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Dynamical Phases of Higher Dimensional Floquet CFTs
by Diptarka Das, Sumit R. Das, Arnab Kundu, and Krishnendu Sengupta
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Krishnendu Sengupta |
| Submission information | |
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| Preprint Link: | scipost_202506_00028v1 (pdf) |
| Date submitted: | June 13, 2025, 3:32 a.m. |
| Submitted by: | Krishnendu Sengupta |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
This paper investigates the dynamical phases of Floquet Conformal Field Theories (CFTs) in space-time dimensions greater than two. Building upon our previous work which introduced quaternionic representations for studying Floquet dynamics in higher-dimensional CFTs, we now explore more general square pulse drive protocols that go beyond a single SU(1, 1) subgroup. We demonstrate that, for multi-step drive protocols, the system exhibits distinct dynamical phases characterized by the nature of the eigenvalues of the quaternionic matrix representing time evolution in a single cycle, leading to different stroboscopic responses. Our analysis establishes a fundamental geometric interpretation where these dynamical phases directly correspond to the presence or absence of Killing horizons in the base space of the CFT and in a higher dimensional AdS space on which a putative dual lives. The heating phase is associated with a non-extremal horizon, the critical phase with an extremal horizon which disappears in the non-heating phase. We develop perturbative approaches to compute the Floquet Hamiltonians in different regimes and show, how tuning drive parameters can lead to horizons, providing a geometric framework for understanding heating phenomena in driven conformal systems.
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The paper analyses higher-dimensional CFTs out of equilibrium, in particular systems evolved under exactly solvable Floquet drives. Similar drives have already been studied in detail for (1+1)d CFTs, and the authors extend these results to higher-dimensional CFTs using quaternionic representations. They then connect various dynamical phases of such drives with the presence or absence of Killing horizons in AdS$_d$, as a putative dual to a CFT$_d$.
The mathematical analysis presented in the paper is mathematically sound, and it extends known results on driven (1+1)d CFTs, which is interesting both theoretically and experimentally. However, I have several comments regarding the conceptual structure and presentation of the manuscript.
1) The introduction is hard to follow: it is quite long and presents several technical details on both the broader field of driven CFTs and the manuscript itself, leaving little space for the underlying physics. Likewise, Section 2 delves immediately into the technical development. A softer start would benefit accessibility.
2) In Section 3.5, the Authors state that $\beta_0 = 1$ "leads to linear growth or decay of correlators." Perhaps they meant 'polynomial' behavior of correlators? I find it counterintuitive for correlators to behave linearly.
3) In Section 4.2.2, the Authors find a “phase with damped oscillatory response”. Could they clarify the physics behind this phase? In these systems one typically finds heating (energy/entropy growth) or non-heating (oscillations), with a critical phase in between. Damped oscillations are usually a sign of non-unitary evolution; however, the evolution here appears unitary. What, then, is the origin of the damping?
4) Related to the previous point, what is the main difference between the higher-dimensional case and (1+1)-d CFTs? While the group structure differs (which I agree with), the physical differences were unclear to me. It would help to highlight more explicitly what changes in higher dimensions.
5) Section 6 on holography and AdS space feels somewhat detached. The earlier CFT methods rely only on conformal symmetry and thus apply more generally than to holographic CFTs alone. Please clarify what the holographic perspective adds, particularly to the central goal of understanding higher-dimensional driven CFTs.
6) In the Discussion the Authors write that the AdS interpretation “reveals heating as a generalised Unruh effect.” Given that the methods are not restricted to holographic CFTs, wouldn’t this then hold for any higher-dimensional CFT, not just holographic ones? Why is the geometric picture needed to interpret it as a generalised Unruh effect?
Furthermore, the manuscript presents several typos.
i) In eq. (1) there is an additional "dt".
ii) In eq. (4.28), perhaps the third Hamiltonian should be H_2(\beta_2)? And the time cycles should be $nT < t < nT + T_0$ , $ nT + T_0 < t < nT + T_0 + T_1$ , $ nT + T_0 + T_1 < t < (n+1)T$ ? In the present form they do not make sense to me.
iii) Above eq. (4.39) there are two colons.
iv) Some "Eq. ..." are without parenthesis, while some are with parenthesis but without space between "Eq." and the reference. Please standardise.
v) Above Section 6.3, "plug-in" should be written "plug in".
Overall, the manuscript currently reads more like a collection of notes and results than a cohesive work with clear physical motivation and a well-defined goal. I worry that the results may not reach the community effectively in this form.
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I recommend publication after the following issues are addressed:
-- Choice of parameter space. Throughout the paper, the dynamical phase diagrams are presented in terms of the parameters T1 and T2. This choice may not be the most natural. In (1+1)d driven CFTs, an additional parameter "L" characterizes the length scale of the deformed Hamiltonian, and the phase diagram is best expressed in terms of the dimensionless ratios T1/L and T2/L. For higher-dimensional driven CFTs, is there an analogous way to define a dimensionless parameter space? Clarifying this point would strengthen the physical interpretation of the results.
-- Physical intuition and initial states. The current analysis considers the ground state as the initial state, for which the time evolution is relatively trivial. If one instead starts from other states, such as primary excited states or thermal states, the dynamics would become richer. What qualitative features of physical observables (e.g., entanglement entropy evolution) might be expected in different phases under these alternative initial conditions? Adding some intuitive discussion here would broaden the scope of the work.
-- Geometric or group-theoretic perspective. The identification of different dynamical phases relies on analyzing the eigenvalues of matrices at each step of the driving protocol. Is there a possible geometric understanding of this classification? In (1+1)d, for instance, different types of Möbius transformations correspond to distinct features in the time evolution of operators. A comment on whether a similar geometric or group-theoretic picture exists in higher dimensions would be very valuable.
-- Some refereces should be cited at the appropriate positions. For example, in the discussion section, it is mentioned that the dynamics starting from a thermal state has been done in 1+1 dimensions. Relevant references added here will be helpful for curious readers. There are also some other places with similar issues.
-- There are some typos throughout the work. The authors need to go through the writing carefully.
Recommendation
Ask for minor revision
- {\bf Choice of parameter space}
All our time periods are in units where the radius of the spatial sphere has been set to unity, hence are dimensionless. We have explicitly stated this before Eq.(2.1).
- {\bf Physical intuition and initial states.}
We agree that dynamics gets richer when the state is excited. We have now pointed to references 24, 38 in our Discussions section where some work (involving one of us) has been done in this regard in $1+1$ dimensions.
- {\bf Geometric or group-theoretic perspective. }
This point was already mentioned in the previous version. In the revised version we have clarified this point further in Section 2 (page 5), in section 5.2 and in the Discussion section (second para) . While in two dimensions the Lorentzian and Euclidean conformal group invariants have a one-to-one relationship, our work indicates that this is not true in higher dimensions. Therefore, a group theoretic classification will require construction of quaternionic invariants of the Lorentzian conformal group SO(d,2) : our work will play an important role to check against this construction in the future.
- {\bf Some references should be cited at the appropriate positions.}
We have taken care of this, the revised draft has also been given a new section 2, in order to increase clarity and improve readability. References to work on thermal states in $1+1$ dimensions have been added in the Discussions section
- {\bf There are some typos throughout the work. }
Indeed we have been able to spot various typos, which are now corrected.
Author: Krishnendu Sengupta on 2025-12-22 [id 6168]
(in reply to Report 2 on 2025-10-15)We have now added an intermediary section 2 which is dedicated to discuss the setup as well as summarizes the results within a broader context to improve readability. At the same time the Introductory section has been made 'softer'.
We have rectified this error. There is no linear growth. The entries in the evolution matrix do depend on the number of steps linearly. However, as the referee points out, this leads to a power law growth in the response.
Our characterization of this phase was indeed incorrect. In the revised version we call this a Hybrid phase. We have now been more explicit regarding the hybrid phase, as this is a genuinely higher dimensional effect. In a new appendix B the $4 X 4$ evolution matrix $V_n$ has been explicitly written down wherein one can see terms that both oscillate as well as exponentially grow as function of stroboscopic time 'n' in the hybrid phase. We have also computed the unequal time 2 point function for this case and added this in the new Figure 4. One can see the exponential decay along with oscillations on top of it as a function of the stroboscopic time.
The main difference from $1+1$ dimensions is the fact that once the drives go beyond $SO(2,2)$ the oscillatory phase becomes non-generic. We have emphasized the differences in section 2 as well as in our Discussions section. Also unlike 1+1 dimensions, there is no direct relationship among the conjugacy classification of the Euclidean vs. the Lorentzian conformal group in higher dimensions.
We have emphasized the relevance of the holographic description in Section 2 and in the Discussion section. For time evolution with a single non-standard Hamiltonian of a holographic CFT, such a bulk horizon is the horizon for a bulk vector field representing the corresponding time evolution. In Floquet dynamics the same is true for the Floquet Hamiltonian. As of now this is a kinematic construct and provides AdS perspective of the dynamics. A bulk observer would experience a generalized Unruh effect, pretty much like a boundary observer. We also believe that this construction will be useful in understanding the dynamical bulk dual to a driven holographic CFT, where the horizons will be time dependent and exhibit novel gravitational phenomena.
Indeed, this effect is not limited to holographic CFTs, as the Referee has pointed out. In the paper we have made a distinction between horizons in the base space-time of the CFT (which we called Killing Horizons) in section 3.2 and Bulk Horizons in section 3.3. The former holds for all CFT's, but the latter is relevant only for holographic CFT's
We have taken care of the typos pointed out by the referee.