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The spectral form factor in the `t Hooft limit -- Intermediacy versus universality
by W.L. Vleeshouwers, V. Gritsev
|As Contributors:||Ward Vleeshouwers|
|Arxiv Link:||https://arxiv.org/abs/2201.07841v2 (pdf)|
|Date submitted:||2022-05-18 10:37|
|Submitted by:||Vleeshouwers, Ward|
|Submitted to:||SciPost Physics|
The Spectral Form Factor (SFF) is a convenient tool for the characterization of eigenvalue statistics of systems with discrete spectra, and thus serves as a proxy for quantum chaoticity. This work presents an analytical calculation of the SFF of the Chern-Simons Matrix Model (CSMM), which was first introduced to describe the intermediate level statistics of disordered electrons at the mobility edge. The CSMM is characterized by a parameter $ 0 \leq q\leq 1$, where the Circular Unitary Ensemble (CUE) is recovered for $q\to 0$. The CSMM was later found as a matrix model description of $U(N)$ Chern-Simons theory on $S^3$, which is dual to a topological string theory characterized by string coupling $g_s=-\log q$. The spectral form factor is proportional to a colored HOMFLY invariant of a $(2n,2)$-torus link with its two components carrying the fundamental and antifundamental representations, respectively. We check that taking $N \to \infty$ whilst keeping $q<1$ reduces the connected SFF to an exact linear ramp of unit slope, confirming the main result from arXiv:2012.11703 for the specific case of the CSMM. We then consider the `t Hooft limit, where $N \to \infty$ and $q \to 1^-$ such that $y = q^N $ remains finite. As we take $q\to 1^-$, this constitutes the opposite extreme of the CUE limit. In the `t Hooft limit, the connected SFF turns into a remarkable sequence of polynomials which, as far as the authors are aware, have not appeared in the literature thus far. A gap opens in the spectrum and, after unfolding by a constant rescaling, the connected SFF approximates a linear ramp of unit slope for all $y$ except $y \approx 1$, where the connected SFF goes to zero. We thus find that, although the CSMM was introduced to describe intermediate statistics and the `t Hooft limit is the opposite limit of the CUE, we still recover Wigner-Dyson universality for all $y$ except $y\approx 1$.
Author comments upon resubmission
This paper has 30 pages (in its arxiv version) and 6 figures
List of changes
The changes listed here are described in more detail in our responses to the referees.
1. Added a treatment of unfolding and revised our conclusions, changed title, abstract, introduction, and conclusion to align with revised conclusions
2. Removed comparison with linear fit to connected SFF, including figure
3. Changed commas to decimal points
4. Added figures on level density and unfolded SFF
5. Added references
6. Corrected typos
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 2022-6-10 (Invited Report)
After a more careful analysis the authors reached the opposite conclusion that the eigenvalue stats are Wigner-Dyson after
the eigenvalues have been rescaled.
Figs. 1, 2 and 4 are for spectra that have not been unfolded. This
is misleading and suggests that the spectral statistics are not
The unfolding was done the poor man's way by just rescaling.
The level density is close to a semicircle, and it is simple
to do an exact numerical unfolding.
Figures 1, 2 and 4 need to be replaced by figures with the
connected spectral form factor for unfolded eigenvalues.
The level density is close to a semi-circle, and it is easy to
unfold the eigenvalues by fitting a semicircle times a
low order even polynomial to the spectral density. Although
N=10 or 20 is not large, I think that the result will already
be close to the Wigner-Dyson result. Numerically, it should
be no problem to go to N=100 and in that case the agreement
should be very good. The authors could keep the original
figures, but then should add the figures with unfolded spectra
next to it.
With regards to integrable systems there are many that do not
have Poisson statistics. For example harmonic oscillators with
commensurate frequencies, or many-body systems with energies given by the sum of single particle energies.
Already mentioned that above