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Internal Levin-Wen models
by Vincentas Mulevičius, Ingo Runkel, Thomas Voß
Submission summary
| Authors (as registered SciPost users): | Vincentas Mulevičius |
| Submission information | |
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| Preprint Link: | scipost_202310_00013v2 (pdf) |
| Date submitted: | 2024-06-03 21:21 |
| Submitted by: | Mulevičius, Vincentas |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Levin-Wen models are a class of two-dimensional lattice spin models with a Hamiltonian that is a sum of commuting projectors, which describe topological phases of matter related to Drinfeld centres. We generalise this construction to lattice systems internal to a topological phase described by an arbitrary modular fusion category $\mathcal{C}$. The lattice system is defined in terms of an orbifold datum $\mathbb{A}$ in $\mathcal{C}$, from which we construct a state space and a commuting-projector Hamiltonian $H_{\mathbb{A}}$ acting on it. The topological phase of the degenerate ground states of $H_{\mathbb{A}}$ is characterised by a modular fusion category $\mathcal{C}_{\mathbb{A}}$ defined directly in terms of $\mathbb{A}$. By choosing different $\mathbb{A}$'s for a fixed $\mathcal{C}$, one obtains precisely all phases which are Witt-equivalent to $\mathcal{C}$. As special cases we recover the Kitaev and the Levin-Wen lattice models from instances of orbifold data in the trivial modular fusion category of vector spaces, as well as phases obtained by anyon condensation in a given phase $\mathcal{C}$.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
• For personal convenience we start with change B, which requested a discussion on what models can be constructed given the initial phase C, perhaps for a concrete simple choice of C. In the previous version this was briefly addressed in the introduction: the phases, realized as ground states spaces of our models, are exactly the phases D which are Witt-equivalent to C, i.e. the product CxD’ has to be a Drinfeld centre. Admittedly this was too brief, to remedy this we added a new section “Universality of Internal Levin—Wen models” to the introduction. In it we explain how our models are capable of performing both anyon condensation as well as the opposite procedure: de-condensation. We also sketch a particular example of this: su(2)_10 phase can be condensed into the Ising phase, so there exist an input for our model, which would combine Ising anyons in a way such that su(2)_10 would emerge (see newly added diagram (1.2)). We do think however that a detailed discussion of this and other similar examples would significantly expand the volume of this already lengthy paper, so, with referees’ permission, we would like to postpone this for a future work.
• Change A requested to explain more the relation between the models introduced in the paper and the models living at a boundary of a Walker--Wang (WW) lattice models. To address this, we expanded the subsection in the introduction where WW models are discussed – we now emphasise that WW provide one with a way to realise topological phases which possibly are not Drinfeld centres, whereas our models show how to entangle and fuse/braid anyons in an existing system so that a new phase emerges independently on what these systems are. For example, the system supporting the initial anyons can come from a boundary WW model, but they can also be realised in other ways (e.g. Ising anyons as Majorana fermions).
Besides these changes, we also corrected a number of typographic errors.