Phases of matter can be defined as equivalence classes of physical systems that share common properties. But which equivalence classes are physically relevant? A guiding principle to answer this question is that properties of a system are governed by the way the microscopic constituents are arranged. Different arrangements, or orders, then lead to different phases of matter. In the late '80s, a new kind of quantum order was discovered, which was coined "topological order". A system with topological order typically possesses more than one ground state, and the number of such states precisely depends on its topology. Remarkably, excited states can be interpreted as quasi-particles with anyonic statistics. Together with their inherent robustness to local perturbations, this makes topological phases prime candidates for quantum computing platforms and quantum memories.

*My research project primarily aims at boosting our understanding of topological order in (3+1)d systems. I am interested in constructing and characterizing lattice models by revealing the underlying algebraic structures. More recently, I have been developing the theory of tensor networks as an analytical and numerical tool for the study of (3+1)d topological phases.*

2021 C. Delcamp

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This manuscript is the result of my effort to get a better grasp on dualities between (3+1)d topological models. Mathematically, such dualities are encoded into the notion of Morita equivalence, which becomes less and less intuitive as we go to higher spatial dimensions, even for simple models such as the toric code. Yet the electromagnetic duality of the toric code has a simple origin from a physical standpoint. One incarnation of this duality is the existence of two inequivalent tensor network representations of the ground state subspace. These representations, which are obtained via an elementary procedure, are characterised by symmetry conditions with respect to string- and membrane-like operators, respectively. One goal of this paper was to clarify how the existence of these representationsâ€”and a fortiori the electromagnetic dualityâ€”is related to Morita equivalence. Relying on category theoretical concepts, I propose a systematic framework to construct families of tensor network representations for finite group generalisations of the toric code. Within this framework, it is apparent that two canonical representations satisfy symmetry conditions with respect to operators encoded into the fusion 2-categories \({\rm 2Vec}_{G}\) and \({\rm 2Rep}(G)\), respectively. These 2-categories are then explicitly checked to be Morita equivalent. Specialising to \(G=\mathbb Z_2\), we recover the aforementioned representations of the toric code.

Topological lattice models in (2+1)d can be defined as Hamiltonian realizations of the Turaev-Viro-Barrett-Westbury topological quantum field theory. In this formulation, the categorical structure encoding the anyonic excitations and their statistics corresponds to the quantum invariant the theory assigns to the circle. In this article, we explain with A. Bullivant that the higher-categorical structure encoding string-like excitations in (3+1)d gauge models of topological phases analogously corresponds to the quantum invariant the topological quantum field theory assigns to the circle. In contrast, the quantum invariant assigned to the torus encodes the loop-like excitations. We then compute the "crossing with the circle conditions", which establish that the dimension, or categorification thereof, of the quantum invariant assigned to a closed manifold \(\Sigma\) is equivalent to that assigned to the manifold \(\Sigma \times \mathbb S_1\). This computation formalizes the idea that loop-like excitations can be obtained by gluing string-like excitations along their endpoints. Finally, we exploit this result in order to revisit the statement that loop-like excitations define representations of the linear necklace group as well as the loop braid group.

Tensor networks provide a very powerful analytical and numerical framework for the study of strongly correlated quantum many-body systems. In particular, they have been exploited to encode (2+1)d topological orders and their anyonic excitations, whereby the characteristic long-range correlation patterns are built up by contracting entanglement degrees of freedom of individual tensors. In these two articles, we initiated a systematic generalization of this approach for the study of (3+1)d topological orders. In the first manuscript, N. Schuch and I constructed and studied two isometric tensor network representations of the (3+1)d toric code. Moreover, we highlighted how the duality relation between these representations encodes the duality between a boundary (2+1)d Ising model and Wegner's \(\mathbb Z_2\) gauge theory. In the second manuscript with D. Williamson and F. Verstraete, we showed that one of these representations is stable to arbitrary local tensor perturbations, including those that do not map to local operators on the physical Hilbert space, and conjectured a relation between stability and so-called virtual symmetries for a given tensor network representation. This result provides promising evidence that (3+1)d topological tensor networks form a set of positive measure and further motivates the study of dual representations.

A salient feature of three-dimensional topological lattice models is the existence of loop-like excitations, where "loop-like" here refers to the topology of a defect whose tubular neighbourhood is a region of the physical system with energy higher than that of the ground state. In general, such a defect supports composite excitations consisting of loop-like fluxes, to which point-like charges are attached, while being threaded by other fluxes. In the same vein, (open) string-like excitations can be considered, which occur for instance when loop-like excitations are brought in contact with gapped boundaries. We propose in this paper a classification and a characterization of such string-like excitations for a class of topological models that have a lattice gauge theory interpretation. Our derivations rely on a generalization of the so-called "tube algebra" approach, which consists in revealing the algebra, or categorification thereof, underlying the excitations and derive the elementary excitations as the corresponding simple modules. Among other things, we find that these string-like excitations are organized into a mathematical structure known as a bicategory, which is equivalent to the so-called monoidal center of another bicategory that serve as input data of the theory.

Partition functions of statistical systems can be written as tensor networks, i.e. collections of tensors that are contracted together according to patterns dictated by graphs, so that computing a partition function boils down to contracting the corresponding tensor network. In the thermodynamic limit, such tensor networks can only be contracted approximately due to the exponential growth of degrees of freedom. In 2017, we introduced with M. Hauru and S. Mizera a simple algorithm coined "gilt-TNR" that performs this approximate contraction very efficiently and, as a by-product, computes the renormalization group of the theory in the space of tensors. In this manuscript, A. Tilloy and I applied the same algorithm to compute the renormalization group flow of the \(\phi^4\) field theory. This was made possible by regularizing space into a fine-grained lattice and discretizing the scalar field in a controlled way. Aside from precious qualitative insights, we were able to compute the critical coupling of the theory in the continuum limit, which is a difficult problem (strongly coupled and non-perturbative). At the time of writing the paper, it was the best estimate for this constant, thus showcasing the power of tensor network renormalization techniques.

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