Fermionic properties of two interacting bosons in a two-dimensional harmonic trap
Pere Mujal Artur Polls Bruno Juliá-Díaz
cond-mat.quant-gas





§ INTRODUCTION

In one dimension, the Bose–Fermi mapping <cit.> theorem establishes a relation between 
the ground state energy and the wave function of strongly interacting bosons 
with those corresponding to noninteracting fermions in the same trapping 
potential. Both systems have the same energy, and the ground 
state wave function of the interacting bosonic system can be obtained by 
symmetrizing the noninteracting fermionic one by taking the absolute value. Several 
works have discussed the onset of the Tonks–Girardeau phase in 
one dimension <cit.>.

In contrast, in two or more dimensions, the theorem does not apply because it is 
based on the  principle, valid in one dimension, that if a particle is fixed at a point, the space 
becomes completely separated in two pieces <cit.>. Therefore, another 
particle cannot access the whole space without encountering the one that is 
fixed, which is not the case with more dimensions. In contrast, the mechanism 
described by Girardeau whereby the particles can avoid feeling the interaction  remains a possibility. In our previous work <cit.>, 
considering two, three, and four particles <cit.> in a two dimensional harmonic trap <cit.>, we showed that interacting bosons avoid feeling the interaction by becoming 
correlated in such a way that the probability of two particles being at the same 
position vanishes <cit.>. 

In the present paper, we compare the numerical calculations for the ground state 
of two interacting bosons in two dimensions with a short-range interaction with the properties 
obtained from the analytical wave functions that describe two noninteracting 
bosons, two noninteracting fermions, and the corresponding symmetrized wave function. 
We show that some of the properties of the interacting two-boson system resemble 
the noninteracting fermionic ones. This work is organized in the following way. 
First, in Section <ref>, we present the considered Hamiltonian. In Section <ref>, 
we present the analytic wave functions mentioned above. In Section <ref>, we discuss 
how the ground state energy of the two-boson system changes as we vary the interaction. 
The corresponding effect on the density profiles and pair correlations is 
presented in Section <ref>. In Section <ref>, we briefly describe our 
numerical methods.  Finally, in Section <ref>, we summarize the main 
conclusions of our work. 

 



§ THE HAMILTONIAN



The system of two interacting identical bosons in a two-dimensional isotropic 
harmonic trap is described by the Hamiltonian,


    ℋ=ℋ_0
    + V(|x⃗_2-x⃗_1|)


where the noninteracting part of the Hamiltonian reads


    ℋ_0=∑_i=1^2 
     ( -ħ^2 /2m∇^2_i+1/2mω^2 |x⃗_i|^ 2 )  ,


and the interaction is modeled by means of a Gaussian-shaped potential <cit.>,


    V(x⃗_1,x⃗_2)= g/π s^2e^-|x⃗_2-x⃗_1|^2/s^2 .


The interaction strength is controlled by g, while s defines the interaction 
range. In this work, we always consider purely repulsive interactions, g≥ 0. A comparison between zero-range models and the Gaussian-shape potential has been studied in <cit.>. In the present work, we concentrate on the dependence on both  the strength g and the range s of the interaction. Notice, however, that we will not consider the regime of large scattering length, as in our case, with a purely repulsive interaction, the scattering length will be of the order of the range s <cit.>. The Hamiltonian can be split into the center-of-mass part and the relative part,


    ℋ_ cm   =   
    -ħ^2/2ℳ∇^2_R⃗+1/2ℳω^2 R^ 2 , 
    ℋ_ r   =   -ħ^2/2μ∇^2_r⃗
    +1/2μω^2 r^ 2+g/π s^2e^-r^2/s^2


where R⃗≡1/2(x⃗_1+x⃗_2) is the center-of-mass 
coordinate, r⃗≡x⃗_1-x⃗_2 is the relative coordinate, 
ℳ≡ 2m is the total mass, and μ≡ m/2 is the reduced mass. 
Hereafter, we will use harmonic oscillator units, i.e. the length in units of 
√(ħ/ (mω)) and the energy in units of ħω. The two 
pieces of the Hamiltonian in those units read


    ℋ_ cm   =   
    -1/4∇^2_R⃗+R^ 2=𝒦_ cm+𝒱^cm_ ho
    ℋ_ r   =   -∇^2_r⃗
    +1/4 r^ 2+g/π s^2e^-r^2/s^2
    =𝒦_r+𝒱^r_ ho+𝒱_ int


where we identify each part: the center-of-mass kinetic energy, 
𝒦_cm=-(1/4)∇^2_R⃗, the relative kinetic energy, 
𝒦_r=-∇^2_r⃗, the center-of-mass harmonic potential, 
𝒱^cm_ho=R^ 2, the relative part of the harmonic potential, 
𝒱^r_ho=(1/4)r^ 2, and the interaction term, 
𝒱_int=(g/(π s^2))exp(-r^2/s^2), which has been properly 
transformed to the harmonic oscillator units.





§ ANALYTIC WAVE FUNCTIONS TO COMPARE WITH THE GROUND STATE



We are interested in understanding how the system changes its structure as we 
increase the interaction strength. In particular, we want to discern whether any 
fermionization takes place in the strong interaction limit, akin to the 1D case, 
despite the fact that the Bose–Fermi mapping theorem does not apply in 2D. To 
enlighten this discussion, we will compare the properties of our numerically 
obtained two-boson system with those of  (1) the wave function of the ground state 
of two bosons in the noninteracting limit, which will correspond to first 
order perturbation theory, (2) the wave function of the ground state 
of two fermions in the noninteracting limit, and (3) the wave function obtained by 
symmetrizing the previous one by taking its absolute value. We will use polar 
coordinates to express the wave functions.



 §.§ First-order Perturbation Theory


The ground state of two noninteracting bosons under the Hamiltonian 
in (<ref>) is the nondegenerate state, 


    Ψ_B(R,φ_R,r,φ_r)=1/πe^-R^2-r^2/4,


which is a state with zero angular momentum. Its energy is computed taking into account 
that it is an eigenfunction of the Hamiltonian in the noninteracting case, 
ℋ_0Ψ_B=2Ψ_B. The expectation value of the 
interaction term is


    ⟨𝒱_ int⟩_Ψ_B=
    g/π s^24π^2∫_0^∞ R  dR ∫_0^∞ r  dr 1/π^2e^-2R^2-r^2/2e^-r^2/s^2=g/π (2+s^2).


The total energy reads


    E_B=2+g/π (s^2+2),


which is the first-order perturbation theory prediction for the energy of the 
system <cit.>. It is worth mentioning that the center-of-mass wave 
function contained in Equation (<ref>) is an eigenfunction of ℋ_ cm 
with an eigenvalue of 1 energy unit. 



 §.§ The Non-Interacting Two-Fermion System


For the noninteracting two-fermion system, the ground state would be 
two-fold degenerate, with a zero center-of-mass angular momentum, and the 
relative angular momentum equal to 1 or -1:


    Ψ^±_F(R,φ_R,r,φ_r)=1/π√(2)e^-R^2-r^2/4r e^± iφ_r .


The previous states are also eigenstates of 
ℋ_0, ℋ_0Ψ^±_F=3Ψ^±_F. The expectation 
value of the interaction energy in this case is


    ⟨𝒱_int⟩_Ψ_F=
    g/π s^24π^2∫_0^∞ R  dR ∫_0^∞ r  dr 1/2π^2e^-2R^2-r^2/2r^2e^-r^2/s^2=gs^2/π (2+s^2)^2,


and the total energy is


    E_F=3+gs^2/π (s^2+2)^2 .


Let us note that this contribution vanishes, as it should, for zero-range interactions, s=0. 



 §.§ Bosonized Two-fermion System


If we symmetrize the previous wave functions by taking their absolute value, 
|Ψ^±_F|=Ψ_|F|, we obtain bosonic wave functions which, as in 1D, do not allow
bosons to sit at the same position. Notice that both fermionic wave functions, (<ref>), are transformed into the same symmetric one:


    Ψ_|F|(R,φ_R,r,φ_r)=1/π√(2)e^-R^2-r^2/4r,


which has no angular dependence. The main effect of this symmetrization is that Ψ_|F| 
is not an eigenfunction of ℋ_0, and the expectation value of the energy in the 
noninteracting case is ⟨ℋ_0⟩_Ψ_|F|=5/2. The interaction energy 
is the same as before, ⟨𝒱_int⟩_Ψ_|F|=(g s^2)/(π (s^2+2)^2), 
and the expectation value of the total energy in this case is


    E_|F|=5/2+gs^2/π (s^2+2)^2 .


The contribution of ⟨ℋ_0 ⟩ to the previous expectation 
value, 5/2, is not equally distributed between the kinetic and harmonic 
potential energy. Therefore, they do not fulfill the virial theorem, which is not necessary since Ψ_|F| is not an eigenfunction of ℋ_0. The expected values for the center-of-mass kinetic and harmonic 
potential energies are 1/2 each one, in agreement with the fact that the center-of-mass 
part of Ψ_|F| is an eigenfunction of ℋ_ cm and it therefore
verifies the virial theorem, 
⟨𝒦_ cm⟩_Ψ_|F|=⟨𝒱^ ho_ cm⟩_Ψ_|F|=1/2.

For the kinetic energy of ℋ_ r , we have to apply to the wave function the operator


    𝒦_r=-∂^2/∂ r^2
    -1/r∂/∂ r-1/r^2∂^2/∂φ^2_r ,

which results in


    𝒦_rΨ_|F|=-e^-R^2-r^2/4(4-8r^2+r^4)/4π√(2)r .


The expectation value then is


    ⟨𝒦_r⟩_Ψ_|F|=
    -4π^2∫_0^∞ r dr ∫_0^∞ R dR e^-2R^2-r^2/2(4-8r^2+r^4)/8π^2=1/2,


and, for the harmonic potential relative energy,


    ⟨𝒱^r_ ho⟩_Ψ_|F|=
    4π^2∫_0^∞ r dr ∫_0^∞ R dR 1/2π^2e^-2R^2-r^2/2r^2r^2/4=1.


To sum up, the total kinetic energy is ⟨𝒦⟩_Ψ_|F|=1, 
and the total harmonic potential energy is ⟨𝒱_ ho⟩_Ψ_|F|=3/2. 

Let us stress that Ψ_|F| can be shown to be the best variational wave function, 
in the strong interacting limit, of the form


    Φ_a(R,φ_R,r,φ_r)=1/π√(2^aΓ(1+a))e^-R^2-r^2/4r^a.

For a=1, we recover Ψ_|F|=Φ_1. The set of functions are eigenfunctions of 
the center-of-mass part, H_ cmΦ_a=1Φ_a. The relative kinetic energy 
is independent of the parameter a and can be shown as ⟨𝒦_r⟩_Φ_a=1/2, 

which coincides with ⟨𝒦_r⟩_Ψ_|F|=⟨𝒦_r⟩_Φ_a. 
The expected value of the relative harmonic oscillator potential energy is 
⟨𝒱^r_ ho⟩_Φ_a=1+a/2. The interaction energy reads


    ⟨𝒱_ int⟩_Φ_a=
    g(1+2/s^2)^-a/π(2+s^2),


which tends to zero when s→ 0 only if a≥ 1. The total variational 
energy depending on a and s is


    E(a,s)_Φ_a=1+ 1/2+1+a/2+g(1+2/s^2)^-a/π(2+s^2).


The energy minimum, in the short-range limit (s→ 0) and for strong 
interactions (g→∞), is reached when a=1, which is precisely the 
case Ψ_|F|=Φ_1 and the energy is E(a=1,s→ 0)_Φ_1=E_|F|(s→ 0)=5/2.







§ THE INTERACTION EFFECT IN THE GROUND STATE ENERGY




 §.§ The Energy Contributions


The ground state energy of the interacting two-boson system is obtained numerically 
following the procedure described in <cit.>, an exact diagonalization 
of the Hamiltonian using the ARPACK implementation of the Lanczos algorithm. Here we 
supply additional information by splitting the total energy of the ground state 
into the kinetic, the harmonic potential, and the interaction contributions, in 
order to study, separately, the effect of increasing the interaction strength.



In Figure <ref>, we see that, in the noninteracting limit, g=0, 
since the wave function of the ground state is the one given in (<ref>), the 
total energy is 2, arising from two equal contributions from the kinetic and 
potential energy. Increasing the interaction strength slightly affects  the 
interaction part of the energy. At the beginning, it starts to increase, but 
it remains mostly constant, decreases for larger interaction strengths, and 
becomes approximately 0. This reflects that the particles avoid feeling the 
interaction by building dynamical quantum correlations in the wave function, 
as discussed in <cit.>. The most relevant contribution to the energy 
increase for a larger interaction strength comes from the harmonic potential part. 
The kinetic energy remains approximately constant and equal to 1. Therefore, 
we can say that the two particles avoid feeling the interaction by separating 
one from the other. This reflects in an increase of the harmonic potential energy 
because they are further away from the center of the trap. This is consistent with 
the density profiles and pair correlation functions computed in <cit.> 
for different interaction strengths.





 §.§ Exploring the Strongly Interacting Limit for a Short-range Interaction


In the strong interacting limit, in Figure <ref>, we see that 
the total energy, the kinetic energy, and the potential energy tend to values 
corresponding to the symmetrized wave function of Equation (<ref>). Clearly, 
the way the energy is distributed for the strongly interacting two-boson system in 
the zero-range limit does not coincide with the noninteracting two-fermion system 
but with the energy decomposition provided by Ψ_|F|. In addition, the value of the ground state energy seems to tend to 2.5 energy units, 
which is the expected value of the energy for the variational wave function Ψ_|F|.

In Figure <ref>, we see how the ground state energy computed numerically 
goes from E_B for g≃0 to be very close to E_|F| when g becomes large for 
a small fixed range, s=0.01. E_B, E_F and E_|F| depend linearly on the 
interaction strength g (see, respectively, Equations (<ref>), (<ref>), (<ref>)). 
However, the slope of the lines corresponding to E_F and E_|F| in Figure <ref> 
is positive but very small, due to the small value of s. 






 §.§ Exploring the Short-Range Limit for a Strong Interaction Strength


The energy dependence on the range of the interaction is shown in Figure <ref>, 
where we compare the ground state energy of the interacting two-boson system computed 
numerically with the expected values given by our analytic wave functions. For a 
fixed and strong interaction strength, g=20, reducing the range of the interaction 
results in a decrease of the ground state energy of the system. The same kind of behavior 
is observed for Ψ_F and its symmetrized wave function, Ψ_|F|, since their 
dependence on the interaction is the same and the shift in energy between them is due 
to the noninteracting part of the Hamiltonian. In the short-range limit, their interaction 
energy tends to zero, and E_0 approaches E_|F|. Differently, the interaction energy 
of Ψ_B does not vanish in the zero-range limit. In fact, the energy for this wave 
function increases for decreasing s.





In Figure <ref>, we observe that not only is E_0 very close to E_|F| 
when s → 0, but the way the energy is distributed also coincides; i.e., 
the expected values of the kinetic energy, the interaction energy, and the harmonic 
potential energy for a strong interaction in the short-range limit for the ground state 
of the interacting two-boson system tend to values 
 computed for Ψ_|F|.

 In addition, we show that the most relevant contributions to the energy when the range 
s varies comes from the harmonic potential energy, because the kinetic and interaction 
parts are mainly independent of s and remain constant in Figure <ref>.





§ THE DENSITY PROFILE AND THE TWO-BODY CORRELATIONS



In this section, we will compare the density profile and the pair correlation 
function of the ground state of the trapped two boson system, with and without 
interactions, with the ones provided by Ψ_F and Ψ_|F|.

The density operator, acting on a system of N particles, is defined as


    ρ̂(x⃗)≡1/N∑_i=1^N δ(x⃗-x⃗_i),

and the pair correlation operator reads


    η̂(x⃗,x⃗')≡1/N(N-1)∑_i=1^N∑_j≠ i^N δ (x⃗-x⃗_i)δ(x⃗'-x⃗_j)


where both operators are normalized to unity. Since we consider the case of two 
identical particles, we compute the density profile and the pair correlation 
function for a given state Ψ(x⃗_1,x⃗_2), respectively, as follows:
ρ(x⃗)=∫ dx⃗_2 | Ψ(x⃗,x⃗_2) |^2, and 
η (x⃗,x⃗') = |Ψ (x⃗,x⃗') |^2 . 

Using Cartesian coordinates, the wave functions of Section <ref> read


    Ψ_B (x⃗_1,x⃗_2)   =   1/πe^-1/2(x_1^2+y_1^2+x_2^2+y_2^2) , 
    Ψ_F^± (x⃗_1x⃗_2)   =   1/π√(2)e^-1/2(x_1^2+y_1^2+x_2^2+y_2^2)((x_1-x_2)± i(y_1-y_2)) ,
    Ψ_|F|(x⃗_1x⃗_2)    =   1/π√(2)e^-1/2(x_1^2+y_1^2+x_2^2+y_2^2)√((x_1-x_2)^2+(y_1-y_2)^2) .


The corresponding density profiles and pair correlations read


    ρ_B(x⃗)                       =   1/πe^-(x^2+y^2)
    η_B(x⃗,x⃗')   =   1/π^2e^-(x^2+y^2+x'^2+y'^2)
    ρ_F(x⃗)                       =   ρ_|F|(x⃗)=1/2πe^-(x^2+y^2)(1+x^2+y^2) 
    η_F(x⃗,x⃗')   =   η_|F|(x⃗,x⃗')=1/2π^2e^-(x^2+y^2+x'^2+y'^2)((x-x')^2+(y-y')^2)  .

Note that both the fermionic and bosonized wave functions give the same density and pair correlation 
as |Ψ_|F|(x⃗_1,x⃗_2)|^2=|Ψ_F^± (x⃗_1,x⃗_2)|^2. 

From the pair correlation function and the density profile, we compute the probability 
of finding a particle at a distance X≡√(x^2+y^2) once we have found the 
other at the origin, that is,


    P(X;0)≡η(x⃗,0⃗)/ρ(0⃗),


which for the two previous discussed cases is, respectively, 
P_B(X;0)=η_B(x⃗,0⃗)/ρ_B(0⃗)=(1/π) e^-X^2, and 
P_F(X;0)=η_F(x⃗,0⃗)/ρ_F(0⃗)=(1/π)e^-X^2X^2.

In Figure <ref>a, we compare the density profiles obtained 
numerically for the ground state of trapped interacting two-boson system with 
the density profile corresponding to the noninteracting case and to the 
noninteracting two-fermion system. We show that, given an interaction range, 
there is an interaction strength such 
that the density profile of the interacting two-boson system is very well approximated 
by the two-fermion density profile. The smaller the interaction range, s, is, in which the density profiles coincide, the 
greater the interaction strength, g, will be. 
Therefore, in the short-range limit and for strong interactions, 
the density profile of two interacting bosons tends toward the noninteracting two-fermion profile.



In the case of the probability of finding a particle in space once we have 
found the other at the origin, we observe, in Figure <ref>b, 
that the numerically computed ones for the interacting two-boson system 
resemble the corresponding   two noninteracting fermions. However, in this case, 
the maximum peak does not coincide, and it is closer to the center of the trap 
for two noninteracting fermions. The probability of finding the two bosons at 
the same position, i.e., at X=0, tends toward zero when the interaction range 
vanishes and the interaction strength increases, as expected.

The effect of decreasing the interaction range for a fixed interaction strength 
is shown in Figure <ref>. Regarding the density profile, in panel (a), 
the interaction strength is strong enough in all cases so that they are closer 
to ρ_F than to ρ_B. The density peak increases 
 for a smaller range, 
although there is practically no effect in the density profile for X>1, and 
all of the numerical profiles fit well with ρ_F. In Figure <ref>b, we observe that the effect of decreasing the range is such that the 
maximum in the probability approaches the center of the trap and the maximum 
value increases. However, in any case the probability vanishes at X=0, as 
it does for ρ_F, which would happen for greater g values and in the 
short-range limit, as  shown in Figure <ref>b.






§ NUMERICAL METHOD


The numerical method was previously developed by the authors and is described in 
detail in <cit.>. First, we write the many-body Hamiltonian in 
a second quantized form. Then we define the set of single particle states, which 
are taken to be the lowest energy eigenstates of the single particle Hamiltonian. 
Once we fix the number of particles, we can define the corresponding Fock basis for 
the problem. 
The final results should be independent of the number of single particle states. In 
the numerical results reported in this paper, we have used up to 200 single particle states. 
The resulting Hamiltonian is then 
diagonalized using the ARPACK implementation of the Lanczos algorithm in order 
to compute the low-energy spectrum of the system and, in particular, the ground state.




§ CONCLUSIONS



In two dimensions, the ground state energy of two strongly interacting bosons 
in a harmonic trap for a short-range interaction is not equal to the noninteracting 
two-fermion system in the same potential, as it was in one dimension. However, the wave function resulting from 
symmetrizing the corresponding noninteracting fermionic ground state is found to be a very good variational trial wave function. It provides an upper-bound very close to the ground state energy obtained by exact diagonalization in the strong interacting and short-range limit. Even more, the distribution of the energy between 
the kinetic and harmonic potential parts in that limit also coincides. The increase of the energy with the interaction strength comes, mostly, from the harmonic potential 
contribution to the energy since the kinetic energy remains almost constant and the interaction 
energy approximately zero. The bosons avoid feeling the interaction by being 
more separate and, therefore, further from the center of the trap, which is 
also reflected in the density profiles obtained. The correlations induced by 
the interaction cause the density profile of the strongly interacting bosons to tend toward 
 the noninteracting two-fermion profile. For the pair correlation function, 
we have found qualitatively the same kind of behavior.



 






The authors thank S. Pilati for his comments and discussions about the scattering length and the regularized pseudopotential in two dimensions. The authors acknowledge financial support of grants 2014SGR-401 from Generalitat de Catalunya and FIS2014-54672-P from the MINECO (Spain). P.M. was supported by an FI grant from Generalitat de Catalunya.


Main ideas were conceived by Pere Mujal, Artur Polls, and Bruno Juliá-Díaz. Pere Mujal and Artur Polls performed all the calculations. Pere Mujal wrote a first version of the manuscript. Pere Mujal, Artur Polls and Bruno Juliá-Díaz discussed the results and wrote the final version of the manuscript.




The authors declare no conflict of interest.

 




















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[Mujal(2017)]PRA96.043614
Mujal, P.; Sarlé, E.; Polls, A.; Juliá-Díaz, B. Quantum correlations and degeneracy of identical bosons in a two-dimensional harmonic trap. Phys. Rev. A 2017, 96, 043614.


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[Jeszenski(2018)]Jeszenski Jeszenski, P.; Cherny, A.Y.; Brand, J. The s-wave scattering length of a Gaussian potential. arXiv 2018, arXiv:1802.07063































