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Low temperature quantum bounds on simple models
by Silvia Pappalardi, Jorge Kurchan
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Submission summary
| Authors (as registered SciPost users): | Silvia Pappalardi |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2106.13269v2 (pdf) |
| Date submitted: | 2021-12-23 15:41 |
| Submitted by: | Pappalardi, Silvia |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In the past few years, there has been considerable activity around a set of quantum bounds on transport coefficients (viscosity, conductivity) and chaos (Lyapunov exponents), relevant at low temperatures. The interest comes from the fact that black-hole models seem to saturate all of them. To gain physical intuition, we focus on the simplest case where these bounds act non trivially: bosonic systems whose lowest energy is a degenerate manifold. In particular, we consider free motion on curved space, for which the Hamiltonian is just the Laplace-Beltrami operator. Thermodynamic examples of these include quantum hard-spheres and quantum spin liquids. In this context, the bounds are approached and are consequences of the uncertainty principle, and one understands the mechanisms whereby quantum mechanics enforces them. For a system to saturate the bound at $T = 0$, it appears as a necessary condition that at each temperature there are some degrees of freedom that are still classical, and some are on the verge of being affected by quantum effects.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022-3-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2106.13269v2, delivered 2022-03-15, doi: 10.21468/SciPost.Report.4694
Strengths
1- The paper deals with bounds on transport and chaos, which have attracted a lot of attention in recent years across different communities
2- It relates these bounds to the fundamental concepts of quantum mechanics, such as the uncertainty principle
3- It adds to the existing literature by offering a pedagogical and intuitive insight on how the bounds arise at low temperatures in systems of free particles propagating in the curved space-time
Report
This work considers how the bounds on chaos and transport emerge at low temperatures.It is shown that classically the bounds are violated at low temperatures for free particles on curved manifolds. Different quantum mechanisms that restore the bounds are identified, as well as a potential scenario where the bound on chaos is violated.
The paper is considering a timely topic of broad interest, offering new insights. The presentation is clear and pedagogical. Therefore I suggest the publication of the paper in SciPost physics after a small revision addressing the points below (point 3 in particular).
Requested changes
1- For completeness it would be useful to write what N in the large N limit is
2- In equation 14b the expression should be divided by the partition sum
3- If I am not mistaken, the Helfand functions should correspond to the time integral of the current divided by sqrt[VT] instead of the time derivative, in order for expressions 14 and 15 to match. Is this accounted for in other results?
4- I would suggest using some other letter instead of s in equation 23, which was already used as the entropy density
5- There are some typos:
boundS at T=0 in abstract,
useD the definition on page 32
From this expresion IT is clear on page 32
missing equation reference on page 38
Missing reference in ref [23]
Report #1 by Anonymous (Referee 1) on 2022-2-26 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2106.13269v2, delivered 2022-02-26, doi: 10.21468/SciPost.Report.4529
Strengths
1. The paper appears to contain some interesting numerical results for the classical and quantum billiard problem on a pseudosphere.
Weaknesses
1. The paper is not readable.
Report
I do not recommend publication. While it is possible that the results are significant and interesting, the work is not written in a way that clearly conveys what was done.
As far as I understand it, the authors considered quantum and classical billiard problems involving a pseudosphere. They look at the pseudosphere glued to a cylinder, the pseudosphere cut-off with a hard-wall, and what I would call a "banded'' pseudosphere where rings of pseudosphere with different radii of curvature are glued together. Furthermore, they compute analogs for these systems of the out of time ordered correlation (OTOC) function of [4]; importantly for this OTOC there is a bound on its exponential behavior which has been argued to be a bound on chaos. I am not 100% certain, but I believe Kurchan and Pappalardi's central thesis in this paper is that this bound comes closest to being saturated in their models at the cross-over between quantum and classical behavior. I am not certain because I have not been able to pinpoint where in the draft they actually show and/or argue for the details of this thesis.
My problems with the manuscript start with the abstract, which instead of conveying the details of what is done, fail to mention the pseudosphere.
Given the juxtaposition of "we consider free motion on curved space" with "Thermodynamic examples of these include quantum hard-spheres and quantum spin liquids," one might be forgiven for assuming the authors will consider quantum hard-spheres and quantum spin liquids in detail.
Figure 1 and the brief discussion in section 7 instead present only some history and speculation about these topics.
Further, to discuss saturating the bound at $T=0$ I thought was misleading as well. I think the authors mean saturate in the low temperature limit, or perhaps more accurately low energy limit since they never really work with a thermal ensemble and instead assume equipartition $E = NT/2$.
I found the following two sentences ludicrous: "This paper aims at being pedagogical. For this reason, all the details of the calculations are reported in the appendix." I found the only pedagogical material to be in the appendices. The main part of the paper was very difficult to follow. Often quantities were introduced which I only later found were defined in the appendices, for example $N_{cut}$ and $m_n$.
Requested changes
1) (1) is not actually a bound. It is known to be violated in certain circumstances.
2)
"On the other hand, the bound to chaos – depending only on temperature and the Planck constant – corresponds to a genuine quantum effect.''
What is the "bound to chaos''?
3)
"The goal of this work'' should be indented.
4)
What does it mean for a bound to be "effective non-trivially''?
5)
"The first is that $\hbar / T$ be finite, which holds in the semiclassical limit for very low temperatures corresponding to the lowest classical energies of the system.'' Surely this is $\hbar/T$ is finite provided $T>0$?
6)
"In the presence of ground-state degeneracies or quasi-degeneracies at the lowest energies; the system may instead be non-harmonic (and chaotic) even "at the bottom of the well'' ''$\to$ "In the presence of ground-state degeneracies, or quasi-degeneracies at the lowest energies, the system may instead be non-harmonic (and chaotic) even "at the bottom of the well'' ''
7)
"In Sec. 6 we discuss how the bound holds in the limit of zero temperature in presence of a hierarchy of length scales'' Which bound? The authors need to make it clearer they focus only on (2).
8)
The authors claim to find a violation of the bound (2) at the end of section 2 and again on pp 19-20 where the discrepancy is attributed to "regulation'' issues. I am upset with such a light treatment of a potentially significant result. It seems that this entire manuscript was built around comparing their results to the bound in (2). If they are comparing apples and oranges, then I need to understand in greater detail the map from apples to oranges. If they are comparing apples to apples, then I need to know why (2) is wrong or this manuscript is wrong.
9)
"reflecting wall at $\tau = \tau_x$'' $\to$ "reflecting wall at $\tau = \tau_L$''
10)
On p 14, the authors claim the high energy particles spend most of their time on the cylinder and then later assume ergodicity, which seems mutually inconsistent.
11)
At the top of p 15, the authors refer to the Loschmidt echo, which I thought was a purely quantum phenomenon, in a section on the classical behavior of their billiard. I found this confusing.
12)
In (49), I did not understand on what the minus sign on the far left was supposed to act, and (50) looked potentially unbounded below.
13)
In gluing the cylinder to the pseudosphere, one could get rid of the step function by adding a constant potential to the cylinder. Did the authors consider this possibility?
14)
In figure 6a, it looks like the data is growing a bit faster than the green line, i.e. violating the bound?
15)
"In this case, the repulsive potential $\delta$ is absent, as for a finite portion of the hyperboloid with a wall at some $\tau_x$.'' I think the authors mean to introduce the example of a pseudosphere with a hard wall, but it's not clear from the text.
16)
"In the second line, we have used...'' but there is no second line in the preceding equation
17)
On p 38, there is an "Eq.(??),"
18) The list of small issues that I found is longer than my patience to describe them. I'll conclude by saying that there were a number of very creative and mutually inconsistent spellings of pseudosphere, Schr\"odinger, Loschmidt and dimensional.