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Time evolution of effective central charge and 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: | http://arxiv.org/abs/1707.05671v1 (pdf) |
| Date submitted: | 2017-07-27 02:00 |
| 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 effective charge is seen to interpolate between $c_{\rm eff}=0$ at $t=0$, and $c_{\rm eff}\approx1$ 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. As a consequence of renormalization group irreversibility, 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.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2017-9-22 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1707.05671v1, delivered 2017-09-22, doi: 10.21468/SciPost.Report.243
Strengths
(1) Clear presentation of the results and workings.
(2) A potentially quite interesting quantity is introduced.
Weaknesses
(1) Limited range of applicability
(2) Work is rather speculative.
Report
The author considers quantum quenches in massive deformations of 1+1 dimensional CFTs. For a very particular (fine tuned) choice
of initial state he defines a quantity c(t), which reduces to the effective central charge at times t=0 and t=\infty. The time dependence of c(t) is studied for several examples.
While I find the basic idea of introducing a quantity c(t) that interpolates between the effective central charges in the initial and steady states quite interesting, there are a number of issues with the proposal put forward. Firstly, in my understanding it is rather restrictive as it basically works only in the particular case where the system thermalizes. For the integrable QFTs of interest here this is a highly fine-tuned and unrepresentative situation as one generically needs to deal with generalized Gibbs ensembles. Moreover, it is not clear to me how to generalize the construction of c(t) to deal with the generic GGE case. However, given that the case considered is the simplest possible one it nevertheless makes sense to consider it first.
I am also unclear about what the interpretation of c(t) is away from its limiting values at t=0 and t=\infty. I suppose the fact that it is generally oscillating in time should make possible interpretations in terms of RG flows difficult if not impossible?
More generally, the connection of the proposed c(t) to RG flows and the utility of knowing the behavior of c(t) are not clear to me.
A couple of minor comments: the reference to the thermodynamic limit above eqn (15) is confusing. Supposedly the authors has in mind that \tau_0,\tau\ll L. I also think that Cardy's work on the return amplitude in the context of quantum revivals should be cited. When discussing the choice of initial state the papers by Cardy and by Mandal et al and possibly others should be cited, as they deal with more generic situations.
Requested changes
(1) I think it would be useful to explain more clearly in precisely what sense the proposed c(t) can be interpreted in terms of RG flows.
To me the connection appears to be somewhat tenuous.
(2) The GE vs GGE issue should be addressed clearly and prominently at an early stage (e.g. when the initial state is introduced), and it should be stated that the current investigation focusses only on the simplest case in which the system thermalizes.
Report #1 by Anonymous (Referee 1) on 2017-8-29 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1707.05671v1, delivered 2017-08-29, doi: 10.21468/SciPost.Report.231
Strengths
1- A new theory is developed and the models analysed match well the expected physics
2- The discussion is quite clear
2- Interesting connection between RG irreversibility and quantum quench irreversibility
Weaknesses
1- The theory is, I think, of relatively limited applicability; it is not clear how it may be applied to quenches with nontrivial post-quench GGEs
2- No numerics is presented for the actual function of time $c(t)$, the main object of study
Report
In this manuscript, the author defines a sort of effective central charge which applies to certain quantum quenches in integrable models. The quantity is then studied in various models and quenches, and showed to behave in a physically sensible way. It allows one to make the relation between irreversibility of the renormalization group, and irreversibility of quantum quenches.
I think the concepts introduced are interesting, and the analysis is done quite well. The results are quite convincingly agreeing with the general physics that the author discusses.
The main quantity defined and studied, eq 19, has however many drawbacks: it is based on the theory of Calabrese and Cardy for quenches in CFT, which itself is relatively limited. For instance, as the author mentions, the meaning of the effective temperature is unclear in most situations. Also, all this is based on a thermal understanding of the post-quench state, and it has been understood, as the author explains well, that GGEs are rather the correct ensembles to consider. It is not clear how formula 19 can work in cases where nontrivial GGEs are reached post-quench, as in particular there is no clear evidence of a central-charge interpretation (even as a TBA effective central charge) of any object considered e.g. on page 7. These drawbacks are not those of the author's theory, but rather those of the theory of Ref 11 on which it is based. However, I find it difficult to see how one can go beyond. Thus, for instance it is not clear how GGEs can be studied or characterized within this framework as is proposed in the conclusion (what would be the ``effective central charge" of a GGE?).
The idea that irreversibility of RG is related to quench irreversibility is interesting, although, again, since the quantity proposed has, it seems, limited applicability, then also this connection is limited to certain situations. Note also that there are very general results for irreversibility of quantum quenches (basically having to do with 2nd law of thermodynamics), see for instance the discussion in Takashi Mori, Extensive increase of entropy in quantum quench, J. Phys. A 49, 444003 (2016).
This being said, the author considers only situations where CFT and its thermal interpretation are applicable post-quench (that is, $m_0\gg m$), and thus everything works well, and the RG interpretation makes it interesting - for instance, it makes it clear why it is not possible to do a quench displaying IM $\to$ TIM properties.
The paper can be accepted for publication, but, besides adding the appropriate citations concerning irreversibility of quantum quenches, I'd like to see two things done:
1) There are figures displaying the oscillatory behaviour of integrands, to make the point that under integration these give zero. However, there is no numerical results for the function $c(t)$ itself! It would be very interesting to have graphs of this function, as evaluated using the TBA formalism.
2) It would be very good if the author can give us a more precise idea of how the effective central charge consider is defined when the post-quench is a nontrivial GGE, and why it is a good quantity (e.g. how it relates to a CFT central charge, and thus to RG irreversibility). If there is no clear idea there, then I would suggest making the statements less strong as to the applicability of the present theory.
Requested changes
1) There are figures displaying the oscillatory behaviour of integrands, to make the point that under integration these give zero. However, there is no numerical results for the function $c(t)$ itself! It would be very interesting to have graphs of this function, as evaluated using the TBA formalism.
2) It would be very good if the author can give us a more precise idea of how the effective central charge consider is defined when the post-quench is a nontrivial GGE, and why it is a good quantity (e.g. how it relates to a CFT central charge, and thus to RG irreversibility). If there is no clear idea there, then I would suggest making the statements less strong as to the applicability of the present theory.