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Internal LevinWen models
by Vincentas Mulevičius, Ingo Runkel, Thomas Voß
Submission summary
Authors (as registered SciPost users):  Vincentas Mulevičius 
Submission information  

Preprint Link:  scipost_202310_00013v2 (pdf) 
Date submitted:  20240603 21:21 
Submitted by:  Mulevičius, Vincentas 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
LevinWen models are a class of twodimensional 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 commutingprojector 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 Wittequivalent to $\mathcal{C}$. As special cases we recover the Kitaev and the LevinWen 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 $\mathcal{C}$.
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 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 multipronged followup work
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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 Wittequivalent 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: decondensation. 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 WalkerWang (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.