Sorting Fermionization from Crystallization in Many-Boson Wavefunctions

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 one-dimensional bosonic systems by exploring their ground state properties using the one- and two-body density matrix. We solve the N-body Schrödinger equation accurately and from first principles using the multiconfigurational time-dependent Hartree for bosons (MCTDHB) and for fermions (MCTDHF) methods. Using the one- and two-body density, fermionization can be distinguished from crystallization in position space. For N interacting bosons, a splitting into an N-fold pattern in the one-body and two-body 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 two-body density.

Figure 1

One-body density of N = 4 bosons as a function of contact [(a,b)] and dipolar [(c,d)] interparticle interaction strength. For contact interactions, the density becomes flatter and broader as the repulsion increases [panel (a) and (b) for λ ≤ 1]. For even larger interaction strengths [panel (a) and (b) for $\lambda \gtrsim 10$], four distinct but not isolated peaks appear and the density gradually converges to the density of four non-interacting fermions as λ → ∞. Due to this convergence, the spread of the density seizes to increase [panel (d)]. For dipolar interactions, the one-body density is clustered at the center of the trap for small interactions [panels (c,d) for $g_d\lesssim 1$]. As g_d increases, the density develops a fourfold splitting [panel (c) and (d) for $g_d\gtrsim 1$]. As a function of increasing interaction strength, the spread of the density continues to increase [panel (d)] and the fourfold spatial splitting intensifies to form four almost completely isolated peaks in the density for sufficiently strong dipolar interactions: crystallization emerges [panels (c,d) for $g_d\gtrsim 10$]. All quantities shown are dimensionless.

Figure 2

Spread of the density ρ(x) as a function of the interaction strength for N = 2, 3, 4, 5 (bottom to top curve, respectively) bosons with contact interparticle interactions. The spread of the density, according to the fitted curves (solid lines) converges exponentially as A_N[exp(−λ/B_N )− 1] to the fermionization limit as λ → ∞ which is shown by the arrows labeled “2F”, “3F”, “4F”, “5F” on the right hand side of the plot. The fit parameters for N = 2, 3, 4, 5 are, respectively, (A₂ = −0.701491, B₂ = 6.45191), (A₃ = −1.25018, B₃ = 6.50518), (A₄ = −1.71554, B₄ = 6.8185), (A₅ = −2.10423, B₅ = 8.63662). All quantities shown are dimensionless.

Figure 3

Two-body density of N = 4 bosons as a function of contact (a) and dipolar [(b)] interparticle interaction strength. For contact interactions, the atoms are clustered at the center (x₁ = x₂ = 0) for small interaction strengths, [panel (a) for λ = 0.1]. As λ increases, the two-body density starts to spread due to the repulsion between the bosons [panel (a) for λ = 1]. For stronger interaction strengths, λ = 10 and λ = 30 in (a), the diagonal, ρ⁽²⁾(x, x), is practically 0: the bosons completely avoid to be at the same position and a “correlation hole” develops. For dipolar interactions, the atoms cluster at the center (x₁ = x₂ = 0) for small interaction strengths, see panel (b) for g_d = 0.1. As g_d increases, the diagonal part, ρ⁽²⁾(x, x) starts to be depleted because the long-range interactions start to dominate the physics [panel (b) for $\gtrsim 1$]. At stronger interaction strengths, the diagonal correlation hole spreads, i.e., the area in the vicinity of x₁ ≈ x₂ for which ρ⁽²⁾(x₁, x₂) ≈ 0 holds is enlarged as a function of g_d [compare panel (b) for g_d = 1.0, 10, and 30]. In contrast to contact interactions, even the off-diagonal (x₁ ≠ x₂) of ρ⁽²⁾(x₁, x₂) forms a complete correlation hole, compare panel (a) for λ = 30 and panel (b) for g_d = 30. All quantities shown are dimensionless.

Figure 4

Spread of the density for N = 2, 3, 4, 5, 6 bosons (bottom to top curve, respectively) with dipole-dipole interactions as a function of the interaction strength g_d. The spread of the density, according to the fitted curves (solid lines) diverges as a power law, $C_N{x}^{D_N}$, in the limit of large interactions g_d → ∞. The fit parameters for N = 2, 3, 4, 5, 6 are, respectively, (C₂ = 0.926851, D₂ = 0.152459), (C₃ = 1.61556, D₃ = 0.151243), (C₄ = 2.13826, D₄ = 0.162034), (C₅ = 2.62615, D₅ = 0.161553), (C₆ = 3.03802, D₆ = 0.165883). Importantly, the power of the divergence of the spread, D_N, seems to be independent of the number of particles N. All quantities shown are dimensionless.

Figure 5

Tracing fermionization and crystallization in the spread of the density (a), the energy (b), and the natural occupations (c,d) as a function of the interaction strength. (a) The spread of the density is quantified by the position of the outermost peak in the density ρ(x). The spread is bounded for contact interactions and unbounded for dipolar interactions. The fits shown suggest that the spread of the density ρ(x) for dipolar interactions diverges with a power law, 2.138g_d^0.162, and for contact interactions it converges as −1.71554[exp(−λ/6.8185) − 1] to the fermionization limit (fit obtained with more points than actually shown, see Appendix 7). (b) The energy as a function of interaction strength is bounded for contact interactions and unbounded for dipolar interactions. The fits suggest that the energy diverges with a power law 10.51g_d^0.277 for dipolar interactions and converges to the fermionization limit exponentially −5.84exp(−λ/6.023) + 8.133 for contact interactions. The thin yellow lines indicate the energy of non-interaction fermions E_λ→∞ and the energy for dipolar fermions E_d−fermion. (c,d) The eigenvalues of the reduced density matrix, i.e., the natural occupations ρ_i^(NO), exhibit depletion for contact interactions (many small ρ_i^(NO) with i > 1 emerge) and full-blown N-fold fragmentation for dipolar interactions (all ρ_i^(NO) with i ≤ N contribute equally), the black dashed lines show the four most populated natural orbitals for dipolar fermions. The ρ_i^(NO) are ordered in decreasing order starting from i = 1. All quantities shown are dimensionless.

Figure 6

Comparison of MCTDHB and ED for N = 4 bosons with dipole-dipole interaction strength g_d = 30. The plot shows the relative error in energy with respect to an MCTDHB computation with M = 32 orbitals as a function of the number of orbitals for the ED and MCTDHB approaches. Due to the variationally optimized basis in MCTDHB computations it features a much smaller error for any number of orbitals. All quantities shown are dimensionless.

Figure 7

Energy as a function of inverse interaction strength, −1/λ, for N = 2, 3, 4, 5 (bottom to top curve, respectively) bosons. Our results are consistent with the analysis in Ref. : the energy linearly converges to the fermionization limit, i.e., when −λ⁻¹ → 0. All quantities shown are dimensionless.

Figure 8

Relative height of the outermost peaks in the density of N = 2, 3, 4, 5 bosons with contact interactions (points, top to bottom, respectively) and relative height of the innermost peaks in the density of N = 4, 3, 5 bosons with contact interactions (lines, top, to bottom, respectively). The relative peak height is consistently smaller for the outermost peak as compared to the innermost peak in the density for all interaction strengths depicted. All quantities shown are dimensionless.

Figure 9

Relative peak height for N = 2, 3, 4, 5, 6 bosons with dipolar interactions as a function of interaction strength. The relative peak height converges towards unity similarly for all particle numbers investigated here as the interaction strength increases. All quantities shown are dimensionless.