Matrix product states for anyonic systems and efficient simulation of dynamics

Abstract

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 anyonic TEBD algorithm, we used it to simulate (i) the ground state (using imaginary time evolution) of an infinite 1D 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 Hubbardlike 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.

DMRG investigation of constrained models: from quantum dimer and quantum loop ladders to hard-boson and Fibonacci anyon chains

“…It is worth mentioning that the approach we suggest is not the first attempt to target the reduced Hilbert space of a constrained model with tensor network algorithms. In the context of anyons, the anyonic fusion tree has been encoded explicitly first into MERA [21,22] and more recently into TEBD [23,24] through an auxiliary 'fusion tensor'. The role of this tensor is to select the states from the MPS tensor that satisfy the constraint and to discard all the other ones.…”

Simulation of braiding anyons using matrix product states

“…It should be noted that this anyonic MPS has been drawn with site indices going upwards, to make apparent the visual similarity with anyonic fusion tree diagrams, but it is essentially the same ansatz given in Ref. 43. Due to the iterative fusion process down the tree of the anyonic MPS the dimensions of the tensors Γ [i] required to exactly construct an arbitrary state will vary, but in practice an upper bound is imposed on the bond dimension χ ahead of time.…”

“…), one can construct an MPS ansatz for anyonic systems at any fixed rational filling. This ansatz may then be used to construct an an approximation to the ground state of a system by means of anyonic algorithms such as the anyonic TEBD algorithm proposed by Singh et al 43 or the anyonic DMRG. 44 Ground state properties such as entropy scaling and correlation functions can be computed using approaches similar to those for conventional tensor networks, but modified to account for anyonic statistics by normalizing vertices, removing loops and bending anyonic charge lines in accordance with the prescriptions given in Ref.…”

“…[40][41][42] In particular, anyonic versions of the Matrix Product States (MPS), and of the TEBD and DMRG algorithms have been proposed and tested with a high degree of accuracy for anyonic chains. 43,44 Tensor network algorithms are adapted to anyons by explicitly hardwiring the constraints implied by the fusion rules of the anyon model into the tensor network ansatz. This provides two important advantages.…”

Transport properties of anyons in random topological environments

“…Full details of the numerical simulation are given in Ref. [25]. Thus, while the position of the walker and the fusion degrees of freedom of the anyons are entangled during the time evolution, the algorithm preserves the distinction between them.…”