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.
Highlights
The physics of the ultracold Bose gas in one spatial dimension is strongly different from that of its three-dimensional counterpart[1,2]
We provide a direct comparison with the ground state properties of non-interacting fermions computed with the multiconfigurational time-dependent Hartree method for fermions (MCTDHF), see Fig. 2
In this paper we highlight the key characteristics of the many-body wavefunction that reveal the difference between the fermionized bosons with contact interactions and crystallized bosons with dipolar interactions
Summary
The physics of the ultracold Bose gas in one spatial dimension is strongly different from that of its three-dimensional counterpart[1,2]. The long-ranged and anisotropic nature of the dipolar interaction potential results in a variety of interesting effects and phenomena[29], like crystallization in one- and two-dimensional 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. Our work complements the findings in Ref.[39] by providing a comparison of single-component (“spin-0”) bosons with contact and long-ranged dipolar interactions in continuous space without a lattice and without resorting to a Hubbard-tight-binding-description. While the MCTDHB method aims at solving the time-dependent Schrödinger equation for a many-body system, using imaginary time-propagation provides the ground-state of the system variationally, equivalent to the work of Ref.[55]
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