Francuz A. Determining topological order with tensor networks

Kraków: Jagiellolonian University, 2021. — 100 p.Exotic phases of matter, which fall beyond the Landau paradigm of phases and phase
transitions, emerged as one of the main directions of research in the field of condensed
matterphysicsinthelastfewdecades. Thisleadtobothadvancementintheirexperimental
realizationsandprogressintheiranalysis, thankstopowerfulnumericalmethodsliketensor
networks. Among those exotic phases there are topologically ordered phases, the analysis
of which is especially hard due to degeneracy of the ground state and no local order
parameter. Topological order gained recognition after it was realized, thanks to Alexei
Kitaev, that quantum computational models can be written in the language of condensed
matter systems. However, apart from few exactly solvable models, the analysis of lattice
Hamiltonians for the occurrence of topological order was considered a very hard problem.
This thesis provides a basic overview of theories aiming at classifying topologically
ordered states and novel numerical approaches to determine both Abelian and non-Abelian
topological order starting from a lattice Hamiltonian. The numerical method of choice
in the study of strongly correlated two-dimensional systems, like topologically ordered
systems, is projected entangled pair states (PEPS), as it allows analysing states which
were not achievable by the state-of-the-art 2D DMRG algorithms due to long correlation
length.
In this thesis, numerical methods of analysing the optimized infinite PEPS (iPEPS) are
presented, allowing to extract the information about the topological order. The key idea
is to find the infinite matrix product operator (iMPO) symmetries of the iPEPS, whose
existence is a necessary condition for the TN state to exhibit topological order. The iMPO
symmetriescanbelaterusedtoobtaintopologicalS andT matrices, which(inmostknown
cases) can be considered as a non-local order parameter of topologically ordered phases,
in the sense that they give us unambiguous information about the model along with its
excitations and their statistics. The method is immune to any small perturbations of the
tensors, which had been a long feared problem due to numerical inaccuracies which may
ariseduringthegroundstateoptimization. Furthermore, findingiMPOsymmetriesenables
an elegant description of the model in terms of the mathematical structure underlying the
topologically ordered phases of matter – modular tensor category.Oświadczenie
Abstract
AcknowledgementsTopological order
Long range entanglement
Quantum statistics
String-nets and fusion category
Tensor networks
Variational ansatz
Algorithms
Entanglement
Gauge transformations and symmetries
Topological order with tensor networks
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