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arXiv:2403.05084 · Physical review. B./Physical review. B

Trapping hard-core bosons in flat-band lattices

Sanghoon Lee, Alexei Andreanov, Tigran Sedrakyan, Sergej Flach A

Published Jun 27, 2024• 5 citations
arXivHTMLPDF

Abstract

We investigate one-dimensional (1D) and 2D cross-stitch lattices with hard-core bosons and analytically construct exact ground states that feature macroscopic degeneracy. The construction relies on the presence of a flat band in the single particle spectrum and the orthogonality of the associated compact localized states (CLS). Up to filling fraction $\ensuremath{\nu}=1/2$, the ground state is constructed by occupying the CLS. Exactly at $\ensuremath{\nu}=1/2$, the ground state becomes a Wigner crystal. For higher filling fractions, the ground state is constructed by filling the CLS sites completely one by one. Macroscopic degeneracy arises from the multiple choices available when occupying or filling the CLS sites. An occupied CLS acts as an impenetrable barrier for bosons both in 1D and 2D, leading to Hilbert space fragmentation. A similar phenomenology also holds for hard-core bosons on the diamond chain and its higher dimensional generalizations. We also discuss the mapping of these hard-core models onto spin models with quantum many-body scars.

DOI
10.1103/physrevb.109.245137

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We investigate one-dimensional (1D) and 2D cross-stitch lattices with hard-core bosons and analytically construct exact ground states that feature macroscopic degeneracy. The construction relies on the presence of a flat band in the single particle spectrum and the orthogonality of the associated compact localized states (CLS). Up to filling fraction \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2, the ground state is constructed by occupying the CLS. Exactly at \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2, the ground state becomes a Wigner crystal. For higher filling fractions, the ground state is constructed by filling the CLS sites completely one by one. Macroscopic degeneracy arises from the multiple choices available when occupying or filling the CLS sites. An occupied CLS acts as an impenetrable barrier for bosons both in 1D and 2D, leading to Hilbert space fragmentation. A similar phenomenology also holds for hard-core bosons on the diamond chain and its higher dimensional generalizations. We also discuss the mapping of these hard-core models onto spin models with quantum many-body scars.
In recent years, there has been a growing interest in flatband systems which exhibit macroscopic degeneracies. These systems offer a valuable mathematical framework for the extreme sensitivity to perturbations and interactions. This sensitivity unveils a wide variety of exotic and unconventional physical phenomena. Moreover, the progress in their experimental realization contributes to the expanding landscape of exploration in this field. This thesis aims to summarize all the works throughout the Ph.D. program. Firstly, an in-depth exploration was conducted on the impact of weak quasiperiodic perturbations on one-dimensional two-band all-bands-flat lattices. These tight-binding Hamiltonian are diagonalized through a sequence of local unitary transformations. By adjusting the quasiperiodic potential parameters, the key achievement involves finding a critical-to-insulator transition and identifying fractality edges in the flatband systems with quasiperiodic perturbations. Next, the investigation delved into the effects of on-site interactions among hard-core bosons in one- and two-dimensional cross-stitch lattices. One key finding is that groundstate energy primarily depends on compact localized states. Moreover, the presence of barriers of compact localized states trap bosons, leading to the emergence of non-ergodic excitation and Hilbert space fragmentation. Lastly, a compact localized eigenstate of the one-dimensional diamond chain using an electric circuit was successfully generated via local (linear and non-linear) driving. This achievement opens up a versatile circuit platform for the generation of flatbands and holds promise for potential applications in the field of quantum information. I hope these collective efforts have expanded the frontiers of the field and made a meaningful contribution to the scientific community.
We study the hard-core bosons in one-dimensional (1D) interacting topological bands at different filling factors using exact diagonalization. At the filling factor $\ensuremath{\nu}=1$ and in the presence of the on-site Hubbard interaction, we find no sign of the existence of the bosonic topological phase, which is in contrast to the fermionic case. Instead by studying the momentum distribution and the condensate fraction we find a superfluid to Mott-insulator transition driven by the Hubbard interaction. At the filling factor $\ensuremath{\nu}=1/3$ and in the presence of longer-ranged interactions, we identify the bosonic fractional topological phase whose ground states are characterized by a threefold degeneracy and a quantized total Berry phase, which is very similar to the fermionic case. Finally we discuss the reason for the different behaviors of hard-core bosons at different filling factors by mapping them to spinless fermions. Our results can be realized in cold-atom experiments.
Bosons interacting repulsively on a lattice with a flat lowest band energy dispersion may, at sufficiently small filling factors, enter into a Wigner-crystal-like phase. This phase is a consequence of the dispersionless nature of the system, which in turn implies the occurrence of single-particle localized eigenstates. We investigate one of these systems---the sawtooth lattice---filled with strongly repulsive bosons at filling factors infinitesimally above the critical point where the crystal phase is no longer the ground state. We find, in the hard-core limit, that the crystal retains its structure in all but one of its cells, where it is broken. The broken cell corresponds to an exotic kind of repulsively bound state, which becomes delocalized. We investigate the excitation spectrum of the system analytically and find that the bound state behaves as a single particle hopping on an effective lattice with reduced periodicity, and is therefore gapless. Thus, the addition of a single particle to a flat-band system at critical filling is found to be enough to make kinetic behavior manifest.
Recent proposals of topological flat band models have provided a new route to realize the fractional quantum Hall effect without Landau levels. We study hard-core bosons with short-range interactions in two representative topological flat band models, one of which is the well-known Haldane model (but with different parameters). We demonstrate that fractional quantum Hall states emerge with signatures of an even number of quasidegenerate ground states on a torus and a robust spectrum gap separating these states from the higher energy spectrum. We also establish quantum phase diagrams for the filling factor 1/2 and illustrate quantum phase transitions to other competing symmetry-breaking phases.
We demonstrate that a complete class of flat-band lattices with underlying commutative local symmetries exhibit a locally fragmented Hilbert space. The equitable partition theorem ensures distinct parities for the compact localized states (CLSs) present in this class of flat-band lattices and the extended eigenstates of the system. In the presence of on-site bosonic interactions, such models exhibit a conserved quantity, the parity of the number of particles located in all the CLSs in a unit cell. As a consequence, the Hilbert space presents local fragmentation, which is only revealed upon rotating the basis of the Hamiltonian that decouples the CLSs at the single-particle level. We find that the fragmentation is strong and also robust to the addition of long-range interactions. As an example, we numerically analyze the fragmentation of the one-dimensional pyrochlore chain, which exhibits nonintegrable sectors, effective single-particle sectors, and frozen states. We also show that the entanglement entropies form a nested-dome structure typical of these fragmented systems and that thermalization is restricted to each subsector.
Strongly interacting bosons that live in a lattice with degeneracy in its lowest-energy band experience frustration that can prevent the formation of a Bose-Einstein condensate. Such systems form an ideal playground to investigate spin-liquid behavior. We use the variational principle and the Chern-Simons technique of fermionization of hard-core bosons on the kagome lattice to find that below lattice filling fraction $\ensuremath{\nu}=\frac{1}{3}$ the system favors a topologically ordered chiral spin-liquid state that is gapped in bulk, spontaneously breaks time-reversal symmetry, and supports massless chiral bosonic edge mode. We construct the many-body variational wave function of the state and show that the corresponding energy coincides with the energy of the flat band. This result proves that the ground state of the system cannot stabilize a Bose condensate below $\ensuremath{\nu}=\frac{1}{3}$. The fermionization and variational scheme we outline apply to any non-Bravais lattice. We distinguish between the roles played by the Chern-Simons gauge field in lattices with a flat band and those exhibiting a moatlike dispersion (which is degenerate along a closed contour in the reciprocal space). We also suggest experimental probes to differentiate the proposed ground state from a condensate.