Abstract
Fermionization is what happens to the state of strongly interacting repulsive bosons interacting with contact interactions in one spatial dimension. Crystallization is what happens for sufficiently strongly interacting repulsive bosons with dipolar interactions in one spatial dimension. Crystallization and fermionization resemble each other: in both cases – due to their repulsion – the bosons try to minimize their spatial overlap. We trace these two hallmark phases of strongly correlated onedimensional bosonic systems by exploring their ground state properties using the one and twobody density matrix. We solve the Nbody Schrödinger equation accurately and from first principles using the multiconfigurational timedependent Hartree for bosons (MCTDHB) and for fermions (MCTDHF) methods. Using the one and twobody density, fermionization can be distinguished from crystallization in position space. For N interacting bosons, a splitting into an Nfold pattern in the onebody and twobody density is a unique feature of both, fermionization and crystallization. We demonstrate that this splitting is incomplete for fermionized bosons and restricted by the confinement potential. This incomplete splitting is a consequence of the convergence of the energy in the limit of infinite repulsion and is in agreement with complementary results that we obtain for fermions using MCTDHF. For crystalline bosons, in contrast, the splitting is complete: the interaction energy is capable of overcoming the confinement potential. Our results suggest that the spreading of the density as a function of the dipolar interaction strength diverges as a power law. We describe how to distinguish fermionization from crystallization experimentally from measurements of the one and twobody density.
Introduction
The physics of the ultracold Bose gas in one spatial dimension is strongly different from that of its threedimensional counterpart^{1,2}. In one spatial dimension, quantum effects are much more prominent. When the interactions are strong, quantum fluctuations are enhanced. Experimentally, in cold atom systems, the dimensionality can be manipulated using a tight transversal confinement that essentially freezes the radial degrees of freedom^{3,4}. Such quasione dimensional systems display intriguing physics that cannot be realized for threedimensional systems: Fermionization, occurs for strongly interacting bosons with contact interactions^{5,6,7,8,9,10,11,12} and crystallization emerges for sufficiently strongly interacting bosons with dipoledipole interactions^{13,14,15,16,17,18,19}. For bosons with contact interactions, fermionization leads to the formation of the TonksGirardeau (TG) gas when the interaction strength tends to infinity. This is a consequence of the BoseFermi mapping^{7,20,21,22} which implies that strongly interacting bosons and noninteracting spinless fermions have the same (onebody) density in position space. With increasing interaction strength, not only the density, but also the energy of fermionized bosons saturates to the energy of noninteracting fermions.
In the case of dipolar interactions, the remarkable phenomenon of crystallization occurs when the interaction strength is sufficiently large. Bosons interacting via a dipoledipole interaction potential have become the primary cold atom system to investigate the manybody physics triggered as consequence of longrange interactions^{23,24,25,26,27,28}. The longranged and anisotropic nature of the dipolar interaction potential results in a variety of interesting effects and phenomena^{29}, like crystallization in one and twodimensional systems^{13,14,15,16,17,18,30,31,32}, that are completely different from the emergent phenomena in the case of strong contact interactions. Crystallization is a consequence of the repulsive and longranged tail of the dipolar interactions dominating the physics^{33,34}: the bosons maximally separate and minimize their spatial overlap. Unlike in the fermionization of bosons with strong contact interactions, the energy of crystallized bosons does not saturate. We note here that it is formally possible to define and measure an order parameter which is a function of the eigenvalues of the reduced onebody density matrix that allows to unequivocally identify the crystal phase of onedimensional dipolar bosons^{34}. Furthermore, we note that the formation of a crystal state is a generic feature of manybody systems of particles with longranged interactions. Fermions with longranged interactions, for instance, form a socalled Wigner crystal^{19,35}.
In this work, we analyze the differences between fermionized and crystallized bosons’ wavefunctions using the energy as well as the onebody and twobody reduced density matrix. We demonstrate how fermionization can be distinguished from crystallization by quantifying the (experimentally accessible) spread of the onebody and twobody densities. The different spreading characteristics of the onebody and twobody densities for fermionized as compared to crystallized bosons are a direct consequence of the different behavior of the energy as a function of interaction strengths. For dipolar interactions the energy as a function of interaction strength is unbounded; this is in stark contrast to the bounded energy as a function of interactions for contact interactions. Current experimental setups, for instance, for Erbium^{36}, ErbiumErbium molecules^{37}, or SodiumPotassium molecules^{38}, enable the experimental exploration of systems with dominant dipoledipole interactions needed for probing the physics of crystallization.
The (momentum) densities of bosons with dipolar interactions have been compared to those of fermions with dipolar interactions in Ref. ^{18}. Going beyond Ref. ^{18}, we compare and quantify the spreading of the full density matrices of bosons with dipolar interactions to the spreading of the density matrices of bosons with contact interactions. We note that Ref. ^{39} discusses and compares the physics of spin1/2 fermions with contact and with longranged interactions in lattices using a Hubbard description. Our work complements the findings in Ref. ^{39} by providing a comparison of singlecomponent (“spin0”) bosons with contact and longranged dipolar interactions in continuous space without a lattice and without resorting to a Hubbardtightbindingdescription.
Fermionization and crystallization entail the breakdown of meanfield approaches like the timedependent GrossPitaevskii (GP) equation^{40,41,42}. To go beyond the GP approximation, multiconfigurational methods are employed^{43,44,45,46,47,48,49}. Variational calculations using parametrized Gaussian functions as singleparticle states have been successfully applied to investigate the crystallization of few particles in two dimensions^{30,50,51}. Here, we use the multiconfigurational timedependent Hartree for bosons (MCTDHB)^{43,44} and fermions (MCTDHF)^{46} methods implemented in the MCTDHX software package^{46,52,53,54} to compute the ground state of the fewparticle Schrödinger equation, see Ref. ^{49} for a Review. While the MCTDHB method aims at solving the timedependent Schrödinger equation for a manybody system, using imaginary timepropagation provides the groundstate of the system variationally, equivalent to the work of Ref. ^{55}.
We illustrate our findings with computations for N = 4 bosons in a parabolic trapping potential and trace the complete range of dipolar and contact interaction strengths by obtaining highly accurate results with MCTDHB.
This paper is structured as follows: in Sec. 2, we discuss the Hamiltonian and quantities of interest, in Sec. 3, we introduce the numerical method, MCTDHB, that we use for obtaining solutions of the fewbody Schrödinger equation, in Sec. 4 we analyze fermionized and crystallized bosonic fewbody states and discuss how they can be sorted from each other and we conclude our paper in Sec. 5. Results for other observables and an assessment of the accuracy of our computations with the exact diagonalization and MCTDHF approaches are collected in the Appendices 6 and 7.
Hamiltonian, One and TwoBody Density
In order to discuss the stationary properties of the ground state (GS) of crystalline and fermionized bosons, we consider the timeindependent manybody Schrödinger equation,
Here, Ψ〉 is the manybody ground state, E its energy, and \(\hat{H}\) the Nparticle Hamiltonian in dimensionless units^{56},
where we set \(V({x}_{i})=\frac{1}{2}{x}_{i}^{2}\) to be the external harmonic trap. The term W(x_{i} − x_{j}) is the interaction potential. All quantities are dimensionless and expressed in harmonic oscillator units. To ensure that the system is in the quasi1D regime, we assume strong confinement in the transversal direction, providing a cigarshaped atomic density. The contact interactions read,
where λ is the interaction strength determined by the scattering length a_{s} and the transverse confinement frequencies^{57}. For longranged dipolar interactions, we have
where g_{d} is the dipolar interaction strength and α is a shortrange cutoff to avoid the divergence at x_{i} = x_{j}. Repulsive interactions could be obtained in a quasi1D BEC by imposing an external magnetic field to align all the dipole moments of the atoms^{18}. This simple approximation to the onedimensional dipoledipole interaction potential in Eq. (4) is justified for the moderate to large interaction strengths and large interparticle distance with respect to the harmoniclength of the transversal confinement^{14,15,18,19,58}, that we focus on in the present work. For such interaction strengths the dipoledipole interaction potential is wellapproximated by the x_{i} − x_{j}^{−3} tail in Eq. (4), see^{59}. Moreover, we have verified the consistency of the approximation in Eq. (4) for the same choice of cutoff parameter, α = 0.05, by a direct comparison to a dipoledipole interaction augmented with an additional contact interaction potential, see Ref. ^{34}. A rigorous discussion of the dipoledipole interaction potential in one and two spatial dimensions can be found in Ref. ^{59}. Here, for the sake of simplicity, we will focus on quasionedimensional systems, with N = 4 interacting bosons for all our calculations and consider repulsive interactions, i.e., λ > 0 and g_{d} > 0, exclusively.
In the following we discuss the reduced onebody density matrix, defined as
Its diagonal,
is simply the onebody density. As a precursor of correlation effects that may be present in the state Ψ〉 of the system, we use the eigenvalues ρ_{i}^{(NO)} of the reduced onebody density matrix ρ^{(1)} in Eq. (5). For this purpose, we write ρ^{(1)} in its eigenbasis:
The eigenvalues ρ_{i}^{(NO)} and eigenfunctions Φ_{i}(x) are referred to as natural occupations and natural orbitals, respectively. If only a single eigenvalue ρ_{i}^{(NO)} is macroscopic, then the state Ψ〉 describes a BoseEinstein condensate^{60}. The case when multiple eigenvalues ρ_{i}^{(NO)} are comparable to the number of particles N is referred to as fragmentation^{43,44,53,54,61,62,63}.
In the following, we will also use the twobody density ρ^{(2)} to characterize crystallization and fermionization. It is defined as
The twobody density quantifies the probability to detect two particles at positions x_{1} and x_{2}.
Numerical Method
The computation of the exact manybody wave function is a difficult problem. To attack the manybody Schrödinger equation, Eq. (1), we use the timedependent Schrödinger equation,
with a Wick’s rotation t → −iτ, i.e., a propagation with imaginary time. We expand the manybody wavefunction Ψ〉 of N interacting bosons in a complete set of timedependent permanents \(\overrightarrow{n};t\rangle ={n}_{1},\,\mathrm{...,}{n}_{M};t\rangle \) with at most M singleparticle states or orbitals. The MCTDHB ansatz for the manybody wave function is thus
Here, the permanents \(\overrightarrow{n};t\rangle \) are symmetrized bosonic manybody states that are also referred to as “configurations”. The sum in Eq. (10) runs on all configurations \(\overrightarrow{n}\) of N particles in M orbitals. The number of permanents and coefficients \({C}_{\overrightarrow{n}}(t)\) is N_{conf} = \((\begin{array}{c}N+M1\\ N\end{array})\). In second quantized representation the permanents are given as
Here \({\hat{b}}_{k}^{\dagger }(t)\) is the bosonic creation operator which creates a boson in the timedependent single particle state \({\varphi }_{k}(x,t)\). Equation (10) spans the full Nbody Hilbert space in the limit of M → ∞. For practical computations, we restrict the number of orbitals and require the convergence of our observables, like the one and twobody density matrix, with respect to the number of singleparticle states M.
A set of coupled equations of motion for, both, the timedependent expansion coefficients \({C}_{\overrightarrow{n}}(t)\) and the timedependent orbitals \({\varphi }_{k}(x,t)\) are obtained by requiring the stationarity of the action of the timedependent Schrödinger equation^{43,44} under variations of \({C}_{\overrightarrow{n}}(t)\) and \({\varphi }_{k}(x,t)\). Using MCTDHB, both, the coefficients and orbitals are variationally optimized^{64}. MCTDHB is thus fundamentally different from exact diagonalization, i.e., an ansatz built with timeindependent orbitals. It can be demonstrated that MCTDHB delivers solutions of the Schrödinger equation at a significantly increased accuracy in comparison to exact diagonalization approaches when the same number of singleparticle basis states is employed, see Refs. ^{53,65}. for a demonstration with the harmonic interaction model and Appendix A for a demonstration with dipoledipole interactions, i.e., the Hamiltonian in Eqs. (2) and (4). Despite the accuracy of MCTDHB for weakly interacting particles, for strong interactions a large number of orbitals is required to describe the system accurately. In the case λ → ∞, the BoseFermi mapping provides analytical solutions that can be compared to the numerical results. These difficulties to converge MCTDHB results for strong contact interactions may be an instance of the discussion provided in Ref. ^{66}.
We solve the set of coupled MCTDHB equations using the MCTDHX software^{46,52,53,54}. When imaginary time is used, the propagation of an initial guess function converges to the ground state of the system, and the stationary properties of the system can be investigated.
Fermionization vs Crystallization
We now discuss our findings on the fermionized and the crystalline state of parabolically trapped onedimensional ultracold bosons. We first independently characterize fermionization and crystallization from a “manybody point of view”, see Sec. 4.1 and Sec. 4.2, respectively. Thereafter, we investigate how to sort the one, fermionization, from the other, crystallization, in Sec. 4.3. Here and in the following, we used the term “manybody point of view” to highlight that our considerations go beyond an effective singleparticle or meanfield description of the state.
Fermionization
Bosons fermionize when they feel an infinitely repulsive contact interaction in one spatial dimension. For fermionized bosons, the total energy E and the density [Eq. (6)] of the system become exactly equal to the energy and the density of noninteracting spinless fermions, respectively. For our showcase of fewbosons systems (N = 2 to N = 5) in a harmonic trap with frequency one, \(V(x)=\frac{1}{2}{x}^{2}\), the limiting value is thus \({E}_{\lambda \to \infty }^{N}=\frac{{N}^{2}}{2}\).
We start our investigation with the onebody density as a function of the interaction strength λ [Fig. 1(a–b)]. For comparatively weak repulsion, the density is clustered at the center of the trap, but becomes flatter and broader when λ increases. For stronger repulsion, the density gradually acquires modulations and the number of humps finally saturates to the number of bosons in the system; four humps for N = 4 bosons are clearly visible when the interaction strength goes above λ ~ 10. The emergence of N maxima in the density indicates that the TG regime is approached. The density modulations/humps are more pronounced in the center of the trap, where the potential is close to zero. For larger distances from the origin, the humps in the density are less pronounced due to the nonzero value of the confinement potential. Importantly, the outermost density modulation also becomes less pronounced if the number of particles N is increased. See also Appendix B for a direct comparison of the relative height of innermost and outermost peaks for different particle numbers N.
We note that the density’s maxima in the TonksGirardeau regime are distinct but not isolated. We also observe that, once the TG regime is reached, the density does not broaden further with increasing values of λ for all particle numbers. We also provide a direct comparison with the ground state properties of noninteracting fermions computed with the multiconfigurational timedependent Hartree method for fermions (MCTDHF), see Fig. 2. We note that the results for noninteracting fermions can be obtained analytically, i.e., here, we use the heavy MCTDHF method only for the sake of computational convenience.
We now move to discuss the twobody densities ρ^{(2)} of bosons with contact interactions [see Fig. 3(a)]. For weak interaction strength, λ = 0.1, the bosons are clustered near the center, i.e, at x_{1} = x_{2} = 0 [Fig. 3(a)]. As the interaction strength increases, ρ^{(2)} spreads out to the offdiagonal (x_{1} ≠ x_{2}) while the diagonal (\({x}_{1} \sim {x}_{2}\)) is depleted [see Fig. 3(a) for λ = 1].
For stronger repulsion a socalled “correlation hole” in the twobody density forms on the diagonal, ρ^{(2)}(x, x) → 0 [see Fig. 3(a) for λ = 10 and λ = 30]. The probability of finding two bosons at the same position tends towards zero. In the limit of infinite repulsion the correlation hole persists in ρ^{(2)}. In analogy, however, to the boundedness of the energy as a function of the interaction strength, the width of twobody density on its antidiagonal [ρ^{(2)}(x, −x)] is also bounded, i.e., the spread of ρ^{(2)} converges in the fermionization limit when λ → ∞.
Similar to the onebody density, the maxima which are formed in the offdiagonal of the twobody density are distinct but not isolated [see Fig. 3(a) for λ = 10 and λ = 30].
We infer that the correlation hole along the diagonal and the confined spread are the unique signatures of the twobody density of a fermionized state.
Crystallization
For bosons with dipoledipole interactions, crystallization occurs when the longrange tail of the interaction [see Eq. (4)] becomes dominant^{34}: the bosons form a lattice structure which allows them to minimize their mutual overlap. To characterize crystallization we analyze the onebody and twobody density for bosons with dipolar interaction of strength g_{d}. We choose the cutoff parameter α = 0.05 in Eq. (4) such that the effective interaction features the same physical beahavior as the “real” dipolar interaction that additionally contains a contactinteraction term (see Ref. ^{34} for a direct comparison).
We plot the onebody density of N = 4 bosons as a function of g_{d} in Fig. 1(c,d). The system is condensed at the center of the trap for small g_{d}. As g_{d} increases, the density starts to exhibit a fourhump structure (see Fig. 1(c,d) for \({g}_{d}\in [\sim \,1,\sim \,\mathrm{5]}\)) similar to the density observed for the fermionization of bosons with contact interactions [Fig. 1(a)].
This attempted fermionization results from a dominant contribution of the shortrange part of the dipolar interaction potential, see also Ref. ^{18}. However, this fermionizationlike behavior is only a precursor to the crystal transition that takes place when the longrange nature of the interaction starts to dominate the physics of the system for larger interaction strengths [Fig. 1(c,d) for \({g}_{d}\gtrsim 10\)]. For crystallized dipolar bosons at sufficiently large g_{d}, the value of the density at its minima between the humps tends to zero while the spreading of the density profile diverges as g_{d} increases, see Fig. 4. At g_{d} = 30.0, we observe four wellisolated peaks heralding the crystallization of the N = 4 bosons. We collect results for other numbers of bosons (N = 2, 3, 5, 6) with dipoledipole interactions – including the relative height of the peaks in the density that shows that the peaks are wellisolated in comparison to particles with contact interactions – in Appendix B. A comparison of MCTDHB results with exact diagonalization is shown in Appendix A.
We now analyze the twobody density for dipoledipole interactions [Fig. 3(b)]. For small interaction strength, g_{d} = 0.1, the atoms are clustered together at the center of the trap. As g_{d} increases, a correlation hole develops: ρ^{(2)}(x, x) tends to zero [Fig. 3(b) for g_{d} ≥ 1]. Thus, due to the longrange interaction, the probability of finding two bosons in the same place is strongly reduced. In the crystalline phase [Fig. 3(b) for g_{d} ≥ 10]: the bosons escape their spatial overlap entirely and even the offdiagonal peaks of ρ^{(2)} become isolated. We term this behavior the formation of an offdiagonal correlation hole. For crystallized bosons, the spread of the antidiagonal of the twobody density, ρ^{(2)}(x, −x), is diverging as g_{d} is increasing [compare Fig. 3(b) for g_{d} = 10 to Fig. 3(b) for g_{d} = 30].
We assert that the correlation hole along the diagonal and the offdiagonal and the unbounded spreading are the unique signatures of the twobody density of a crystalline state of dipolar bosons.
Sorting crystallization from fermionization
We now discuss how to distinguish fermionized from crystallized manybody states. One clear distinction is given by the spread of the one and twobody densities: for bosons with contact interactions it is bounded, while for bosons with dipoledipole interactions it diverges as a function of the interaction strength. We assert that, (1) the bounded spreading of the density for contact interactions is a consequence of the bounded energy as the interaction strength tends to infinity. Similarly, we assert, (2) that the unbounded spreading of the density for dipoledipole interactions is a consequence of the unbounded energy as the interaction strength g_{d} tends to infinity. To validate the assertions (1) & (2), we quantify the spreading of the density as a function of interactions and plot the position of its outermost peak as a function of the interaction strength in Fig. 5(a) and compare it to the energy in Fig. 5(b), for N = 4.
From fitting the energy in Fig. 5(b) we can infer that the energy as a function of contact interaction strength approaches the fermionization limit exponentially, a power law does not fit as accurately the data. For very large interactions, in the limit of λ^{−1} → 0, our results are in agreement with the analysis in Ref. ^{67}, see Appendix B. For dipolar interactions, the growth of the energy as a function of the interaction strength is fitting well to a power law.
Indeed, the comparison of Fig. 5(a,b) corroborates our assertions (1) & (2), and holds for different number of particles.
We thus conclude that the crystalline phase can be distinguished from the TG regime gas by virtue of the behavior of its density profile as a function of the strength of the interparticle interactions: The width of the density distribution converges for an increasing strength of contact interactions, but it continuously spreads for an increasing strength of longrange interactions [compare Fig. 1(b) with Fig. 1(d) as well as Fig. 5(a) with Fig. 5(b)]. In Appendix B, we demonstrate that the exponent of the power law of the spreading of the density as a function of the strength of the interaction is independent of the particle number N.
In the case of longranged interactions, the unbounded spreading of densities as a function of increasing interaction strength and the formation of wellisolated peaks are in sharp contrast to the bounded spreading of densities and the nonisolated peaks in the case of contact interactions in the TG regime [cf. Figs 1 and 5(a–b)].
We now turn to analyze the eigenvalues of the reduced onebody density matrix, the socalled natural occupations^{60}, as a function of the interaction strength between the particles [Fig. 5(c,d)]. As expected^{8,33,34}, when the value of the interaction strength increases, the occupation of the first natural orbital decreases while the other orbitals start to be occupied. For contact interactions, mostly one natural occupation, n_{1}, dominates, while the other occupations n_{k}, k > 1 remain comparatively small even for large values of λ: depletion emerges as the fermionized state is reached [Fig. 5(c)], see also Ref. ^{8}. For longrange interactions, however, all occupations ρ_{k}^{(NO)} for k ≤ N contribute on an equal footing for large values of g_{d}. This fullblown Nfold fragmentation emerges as the crystal state is reached [Fig. 5(d)], see also Ref. ^{34}. In the crystal state the bosons behave similar to distinguishable particles^{68}, and the particle statistics does not influence the physical observables considered in Fig. 5(b,d): the energy and the natural occupations for bosons and fermions converge to the same values. Thus, the finding of Ref. ^{68} for two particles may be extended to larger number of particles.
The emergence of complete fragmentation is a consequence of longranged interactions and in sharp contrast to the emergent depletion in the case of contact interactions.
Conclusions
In this paper we highlight the key characteristics of the manybody wavefunction that reveal the difference between the fermionized bosons with contact interactions and crystallized bosons with dipolar interactions.
In the case of fermionization, the one(two)body density shows a modulation with a number of maxima corresponding to the number of particles. The maxima are confined but not completely separated. The incomplete separation is a consequence of the representability of momentum distribution of fermionized bosons using a basis set: infinitely many basis states are necessary to accurately resolve the cusp – a fact that is reflected by the depletion of the state which we quantified by the eigenvalues of the reduced onebody density matrix. We found that the peaks in the density as well as the energy as a function of the interaction strength approach the fermionization limit exponentially.
In the case of crystallization, the one(two)body density shows wellseparated peaks whose distances diverge as a function of the interaction strength as a power law. This completed separation is the consequence of the formation of a Mottinsulatoralike manybody state where the “lattice potential” is replaced by the longranged interparticle interactions and the “lattice constant” is dictated by the strength of the interparticle interactions.
We close by stating that all the signatures that distinguish crystalline bosons from fermionized bosons can be measured experimentally using singleshot absorption imaging^{69,70,71,72,73}. From experimental absorption images, the onebody and twobody density are available as averages of many singleshot images. Thus, a direct verification of our results for the spread of the onebody and twobody density can be performed. Furthermore, Refs. ^{34,63} suggest that the natural occupations can be inferred from the integrated variance of singleshot images, at least at zero temperature. It is, of course, an open question how thermal fluctuations affect the variance in absorption images and up to which temperature it is still possible to determine the fragmentation of the system.
Appendix A: Comparison of MCTDHB and exact diagonalization
In this Appendix, we demonstrate that MCTDHB yields solutions to the Schrödinger equation at a larger accuracy as compared to the exact diagonalization approach (ED). As is conventional, we use the eigenfunctions of the noninteracting system as the singleparticle basis states for the ED. We solve the same system as shown in Fig. 1(c),(d) for an interaction strength of g_{d} = 30 and compare the energies obtained with MCTDHB and ED, see Fig. 6. Due to the variationally optimized singleparticle basis in MCTDHB computations it features a much smaller error than the ED computations with an unoptimized singleparticle basis for the same number of orbitals. This observation is in agreement with other works that benchmark the MCTDHB and the MCTDHF approaches against ED, see Ref. ^{65} and Ref. ^{46}, respectively.
Appendix B: Different Particle Numbers
In this Appendix, we corroborate our results in the main text by studying different particle numbers.
Contact interactions
The results of the manuscript have been obtained with MCTDHB with M = 12, 14, 20, 22 orbitals for N = 2, 3, 4, 5 bosons, respectively, with a contact interaction strength up to λ = 1000. In Fig. 2 of the main text, our results are consistent with fits of an exponential function A_{N}[exp(−λ/B_{N}) − 1], see caption of Fig. 2 for the fitting parameters A_{N} and B_{N}. Furthermore, we assess the convergence of the spread of the density as a function of the interaction strength to the spread of the density of the noninteracting fermionic system, see arrows labeled “2F”,“3F”,“4F”, and “5F” in Fig. 2.
To compare our results for the energy in the fermionization limit to analytical predictions for very large contact interaction strengths in Ref. ^{67}, we plot the energies as a function of −λ^{−1} in Fig. 7. We find that our results are consistent with the linear limit for the energy as a function of −λ^{−1} of Ref. ^{67}.
We now turn to the relative height of the innermost and outermost peak(s),
in the density. Here, ρ_{max} refers to the value of the density ρ(x) at the peak position and ρ_{min} refers to the value of the density ρ(x) at the position of the minimum to the left to the considered peak. See Fig. 8 for a plot of Δρ(x) for N = 2, 3, 4, 5 bosons. It is clearly seen that, for fixed N, the outermost peaks’ relative height is much smaller than the relative height of the innermost peaks.
Dipolar interactions
Here, we assess the validity of the powerlawlike unbounded spreading of the density as a function of the strength of dipoledipole interactions, that we have shown in Fig. 4 of the main text for N = 4 particles. In Fig. 4 We plot the spread of the density for N = 2, 3, 4, 5, 6 dipolar bosons obtained with MCTDHB with M = 16, 16, 22, 28, 26 orbitals, respectively, and fit it with a power law \({C}_{N}{g}_{x}^{{D}_{N}}\). We find that the exponent in the power law is almost identical for all particle numbers studied here, i.e., D_{N} ≈ 0.16 for N = 2, 3, 4, 5, 6.
We now discuss the relative peak height Δρ(x), see Eq. (B1), of the outermost peak as a function of the dipolar interaction strength, see Fig. 9 for a plot for N = 2, 3, 4, 5, 6. As hinted by Fig. 1(c) in the main text, the relative peak height for the case of the dipoledipole interactions converges towards unity as the strength of interactions g_{d} increases, because the values of the minimum, ρ_{min} in Eq. (B1), tends to zero: the peaks in the crystal state are wellisolated in comparison to the peaks in the fermionization limit for bosons with contact interactions (compare magnitude of relative peak heights in Figs. 8 and 9).
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Acknowledgements
S. Bera acknowledges DST (Govt. of India) for the financial support through INSPIRE fellowship [2015/IF150245] and Rhombik Roy for some useful discussions. B. Chakrabarti acknowledges FAPESP (grant No. 2016/196220). AG and MCT acknowledge FAPESP and AG thanks CNPq for financial support. B. Chatterjee acknowledges financial support from the Department of Science and Technology, Government of India under the DST Inspire Faculty fellowship. AUJL and CL acknowledge financial support by the Austrian Science Foundation (FWF) under grant Nos. P 32033 and M 2653, respectively, and the Wiener Wissenschafts und TechnologieFonds (WWTF) project No. MA16066.
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S.B. conducted the numerical and theoretical investigations, S.B., C.L. and A.U.J.L. wrote the manuscript, A.U.J.L. and C.L. performed complementary calculations, S.B., B. Chak., A.G., M.C.T., M.L.L., B. Chat., C.L., and A.U.J.L. conceived the idea for the project and supported the writeup.
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Bera, S., Chakrabarti, B., Gammal, A. et al. Sorting Fermionization from Crystallization in ManyBoson Wavefunctions. Sci Rep 9, 17873 (2019). https://doi.org/10.1038/s41598019531791
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