Topic bram

_thesistopics/2024/BVancraeynest1.md

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+---
+title: "Boundaries of string-net models"
+promoter: Frank Verstraete
+supervisor: Bram vancraeynest-De Cuiper
+contact: Bram vancraeynest-De Cuiper
+topic: "Condensed Matter Theory"
+year: "2024"
+---
+
+##### Context
+
+Topological phases of matter are highly-entangled quantum phases which are characterized by a robust ground state degeneracy and exotic anyonic excitations. Their robustness against local noise makes these topological systems prime candidates for storage and manipulation of quantum information.
+
+It is believed that all non-chiral (2+1)d topological orders can be microscopically realized by a mechanism called string-net condensation. This mechanism was introduced by Levin and Wen in 2004 [1]. Soon after their introduction it was appreciated that these systems are efficiently represented in terms of tensor networks [1,2]. In essence, these networks assign to each physical spin a tensor which is connected via virtual bonds to its neighbors. These bonds capture the underlying entanglement patterns of the state. The Ghent quantum group has been at the forefront of both theoretical and numerical research of tensor networks.
+
+The advantages of studying Levin-Wen models in terms of tensor networks are manifold. They not only allow for an elegant computation and representation of the different anyonic excitations but also allow for a characterization of gapped boundaries and edge modes of Levin-Wen models. Indeed, it is understood that the computational capabilities of non-chiral topological orders can be severely influenced by the presence of boundaries.
+
+Two distinct but equivalent approaches to characterize the gapped boundaries and their corresponding Hamiltonians governing the edge dynamics have been proposed in [4] and [5]. Whereas the first approach can be directly interpreted in terms of tensor network representations [6], the second approach is expected to be generalizable to (3+1)d.
+
+#### Goals
+
+The first goal of this thesis is to compare the approaches in [4] and [5] by rephrasing the approach of [5] in terms of explicit tensors. In particular, it would be interesting to study the edge modes of Levin-Wen models in this language.
+
+A second more ambitious goal would be to generalize the approach presented in [5] to (3+1)-dimensional Walker-Wang models [7]. The models form a generalization of Levin-Wen models to (3+1)d and a systematic study of their gapped boundaries is currently lacking.
+
+This project is suited for a student with a strong interest in theoretical physics and abstract mathematical techniques applied to quantum lattice models.
+
+[1] J. I. Cirac, D. Pérez-García, N. Schuch, F. Verstraete, [arXiv:2011.12127](https://arxiv.org/abs/2011.12127)
+
+[2] Buerschaper et al., [arXiv:0809.2393](https://arxiv.org/abs/0809.2393)
+
+[3] Levin, Wen, [arXiv:0404617](https://arxiv.org/abs/cond-mat/0404617)
+
+[4] Kitaev, Kong, [arXiv:1104.5047](https://arxiv.org/abs/1104.5047)
+
+[5] Hu, et al., [arXiv:1706.03329](http://arxiv.org/abs/1706.03329)
+
+[6] Lootens, et al., [arXiv:2008.11187](https://arxiv.org/abs/2008.11187)
+
+[7] Walker, Wang, [arXiv:1104.2632](https://arxiv.org/abs/1104.2632)