Matrix product states (MPS) have proven to be a very successful tool to study lattice systems with local degrees of freedom such as spins or bosons. Topologically ordered systems can support anyonic particles which are labeled by conserved topological charges and collectively carry nonlocal degrees of freedom. In this paper we extend the formalism of MPS to lattice systems of anyons. The anyonic MPS is constructed from tensors that explicitly conserve topological charge. We describe how to adapt the time-evolving block decimation (TEBD) algorithm to the anyonic MPS in order to simulate dynamics under a local and charge-conserving Hamiltonian. To demonstrate the effectiveness of the anyonic TEBD algorithm, we used it to simulate (i) the ground state (using imaginary time evolution) of an infinite one dimensional critical system of (a) Ising anyons and (b) Fibonacci anyons, both of which are well studied, and (ii) the real time dynamics of an anyonic Hubbard-like model of a single Ising anyon hopping on a ladder geometry with an anyonic flux threading each island of the ladder. Our results pertaining to (ii) give insight into the transport properties of anyons. The anyonic MPS formalism can be readily adapted to study systems with conserved symmetry charges, as this is equivalent to a specialization of the more general anyonic case.Matrix product states (MPS) have proven to be a very successful tool to study lattice systems with local degrees of freedom such as spins or bosons. Topologically ordered systems can support anyonic particles which are labeled by conserved topological charges and collectively carry nonlocal degrees of freedom. In this paper we extend the formalism of MPS to lattice systems of anyons. The anyonic MPS is constructed from tensors that explicitly conserve topological charge. We describe how to adapt the time-evolving block decimation (TEBD) algorithm to the anyonic MPS in order to simulate dynamics under a local and charge-conserving Hamiltonian. To demonstrate the effectiveness of the anyonic TEBD algorithm, we used it to simulate (i) the ground state (using imaginary time evolution) of an infinite one dimensional critical system of (a) Ising anyons and (b) Fibonacci anyons, both of which are well studied, and (ii) the real time dynamics of an anyonic Hubbard-like model of a single Ising anyon hopping on a ladder geometry with an anyonic flux threading each island of the ladder. Our results pertaining to (ii) give insight into the transport properties of anyons. The anyonic MPS formalism can be readily adapted to study systems with conserved symmetry charges, as this is equivalent to a specialization of the more general anyonic case.

arXiv:1311.0967Physical Review BFeb 13, 201423 citations

PHASES IN THE PHYSICS OF ANYONS

Carlo A. Trugenberger

Foundations of Quantum MechanicsApr 1, 19920 citations

Symmetry reduction induced by anyon condensation: A tensor network approach

José Garre-Rubio, S. Iblisdir, David Pérez-Garcı́a

Topological ordered phases are related to changes in the properties of their quasiparticle excitations (anyons). We study these relations in the framework of projected entanglement pair states and show how condensing and confining anyons reduces a local gauge symmetry to a global on-site symmetry. We also study the action of this global symmetry over the quasiparticle excitations. As a byproduct, we observe that this symmetry reduction effect can be applied to one-dimensional systems as well and brings about appealing physical interpretations on the classification of phases with symmetries using matrix product states. The case of ${\mathbb{Z}}_{2}$ on-site symmetry is studied in detail.Topological ordered phases are related to changes in the properties of their quasiparticle excitations (anyons). We study these relations in the framework of projected entanglement pair states and show how condensing and confining anyons reduces a local gauge symmetry to a global on-site symmetry. We also study the action of this global symmetry over the quasiparticle excitations. As a byproduct, we observe that this symmetry reduction effect can be applied to one-dimensional systems as well and brings about appealing physical interpretations on the classification of phases with symmetries using matrix product states. The case of ${\mathbb{Z}}_{2}$ on-site symmetry is studied in detail.

arXiv:1702.08759Physical review. B./Physical review. BOct 16, 201715 citations

Quantum Ising Model on $(2+1)-$ Dimensional Anti $-$ de Sitter Space using Tensor NetworksQuantum Ising Model on $(2+1)-$Dimensional Anti$-$de Sitter Space using Tensor Networks

Abhishek Samlodia, Simon Catterall, Alexander F. Kemper +2 more

We study the quantum Ising model on (2+1)-dimensional anti-de Sitter space using Matrix Product States (MPS) and Matrix Product Operators (MPOs). We explore the bulk phase diagram of the theory on regular tessellations of hyperbolic space with coordination number seven and find disordered and ordered phases separated by a phase transition. We find that the boundary-boundary spin correlation function exhibits power law scaling deep in the disordered phase of the Ising model consistent with holography. At the critical point, we find the boundary entanglement entropy scales logarithmically with subsystem size but away from this, we see a linear scaling. In comparison, the full system exhibits a volume law scaling, which is expected in chaotic and/or highly connected systems. We also measure Out of time Ordered Correlators (OTOCs) to explore the scrambling behavior of the theory.We study the quantum Ising model on (2+1)-dimensional anti-de Sitter space using Matrix Product States (MPS) and Matrix Product Operators (MPOs). We explore the bulk phase diagram of the theory on regular tessellations of hyperbolic space with coordination number seven and find disordered and ordered phases separated by a phase transition. We find that the boundary-boundary spin correlation function exhibits power law scaling deep in the disordered phase of the Ising model consistent with holography. At the critical point, we find the boundary entanglement entropy scales logarithmically with subsystem size but away from this, we see a linear scaling. In comparison, the full system exhibits a volume law scaling, which is expected in chaotic and/or highly connected systems. We also measure Out of time Ordered Correlators (OTOCs) to explore the scrambling behavior of the theory.

hep-latCross-listedarXiv:2512.20838Physical review. D/Physical review. D.Dec 25, 20250 citations

Disordered Pinned Anyons in Two Dimensions

Chris R. Laumann, David A. Huse, Andreas Ludwig +3 more

Bulletin of the American Physical SocietyMar 23, 20110 citations

{\mathbb{Z}}_{2}

on-site symmetry is studied in detail.

Quantum Ising Model on

(2+1)-

Dimensional Anti

-

de Sitter Space using Tensor Networks