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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 | |
|---|---|
| Preprint Link: | http://arxiv.org/abs/1707.05671v2 (pdf) |
| Date submitted: | 2017-11-27 01:00 |
| Submitted by: | Cortes Cubero, Axel |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| 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 $c_{\rm eff}\sim 1$ 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. 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.
Author comments upon resubmission
I will point out that the most important modification in this version, which took some time to rewrite, is that I corrected a significant error that I discovered after the first submission. In this version I have a slight modification on the definition of the time-dependent effective central charge, after I realized the previous definition was problematic in some cases. The arising issues were resolved by modifying the definition with an additional time-averaging step which is described in the text (Eq. 21).
I expanded the discussion about the initial states of CFT quenches, and the range of validity of our proposal, addressing an issue that was raised by both referees. The starting point for the proposal are the Calabrese-Cardy initial states, which are known to lead to effective thermalization at late times, and not to a nontrivial GGE. Despite the simplicity of these states, these are known to describe accurately massive-theory-to-CFT quenches, which is the motivation to consider them. We later consider quantum quenches into non-conformal theories and propose a definition for effective central charge based on the general form of the return amplitude, by comparing it to a CFT quench (with Calabrese-Cardy initial states). Our proposal is then expected to be valid and reasonable for quantum quenches which reduce to the Calabrese-Cardy quench in an appropriate limit (taking the post-quench massive parameter to zero). If when we reduce the ratio $m/m_0$, we see that the late-time dynamics become thermal, as is expected in the CFT-limit, then we expect our proposal to be valid. As we discuss in the text, this is enough to describe a reasonably wide range of quantum quenches of massive theories.
While other more exotic quench protocols might require a modification to the Calabrese-Cardy initial state, which would result on a redefinition of the effective central charge, we feel our proposal describes a wide enough range of of physically relevant quantum quenches. It would be interesting to study extensions to other more possible quenches in the future, but this is beyond the scope of this first article on the subject.
As suggested by Referee 1, I have added a plot for the time evolution of the effective central charge, where one can observe the behavior we propose. Namely, the central charge seems to interpolate between the expected values at $t=0$ and $t\to\infty$. At finite times the charge can oscillate as it reaches its asymptotic value. Any such numerical evaluation is, however, limited to short times, as the numerical precision is affected at late times.
Both referees raised the issue of the definition and interpretation of the central charge at late times, when the state is described by a GGE. I have added some discussion on this subject in the text. This interpretation is perhaps clarified by understanding the GGE as a thermal-like state, where each momentum mode can have a different temperature (with the typical constraint that the zero-momentum mode has an equal or higher temperature than the other modes). Given some momentum-dependent temperature, one can then easily define and understand an effective central charge in direct analogy to the thermal one. This GGE central charge has to be smaller or equal to the thermal charge at the same effective temperature (since all momentum-modes then have the same constant temperature, which is the highest value corresponding to the zero-mode temperature). It can be clearly seen that the GGE central charge reduces to the thermal charge in the large-quench limit, where the effective temperature acquires a constant value.
Responding to issues raised by Referee 2, I did not include a deep discussion on revivals, as indeed I am considering the limit where system size, L, is much larger than the length scales given by the effective temperature. The CFT partition function we present is computed on an infinite strip geometry, while revival effects are seen by considering a finite ring geometry. I have not added a discussion of the effects of revivals, since this is presently beyond the scope of my proposal.
Finally, responding to Referee 2, to avoid the risk of too much speculation, I currently do not offer a full interpretation of the oscillating finite-times dynamics of the effective central charge, and its RG interpretation. Studying the time evolution, as in Figure 2b, it is very tempting to think this quantity somewhat describes how the system evolves from one value of central charge to the other, and provides a description of more exotic configurations that are visited in the finite time non-equilibrium evolution. Aside from a few comments in the Conclusions, I refrain from making any strong statements about the finite time evolution, as this would require perhaps a deeper numerical study in a wider variety of scenarios. The only interpretation I can offer in all honesty is that between the relation of the $t=0$ and $t\to\infty$ values of the charge, which is discussed at length in the paper.
List of changes
-The most significant modification is a change in the definition of the effective central charge (introducing an additional time-averaging step), which fixes some issues with the previous definition.
-Discussion about the Calabrese-Cardy initial states and the range of validity of our proposal is largely enhanced. The issue is now addressed at length in the introduction, an in the section about CFT quenches.
-Discussion was added on the interpretation of an effective central charge corresponding to a GGE state, particularly in Section 6, concerning the free massive boson. The discussion of the GGE vs GE was also enhanced in the introduction.
-A numerical plot was added showing explicitly the time evolution of the effective central charge for the free massive boson quench.
-References suggested by the referees were added, as well as brief corresponding discussions. Other smaller typos were corrected.
Current status:
Reports on this Submission
Anonymous Report 2 on 2018-1-2 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1707.05671v2, delivered 2018-01-02, doi: 10.21468/SciPost.Report.315
Report
Second Report on "Time evolution of effective central charge and RG
irreversibility after a quantum quench"
I think that some of the main points of my first report have not been
addressed in a satisfactory way. In particular it remains unclear what
the connection of the proposed c(t) to RG flows is, and what useful
information can be extracted from the knowledge of c(t).
I therefore do not think that the paper can be published in its
current form. I would suggest the author to rewrite the manuscript
and focus on the calculations he has done, but drop unsubstantiated
speculation (except perhaps in the conclusions).
My more detailed comments and requested changes are listed below.
Requested changes
1. "RG irreversibility" should be dropped from the title
and elsewhere in the manuscript. As far as I can see the paper
contains no information on the relation to RG flows and
irreversiblity. Sentences like "In particular, we are interested in
the consequences of the irreversibility of RG flow in the
non-equilibrium evolution" or "The irreversibility of RG flow is
reflected in the fact that these quenches can only be performed in one
direction..." are as far as I can see inappropriate. If the author
wants to speculate on the relation of his results to RG flows and
irreversibility he could do so in the conclusions, but clearly identify
such comments as speculations.
2. As I noted in my previous report, the steady state after
quenches to an integrable theory is always described by a GGE and
only reduces to the Calabrese-Cardy initial state in an (unphysical)
limit. For the case of the Ising field theory this was shown
explicitly in P. Calabrese et al, J. Stat. Mech. (2012) P07022 section
2.5 and a comprehensive discussion in a CFT setting was given by
J. Cardy in Ref. [13]. As I said in my previous report, as a first
step it is entirely reasonable to work with the CC initial state here,
but the author should clearly state that this is an approximation:
for certain quenches the steady state is approximately thermal in the
sense that the GGE is very close to a GE (in the free case the
mode-dependent temperatures then become almost equal). I take
objection to the author's reply to my first report that
"While other more exotic quench protocols might require a modification
to the Calabrese-Cardy initial state redefinition of the effective
central charge, we feel our proposal describes a wide enough range of
of physically relevant quantum quenches."
The fact is that generic quenches require a modification of the CC
initial state. The author's proposal applies to an approximate
description of an interesting class of quantum quenches.
When introducing the CC state in (15) a discussion along these lines
should be added.
3. I believe the discussion below (12) to be misleading. The return
amplitude corresponds to the partition function of a
Stat. Mech. system with boundaries, where an analytic continuation in
system size rather than temperature has been carried out (the inverse
temperature of the Stat. Mech. problems maps onto the system size of
the quench problem). This correspondence follows from Fig. 1a by
considering the two different transfer directions.
The strip geometry shown in Fig. 1b corresponds to the situation found
for CFTs where analytic continuation in time has been performed and an
appropriate regularization procedure (that leads to a finite width of
the slab) has been employed. In my understanding these are two
entirely separate issues, and the first correspondence holds true
much more generally than the second.
I think the discussion around eqns (13) and (14) follows from the
first correspondence (the system size in the Stat Mech problem maps
onto imaginary time in the quench setting).
4. After (21) there is a discussion that suggests that thermalization
can be expected after e.g. mass quenches. This should be changed,
cf. my comment (2) above.
5. Below (27) the author refers to a regime m_0\gg m when he actually
means the limit \lim_{(m_0/m)\to\infty}. It is only in this (unphysical)
limit where thermalization occurs. For m_0\gg m the system thermalizes
approximately. I suggest to refer to approximate thermalization here.
6. Appropriate axis labels should be introduced in Fig. 5.
7. The Conclusions section should be rewritten. As the paper does not
establish any connection to RG flows, comments speculating on such
connections should be phrased much more carefully than is currently
done. Fig. 2 shows that the introduced effective central charge is
still oscillatory in time, which seems to constitute a problem
with regards to notions of irreversibility. I also think that when
discussing the notion of irreversibility it would be useful to have a
clear and explicit discussion of what precisely the author means by
this term in the quench context, as some works in the literature
(e.g. arXiv:1711.00015) ascribe a rather different meaning to it.
The "RG" in the sentence stating that an interpretation of the
effective central charge at finite times is currently missing should
be removed: as far as I can see there presently is no physical or
other interpretation of this object.
Anonymous Report 1 on 2017-12-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1707.05671v2, delivered 2017-12-14, doi: 10.21468/SciPost.Report.295
Report
I think the referee has addressed the points correctly. The discussion now puts the results in a better way into the general context. The ideas are interesting. Also I think the author for correcting the mistake found - it is indeed often important to average out oscillations that occur at large times.