SciPost Submission Page
Minimal Areas from Entangled Matrices
by Jackson R. Fliss, Alexander Frenkel, Sean A. Hartnoll, Ronak M Soni
Submission summary
| Authors (as registered SciPost users): | Ronak Soni |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2408.05274v1 (pdf) |
| Date submitted: | 2024-09-09 18:12 |
| Submitted by: | Soni, Ronak |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
We define a relational notion of a subsystem in theories of matrix quantum mechanics and show how the corresponding entanglement entropy can be given as a minimisation, exhibiting many similarities to the Ryu-Takayanagi formula. Our construction brings together the physics of entanglement edge modes, noncommutative geometry and quantum internal reference frames, to define a subsystem whose reduced state is (approximately) an incoherent sum of density matrices, corresponding to distinct spatial subregions. We show that in states where geometry emerges from semiclassical matrices, this sum is dominated by the subregion with minimal boundary area. As in the Ryu-Takayanagi formula, it is the computation of the entanglement that determines the subregion. We find that coarse-graining is essential in our microscopic derivation, in order to control the proliferation of highly curved and disconnected non-geometric subregions in the sum.
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
Current status:
Reports on this Submission
Report
Summary:
This paper is a continuation of previous efforts to define a Hilbert space factorization in matrix models that leads to an emergent area law entanglement entropy. The matrix model consists of NxN Hermitian matrices subject to a gauge symmetry given by conjugation by U(N) matrices. It is known that factorization in gauge theories requires a lifting of the Gauss law constraint, leading to an extended Hilbert space on which certain gauge symmetries are promoted to physical ``edge mode" symmetries. Physical states satisfying the Gauss law constraint are singlets under these edge mode symmetries, and their presence leads to a Gauss law entanglement entropy that counts the corresponding edge modes. These transforms under different representations of the symmetry, and the entropy involves dimensions of these representations.
For a generic matrix model, a preliminary (partially gauge fixed) notion of a subsystem can be given by a fixed MxM block, in which case the edge mode symmetry is U(M)xU(N-M). It was explained in previous work that for semi-classical states Fuzzy sphere state of the mini-BMN model, the U(N) symmetry has a low energy description in terms of area preserving diffeomorphisms of a two-sphere, and the MxM block can be viewed as a subregion of a two sphere. In the low energy description, the edge mode symmetry U(M)xU(N-M) are volume preserving diffeos that fix the subregion. The saddle point approximation to the entropy produces an area law on the sphere ( in this case area= perimeter) , which counts the dimensions of the dominant irrep.
In this work, the authors incorporate the edge mode symmetries U(N)/U(M)xU(N-m) that do not fix the MxM block: these ``wiggle" modes move the subregion on the sphere around in an volume preserving away. A gauge fixed subsystem then corresponds to any region of fixed volume. The authors show that reduced density matrix
involves direct sum over such subregions of fixed volume, and a saddle point approximation pics out the region with minmimal area. This is then compared to the area minimization of the Ryu-Takayanagi formula in holographic theories.
Recommendation:
The paper provides a compelling proposal for the factorization map and extended Hilbert space construction for matrix models, and illustrates in detail how this works in the mini-BMN model. I am happy to recommend its publication, provided the questions below are answered.
1.) Section 2.3 on the interpretation discussion of quantum reference frames is interesting but I would like to clarify how it goes beyond the usual paradigm of edge modes as ``Stuckelberg" degrees of freedom
For example consider the discussion around eq 2.16, that is supposed to make the matrix elements of X a gauge invariant quantity, despite that fact it manifestly transforms under conjugation by a U(N) matrix U. In the "Stuckelberg" edge mode paradigm, we would say that to make X gauge invariant, we need to introduce edge modes g in U(N), which is a choice of gauge promoted to a physical degree of freedom. Then we dress X by these edge modes by introducing gXg^{-1} as the new gauge invariant degree of freedom, where the gauge transformations map X to UXU^{-1} and g to gU^{-1}. Then gXg^{-1} is gauge invariant, but only after introducing g as a physical edge modes. Do the authors agree that what I said above is equivalent to the construction below 2.16?
If so, is the quantum reference frame the Stuckelberg formalism ?
2.) I am puzzeld by the definition of F in 2.38. According to the first two definitions, H and H' are both in G. What does it mean to take there direct product and then delete G?
Or did you mean by G\cap U(M) something different than the intersection of two sets? Like taking elements in G and restricting to their action on the first M indices?
Also Flag manifolds are quotients, did you mean HxH'/G instead of HxH'\G?
3.) Equation 2.39 is difficult to parse. Can an explicit definition of |\psi_{V}> be given?
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)