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arXiv:1905.07454 · Scientific Reports

Braiding properties of worldline configurations in hardcore lattice bosons

Fabio Lingua, Wei Wang, Liana Shpani, Barbara Capogrosso-Sansone

Published Oct 29, 2022• 0 citations
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Abstract

In this manuscript, we study braiding properties of worldline configurations for a variety of ground-states of hardcore Bose-Hubbard models in two dimensions. Configurations are collections of particle paths and result from the path-integral formulation of statistical mechanics. For hard-core bosons, configurations can be seen as geometric braids and therefore can be assigned a certain topological structure, i.e. a way to classify braiding events among worldlines. By means of Monte Carlo calculations, we study superfluid phase and a variety of insulating phases and observe that ground-states of different quantum phases correspond to different probability distributions of braiding properties.

DOI
10.1038/s41598-022-22894-7

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In this manuscript, we study braiding properties of worldline configurations for a variety of ground-states of hardcore Bose-Hubbard models in two dimensions. Configurations are collections of particle paths and result from the path-integral formulation of statistical mechanics. For hard-core bosons, configurations can be seen as geometric braids and therefore can be assigned a certain topological structure, i.e. a way to classify braiding events among worldlines. By means of Monte Carlo calculations, we study superfluid phase and a variety of insulating phases and observe that ground-states of different quantum phases correspond to different probability distributions of braiding properties.
In this manuscript, we present a proposal to relate topological structure of worldline configurations to multipartite entanglement. Configurations result from the path-integral formulation of the density matrix in the limit of zero temperature. We consider hard-core bosons for which configurations, i.e. collections of particle paths, can be seen as geometric braids with a certain topological structure. We propose that properties of worldline configurations may realize a comprehensive deciphering of multipartite entanglement. By means of Monte Carlo calculations, we study checkerboard, stripe, valence-bond solids, Z2\mathbb{Z}_2Z2​ topologically ordered spin liquid, and superfluid phase. We find that each ground-state is characterized by a certain `topological spectrum' which can be used to differentiate among different ground-states.
The superfluid-insulator transition in systems of lattice bosons is usually analyzed in the framework of the Bose-Hubbard model, and has been extensively studied by theory and simulations. Less attention has been paid to the remnants of the transition in truncated lattices, with or without periodic boundary conditions. Here we consider the hard-core limit of the extended Bose-Hubbard model on small square and triangular grids -- i.e., sections of the square and triangular lattices containing up to 13 sites. By mapping out the zero-temperature phase diagram through exact diagonalization, we find ground-state characteristics that are markedly different from those emerging in the thermodynamic limit, together with similarities. The dichotomy between superfluid-like and insulating-like behavior is then investigated in two-dimensional systems of a few interacting bosons in the continuum, subject to confining and optical-lattice potentials mimicking the 3×33\times 33×3 square grid. Using path-integral Monte Carlo simulations, we compute kinetic and potential energies, as well as superfluidity and exchange-cycle statistics, finding hints of Bose-Hubbard behavior even in systems of just five particles.
We study the ground state properties of the extended hard-core Bose–Hubbard model in a two-dimensional Shastry–Sutherland lattice by quantum Monte Carlo simulations. The effects of the geometric frustration, described by the ratio t ′ / t between hopping integrals along the nearest-neighbor links and the diagonal links, are studied. We find that for small values of frustration amplitude t ′ / t , a checkerboard solid state emerges at ρ= 1/2; for large values of frustration amplitude t ′ / t , a solid state emerges at ρ= 1/2 with one boson per dimer (dimers on diagonal links); for intermediate values of frustration amplitude t ′ / t , besides the ρ= 1/2 solid state, the ρ= 1/3,2/3 solid states emerge. No stable supersolid state is observed in our calculations. At half filling, we find solid-superfluid-solid phase transition as the frustration amplitude t ′ / t is increased.
A ground state path integral quantum Monte Carlo algorithm is introduced that allows for the study of entanglement in lattice bosons at zero temperature. The Rényi entanglement entropy between spatial subregions is explored across the phase diagram of the one dimensional Bose-Hubbard model for systems consisting of up to L=256 L = 256 sites at unit-filling without any restrictions on site occupancy, far beyond the reach of exact diagonalization. The favorable scaling of the algorithm is demonstrated through a further measurement of the Rényi entanglement entropy at the two dimensional superfluid-insulator critical point for large system sizes, confirming the existence of the expected entanglement boundary law in the ground state. The Rényi estimator is extended to measure the symmetry resolved entanglement that is operationally accessible as a resource for experimentally relevant lattice gases with fixed total particle number.
We study the ground state properties of the extended hard-core Bose–Hubbard model in a two-dimensional anisotropic triangular lattice by quantum Monte Carlo simulations. We present the ground state phase diagram at half filling. Four kinds of phases, including two solid phases, a superfluid phase and a supersolid phase, are observed. The supersolid phase exists only in a narrow region close to the isotropic limit. Away from half filling, we find that the results are strongly dependent on the phase at half filling. No supersolid phase exists if the ground state at half filling is a superfluid or a solid.
Motivated by some inconsistencies in the way quantum fluctuations are included beyond the classical treatment of hard-core bosons on a lattice in the recent literature, we revisit the large-S semi-classical approach to hard-core bosons on the square lattice at T=0. First of all, we show that, if one stays at the purely harmonic level, the only correct way to get the 1/S correction to the density is to extract it from the derivative of the ground state energy with respect to the chemical potential, and that to extract it from a calculation of the ground state expectation value of the particle number operator, it is necessary to include 1/\sqrt{S} corrections to the harmonic ground state. Building on this alternative approach to get 1/S corrections, we provide the first semiclassical derivation of the momentum distribution, and we revisit the calculation of the condensate density. The results of these as well as other physically relevant quantities such as the superfluid density are systematically compared to quantum Monte Carlo simulations. This comparison shows that the logarithmic corrections in the dilute Bose gas limit are only captured by the semi-classical approach if the 1/S corrections are properly calculated, and that the semi-classical approach is able to reproduce the 1/k divergence of the momentum distribution at k=0. Finally, the effect of 1/S^2 corrections is briefly discussed.