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arXiv:2004.11880 · Physical review. B./Physical review. B

Quantum caging in interacting many-body all-bands-flat lattices

Carlo Danieli, Alexei Andreanov, Thudiyangal Mithun, Sergej Flach A

Published Aug 17, 2021• 38 citations
arXivHTMLPDF

Abstract

We consider translationally invariant tight-binding all-bands-flat networks which lack dispersion. In a recent work [C. Danieli et al., Phys. Rev. B 104, 085131 (2021)], we identified the subset of these networks that shows nonlinear caging in the presence of Kerr-like local nonlinearities, i.e., it preserves nonexcited network sites and therefore keeps compact excitations compact. Here we replace nonlinear terms by Bose-Hubbard interactions and study quantum caging. We identify the quantum caging conditions that are related to the nonlinear caging conditions and that guarantee the existence of an extensive set of conserved quantities in any lattice dimension, as first revealed in Tovmasyan et al. [Phys. Rev. B 98, 134513 (2018)] for a set of specific networks. Consequently, transport is realized through moving pairs of interacting particles that break the single-particle caging. We further prove the existence of degenerate energy renormalized compact states for any finite number $M$ of participating particles in one- and higher-dimensional lattice---results that explain and generalize previous observations for two particles on a diamond chain [Vidal et al., Phys. Rev. Lett. 85, 3906 (2000)].

DOI
10.1103/physrevb.104.085132

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We consider translationally invariant tight-binding all-bands-flat networks which lack dispersion. In a recent work [C. Danieli et al., Phys. Rev. B 104, 085131 (2021)], we identified the subset of these networks that shows nonlinear caging in the presence of Kerr-like local nonlinearities, i.e., it preserves nonexcited network sites and therefore keeps compact excitations compact. Here we replace nonlinear terms by Bose-Hubbard interactions and study quantum caging. We identify the quantum caging conditions that are related to the nonlinear caging conditions and that guarantee the existence of an extensive set of conserved quantities in any lattice dimension, as first revealed in Tovmasyan et al. [Phys. Rev. B 98, 134513 (2018)] for a set of specific networks. Consequently, transport is realized through moving pairs of interacting particles that break the single-particle caging. We further prove the existence of degenerate energy renormalized compact states for any finite number MMM of participating particles in one- and higher-dimensional lattice---results that explain and generalize previous observations for two particles on a diamond chain [Vidal et al., Phys. Rev. Lett. 85, 3906 (2000)].
We generate translationally invariant systems exhibiting many-body localization from all-bands-flat single-particle lattice Hamiltonians dressed with suitable short-range many-body interactions. This phenomenon, dubbed many-body flatband localization, is based on symmetries of both single-particle and interaction terms in the Hamiltonian, and it holds for any interaction strength. We propose a generator of corresponding Hamiltonians which covers both interacting bosons and fermions for arbitrary lattice dimensions, and we provide explicit examples of such models in one and two lattice dimensions. We also explicitly construct an extensive set of local integrals of motion for this set of models. Our results can be further generalized to long-range interactions as well as to systems lacking translational invariance.
Flat band physics is a central theme in modern condensed matter physics. By constructing a tight--binding single particle system that has vanishing momentum dispersion in one or more bands, and subsequently including more particles and interactions, it is possible to study physics in strongly interacting regimes. Inspired by the glued trees that first arose in one of the few known examples of quantum supremacy, we define and analyze two infinite families of tight binding single particle Bose--Hubbard models that have only flat bands, and only compact localized states despite having any nonnegative number of translation symmetries. The first class of model that we introduce is constructed by replacing a sufficiently large fraction of the edges in a generic countable graph with glued trees modified to have complex hoppings. The second class arises from thinking of complex weighted glued trees as rhombi that can then be used to tile two dimensional space, giving rise to the familiar dice lattice and infinitely many generalizations thereof, of which some are Euclidian while others are hyperbolic.
Translationally invariant flatband Hamiltonians with interactions lead to a many-body localization transition. Our models are obtained from single-particle lattices hosting a mix of flat and dispersive bands, and equipped with fine-tuned two--body interactions. Fine-tuning of the interaction results in an extensive set of local conserved charges and a fragmentation of the Hilbert space into irreducible sectors. In each sector, the conserved charges originate from the flatband and act as an effective disorder inducing a transition between ergodic and localized phases upon variation of the interaction strength. Such fine-tuning is possible in arbitrary lattice dimensions and for any many-body statistics. We present computational evidence for this transition with spinless fermions.
The Bose-Einstein condensation of noninteracting particles restricted to move on the sites of hierarchical diamond lattices is investigated. Using a tight-binding single-particle Hamiltonian with properly rescaled hopping amplitudes, we are able to employ an orthogonal basis transformation to exactly map it on a set of decoupled linear chains with sizes and degeneracies written in terms of the network branching parameter q and generation number n. The integrated density of states is shown to have a fractal structure of gaps and degeneracies with a power-law decay at the band bottom. The spectral dimension d(s) coincides with the network topological dimension d(f) = ln(2q)/ln(2). We perform a finite-size scaling analysis of the fraction of condensed particles and specific heat to characterize the critical behavior of the BEC transition that occurs for q > 2 (d(s) > 2). The critical exponents are shown to follow those for lattices with a pure power-law spectral density, with non-mean-field values for q < 8 (d(s) < 4). The transition temperature is shown to grow monotonically with the branching parameter, obeying the relation 1/T(c) = a + b/(q - 2).
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 investigate the properties of a few interacting bosons in a Creutz ladder, which has become a standard model for topological systems, and which can be realized in experiments with cold atoms in optical lattices. At the single-particle level, this system may exhibit a completely flat energy landscape with nontrivial topological properties. In this scenario, we identify topological two-body edge states resulting from the bonding of single-particle edge and flat-band states. We also explore the formation of two- and three-body bound states in the strongly interacting limit, and we show how these quasiparticles can be engineered to replicate the flat-band and topological features of the original single-particle model. Furthermore, we show that in this geometry perfect Aharonov-Bohm caging of two-body bound states may occur for arbitrary interaction strengths, and we provide numerical evidence that the main features of this effect are preserved in an interacting many-body scenario resulting in many-body Aharonov-Bohm caging. Published by the American Physical Society 2024