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Toward a 3d Ising model with a weakly-coupled string theory dual
by Nabil Iqbal, John McGreevy
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It has long been expected that the 3d Ising model can be thought of as a string theory, where one interprets the domain walls that separate up spins from down spins as two-dimensional string worldsheets. The usual Ising Hamiltonian measures the area of these domain walls. This theory has string coupling of unit magnitude. We add new local terms to the Ising Hamiltonian that further weight each spin configuration by a factor depending on the genus of the corresponding domain wall, resulting in a new 3d Ising model that has a tunable bare string coupling $g_s$. We use a combination of analytical and numerical methods to analyze the phase structure of this model as $g_s$ is varied. We study statistical properties of the topology of worldsheets and discuss the prospects of using this new deformation at weak string coupling to find a worldsheet description of the 3d Ising transition.
Published as SciPost Phys. 9, 019 (2020)
Submission & Refereeing History
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Reports on this Submission
- Cite as: Anonymous, Report on arXiv:2003.04349v2, delivered 2020-07-01, doi: 10.21468/SciPost.Report.1795
It has long been conjectured that the 3D Ising model is dual to some sort of string theory, where the domain walls of the ordered phase are supposed to be the image(s) of the string worldsheet. A precise formulation has, alas, proven elusive. The string theory is almost certainly strongly-coupled (|g_s|=1) and there's the delicate matter of how to count the configurations where the string is immersed, rather than embedded.
There are two well-known formulations for dealing with this issue: either one imposes self-avoidance (which is computationally-expensive and not quite right, because domain wall are allowed to touch) or one takes g_s=-1 and hope that contributions from immersions of non-oriented string worldsheets cancel the "bad" configurations.
The authors have embarked on a rather nice Monte-Carlo study of these two alternative formulations, with variable string coupling, hoping to map out the phase structure of the string theory.
This is an interesting, and highly worthwhile avenue to pursue. I recommend this paper for publication.