A hallmark feature of topologically ordered states of matter is the dependence of ground-state degeneracy (GSD) on the topology of the manifold determined by the global shape of the system. Although the topology of a physical system is practically hard to manipulate, recently, it was shown that in certain topologically ordered phases, topological defects can introduce extra topological GSD. Here the topological defects can be viewed as effectively changing the topology of the physical system. Previous studies have been focusing on two spatial dimensions with pointlike topological defects. In three dimensions, linelike topological defects can appear. They are closed loops in the bulk that can be linked and knotted, effectively leading to complex three-dimensional manifolds in certain topologically ordered states. This paper studies the properties of such line defects in a particular context: the lattice dislocations. We give an analytical construction, together with support from exact numerical calculations, for the dependence of the GSD on dislocations of certain doubled versions of the exactly solvable Kitaev's toric code models in both two and three dimensions. We find that the GSD of the 3D model depends only on the total number of dislocation loops, no matter how they are linked or knotted. The results are extended to ${\mathbb{Z}}_{n}$ generalizations of the model. Additionally, we consider the phases in which the crystalline orders are destroyed through proliferation of double dislocations. The resulting phases are shown to host topological orders described by non-Abelian gauge theories.A hallmark feature of topologically ordered states of matter is the dependence of ground-state degeneracy (GSD) on the topology of the manifold determined by the global shape of the system. Although the topology of a physical system is practically hard to manipulate, recently, it was shown that in certain topologically ordered phases, topological defects can introduce extra topological GSD. Here the topological defects can be viewed as effectively changing the topology of the physical system. Previous studies have been focusing on two spatial dimensions with pointlike topological defects. In three dimensions, linelike topological defects can appear. They are closed loops in the bulk that can be linked and knotted, effectively leading to complex three-dimensional manifolds in certain topologically ordered states. This paper studies the properties of such line defects in a particular context: the lattice dislocations. We give an analytical construction, together with support from exact numerical calculations, for the dependence of the GSD on dislocations of certain doubled versions of the exactly solvable Kitaev's toric code models in both two and three dimensions. We find that the GSD of the 3D model depends only on the total number of dislocation loops, no matter how they are linked or knotted. The results are extended to ${\mathbb{Z}}_{n}$ generalizations of the model. Additionally, we consider the phases in which the crystalline orders are destroyed through proliferation of double dislocations. The resulting phases are shown to host topological orders described by non-Abelian gauge theories.

arXiv:1305.0214Physical Review BJul 30, 201336 citations

Topological Phases in Condensed Matter Physics

Saurabh Basu

Springer eBooksJan 1, 20232 citations

Disorder and Defects in Topological Phases

Roderich Moessner, Joel E. Moore

Cambridge University Press eBooksApr 9, 20210 citations

Matter-wave soliton control in optical lattices with topological dislocations

Yaroslav V. Kartashov, Lluís Torner

We address the concept of guiding and transporting matter-wave solitons along the channels of optical lattices made by interference patterns of beams containing topological wavefront dislocations. The local lattice distortions that occur around the dislocations cause solitons to move along reconfigurable paths, a phenomenon that may be used for controlled all-optical manipulation of Bose-Einstein condensates. Multiple dislocations form traps that can capture and hold moving solitons. We also show that suitable lattice topologies may be used to explore matter-wave soliton collisions and interactions in a confined reconfigurable environment.We address the concept of guiding and transporting matter-wave solitons along the channels of optical lattices made by interference patterns of beams containing topological wavefront dislocations. The local lattice distortions that occur around the dislocations cause solitons to move along reconfigurable paths, a phenomenon that may be used for controlled all-optical manipulation of Bose-Einstein condensates. Multiple dislocations form traps that can capture and hold moving solitons. We also show that suitable lattice topologies may be used to explore matter-wave soliton collisions and interactions in a confined reconfigurable environment.

arXiv:physics/0608060Physical Review AOct 25, 20062 citations

Geometry defects in bosonic symmetry-protected topological phases

Yizhi You, Yi‐Zhuang You

In this paper we focus on the interplay between geometry defects and topological properties in bosonic symmetry-protected topological (SPT) phases. We start from eight copies of 3D time-reversal $(\mathcal{T})$ invariant topological superconductors (TSC) on a crystal lattice. We melt the lattice by condensation of disclinations and therefore restore the rotation symmetry. Such a disclination condensation procedure confines the fermion and afterwards turns the system into a 3D boson topological liquid crystal (TCL). The low energy effective theory of this crystalline-liquid transition contains a topological term inherited from the geometry axion response in TSC. In addition, we investigate the interplay between dislocation and superfluid vortex on the surface of TCL. We demonstrate that the $\mathcal{T}$ and translation invariant surface state is a double $[e\mathcal{T}m\mathcal{T}]$ state with intrinsic surface topological order. We also look into the exotic behavior of dislocation in the 2D boson SPT state described by an $O(4)$ nonlinear $\ensuremath{\sigma}$ model $(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ with topological $\mathrm{\ensuremath{\Theta}}$ term. By dressing the $O(4)$ vector with spiral order and gauging the symmetry, the dislocation has mutual semion statistics with the gauge flux. Further reducing the $O(4)\phantom{\rule{4pt}{0ex}}\mathrm{NL}\ensuremath{\sigma}M$ to the Ising limit, we arrive at the Levin-Gu model with stripy modulation whose dislocation has nontrivial braiding statistics.In this paper we focus on the interplay between geometry defects and topological properties in bosonic symmetry-protected topological (SPT) phases. We start from eight copies of 3D time-reversal $(\mathcal{T})$ invariant topological superconductors (TSC) on a crystal lattice. We melt the lattice by condensation of disclinations and therefore restore the rotation symmetry. Such a disclination condensation procedure confines the fermion and afterwards turns the system into a 3D boson topological liquid crystal (TCL). The low energy effective theory of this crystalline-liquid transition contains a topological term inherited from the geometry axion response in TSC. In addition, we investigate the interplay between dislocation and superfluid vortex on the surface of TCL. We demonstrate that the $\mathcal{T}$ and translation invariant surface state is a double $[e\mathcal{T}m\mathcal{T}]$ state with intrinsic surface topological order. We also look into the exotic behavior of dislocation in the 2D boson SPT state described by an $O(4)$ nonlinear $\ensuremath{\sigma}$ model $(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ with topological $\mathrm{\ensuremath{\Theta}}$ term. By dressing the $O(4)$ vector with spiral order and gauging the symmetry, the dislocation has mutual semion statistics with the gauge flux. Further reducing the $O(4)\phantom{\rule{4pt}{0ex}}\mathrm{NL}\ensuremath{\sigma}M$ to the Ising limit, we arrive at the Levin-Gu model with stripy modulation whose dislocation has nontrivial braiding statistics.

arXiv:1603.02694Physical review. B./Physical review. BJun 16, 201615 citations

Topological states of matter

David S. Simon

Institute of Physics eBooksMay 1, 20210 citations

{\mathbb{Z}}_{n}

generalizations of the model. Additionally, we consider the phases in which the crystalline orders are destroyed through proliferation of double dislocations. The resulting phases are shown to host topological orders described by non-Abelian gauge theories.

In this paper we focus on the interplay between geometry defects and topological properties in bosonic symmetry-protected topological (SPT) phases. We start from eight copies of 3D time-reversal

(\mathcal{T})

invariant topological superconductors (TSC) on a crystal lattice. We melt the lattice by condensation of disclinations and therefore restore the rotation symmetry. Such a disclination condensation procedure confines the fermion and afterwards turns the system into a 3D boson topological liquid crystal (TCL). The low energy effective theory of this crystalline-liquid transition contains a topological term inherited from the geometry axion response in TSC. In addition, we investigate the interplay between dislocation and superfluid vortex on the surface of TCL. We demonstrate that the

\mathcal{T}

and translation invariant surface state is a double

[e\mathcal{T}m\mathcal{T}]

state with intrinsic surface topological order. We also look into the exotic behavior of dislocation in the 2D boson SPT state described by an

O(4)

nonlinear $\ensuremath{\sigma}$ model $(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ with topological $\mathrm{\ensuremath{\Theta}}$ term. By dressing the

O(4)

vector with spiral order and gauging the symmetry, the dislocation has mutual semion statistics with the gauge flux. Further reducing the $O(4)\phantom{\rule{4pt}{0ex}}\mathrm{NL}\ensuremath{\sigma}M$ to the Ising limit, we arrive at the Levin-Gu model with stripy modulation whose dislocation has nontrivial braiding statistics.