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Outofequilibrium fullcounting statistics in Gaussian theories of quantum magnets
by Riccardo Senese, Jacob H. Robertson, Fabian H. L. Essler
Submission summary
Authors (as registered SciPost users):  Fabian Essler · Riccardo Senese 
Submission information  

Preprint Link:  scipost_202401_00001v1 (pdf) 
Date submitted:  20240104 10:42 
Submitted by:  Senese, Riccardo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider the probability distributions of the subsystem (staggered) magnetization in ordered and disordered models of quantum magnets in D dimensions. We focus on Heisenberg antiferromagnets and longrange transversefield Ising models as particular examples. By employing a range of selfconsistent timedependent meanfield approximations in conjunction with HolsteinPrimakoff, DysonMaleev, Schwinger boson and modified spinwave theory representations we obtain results in thermal equilibrium as well as during nonequilibrium evolution after quantum quenches. To extract probability distributions we derive a simple formula for the characteristic function of generic quadratic observables in any Gaussian theory of bosons.
Current status:
Reports on this Submission
Strengths
1 detailed comparison of meanfield full counting statistics with exact results
2comparison of different meanfield approaches for bosons
3derivation of analytical formula for FCS of quadratic observables for bosons
4spinwave theory around a timedependent direction
Weaknesses
1The paper lacks a bit of structure as derivations and results as well as equilibrium and nonequilibrium results are mixed.
2Extreme value statistics of staggered magnetization in 2d/3d Heisenberg model lacks explanation. This could be an artifact.
Report
In their manuscript "Outof Equilibrium FullCounting Statistics in Gaussian Theories of Quantum Magnets" the authors derive an analytical formula for the generating function of quadratic operators in Gaussian theories of bosons and apply it to different meanfield theories of quantum magnets in equilibrium and after a quantum quench. The work provides detailed derivations and compares the accuracy of the full counting statistics of the meanfield theory with other published methods, e.g. Monte Carlo results for the 2d Heisenberg model. The relative merits of different meanfield approximations (SchwingerBoson meanfield theory, modified spinwave theory) are assessed. The authors focus on the 2d and 3d Heisenberg model and the longrange transversefield Ising model. The validity of calculating second and fourth cumulants in meanfield theory and generating the FCS from this is investigated.
The phenomenological fit of the PDF for the 2d and 3d Heisenberg model to an extremevalue distribution is an interesting observation, which calls for an explanation. The fact that for both 2d and 3d Heisenberg model the PDF of the staggered magnetization follows a Gumbel distribution is rather suspicious, since the former model has no longrange order at finite temperature while the latter has. Why is the PDF in Fig. 1 at zero temperature not a semicircular distribution like in Fig. 3 of the supplemental material of Ref. [20] ? Is this an artifact of the spinwave expansion around an ordered state with welldefined orientation of the spins ? The PDF for the 2d Heisenberg antiferromagnet in section 3 appears to have the correct Gaussian shape, but this is the same model as in Fig. 1, only the mathematical treatment is different.
The longrange transversefield Ising chain has a finitetemperature KosterlitzThouless floating phase at alpha=2 (See e.g. PRB 64, 184106 (2001); Journal of Computational Physics, vol. 228, 7 (2009) and J. Stat. Mech. (2020) 063105).
Here, indeed one might expect nonGaussian magnetization fluctuations inside a relatively narrow temperature window. This is definitely the case for the related 2d XY model, see Bramwell et al. Nature 396, 552 (1998). An interesting question is whether the authors' spinwave approximation can reproduce this regime.
In the section on quantum quenches (before Eq. 32) it is not stated what the Hamiltonian after the quench is.
In my view, the paper is suitable for publication in Scipost Physics provided the authors can give a more indepth explanation for the occurrence of the extremevalue statistics in the linear spinwave calculation for the 2d and 3d Heisenberg model in Figs. 1 and 2.
The detailed benchmarks with exact results may be very useful to assess the validity of FCS of meanfield theories in regimes where no other methods are available, especially nonequilibrium settings, opening a new pathway for interpretation of numerous experimental results.
Requested changes
1explain occurrence of the Gumbel distribution in Fig. 1
Recommendation
Ask for minor revision
Strengths
1. this topic of this work is timely
2. the results of this work is useful to the study of this field
3. the paper is well organized and clearly written
Report
This paper by Senese et. al. reports a simple formula to calculate the full counting statistics of Gaussian states (i.e. probability distribution of the measurement outcomes of observables in subsystems) from the twobody correlators. Based on such a formula, the authors leverage the wellestablished spin wave theory to calculate the full counting statistics in various spin systems. As the starting point for this study is the Gaussian states, it is expected (at least not surprising) that the full counting statistics can be expressed as a quadratic form of bosonic variables. Despite that, full counting statistics is a timely topic recently. Figuring out the exact expression, as well as several examples to demonstrate the power of this expression, is important to this field. Therefore, I find this work, in principle, suitable to be published in SciPost.
Nevertheless, I would like to invite the authors to address the following point before the paper gets published.
How well the quantum system are described by a Gaussian state basically determines the applicability of the formula proposed by the authors. Indeed, in a few cases, especially in the scenarios of quench dynamics, the authors found large discrepancy between the results calculated with their formula and the strict results. While several of these examples are helpful, it is better if the authors can provide some more general principles and guiding rules for when the formula works the best or fails.
Another related point is that the authors mainly focus on the examples when the quantum systems can thermalize or is already at thermal equilibrium. Recently, full counting statistics in integrable systems also receive quite a lot of interest. I also recommend the authors to comment and discuss the validity of their results in those integrable systems which do not fully thermalize to a Gibbs ensemble.
Recommendation
Ask for minor revision