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Mean string field theory: Landau-Ginzburg theory for 1-form symmetries
by Nabil Iqbal, John McGreevy
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Submission summary
| Authors (as registered SciPost users): | Nabil Iqbal · John McGreevy |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2106.12610v2 (pdf) |
| Date submitted: | 2022-08-10 10:53 |
| Submitted by: | Iqbal, Nabil |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
By analogy with the Landau-Ginzburg theory of ordinary zero-form symmetries, we introduce and develop a Landau-Ginzburg theory of one-form global symmetries, which we call mean string field theory. The basic dynamical variable is a string field -- defined on the space of closed loops -- that can be used to describe the creation, annihilation, and condensation of effective strings. Like its zero-form cousin, the mean string field theory provides a useful picture of the phase diagram of broken and unbroken phases. We provide a transparent derivation of the area law for charged line operators in the unbroken phase and describe the dynamics of gapless Goldstone modes in the broken phase. The framework also provides a theory of topological defects of the broken phase and a description of the phase transition that should be valid above an upper critical dimension, which we discuss. We also discuss general consequences of emergent one-form symmetries at zero and finite temperature.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2022-9-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2106.12610v2, delivered 2022-09-14, doi: 10.21468/SciPost.Report.5695
Report
The paper on “Mean string field theory: Landau-Ginzburg theory for 1-form symmetries” by Iqbal and McGreevy is a beautifully written paper exploring effective field theories for states in which the symmetry-driven organizing principle is based on higher-form symmetries. If a very powerful Landau-Ginzburg framework exists for states “made of” particles, why would it not also exist (and be useful) for states “made of” one-dimensional objects — strings? The authors construct such a theory in the style of Landau and Ginzburg and consider some simple implications of their framework. They even perform some concrete non-trivial calculations despite the complicated machinery that is required to do so. The paper is very interesting and it is written in a clear, explicit, and perhaps most importantly for such a difficult subject with a long and murky history, in an honest way. It is my opinion that the paper certainly deserves to be published by Scipost. I do not insist on any changes to the text but the authors may decide to do so in response to my questions and comments below.
The subject of (fundamental) string field theory is one of the biggest challenges in theoretical physics. Unfortunately, being an incredibly difficult subject, I think that it is fair to say that it has by and large in many ways failed. The subject of the present paper is conceptually unrelated to it, but the techniques required to write a mean string field theory are certainly related. The fact that the authors need to spend many pages discussing a derivative is only one example of why this is so hard. So, regarding this derivative operator, is it clear that the definition they use is the correct/physical/best/unique/… one? While the authors have an extensive discussion of the derivative, the answers to these questions are still not obvious to me.
Is it clear that one can neglect higher derivative terms? Is there some reason that they are suppressed in the IR in a similar way to normal derivative terms in standard EFTs?
I think that a lot of similar questions can be raised here because the scheme by which to organize such an EFT is not clear; which terms one should take, which ones one should neglect, and why. The authors certainly discuss some of this, but perhaps this could be clarified even further. Is it possible in some simple(st) setup to understand why the terms that they take into account are truly the dominant ones?
One other question: above eq. (2.9), what does it mean that the upper critical dimension for U(1) is 4? Why is it related to the group choice and what is it for other groups? It also seems that there is a word (maybe “for”) missing in “(which the U(1) case above is 4)”.
Finally, would it be useful to describe in greater detail why one cannot write down a mean field theory for open strings? Surely, some condensed matter system can be made of open dynamical topological defects.
Report #1 by Anonymous (Referee 2) on 2022-9-5 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2106.12610v2, delivered 2022-09-05, doi: 10.21468/SciPost.Report.5646
Report
The authors take on the ambitious task of developping the Landau-Ginzburg theory for string-like degrees of freedom.
There are a number of motivitions to do this, including:
- in gauge theories, the natural gauge-invariant degrees of freedom are often string-like;
- many theories of interest have higher-form symmetries -- organizing the degrees of freedom in terms of objects charged under these symmetries makes their action manifest, which may be helpful;
- one may hope that putative interacting CFTs in d>4, which cannot be constructed by flowing from free (particle-like) UV CFTs, can instead be obtained by deforming a free string theory in the UV.
Technically, this approach ("mean string field theory"), is complicated by a proliferation of interactions of the form (2.24) (see also Figure 2), which cannot be forbidden based on symmetries. The lack of a scaling or power-counting assignment for these terms makes it difficult to uplift this approach to a systematic effective (string) field theory. Many of the results in this paper, including reproducing Higgs and Coulomb phases, are obtained by neglecting such terms. The authors however make the interesting comment that these terms may be responsible for making certain phase transitions 1st order in dimensions d>4.
The authors are very transparent on the limitations of the effective string approach. Technical difficulties notwithstanding, this approach is interesting and provides a valuable and very different perspective on critical systems. Moreover, this paper is timely, and contributes to several active research directions, such as extensions of the Landau paradigm, importing aspects of the classification of topological phases of matter to richer gapless systems, and the recent focus on line operators in CFTs.
I therefore recommend publication of this manuscript in its current form.