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Statistical Mechanics of Exponentially Many Low Lying States
by Swapnamay Mondal
Submission summary
Authors (as registered SciPost users):  Swapnamay Mondal 
Submission information  

Preprint Link:  https://arxiv.org/abs/2310.12264v2 (pdf) 
Date submitted:  20240708 13:06 
Submitted by:  Mondal, Swapnamay 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
It has recently been argued that for nearextremal black holes, the entropy and the energy above extremality respectively receive a logT and a Tlinear correction, where T is the temperature. We show that both these features can be derived in a low but not too low temperature regime, by assuming the existence of exponentially many low lying states cleanly separated from rest of the spectrum, without using any specific theory. Argument of the logarithm in the expression of entropy is seen to be the ratio of temperature and the bandwidth of the low lying states. We argue that such spectrum might arise in nonsupersymmetric extremal brane systems. Our findings strengthen Page's suggestion that there is no true degeneracy for nonsupersymmetric extremal black holes.
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 multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Current status:
Reports on this Submission
Strengths
1. well motivated
2. well explained
3. important results
4. detailed numerical analysis
Weaknesses
1. comparison with existing results
2. physical reasoning for temperature cutoff
Report
recommended for publication
Requested changes
1. Physical reasoning for temperature cutoff
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report
This paper studies the lowtemperature corrections to entropy and energy by analyzing the statistical mechanics of lowenergy states. These corrections are calculated in the context of a uniform band of lowenergy states that are separated from the highenergy states by a large gap.
I recommend the paper for publication after minor changes that are explained below.
(1) Section 3.2 tries to argue that the partition function of a generic spectrum, $Z_{generic}$, is wellapproximated by choosing the energy levels to be equally spaced, $Z_{equispaced}$.
This is done numerically by comparing $Z_{generic}$ and $Z_{equispaced}$. However, the numerical analysis presented in the paper is not sufficient when the temperature is small. The paper chooses $\Delta = 1$ and the plots in Figures 15 show $T \in [0,1]$. However, most of the relative error plots showing $(Z_{generic}  Z_{equispaced})/Z_{equispaced}$ tends to blow up for small $T$. This is problematic because the paper is very interested in the temperature range $T \ll \Delta$. More evidence is needed to show that $Z_{generic} \approx Z_{equispaced}$ in this temperature range.
(2) Relatedly, it is not clear why $Z_{generic}$ obtained from taking the energy levels to be independent, uniformly distributed random variables is a good ansatz for the cases under consideration.
For instance, if we assume that the energy levels are given by a matrix model then we expect the density of states to have a square root edge, i.e. $\rho(E) \sim \sqrt{E}.$ Moreover, the energy levels are not independent but they level repel. This does not match with the assumptions for $Z_{generic}$. Some comments on this would be helpful.
(3) In section 3.3, case 2 studies the low but not too low temperatures. Some passing comments are made below equation (3.14) but these should be explained better. It would be helpful if the large system limit is explained better. Also, it is not clear why $T \to 0$ is a valid limit in this case.
Recommendation
Ask for minor revision
Report
The subject of the paper is wellaligned with the journal's theme. However, before accepting for publication, some changes need to be incorporated. This is mentioned in detail in the report attached.
Recommendation
Ask for major revision