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Lattice Genons

Zhao Liu, Gunnar Möller, Emil J. Bergholtz

Published Feb 16, 2017• 0 citations
OpenAlex

Abstract

We investigate extrinsic wormholelike twist defects that effectively increase the genus of space in lattice versions of multicomponent fractional quantum Hall systems. Although the original band structure is distorted by these defects, leading to localized midgap states, we find that a new lowest flat band representing a higher genus system can be engineered by tuning local single-particle potentials. Remarkably, once local many-body interactions in this new band are switched on, we identify various Abelian and non-Abelian fractional quantum Hall states, whose ground-state degeneracy increases with the number of defects, i.e, with the genus of space. This sensitivity of topological degeneracy to defects provides a proof of concept demonstration that genons, predicted by topological field theory as exotic non-Abelian defects tied to a varying topology of space, do exist in realistic microscopic models. Specifically, our results indicate that genons could be created in the laboratory by combining the physics of artificial gauge fields in cold atom systems with already existing holographic beam shaping methods for creating twist defects.

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Exotic Non-Abelian Topological Defects in Lattice Fractional Quantum Hall States

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We investigate extrinsic wormholelike twist defects that effectively increase the genus of space in lattice versions of multicomponent fractional quantum Hall systems. Although the original band structure is distorted by these defects, leading to localized midgap states, we find that a new lowest flat band representing a higher genus system can be engineered by tuning local single-particle potentials. Remarkably, once local many-body interactions in this new band are switched on, we identify various Abelian and non-Abelian fractional quantum Hall states, whose ground-state degeneracy increases with the number of defects, i.e, with the genus of space. This sensitivity of topological degeneracy to defects provides a "proof of concept" demonstration that genons, predicted by topological field theory as exotic non-Abelian defects tied to a varying topology of space, do exist in realistic microscopic models. Specifically, our results indicate that genons could be created in the laboratory by combining the physics of artificial gauge fields in cold atom systems with already existing holographic beam shaping methods for creating twist defects.

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Anyonic symmetry and twist defects

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Non-Abelian excitations are sought after because of the promise they hold for topological quantum computation. However, in spite of many proposals of systems in which they can occur intrinsically, observation has proved to be elusive. In this thesis, we study ways in which we can engineer such excitations by combining two parent Abelian theories, and as a result allow for defects which host non-Abelian zero modes. We first formulate the symmetries of the anyonic excitations of Abelian topological states. Then, we study properties of semiclassical point defects, called twist defects, which permute orbiting anyons by their symmetries. We see how to realize these defects at domain walls on the gapped edges of non-chiral fractional quantum Hall states. We have distinct gapped edge phases in one to one correspondence with the anyonic symmetries. Non-Abelian twist defects appear between different gapped phases. We study the twist defects for a variety of bosonic ADE fractional quantum spin Hall systems for which the anyonic symmetries turn out to be given by their Dynkin diagrams. We also look at superconducting heterostructures with fermionic quantum Hall states and show the presence of anyonic symmetries and twist defects which physically correspond to a fermion parity flip. This can result in Majorana bound states at such defects instead of more exotic parafermions as was believed before. This is because the presence of the superconductor necessitates the consideration of a larger bosonic topological order. We then investigate the twist defects further by promoting the anyonic symmetries from global to local. This deconfines the defects and results in non-Abelian topological states called twist liquids. Finally, we study the emergence of new topological order from two constituent systems driven by the condensation of a composite boson. This allows us to study a variety of systems with non-Abelian excitations.

•Oct 13, 2017•0 citations
We investigate extrinsic wormholelike twist defects that effectively increase the genus of space in lattice versions of multicomponent fractional quantum Hall systems. Although the original band structure is distorted by these defects, leading to localized midgap states, we find that a new lowest flat band representing a higher genus system can be engineered by tuning local single-particle potentials. Remarkably, once local many-body interactions in this new band are switched on, we identify various Abelian and non-Abelian fractional quantum Hall states, whose ground-state degeneracy increases with the number of defects, i.e, with the genus of space. This sensitivity of topological degeneracy to defects provides a proof of concept demonstration that genons, predicted by topological field theory as exotic non-Abelian defects tied to a varying topology of space, do exist in realistic microscopic models. Specifically, our results indicate that genons could be created in the laboratory by combining the physics of artificial gauge fields in cold atom systems with already existing holographic beam shaping methods for creating twist defects.
We investigate extrinsic wormholelike twist defects that effectively increase the genus of space in lattice versions of multicomponent fractional quantum Hall systems. Although the original band structure is distorted by these defects, leading to localized midgap states, we find that a new lowest flat band representing a higher genus system can be engineered by tuning local single-particle potentials. Remarkably, once local many-body interactions in this new band are switched on, we identify various Abelian and non-Abelian fractional quantum Hall states, whose ground-state degeneracy increases with the number of defects, i.e, with the genus of space. This sensitivity of topological degeneracy to defects provides a "proof of concept" demonstration that genons, predicted by topological field theory as exotic non-Abelian defects tied to a varying topology of space, do exist in realistic microscopic models. Specifically, our results indicate that genons could be created in the laboratory by combining the physics of artificial gauge fields in cold atom systems with already existing holographic beam shaping methods for creating twist defects.
We present a detailed microscopic investigation of fractional quantum Hall states on effective higher-genus surfaces emerging in a coupled bilayer lattice model featuring holes whose counter propagating chiral edge states are hybridized and gapped out. Although the holes distort the original band structure and lead to ingap remnants of the continuum edge modes, we find that a lowest nearly flat band representing a higher-genus system may naturally form by controlling the local hopping terms that gap out the boundaries. Remarkably, even in the extreme lattice limit where each hole consists of just a single removed site, local interactions in this new flat band lead to various Abelian and non-Abelian fractional quantum Hall states on the emergent higher-genus surfaces which we identify by extracting the non-trivial topological ground-state degeneracies and the fractional statistics of quasiparticles. These results demonstrate that our microscopic lattice model is a promising platform to realize novel fractional quantum Hall states with gapped boundaries, thus enabling a possible new route towards universal topological quantum computation.
We present a detailed microscopic investigation of fractional quantum Hall states with gapped boundaries in a coupled bilayer lattice model featuring holes whose counterpropagating chiral edge states are hybridized and gapped out. We focus on an lattice limit for cold-atom experiments, in which each hole just consists of a single removed site. Although the holes distort the original band structure and lead to in-gap remnants of the continuum edge modes, we find that the lowest nearly flat band representing a higher-genus system may naturally form by controlling the local hopping terms that gap out the boundaries. Remarkably, local interactions in this new flat band lead to various Abelian and non-Abelian fractional quantum Hall states with gapped boundaries residing on emergent higher-genus surfaces, which we identify by extracting the nontrivial topological ground-state degeneracies and the fractional statistics of quasiparticles. These results demonstrate the feasibility of realizing novel fractional quantum Hall states with gapped boundaries even in the extreme lattice limit, thus enabling a possible new route towards universal topological quantum computation.
It has recently been realized that a general class of non-Abelian defects can be created in conventional topological states by introducing extrinsic defects, such as lattice dislocations or superconductor-ferromagnet domain walls in conventional quantum Hall states or topological insulators. In this paper, we begin by placing these defects within the broader conceptual scheme of extrinsic twist defects associated with symmetries of the topological state. We explicitly study several classes of examples, including Z2{Z}_{2}Z2​ and Z3{Z}_{3}Z3​ twist defects, where the topological state with NNN twist defects can be mapped to a topological state without twist defects on a genus $g\ensuremath{\propto}N$ surface. To emphasize this connection we refer to the twist defects as genons. We develop methods to compute the projective non-Abelian braiding statistics of the genons, and we find the braiding is given by adiabatic modular transformations, or Dehn twists, of the topological state on the effective genus ggg surface. We study the relation between this projective braiding statistics and the ordinary non-Abelian braiding statistics obtained when the genons become deconfined, finite-energy excitations. We find that the braiding is generally different, in contrast to the Majorana case, which opens the possibility for fundamentally novel behavior. We find situations where the genons have quantum dimension 2 and can be used for universal topological quantum computing (TQC), while the host topological state is by itself nonuniversal for TQC.
The quest for non-Abelian anyons, exotic quasiparticles fundamental to topological quantum computing, remains a forefront challenge in condensed matter physics. This paper explores the universal topological signatures that characterize non-Abelian anyons when hosted within engineered flat band systems. Flat bands, distinguished by their quenched kinetic energy, dramatically enhance interaction effects, providing a unique platform for stabilizing exotic many-body phases and fostering the emergence of non-Abelian statistics. We theoretically investigate how specific lattice geometries and tuning parameters in various engineered platforms, such as Moiré superlattices, photonic crystals, and ultracold atomic gases, can lead to the formation of topological flat bands conducive to non-Abelian anyonic excitations. Our analysis identifies several universal signatures, including robust ground state degeneracy sensitive to system topology, characteristic patterns in topological entanglement entropy, and distinct braiding statistics manifested through non-Abelian holonomies and fractional quantum numbers. These signatures are shown to be robust against local perturbations and largely independent of microscopic details, providing reliable indicators for experimental detection. We discuss the implications for developing fault-tolerant quantum computation and outline potential experimental methodologies for probing these predicted signatures, emphasizing the transformative potential of engineered flat bands in advancing topological quantum science.
Non-Abelian excitations are sought after because of the promise they hold for topological quantum computation. However, in spite of many proposals of systems in which they can occur intrinsically, observation has proved to be elusive. In this thesis, we study ways in which we can engineer such excitations by combining two parent Abelian theories, and as a result allow for defects which host non-Abelian zero modes. We first formulate the symmetries of the anyonic excitations of Abelian topological states. Then, we study properties of semiclassical point defects, called twist defects, which permute orbiting anyons by their symmetries. We see how to realize these defects at domain walls on the gapped edges of non-chiral fractional quantum Hall states. We have distinct gapped edge phases in one to one correspondence with the anyonic symmetries. Non-Abelian twist defects appear between different gapped phases. We study the twist defects for a variety of bosonic ADE fractional quantum spin Hall systems for which the anyonic symmetries turn out to be given by their Dynkin diagrams. We also look at superconducting heterostructures with fermionic quantum Hall states and show the presence of anyonic symmetries and twist defects which physically correspond to a fermion parity flip. This can result in Majorana bound states at such defects instead of more exotic parafermions as was believed before. This is because the presence of the superconductor necessitates the consideration of a larger bosonic topological order. We then investigate the twist defects further by promoting the anyonic symmetries from global to local. This deconfines the defects and results in non-Abelian topological states called twist liquids. Finally, we study the emergence of new topological order from two constituent systems driven by the condensation of a composite boson. This allows us to study a variety of systems with non-Abelian excitations.