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Time evolution of effective central charge and signatures of RG irreversibility after a quantum quench
by Axel Cortes Cubero
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Submission summary
| Authors (as registered SciPost users): | Axel Cortes Cubero |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/1707.05671v3 (pdf) |
| Date accepted: | 2018-03-19 |
| Date submitted: | 2018-02-26 01:00 |
| Submitted by: | Cortes Cubero, Axel |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
At thermal equilibrium, the concept of effective central charge for massive deformations of two-dimensional conformal field theories (CFT) is well understood, and can be defined by comparing the partition function of the massive model to that of a CFT. This temperature-dependent effective charge interpolates monotonically between the central charge values corresponding to the IR and UV fixed points at low and high temperatures, respectively. We propose a non-equilibrium, time-dependent generalization of the effective central charge for integrable models after a quantum quench, $c_{\rm eff}(t)$, obtained by comparing the return amplitude to that of a CFT quench. We study this proposal for a large mass quench of a free boson, where the charge is seen to interpolate between $c_{\rm eff}=0$ at $t=0$, and $c_{\rm eff}\sim 1$ at $t\to\infty$, as is expected. We use our effective charge to define an "Ising to Tricritical Ising" quench protocol, where the charge evolves from $c_{\rm eff}=1/2$ at $t=0$, to $c_{\rm eff}=7/10$ at $t\to\infty$, the corresponding values of the first two unitary minimal CFT models. We then argue that the inverse "Tricritical Ising to Ising" quench is impossible with our methods. These conclusions can be generalized for quenches between any two adjacent unitary minimal CFT models. We finally study a large mass quench into the "staircase model" (sinh-Gordon with a particular complex coupling). At short times after the quench, the effective central charge increases in a discrete "staircase" structure, where the values of the charge at the steps can be computed in terms of the central charges of unitary minimal CFT models. When the initial state is a pure state, one always finds that $c_{\rm eff}(t\to\infty)\geq c_{\rm eff}(t=0)$, though $c_{\rm eff}(t)$, generally oscillates at finite times. We explore how this constraint may be related to RG flow irreversibility.
Author comments upon resubmission
First, it seems Referee 1 was satisfied with the previous version and no changes were suggested. I am happy to see that in his/her opinion the quality of the manuscript has already improved, and I thank again the referee for their work.
In this version I have incorporated several changes, in attempt to answer for some of the issues raised by Referee 2. Here I will review the modifications step by step, corresponding to the items enumerated in the referee's report (see the list of changes below).
List of changes
1) I have made many changes throughout the paper regarding the discussion of "RG irreversibility", including several mentions in the introduction and the conclusion sections. The main theme has been to soften the tone of some claims. When the value of the effective central charge exhibits some apparent irreversibility properties, c_{\rm eff}(\infty)\geq c_{\rm eff}(0), it is commented that this is suggestive and may point to a connection with the concept of RG flow and irreversibility, by comparing to the known properties of the equilibrium c-function. This is in contrast with previous versions where it may seem like I claim "this IS RG irreversibility", and made it sound like a claim of a rigorous proof.
Accordingly, I have modified the title with the phrase "signatures of RG irreversibility", suggesting that this is not a direct proof of RG irreversibility, but that arguments are presented to suggest what we see may be connected with it. I have opted not to entirely remove "RG irreversibility" from the title, since I feel that would be slightly dishonest to the paper, since this is a subject that is widely discussed, and a central theme in the paper. I hope that this softening the tone of the claims, and the title, will be enough to reflect the nature of what is actually discussed in the paper.
2)The discussion regarding the Calabrese-Cardy initial states has been enhanced. I attempt to make it more clear that this is indeed just a simplified model that is only useful as a first step. A new paragraph augmenting this discussion was added after Eq. 15, just after introducing these states. There is also more discussion added on this at the end of the same section. I have also added in the conclusions section a brief discussion of how one could a attempt a generalization to other kinds of initial states, which do not lead to effective thermalization. This should make it clear that it should be conceptually possible to define similarly an effective central charge for wider classes of quantum quenches, the only limitation is a practical computational one.
3)The discussion of the relation between the return amplitude and the partition function on the strip is clarified. Indeed I agree, that the roles of the space and time coordinates are exchanged in this analogy, and this was not very clear in my previous discussion. This discussion has been modified, and hopefully the relation is now clear.
4) Together with the modifications in the discussion of CC initial states, the perhaps misleading claim has been corrected, along with a more careful discussion of when thermalization is expected to occur.
5) The issue of the limits and conditions which lead effective thermalization has been more carefully addressed, including the limit \lim_{(m_0/m)\to\infty}, throughout the paper.
6)The labels in the plots have been modified.
7) There have been several modifications in the conclusions. Any claims regarding RG irreversibility have been softened. It is explained that the results are suggestive at some connection with this concept, given how the effective central charge is related to the known equilibrium c-function, but any hard claims that make it seem like we have found some rigorous proof are dropped. The discussion regarding the finite-time values of the central charge is also enhanced, making it clear that it is difficult to see if and how it may be connected with RG flow, and making it clear that for now one can only speculate about this. As mentioned before, the discussion of initial states was also enhanced in the conclusions.
Published as SciPost Phys. 4, 016 (2018)