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Manybody chaos and anomalous diffusion across thermal phase transitions in two dimensions
by Sibaram Ruidas, Sumilan Banerjee
Submission summary
As Contributors:  Sumilan Banerjee 
Arxiv Link:  https://arxiv.org/abs/2007.12708v2 (pdf) 
Date submitted:  20201210 12:33 
Submitted by:  Banerjee, Sumilan 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Chaos is an important characterization of classical dynamical systems. How is chaos linked to the longtime dynamics of collective modes across phases and phase transitions? We address this by studying chaos across Ising and KosterlitzThouless transitions in classical XXZ model. We show that spatiotemporal chaotic properties have crossovers across the transitions and distinct temperature dependence in the high and lowtemperature phases which show normal and anomalous diffusions, respectively. Our results also provide new insights into the dynamics of interacting quantum systems in the semiclassical limit.
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In this paper, Ruidas and Banerjee report spatiotemporal properties of dynamical correlation functions and chaos (suitably defined) in a classical spin model across thermal phase transitions and compare between the behaviours of chaos and longtime dynamics of collective modes. For the 2D classical XXZ model which harbours both KosterlitzThouless and Ising transitions at different regimes of the anisotropy, they map out the dynamical properties in the different phases of the models using classical MonteCarlo sampling and time evolution using the classical equations of motion. The underlying motivation is to explore connection between scrambling (or exponential decay of outoftimeordered correlation functions) as a diagnostic of chaos in quantum many body systems and the transport properties. However, in the absence of controlled numerical or analytical control over quantum systems of interest, they focus their study on a numerically tractable classical system, which could be viewed as the semiclassical limit of the quantum model. Their results suggest that, first, both chaos and transport show distinct features in the different thermodynamic phases. However, chaos follows a more universal ballistic behaviour across the phases, and the phase transition manifests only in the subtle form of crossover in the Lyapunov exponent and a minima in the butterfly velocity. Second, the dynamical autocorrelation function has sharper signatures of the phase transition, with diverging timescales and anomalous diffusion in the different phases, with distinct behaviour for KT and Ising transitions. Furthermore, their studies suggest that mechanisms of chaos and spin transport are different at low temperatures, and even at high temperature for the Ising case.
This study is a natural and necessary addition to the literature exploring the connection of many body chaos and transport, and more specifically to the more recent studies of butterfly effect in classical systems. The study also identifies several interesting features which would seed future studies in the field. Hence, I recommend this paper for publication, although I do have a few questions, concerns, and suggestions. I will list them out below. I hope that the authors address these issues before publication.
1. The authors consider two different measures of the butterfly effect in classical spin system, the decorrelator $D(x,t)$ and $\langle \delta S_{r}(t)^2\rangle$. All the reported results of the parameters such as Lyapunov exponent, $t_0$ and butterfly velocity are for the decorrelator, while it is claimed that the results are the same for the other quantity as well. I think this comparison warrants more evidence, if not in the main text, then as a section in the appendix. Also, why these quantities differ is not immediately clear from the text, and a little more exposition would be nice. Since the claim is that these quantities differ in $\mathcal{O}(\epsilon^2)$, and the data to be fitted are $\sim 10^{8} \to 10^{1}$ while $\epsilon \sim 10^{4}$, there should be more clarity on how well the quantities derived from these two quantities match up.
2. The generalized Lyapunov exponent $D(x,t)$ is fitted to a ballistic form $\sim \exp\left[\lambda_L t(1(x/v_{B}(tt0))^\nu)\right] $, and it is claimed that $\nu \sim 2$ for all the cases, but there is no evidence shown for this, and it would be nice to see the goodness of the fit, especially as a function of Temperature. This is relevant, because in local quantum models, it has been shown that the generalized Lyapunov exponent is possibly not ballistic, but has a diffusively broadening wavefront, $\exp\left[\lambda_L t(1(x/v_{B}(tt0)))^{1+p}\right]$, with $p = 1$ definitively shown for random circuit models. From the locus of $\lambda_L(x,t) = 0$, there doesn’t seem to be broadening at these timescales in the model considered here, but it will be interesting the study this near the thermal phase transition, where, as the authors point out, chaotic behaviour could manifest similarly for classical and quantum case. I also think that the lack of wavefront broadening in their study is a significant finding and should be highlighted more.
3. For the easy plane case (KT transition), it is suggested that there is an anomalous subdiffusion for spin transport in the low temperature phase. However, as the authors also point out, the dynamical exponent $\alpha$ has an upward drift with time (Fig. 12 in appendix), so it is not clear whether this persists in the longtime limit. The authors claim that in these timescales, there is already evidence of ballistic chaotic behaviour, which would suggest that the mechanisms of dynamical spin transport and chaos are different. However, there is an onset timescale for chaos, $t_0$ as denoted by the authors, which also seem to diverge at low temperatures. This would suggest a possible conflict, and the nonexponential behaviour of the decorrelator could coincide with subdiffusion at these timescales. Is this a possibility? If it is, then the authors should add a further qualification to the claims.
A few more minor points:
4. Fig 1 seems to have disconnected figures lumped together, which is disorienting in the first read. Fig 1a is perfect for the introduction, but it was not clear if the reader is supposed to pay attention to 1b, c, d, e while going through the introduction. I think it would improve the presentation of the paper if the 1 be is separated from the summary phase diagram and presented along with the technical results for chaos.
5. There are some issues with labelling in the figures. The authors consider two anisotropies  $\Delta = 0.3$ and $\Delta = 1.2$  it will make the reader’s life much easier to label each relevant figure with this so that they are not disoriented. Furthermore, the legends in the plot should be labelled for clarity – for example, in 2c, the different plots are for temperatures, so that should appear in the label in the figures. Same for 1d, 2a, 2b, 3, 4a, 5a, 7, 9, 10 and 11.