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This work proposes a minimal model extending the duality between classical statistical spin systems and fermionic systems beyond the case of free fermions. A Jordan-Wigner transformation applied to a two-dimensional tensor network maps the partition sum of a classical statistical mechanics model to a Grassmann variable integral, structurally similar to the path integral for interacting fermions in two dimensions. The resulting model is simple, featuring only two parameters: one governing spin-spin interaction (dual to effective hopping strength in the fermionic picture), the other measuring the deviation from the free fermion limit. Nevertheless, it exhibits a rich phase diagram, partially stabilized by elements of topology, and featuring three phases meeting at a tricritical point. Besides the interpretation as a spin and fermionic system, the model is closely related to loop gas and vertex models and can be interpreted as a parity-preserving (non-unitary) circuit. Its minimal construction makes it an ideal reference system for studying non-linearities in tensor networks and deriving results by means of duality., Comment: 17 pages, 6 figures

In this work we present a proposal for realising non-Abelian anyons on a NISQ device. In particular we explore the feasibility of implementing the quantum double model $D(D_4)$. We propose techniques to drastically simplify the circuits for the manipulation and measurements of anyons. Numerical simulations with realistic noise models suggest that current NISQ technology is capable of probing signatures of non-Abelian anyons far beyond elemental properties such as the non-commutativity of braids. In particular, we conclude that experimentally measuring the full modular data of the model is feasible., Comment: 40 pages without references, 28 figures, 7 sections and 5 appendices

We consider fixed-point models for topological phases of matter formulated as discrete path integrals in the language of tensor networks. Such zero-correlation length models with an exact notion of topological invariance are known in the mathematical community as state-sum constructions or lattice topological quantum field theories. All of the established ansatzes for fixed-point models imply the existence of a gapped boundary as well as a commuting-projector Hamiltonian. Thus, they fail to capture topological phases without a gapped boundary or commuting-projector Hamiltonian, most notably chiral topological phases in $2+1$ dimensions. In this work, we present a more general fixed-point ansatz not affected by the aforementioned restrictions. Thus, our formalism opens up a possible way forward towards a microscopic fixed-point description of chiral phases and we present several strategies that may lead to concrete examples. Furthermore, we argue that our more general ansatz constitutes a universal form of topological fixed-point models, whereas established ansatzes are universal only for fixed-points of phases which admit topological boundaries., Comment: v4: version accepted for publication in PRB

We propose a unified mathematical framework for classifying phases of matter. The framework is based on different types of combinatorial structures with a notion of locality called lattices. A tensor lattice is a local prescription that associates tensor networks to those lattices. Different lattices are related by local operations called moves. Those local operations define consistency conditions for the tensors of the tensor network, the solutions to which yield exactly solvable models for all kinds of phases. We implement the framework to obtain models for symmetry-breaking and topological phases in up to three space-time dimensions, their boundaries, defects, domain walls and symmetries, as well as their anyons for 2+1-dimensional systems. We also deliver ideas of how other kinds of phases, like SPT/SET, fermionic, free-fermionic, chiral, and critical phases, can be described within our framework. We also define another structure called contracted tensor lattices which generalize tensor lattices: The former associate tensors instead of tensor networks to lattices, and the consistency conditions for those tensors are defined by another kind of local operation called gluings. Using this generalization, our framework also covers mathematical structures like axiomatic (non-fully extended or defective) TQFTs, that do not directly describe phases on a microscopic physical level, but formalize certain aspects of potential phases, like the anyon statistics of 2+1-dimensional phases. We also introduce the very powerful concept of (contracted) tensor lattice mapping, unifying a lots of different operations, such as stacking, anyon fusion, anyon condensation, equivalence of different fixed point models, taking the Drinfel'd centre, trivial defects or interpreting a bosonic model as a fermionic model., Comment: First version of many more