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Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics

by Heidar Moradi, Seyed Faroogh Moosavian, Apoorv Tiwari

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Submission summary

Authors (as registered SciPost users): Seyed Faroogh Moosavian · Heidar Moradi · Apoorv Tiwari
Submission information
Preprint Link: scipost_202305_00030v1  (pdf)
Date accepted: 2023-08-07
Date submitted: 2023-05-18 12:19
Submitted by: Tiwari, Apoorv
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We outline a holographic framework that attempts to unify Landau and beyond-Landau paradigms of quantum phases and phase transitions. Leveraging a modern understanding of symmetries as topological defects/operators, the framework uses a topological order to organize the space of quantum systems with a global symmetry in one lower dimension. The global symmetry naturally serves as an input for the topological order. In particular, we holographically construct a String Operator Algebra (SOA) which is the building block of symmetric quantum systems with a given symmetry $G$ in one lower dimension. This exposes a vast web of dualities which act on the space of $G$-symmetric quantum systems. The SOA facilitates the classification of gapped phases as well as their corresponding order parameters and fundamental excitations, while dualities help to navigate and predict various corners of phase diagrams and analytically compute universality classes of phase transitions. A novelty of the approach is that it treats conventional Landau and unconventional topological phase transitions on an equal footing, thereby providing a holographic unification of these seemingly-disparate domains of understanding. We uncover a new feature of gapped phases and their multi-critical points, which we dub fusion structure, that encodes information about which phases and transitions can be dual to each other. Furthermore, we discover that self-dual systems typically posses emergent non-invertible, i.e., beyond group-like symmetries. We apply these ideas to $1+1d$ quantum spin chains with finite Abelian group symmetry, using topologically-ordered systems in $2+1d$. We predict the phase diagrams of various concrete spin models, and analytically compute the full conformal spectra of non-trivial quantum phase transitions, which we then verify numerically.

List of changes

Below are the list of changes in the new revised version. For the sake of readability, all the newly-added sentences in the resubmitted version are highlighted in brown color. All reference and page numbering in the following refers to the resubmitted version.

(1) We have added a few paragraphs in the Introduction, starting on page 4 and ending on
page 6 right above Summary of Results, explaining (1) similar ideas in different contexts that
appeared in the past, (2) the similarities and differences of our approach to the problem of
unification of phases of matter with other approaches, and contextualizing the work, and
finally (3) emphasizing on the practical aspect and concreteness of our approach to building
up phase diagrams of various models from the ground up.

(2) Added a sentence in the introduction clarifying the relation between the string operator algebra
developed in our work and the patch operators discussed in arXiv:1912.13492 (Wen-Ji) and
arXiv:2203.03596 (Wen-Chatterjee).

(3) We added a few sentences at the beginning of Section 3.1 to describe the content of the section
more concretely.

(4) We have added a paragraph at the end of Section 3.7 to emphasize that the construction in
that section is new and generalizes the one that appeared in [39, 40].

(5) In Section 4.1, we have added a few sentences (as a footnote) after the sentence

"We will call the group structure of a Lagrangian subgroup L the fusion structure of the corresponding gapped phase."

to clarify the confusion regarding the term “fusion structure”.

(6) We added a sentence in the very first paragraph of Section 4.3 to clarify further the content
of the section and to explain better when we can compute the full twisted partition functions
of a transition.

(7) We have added a reference to the classic paper of Kramers and Wannier [41, 42] in Section
3.5, where we are discussing the generalized version thereof applicable to any finite Abelian
group G.

(8) List of added references scattered throughout of the newly-submitted manuscript: [1, 4, 5, 6, 2, 3, 43, 37, 38, 44, 36, 30, 34, 40, 41, 42].

(9) All the other points/questions raised by the referee have been addressed in the above replies.

Published as SciPost Phys. Core 6, 066 (2023)

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