# Investigating ultrafast quantum magnetism with machine learning

### Submission summary

 As Contributors: Giammarco Fabiani Arxiv Link: https://arxiv.org/abs/1903.08482v2 Date submitted: 2019-03-22 Submitted by: Fabiani, Giammarco Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Condensed Matter Physics - Theory

### Abstract

We investigate the efficiency of the recently proposed Restricted Boltzmann Machine (RBM) representation of quantum many-body states to study both the static properties and quantum spin dynamics in the two-dimensional Heisenberg model on a square lattice. For static properties we find close agreement with numerically exact Quantum Monte Carlo results in the thermodynamical limit. For dynamics and small systems, we find excellent agreement with exact diagonalization, while for larger systems close consistency with interacting spin-wave theory is obtained. In all cases the accuracy converges fast with the number of network parameters, giving access to much bigger systems than feasible before. This suggests great potential to investigate the quantum many-body dynamics of large scale spin systems relevant for the description of magnetic materials strongly out of equilibrium.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 1903.08482v2 on 22 March 2019

## Reports on this Submission

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### Strengths

1- The results presented are new.

2- The technicalities are well-covered in the paper.

### Weaknesses

1 - In addition to the energy and the magnetization, the authors could consider other ground state static observables, e.g., magnetic susceptibility, to check the accuracy of the method.

### Report

The paper by Fabiani et. al. investigates the accuracy of a recently proposed quantum many-body variational method based on the restricted Boltzmann Machine (RDM) neural network. As point out in the introduction, this method has the potential to efficiently simulate static and dynamic properties of many-body wave functions in any dimension. However, the efficiency of the RDM method in simulations of dynamical properties was not tested in dimensions higher then one. The introduction motivates well this point.

In the paper, the RDM method is applied to study static and some dynamical properties of the prototypical two-dimensional Heisenberg model (HM); the results are then validated with other exact (or approximated) methods.

The results presented by the authors provide relevant information about the the efficiency of the restricted Boltzmann Machine in two-dimensions.

### Requested changes

1 - The authors mention that for larger systems, already for $\alpha = 4$ "convergence is reached within Monte Carlo error". Is this a general feature of the method, i.e., larger systems requires smaller $\alpha$ for convergence? or just a numerical observation for this specific case? I think the authors should comment about this on the manuscript.

• validity: good
• significance: high
• originality: ok
• clarity: good
• formatting: excellent
• grammar: good

### Strengths

1- Timeless investigation of RBM states for strongly-correlated systems.

2- Accurate numerical calculations

3- Well written

### Weaknesses

1- Some minor points should be addressed (see report)

### Report

The paper is nice and well written. Just a few comments

1) I do not like the fact the nomenclature of reinforcement learning'' for the optimization technique. Even though this is a fancy name, the optimisation is just standard, according to the Monte Carlo community (see ref.26).

For the real-time evolution a VMC approach was proposed in Scientific Reports 2, 243 (2012) for a Bose-Hubbard model. I think that this paper should be cited.

2) Are the variational parameters real of complex for the static calculations?
Is the Marshall sign imposed?

3) The standard way to define the energy accuracy is to normalize |E_vmc-E_0| by E_0 and not by E_vmc.

### Requested changes

1) Do not use reinforcement learning'' and add the reference.

2) Specify if W is real or complex for the static calculations.

3) Change the normalization in the accuracy.

• validity: good
• significance: good
• originality: ok
• clarity: high
• formatting: excellent
• grammar: excellent