Low-dimensional topological objects, such as knots and braids, have become prevalent in multiple areas of physics, such as fluid dynamics, optics, and quantum information processing. Such objects also now play a role in cryptography, where a framed knot can store encoded information using its braid representation for communications purposes. The greater resilience of low-dimensional topological elements under deformations allows them to be employed as a reliable framework for information exchange. Here, we introduce a challenge-response protocol as an application of this construction for authentication. We provide illustrative examples of both procedures showing how framed links and braids may help to enhance secure communication.Low-dimensional topological objects, such as knots and braids, have become prevalent in multiple areas of physics, such as fluid dynamics, optics, and quantum information processing. Such objects also now play a role in cryptography, where a framed knot can store encoded information using its braid representation for communications purposes. The greater resilience of low-dimensional topological elements under deformations allows them to be employed as a reliable framework for information exchange. Here, we introduce a challenge-response protocol as an application of this construction for authentication. We provide illustrative examples of both procedures showing how framed links and braids may help to enhance secure communication.

arXiv:2212.04350Dec 7, 20222 citations

The quest for higher dimensionality

Stanislav S. Straupe, S. P. Kulik

Nature PhotonicsAug 31, 201026 citations

Tell-tale topology

Steven R. White

Nature PhysicsNov 11, 20124 citations

(3+1) -dimensional topological quantum field theory from a tight-binding model of interacting spinless fermions

Mauro Cirio, Giandomenico Palumbo, Jiannis K. Pachos

Currently, there is much interest in discovering analytically tractable $(3+1)$ -dimensional models that describe interacting fermions with emerging topological properties. Towards that end we present a three-dimensional tight-binding model of spinless interacting fermions that reproduces, in the low-energy limit, a $(3+1)$ -dimensional Abelian topological quantum field theory called the BF model. By employing a mechanism equivalent to Haldane's Chern insulator, we can turn the noninteracting model into a three-dimensional chiral topological insulator. We then isolate energetically one of the two Fermi points of the lattice model. In the presence of suitable fermionic interactions, the system, in the continuum limit, is equivalent to a generalized $(3+1)$ -dimensional Thirring model. The low-energy limit of this model is faithfully described by the BF theory. Our approach directly establishes the presence of $(2+1)$ -dimensional BF theory at the boundary of the lattice and it provides a way to detect the topological order of the model through fermionic density measurements.Currently, there is much interest in discovering analytically tractable $(3+1)$-dimensional models that describe interacting fermions with emerging topological properties. Towards that end we present a three-dimensional tight-binding model of spinless interacting fermions that reproduces, in the low-energy limit, a $(3+1)$-dimensional Abelian topological quantum field theory called the BF model. By employing a mechanism equivalent to Haldane's Chern insulator, we can turn the noninteracting model into a three-dimensional chiral topological insulator. We then isolate energetically one of the two Fermi points of the lattice model. In the presence of suitable fermionic interactions, the system, in the continuum limit, is equivalent to a generalized $(3+1)$-dimensional Thirring model. The low-energy limit of this model is faithfully described by the BF theory. Our approach directly establishes the presence of $(2+1)$-dimensional BF theory at the boundary of the lattice and it provides a way to detect the topological order of the model through fermionic density measurements.

arXiv:1309.2380Physical Review BAug 13, 201420 citations

Shadows of anyons and the entanglement structure of topological phases

Jutho Haegeman, Valentin Zauner-Stauber, Norbert Schuch +1 more

The low-temperature dynamics of quantum systems are dominated by the low-energy eigenstates. For two-dimensional systems in particular, exotic phenomena such as topological order and anyon excitations can emerge. While a complete low-energy description of strongly correlated systems is hard to obtain, essential information about the elementary excitations is encoded in the eigenvalue structure of the quantum transfer matrix. Here we study the transfer matrix of topological quantum systems using the tensor network formalism and demonstrate that topological quantum order requires a particular type of 'symmetry breaking' for the fixed point subspace. We also relate physical anyon excitations to domain-wall excitations at the level of the transfer matrix. This formalism enables us to determine the structure of the topological sectors in two-dimensional gapped phases very efficiently, therefore opening novel avenues for studying fundamental questions related to anyon condensation and confinement.The low-temperature dynamics of quantum systems are dominated by the low-energy eigenstates. For two-dimensional systems in particular, exotic phenomena such as topological order and anyon excitations can emerge. While a complete low-energy description of strongly correlated systems is hard to obtain, essential information about the elementary excitations is encoded in the eigenvalue structure of the quantum transfer matrix. Here we study the transfer matrix of topological quantum systems using the tensor network formalism and demonstrate that topological quantum order requires a particular type of 'symmetry breaking' for the fixed point subspace. We also relate physical anyon excitations to domain-wall excitations at the level of the transfer matrix. This formalism enables us to determine the structure of the topological sectors in two-dimensional gapped phases very efficiently, therefore opening novel avenues for studying fundamental questions related to anyon condensation and confinement.

arXiv:1410.5443Nature CommunicationsOct 6, 201583 citations

Topology and Collapse

M. Socolovsky

Advances in Applied Clifford AlgebrasOct 4, 20080 citations

Currently, there is much interest in discovering analytically tractable

(3+1)

-dimensional models that describe interacting fermions with emerging topological properties. Towards that end we present a three-dimensional tight-binding model of spinless interacting fermions that reproduces, in the low-energy limit, a

(3+1)

-dimensional Abelian topological quantum field theory called the BF model. By employing a mechanism equivalent to Haldane's Chern insulator, we can turn the noninteracting model into a three-dimensional chiral topological insulator. We then isolate energetically one of the two Fermi points of the lattice model. In the presence of suitable fermionic interactions, the system, in the continuum limit, is equivalent to a generalized

(3+1)

-dimensional Thirring model. The low-energy limit of this model is faithfully described by the BF theory. Our approach directly establishes the presence of

(2+1)

-dimensional BF theory at the boundary of the lattice and it provides a way to detect the topological order of the model through fermionic density measurements.