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Register a new userLaughlin-Type Topological Order on a Fractal Lattice with a Local Hamiltonian
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Added by
Anne Ersbak Bang Nielsen
Publication date
2021
fields
Physics
and research's language is
English
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Anyons are mainly studied and considered in two spatial dimensions. For fractals, the scaling dimension that characterizes the system can be non integer and can take values between that of a standard one-dimensional or two-dimensional system. Generating Hamiltonians that meet locality conditions and support anyons is not a simple task. Here, we construct a local Hamiltonian on a fractal lattice which realizes physics similar to the fractional quantum Hall effect. The fractal lattice is obtained from a second generation Sierpinski carpet, which has 64 sites, and is characterized by a Hausdorff dimension of 1.89. We demonstrate that the proposed local Hamiltonian acting on the fractal geometry has Laughlin-type topological order by creating anyons and then studying their charge and braiding statistics. We also find that the energy gap between the ground state and the first excited state is approximately three times larger for the fractal lattice than for a standard square lattice with 64 sites, and the model on the fractal lattice is significantly more robust against disorder. We propose a scheme to implement fractal lattices and our proposed local Hamiltonian for ultracold atoms in optical lattices. The discussed scheme could also be utilized to study integer quantum Hall phases and the physics of other quantum systems on fractal lattices.
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We analyze the proposal of achieving a Mott state of Laughlin wave functions in an optical lattice [M. Popp {it et al.}, Phys. Rev. A 70, 053612 (2004)] and study the consequences of considering the anharmonic corrections to each single site potential expansion that were not taken into account until now. Our result is that, although the anharmonic correction reduces the maximum frequency at which the system can rotate before the atoms escape from each site (centrifugal limit), the Laughlin state can still be achieved for a small number of particles and a realistic value of the laser intensity.
We propose the construction of a many-body phase of matter with fractal structure using arrays of Rydberg atoms. The degenerate low energy excited states of this phase form a self-similar fractal structure. This phase is analogous to the so-called type-II fracton topological states. The main challenge in realizing fracton-like models in standard condensed matter platforms is the creation of multi-spin interactions, since realistic systems are typically dominated by two-body interactions. In this work, we demonstrate that the Van der Waals interaction and experimental tunability of Rydberg-based platforms enable the simulation of exotic phases of matter with fractal structures, and the study of a quantum phase transition involving a fractal ordered phase.
Disorder and localization have dramatic influence on the topological properties of a quantum system. While strong disorder can close the band gap thus depriving topological materials of topological features, disorder may also induce topology from trivial band structures, wherein topological invariants are shared by completely localized states in real space. Here we experimentally investigate a fundamentally distinct scenario where a topological phase is identified in a critically localized regime, with eigenstates neither fully extended nor completely localized. Adopting the technique of momentum-lattice engineering for ultracold atoms, we implement a one-dimensional, generalized Aubry-Andre model with off-diagonal quasi-periodic disorder in momentum space, and characterize its localization and topological properties through dynamic observables. We then demonstrate the impact of interactions on the critically localized topological state, as a first experimental endeavour toward the clarification of many-body critical phase, the critical analogue of the many-body localized state.
We report the experimental realization of a topological Creutz ladder for ultracold fermionic atoms in a resonantly driven 1D optical lattice. The two-leg ladder consists of the two lowest orbital states of the optical lattice and the cross inter-leg links are generated via two-photon resonant coupling between the orbitals by periodic lattice shaking. The characteristic pseudo-spin winding in the topologically non-trivial bands of the ladder system is demonstrated using momentum-resolved Ramsey-type interferometric measurements. We discuss a two-tone driving method to extend the inter-leg link control and propose a topological charge pumping scheme for the Creutz ladder system.
A large number of symmetry-protected topological (SPT) phases have been hypothesized for strongly interacting spin-1/2 systems in one dimension. Realizing these SPT phases, however, often demands fine-tunings hard to reach experimentally. And the lack of analytical solutions hinders the understanding of their many-body wave functions. Here we show that two kinds of SPT phases naturally arise for ultracold polar molecules confined in a zigzag optical lattice. This system, motivated by recent experiments, is described by a spin model whose exchange couplings can be tuned by an external field to reach parameter regions not studied before for spin chains or ladders. Within the enlarged parameter space, we find the ground state wave function can be obtained exactly along a line and at a special point, for these two phases respectively. These exact solutions provide a clear physical picture for the SPT phases and their edge excitations. We further obtain the phase diagram by using infinite time-evolving block decimation, and discuss the phase transitions between the two SPT phases and their experimental signatures.
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