system, the eigen-
tical diagonal correlations as in an ideal Bose gas, result- values of the OBDM correspond to the momentum distribu-
ing in what is known as the fermionic Tonks-Girardeau tion allowing us to obtain its exact expression. Moreover,
(FTG) gas [7–10]. Recent experimental advances in spin- we demonstrate that the eigenvalues of the OBDM come in
polarized fermions with p-wave resonances pave the way pairs. In that way, while all the individual fermionic eigen-
for the experimental realization of the FTG gas using states of the OBDM are different, the density of fermionic
confinement-induced resonances [11, 12]. Another pecu- pairs (i.e. composite bosons) remains exactly the same, al-
liarity of the one-dimensional geometry is that quantum though its specific shape depends on the external potential.
fluctuations destroy BEC in gases with finite interactions. Thus, we provide a comprehensible picture of the mecha-
Indeed, in uniform systems, the ODLRO vanishes in power- nism of composite boson formation and how this results in
law decay lim|x−x ′ |→∞ ρ1 (x, x ′ ) → n/(n|x − x ′ |)1/(2K) where the density profile being equal to that of a Bose-Einstein
K is the Luttinger parameter [13, 14]. Formally, the ODLRO condensate.
is restored in the limit of ideal Bose gas (K → ∞). Its Fermionic Tonks-Girardeau gas. FTG gas describes N
fermionic counterpart corresponds to the FTG gas which fermions with coordinates x 1 , · · · , x N interacting via short-
 2

range p-wave attraction tuned in such a way that the By solving the universal eigenproblem (5) for an increas-
ground state wave function ing even number of fermions we find that the eigenvalues
always appear in pairs which leads to crucial physical con-
N
Y sequences, as will be discussed below. We find that the
ψ F (x 1 , . . . , x N ) = ψB (x 1 , . . . , x N ) sgn (x k − x j ) , (1)
eigenvectors obtained for the N = 2 case remain valid for
j<k
larger (and even) numbers of particles. We obtain the fol-
is related to the one of an ideal Bose gas, ψB (x 1 , . . . , x N ) = lowing explicit expressions for eigenvalues (plus-minus sign
QN
denotes the double degeneracy)
i=1 φ0 (x i ), according to Girardeau’s mapping [4] where
φ0 (x) is the single-particle ground state. The coherence 384 48
properties are encoded in the OBDM, λk± = − + (9)
[π(2k − 1)]4 [π(2k − 1)]2
Z
ρ1 (x, x′) = N ψ F (x, x 2 , . . . x N )ψ∗F (x′, x 2 , . . . x N )d x 2 . . . d x N . for N = 4 and
46080 5760 120
λk± = − + (10)
The OBDM can be expressed in terms of the orbitals of the [π(2k − 1)] 6 [π(2k − 1)]4 [π(2k − 1)]2
non-interacting bosons [16]:
for N = 6 fermions, while eigenvalues up to N = 10 are
ρ1 (x, x′) = N φ0 (x)φ0∗ (x ′ )[1 − 2P(x, ′
x )] N −1
, (2) reported in the Supplementary Material. By thoroughly ex-
amining the analytic expressions for the occupations of the
where natural orbitals, we have arrived at an explicit analytic ex-
Z x′ pression for the doubly degenerate eigenvalues,
′
P(x, x ) = |φ0 (z)|2 dz . (3) 
NP/2
x
 (−1)i+1 4i N !

 , even N
[(2k−1)π]2i (N −2i)!
To find its eigenvalues, commonly referred to as occupation λNk± = i=1
(N −1)/2 (11)
P (−1)i+1 4i N !
[2kπ]2i (N −2i)! , odd N.

numbers, and associated eigenvectors, known as natural or- 

i=1
bitals, one has to solve the following eigenproblem:
Z Expressions (11) constitute the main result of our work and
d x ′ ρ1 (x, x ′ )χk (x ′ ) = λk χk (x) . (4) provide valuable insights into the behavior and properties
of the system, shedding light on its fundamental character-
istics. In the absence of an external field, the eigenvalues
Recently, it has been proven that the eigenproblem can
λk of the OBDM can be related to the momentum distribu-
be reduced to a universal form, showing that the occupa-
tion n(k) of a gas on a ring of a circumference L, according
tion numbers are independent of φ0 (x) and, consequently,
to λk = n(k)/L. The allowed momenta as k = ±2kπ/L
completely unaffected by the external potential [15]. The
for odd N and k = ±(2k − 1)π/L for even N and Eq. (11)
demonstration is not limited to the OBDM but is appli-
reproduces the momentum distribution found in Ref. [17].
cable to any N -body density matrix. The universal φ-
For an odd number of fermions, all eigenvalues are dou-
independent eigenproblem reads [15],
bly degenerate except the largest one whose value is always
Z 1 equal to one, λ0 = 1, with v0 ( y) = 1 being the correspond-
d y ′ N (1 − 2| y − y ′ |)N −1 vk ( y ′ ) = λk vk ( y) , (5) ing eigenvector. As for the doubly degenerate eigenval-
0 ues, the eigenvectors are slightly different from the ones
where the actual eigenvectors χk (x) can be obtained from observed in the even case. Specifically, they are given by
vk ( y) using the transformation p
vk+ ( y) = 2 sin[2kπ y], (12)
p
vk− ( y) = 2 cos[2kπ y],
 
χk (x) = φ0 (x)vk F (x) , (6) (13)
Rx
where F (x) = −∞ |φ0 (z)|2 dz. The universal eigenprob- and they remain valid for any number of odd particles.
lem (5) was solved for the simplest system of N = 2 in In particular, the universal eigenproblem (5) can be ap-
Ref. [15] finding that all the eigenvalues are doubly degen- plied to a plain p box with a flat bosonic single-particle
erate and equal to λk± = 8/[π(2k−1)]2 with corresponding state, φ0 (x) = 1/ L. In that case, function (3) expresses
eigenvectors given by as P(x, x ′ ) = |x − x ′ |/L and measures the relative dis-
p tance. In a box of size L, OBDM (2) satisfies ρ(L, 0) =
vk+ ( y) = 2 sin[(2k − 1)π y], (7) (−1)N −1 ρ(0, 0) which corresponds to periodic (antiperi-
p
vk− ( y) = 2 cos[(2k − 1)π y]. (8) odic) boundary conditions for odd (even) N . Such a choice
of boundary conditions is appropriate for forming closed
In the following, we extend the solution of the universal shells in a fermionic gas. It can be verified, as shown in the
eigenproblem for cases where N > 2. Supplementary Material, that the exact eigenstates of the
 3

FIG. 1. Eigenvalues λk± as a function of the pair number k for

different number of fermions N with symbols. The approximation
to Lorentzian shape, see Eq. (14), is shown with a continuous line
for each number of fermions.

FIG. 2. OBDM for N = 2, N = 4 and N = 10 for the case of the

p
pby plane waves χk+ (x) = 2/L cos(kx)
OBDM are given untrapped case, left column, and a harmonic trap, right column.
and χk− (x) = 2/L sin(kx) with allowed momenta, i.e.
k = 0, ±2π/L, ±4π/L, · · · ± 2kπ/L for odd N and k =
in Fig. 2 characteristic examples of the OBDM ρ1 (x, x ′ ) of
±π/L, ±3π/L, · · · ± (2k − 1)π/L for even N . The ther-
an untrapped FTG gas and a FTG gas in a harmonic oscilla-
modynamic limit N → ∞ is taken on an untrapped sys-
tor. The diagonal terms n(x) = ρ1 (x, x) provide the density
tem by increasing the periodicity length L at a fixed den-
profile of the system which is flat in the untrapped case and
sity n = N /L. Quite interestingly, the thermodynamic
has a Gaussian shape in a harmonic oscillator. The antidi-
OBDM can be evaluated explicitly and it exhibits an ex-
agonal terms ρ1 (x, −x) quantify the loss of coherence with
ponential form, ρ1 (x, x ′ ) = n exp(−2n|x − x ′ |) [16], in
x → ∞ asymptotic value equal to zero (i.e. ODLRO is ab-
contrast to the typical power-law behavior found in com-
sent).
pressible systems and predicted by the Luttinger liquid [14].
Composite boson formation. The formation of compos-
Its Fourier transform provides the momentum distribution
ite bosons is evident from the pairing observed in the occu-
n(k) = L/[1 + (k/(2n))2 ] which has a Lorentzian shape.
pation numbers of the OBDM. In the case of an even num-
The relation between the eigenvalues and the momentum
ber of fermions, all occupation numbers exhibit a double
distribution of the untrapped FTG, λk = n(k)/L, allows for
degeneracy whereas for odd N all fermions form composite
a concise approximation for the eigenvalues as
bosons except one, whose eigenvector coincides with the
¨ 1 single-particle ground state with unit eigenvalue. To gain a
1 + (±π(k−1/2) / N )2
, even N
λNk± ≈ 1 (14) deeper insight into the pairing mechanism, it is instructive
1 + (±πk / N )2
, odd N to explicitly construct an anti-symmetric two-particle state
from pairs of eigenvectors of a doubly degenerate eigen-
asymptotically decaying as 1/k2 for k → ∞ as also dis- value, denoted as χk+ (x) and χk− (x),:
cussed in Refs. [17, 18].
We have conducted numerical verification to validate the χk+ (x 1 )χk− (x 2 ) − χk+ (x 2 )χk− (x 1 )
ψk (x 1 , x 2 ) = p . (15)
correctness of the derived expressions for λk± across various 2
values of N . The numerical results confirm the accuracy and
reliability of the analytical findings. The probability of finding a fermion at position x occupying
the paired state is identical for both fermions forming the
Figure 1 reports the universal values of the occupations
composite boson. We denote this probability as Pk (x 1 =
(shown with symbols) of the natural orbitals λk± for the
x) = Pk (x 2 = x) = Pk (x) and it is given by
different number of fermions as compared to the thermo-
dynamic Lorentzian shape (14), shown with lines. We find |χk+ (x)|2 + |χk− (x)|2
that for N ≳ 10, the Lorentzian shape is quite precise al- Pk (x) = . (16)
2
though it fails in a few-body system.
It is crucial to bear in mind that although the occupation This quantity physically corresponds to the density profile
numbers λk are universal and are independent of the spe- of the composite boson with index k. Mathematically, it
cific shape of the external potential [15], the matrices and is formed from two contributions |χk+ (x)|2 and |χk− (x)|2 ,
corresponding eigenvectors are strongly influenced by the each dependent according to Eq. (6) on the bosonic den-
type of external potential used. To show that, we report sity profile |φ0 (x)|2 and two corresponding eigenvectors
 4

FIG. 3. Fermionic natural orbitals and composite boson density FIG. 4. Fermionic natural orbitals and composite boson density
profile of an untrapped FTG gas for an even number of particles. profile in a harmonic trap potential for an even number of parti-
First and second columns, the first six natural orbitals, χk+ (x) and cles. First and second columns, the first six natural orbitals, χk+ (x)
χk− (x), respectively, corresponding to the three largest doubly de- and χk− (x), respectively, corresponding to the three largest dou-
generate eigenvalues k = 1, 2, 3 of the OBDM. Third column, the bly degenerate eigenvalues k = 1, 2, 3 of the OBDM. Third column,
first three composite boson density profiles Pk (x). the first three composite boson density profiles Pk (x).

vk+ ( y) and vk− ( y) defined in Eqs. (7-8) and Eqs. (12- a local quantity, is not affected by the Girardeau mapping
13). The crucial aspect is that potential-dependent function and remains the same in FTG and ideal Bose gas.
F (x) appears as an identical argument in both eigenvectors, To gain further insight into the mechanism of compos-
|vk+ (F (x))|2 and |vk− (F (x))|2 , resulting in a summation of ite bosons formation, we thoroughly examine two distinct
the squares of a cosine and sine functions, which sum up to examples of external potential. In the first example, we
unity. Consequently, all composite bosons have exactly the consider anpuntrapped FTG gas with flat bosonic density,
same density profile given by φ0 (x) = 1/ L, in a ring of perimeter L. The second exam-
ple involves a harmonic oscillator with the single-particle
Pk (x) = |φ0 (x)|2 . (17)
ground state wave function given by a Gaussian,
Note that, any linear combination of the orthonormal eigen- 1
e−x /(2aosc ) ,
2 2
vectors defined in Eqs. (7-8) for the even case and Eqs. (12- φ0 (x) = p (19)
π1/4 aosc
13) for the odd case, would yield exactly the same density
of a fermionic pair. Therefore, the density of each of the p
where aosc = ħ h/(mω) is the harmonic oscillator length, m
composite bosons is independent of the freedom of choice
the particle mass and ω the frequency of the trap. In Figs. 3-
of the eigenvectors that one has since the eigenvalues are
4 we show the first six atomic natural orbitals alongside
doubly degenerated.
the density of the first three composite bosons in both the
The total density sums from identical contributions aris-
untrapped FTG and in the harmonic trap, respectively.
ing from composite bosons, that is 2|φ0 (x)|2 per pair, and a
Notably, while all atomic orbitals differ, as expected for
single boson contribution of |φ0 (x)|2 in the case of an odd
fermions, the density profiles of composite bosons remain
number of particles. Consequently, the total density is given
consistently the same and equal to Pk = |φ0 (x)|2 for any k.
by
This provides a real-space picture of how occupied natural
n(x) = N |φ0 (x)|2 (18) orbitals, i.e. fermions, effectively pair to result in a total
density equivalent to that of the ideal Bose gas.
which is equivalent to the density profile of an ideal Bose In the context of the formation of composite bosons, the
gas. Then, it follows that given a pair of natural orbitals situation recalls the BEC-BCS crossover in two-component
with the same eigenvalue k, if one of them is occupied Fermi gases. Both systems involve attractive fermion pair-
the other is also occupied forming a composite boson with ing, leading to the formation of composite bosonic pairs.
the contribution to the total density of the system equal to The key difference lies in molecular structure: In the BCS-
2|φ0 (x)|2 and the total density of the system is equivalent BEC crossover, each molecule contains one spin-up and one
to that of a Bose-Einstein condensate. Indeed, n(x) being spin-down fermion with s-wave interaction, resulting in the
 5

internal structure which is the same for all molecules. On Excellence María de Maeztu 2020-2023” award to the
the opposite, in a single-component Fermi gas with p-wave Institute of Cosmos Sciences, Grant CEX2019-000918-M
interactions, the Pauli exclusion principle applies leading funded by MCIN/AEI/10.13039/501100011033. We
to a much more intricate scenario, as each composite boson acknowledge financial support from the Generalitat de
has a unique internal configuration while sharing exactly Catalunya (Grants 2021SGR01411 and 2021SGR01095).
the same density. A.R.-F. acknowledges funding from MIU through Grant No.
Conclusions. In this study, we investigate the coherence FPU20/06174.
properties of a fermionic Tonks-Girardeau gas in the pres-
ence of an external potential. We present an analytical ex-
pression (11), for the eigenvalues of the one-body density
matrix, applicable for any number of fermions N and under
[1] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation
an arbitrary external field. For a large number of fermions
(Oxford University Press, Oxford, 2003).
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proach a Lorentzian shape (14). A remarkable feature of [3] O. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956).
the obtained expressions is that they are universal in the [4] M. Girardeau, Journal of Mathematical Physics 1, 516
sense that the occupation numbers depend only on N and (2004).
remain independent of the specific shape of the external [5] T. Cheon and T. Shigehara, Phys. Rev. Lett. 82, 2536 (1999).
potential. The natural orbitals exhibit double degeneracy, [6] M. D. Girardeau, E. M. Wright, and J. M. Triscari, Phys. Rev.
implying that pairs of attractive fermions create composite A 63, 033601 (2001).
[7] M. D. Girardeau, H. Nguyen, and M. Olshanii, Opt. Commun.
bosons. In turn, each composite boson exhibits a particle
243, 3 (2004).
density profile equivalent to that of an ideal Bose gas. This [8] M. D. Girardeau and A. Minguzzi, Phys. Rev. Lett. 96, 080404
physical picture implies that degenerated natural orbitals (2006).
must always be either both occupied or both unoccupied. [9] A. Minguzzi and M. D. Girardeau, Phys. Rev. A 73, 063614
Finally, this pairing results in the fact that the total density (2006).
of the system is equivalent to that of an ideal Bose gas. [10] M. D. Girardeau and G. E. Astrakharchik, Phys. Rev. A 81,
These findings emphasize the significance of composite 043601 (2010).
[11] B. E. Granger and D. Blume, Phys. Rev. Lett. 92, 133202
bosons formation. The universality of the results and
(2004).
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on quantum phenomena occurring in p-wave fermions, Hulet, Phys. Rev. Lett. 125, 263402 (2020).
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We acknowledge helpful and insightful discussions
[16] S. A. Bender, K. D. Erker, and B. E. Granger, Phys. Rev. Lett.
with Joachim Brand that significantly contributed to the 95, 230404 (2005).
development of this Letter. [17] Y. Sekino, S. Tan, and Y. Nishida, Phys. Rev. A 97, 013621
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This work has been funded by Grants No. PID2020- [18] X. Cui, Phys. Rev. A 94, 043636 (2016).
114626GB-I00 and PID2020-113565GB-C21 by
MCIN/AEI/10.13039/5011 00011033 and "Unit of

Supplementary Material

Explicit expressions for the eigenvalues of the OBDM

Even N
 6

8
N =2, λk± = (20)
[π(2k − 1)]2
 
8 1
N =4, λk± = 48 − + (21)
[π(2k − 1)]4 [π(2k − 1)]2
 
384 48 1
N =6, λk± = 120 − + (22)
[π(2k − 1)]6 [π(2k − 1)]4 [π(2k − 1)]2
 
46080 5760 120 1
N =8, λk± = 224 − + − + (23)
[π(2k − 1)]8 [π(2k − 1)]6 [π(2k − 1)]4 [π(2k − 1)]2
 
10321920 1290240 26880 224 1
N = 10 , λk± = 360 − + − + (24)
[π(2k − 1)]10 [π(2k − 1)]8 [π(2k − 1)]6 [π(2k − 1)]4 [π(2k − 1)]2

Odd N

24
N =3, λk± = (25)
[π2k]2
 
24 1
N =5, λk± = 80 − + (26)
[π2k] 4 [π2k]2
 
1920 80 1
N =7, λk± = 168 − + (27)
[π2k]6 [π2k]4 [π2k]2
 
322560 13440 168 1
N =9, λk± = 288 − + − + (28)
[π2k]8 [π2k]6 [π2k]4 [π2k]2

Eigenstates of the OBDM in a box of size L

Even N
The pairs of eigenstates with eigenvalue λk± of the OBDM in a box of size L are
v
(2k − 1)πx
t2  
χk+ (x) = sin , (29)
L L
v
(2k − 1)πx
t2  
χk− (x) = cos . (30)
L L

Defining plane waves ψk+ (x) and ψk− (x) as

1
1
ψk+ (x) ≡ p χk+ + iχk− = p e i(2k−1)πx/L , (31)
2 L
1
1 −i(2k−1)πx/L
ψk− (x) ≡ p χk+ − iχk− = p e , (32)
2 L
and since the eigenproblem to be solved is linear and χk+ (x) and χk− (x) share the same eigenvalue λk± , we prove that
the plane waves ψk+ (x) and ψk− (x) are also eigenstates with eigenvalue λk± . The plane waves just defined have allowed
momenta k = ±(2k − 1)π/L .

Odd N
The prove for an odd number of fermions is really similar but now the eigenstates with eigenvalue λk± of the OBDM are
v  
t2 2kπx
χk+ (x) = sin , (33)
L L
v  
t2 2kπx
χk− (x) = cos . (34)
L L
 7

Thus, the defined plane waves are

1
1
ψk+ (x) ≡ p χk+ + iχk− = p e i2kπx/L , (35)
2 L
1
1 −i2kπx/L
ψk− (x) ≡ p χk+ − iχk− = p e , (36)
2 L
which, by the same reasoning as in the even case, are also eigenstates of the OBDM with eigenvalue λk± . The plane waves
in the odd case have allowed momenta k = ±2kπ/L .