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Theory of oblique topological insulators

by Benjamin Moy, Hart Goldman, Ramanjit Sohal, Eduardo Fradkin

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Authors (as registered SciPost users): Benjamin Moy
Submission information
Preprint Link: scipost_202209_00027v1  (pdf)
Date submitted: 2022-09-14 03:51
Submitted by: Moy, Benjamin
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


A long-standing problem in the study of topological phases of matter has been to understand the types of fractional topological insulator (FTI) phases possible in 3+1 dimensions. Unlike ordinary topological insulators of free fermions, FTI phases are characterized by fractional $\Theta$-angles, long-range entanglement, and fractionalization. Starting from a simple family of $\mathbb{Z}_N$ lattice gauge theories due to Cardy and Rabinovici, we develop a class of FTI phases based on the physical mechanism of oblique confinement and the modern language of generalized global symmetries. We dub these phases oblique topological insulators. Oblique TIs arise when dyons—bound states of electric charges and monopoles—condense, leading to FTI phases characterized by topological order, emergent one-form symmetries, and gapped boundary states not realizable in 2+1-D alone. Based on the lattice gauge theory, we present continuum topological quantum field theories (TQFTs) for oblique TI phases involving fluctuating one-form and two-form gauge fields. We show explicitly that these TQFTs capture both the generalized global symmetries and topological orders seen in the lattice gauge theory. We also demonstrate that these theories exhibit a universal "generalized magnetoelectric effect" in the presence of two-form background gauge fields. Moreover, we characterize the possible boundary topological orders of oblique TIs, finding a new set of boundary states not studied previously for these kinds of TQFTs.

Author comments upon resubmission

We thank the referees for their comments and questions. We give detailed responses to the points raised by each of the referees using the "Reply to the above report" option. Note that all equation and reference numbers correspond to those in the initially submitted manuscript.

List of changes

p. 4: Included a more explicit statement that oblique TIs are not SPTs protected by a zero-form symmetry

p. 6: Highlighted the parafermion modes of the boundary states of the (1+1)-D model discussed in Appendix E

p. 8: Added an explicit statement about charge N becoming energetically suppressed

pp. 9, 21, 29: Corrected group theory misprints noted by Referee 1

p. 12: Added comments on energetic stability of the gapped oblique TI phases

p. 33: Clarified when gauge fixing is allowed in the generalized magnetoelectric effect

p. 43: Added Ref. [65], a related paper posted after we submitted our original version

Appendix E: Included a (1+1)-D analogue of the generalized magnetoelectric effect and commented more about the parafermion boundary modes

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2022-10-4 (Invited Report)


The authors have answered my questions.

  • validity: good
  • significance: ok
  • originality: ok
  • clarity: ok
  • formatting: good
  • grammar: good

Anonymous Report 1 on 2022-9-29 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202209_00027v1, delivered 2022-09-29, doi: 10.21468/SciPost.Report.5799


More explanation is required about deriving the bulk response (6.18). In particular, (6.18) implies that the same bulk low energy effective theory with parameter m or m+2Nn and the same coupling to the background field B of global one-form symmetry gives different response, and thus it is ambiguous. Can the author clarify this?

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Benjamin Moy  on 2022-10-12  [id 2914]

(in reply to Report 1 on 2022-09-29)

As the referee points out, Eq. (6.11) and the bulk topological order have the periodicity $m\sim m+2Nn$, so there is some ambiguity in the response computed using the TQFT. An analogous situation arises in the fractional quantum Hall effect, where the effective Chern-Simons theory can only compute the fractional part of the Hall conductivity since one can always "add Landau levels". We note, however, that for a given $N$, oblique confining states with different topological orders will never have the same response coefficient. Moreover, when the theory is on a manifold with a boundary, the response is unambiguous. In the quantum Hall case, a boundary is required to inject a current and measure the Hall voltage, though the Hall conductivity is still a feature characterizing local response of the quantum Hall fluid regardless of whether there is a boundary. The same considerations should apply to the response for an oblique confining state. We have added a paragraph to the end of Section 6 to clarify these points.

Anonymous on 2023-02-02  [id 3299]

(in reply to Benjamin Moy on 2022-10-12 [id 2914])

In the case of fractional quantum Hall, it does not make sense to discuss fractional Chern-Simons term as it is not gauge invariant. It makes sense to talk about contact term in current two-point function, or relate the current with the background field. One can stack an integer quantum Hall, and that reflects in the contact term or the quantum Hall conductance. In particular, one can make sense of these on manifolds without boundary.

In your setting, however, I don't see a way to define the response on manifolds without boundary. Since the ambiguity modifies the denominator not the numerator of the coefficient, it is also not related to stacking of invertible phases. So it is quite different from the fractional quantum Hall case.

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