SciPost Submission Page
Internal Levin-Wen models
by Vincentas Mulevičius, Ingo Runkel, Thomas Voß
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Vincentas Mulevičius |
| Submission information | |
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| Preprint Link: | scipost_202310_00013v2 (pdf) |
| Date accepted: | 2024-09-10 |
| 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.
Published as SciPost Phys. 17, 088 (2024)
Reports on this Submission
Report #2 by Anonymous (Referee 3) on 2024-8-28 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202310_00013v2, delivered 2024-08-28, doi: 10.21468/SciPost.Report.9669
Strengths
I think that this paper is very interesting, the main result is an important contribution to the field. Moreover, the paper is very well written with many beautiful illustrating pictures.
Weaknesses
I did not find any obvious weakness in this paper.
Report
In recent years, one of the authors Runkel and his collaborators have systematically developed the theory of orbiford constructions in 2+1D TQFT. It is a state-sum construction that largely generalizes the so-called anyon condensation construction, which is reversible. The orbiford construction is reversible and closer to physical reality. As far as I can tell, this work is a Hamiltonian version of this state-sum construction. I think that it is a very interesting and important result, which was long anticipated in this field.
Requested changes
In the orbiford data, an algebra A decorates a surface (1-codimensional). I suspect that this work is the first physical realization of the condensation theory of topological defects of codimension 1 (see arXiv:2403.07813). It is helpful if the authors can add some comments of about it.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Report #1 by Anonymous (Referee 1) on 2024-8-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202310_00013v2, delivered 2024-08-18, doi: 10.21468/SciPost.Report.9608
Strengths
1- Novel lattice construction of topological orders beyond those accessible with standard Levin-Wen construction
2- Clear exposition with many high-quality diagrams that provide intuition for the abstract underlying categorical concepts
Report
Dear authors and editors,
my sincere apologies for the delayed response.
In short, I agree with the previous referees that this is a very valuable addition to the literature and I am happy to recommend it for publication. As far as I can tell, the authors have addressed the comments made by the previous with the improvements in their resubmission.
I have only a few minor comments and a request for clarification that the authors might find useful:
- It is mentioned that the lattice model is non-local, because the state space on which the Hamiltonian is not of tensor product form. As far as I can tell however, the individual terms in the Hamiltonian still only act on a subset of the degrees of freedom that surround a vertex/edge/fase, and does not introduce long-range interactions. Is this correct? If so, the use of "non-locality" is a bit confusing, since this is most often taken to mean a Hamiltonian which includes long-range interactions between degrees of freedom that are far away from each other. As an example, lattice gauge theories (quantum double models in which one manifestly enforces the vertex term) also do not have a tensor product state space, but we still consider these models to be local in that there is a finite (Lieb-Robinson) velocity at which information can propagate due to the locality of the interactions.
- From a practical point of view, realising standard Levin-Wen models is difficult due to the number of degrees of freedom on which the Hamiltonian acts (in the case of no multiplicities, this is a 12-body term). This is a very relevant problem however, since the explicit realisation of e.g. the Fibonacci topological order provides a means to do universal topological quantum computation via the braiding of the topological quasiparticles. The standard Levin-Wen construction can only produce doubled Fibonacci; does the internal Levin-Wen construction of this paper for the Fibonacci topological phase admit an equally complicated Hamiltonian since it lives on top of a conventional Levin-Wen model?
- Given that this construction is able to produce chiral topological order, is there any way to use these lattice models to explore the necessarily gapless boundary theories for these models? Is it easy to see that there cannot exist a gapped boundary theory?
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)