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Investigation of the Néel phase of the frustrated Heisenberg antiferromagnet by differentiable symmetric tensor networks
by Juraj Hasik, Didier Poilblanc, Federico Becca
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|Authors (as Contributors):||Juraj Hasik · Didier Poilblanc|
|Date submitted:||2020-11-10 00:10|
|Submitted by:||Hasik, Juraj|
|Submitted to:||SciPost Physics|
The recent progress in the optimization of two-dimensional tensor networks [H.-J. Liao, J.-G. Liu, L. Wang, and T. Xiang, Phys. Rev. X 9, 031041 (2019)] based on automatic differentiation opened the way towards precise and fast optimization of such states and, in particular, infinite projected entangled-pair states (iPEPS) that constitute a generic-purpose Ansatz for lattice problems governed by local Hamiltonians. In this work, we perform an extensive study of a paradigmatic model of frustrated magnetism, the J 1 − J 2 Heisenberg an- tiferromagnet on the square lattice. By using advances in both optimization and subsequent data analysis, through finite correlation-length scaling, we re- port accurate estimations of the magnetization curve in the Néel phase for J 2 /J 1 ≤ 0.45. The unrestricted iPEPS simulations reveal an U (1) symmetric structure, which we identify and impose on tensors, resulting in a clean and consistent picture of antiferromagnetic order vanishing at the phase transition with a quantum paramagnet at J 2 /J 1 ≈ 0.46(1). The present methodology can be extended beyond this model to study generic order-to-disorder transitions in magnetic systems.
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Reports on this Submission
Anonymous Report 3 on 2020-12-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202011_00009v1, delivered 2020-12-08, doi: 10.21468/SciPost.Report.2265
In the paper "Investigation of the Néel phase of the frustrated Heisenberg antiferromagnet by differentiable symmetric tensor networs" the authors present an estimation of the magnetization curve of the $J_1-J_2$ Heisenberg antiferromagnet on a square lattice.
They report that their $J_2$ at which the phase transition is located agrees with the range given in the literature.
The main results of this paper are the challenging methodological combination of previously described techniques and the resulting magnetization curve for the model at hand.
Both parts are presented in detail and the results can be reproduced easily (due to the fact that the used code is open source).
I believe this work is a notable contribution, especially for benchmarking future developments. Nevertheless I would suggest several (mainly cosmetic and hence debatable) improvements in the presentation prior to publication.
Consider giving (review) references to the mentioned methods (DMRG, fRG, VMC, MPS).
Please move the floating objects closer to their appearance in the text. Table 1 (page 7) is the first floating environment mentioned in the text on page 8, which is fine. But figure 1 (page 5) and figure 2 (page 6) are mentioned afterwards.
In figure 5 basically the same data is shown twice. Maybe consider two different line types for the different extrapolations within only one figure. Especially, since the quadratic extrapolation is supposed to be less trustworthy.
Consider adding an inset in figure 6 for the most interesting part ($J_2=0.4 - 0.5$).
As the error bars do not add that much insight (except in figure 6) they can as well be disabled.
Maybe consider to move figure 4 (or a portion of it) to an appendix, since only the first subfigure ($J_2=0$) is discussed in detail and the main result of the figure is summarized in table 2.
"Once the basic (low D) structure of tensor is established [...]", probably an identifier for the tensor is missing?
"The importance of our findings is twofold." Linguistically only one argument follows.
Anonymous Report 2 on 2020-12-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202011_00009v1, delivered 2020-12-08, doi: 10.21468/SciPost.Report.2263
Ground state properties of square-lattice S=1/2 Heisenberg model is studied by the variational state that is given by an infinite and uniform tensor contraction. The local tensor is expressed as the combination of linearly independent tensors, each of which satisfies the symmetry. This representation enables the systematic and stable numerical optimization of the variational state. The environment tensor is created by the CTMRG method, where the corner transfer matrix dimension is carefully chosen so that the cut-off effect does not affect the variational optimization process. Convergence in both the ground-state energy per site and magnetization is checked from D=2 to D=7. A remarkable point is that the correlation length of the optimized variational state is used for the final scaling analysis, and this physical scaling scheme realizes the good convergence. The technical explanations are well separated from the numerical data analysis, and therefore this article fits both to the experts in tensor network and also to general readers who are interested in 2D magnets. For these reasons, and to avoid disturbing the good structure of sentences, I recommend the publication of this article as it is, except for tiny corrections added by authors only if they are necessary.
Anonymous Report 1 on 2020-12-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202011_00009v1, delivered 2020-12-07, doi: 10.21468/SciPost.Report.2260
In this paper the authors study the ground state of the spin-half
J1-J2 model on the square lattice by means of differentiable
symmetric tensor networks.
This model is a paradigmatic highly frustrated quantum model and
it allows the investigation of frustration driven quantum phase
transitions between semi-classical ordered and quantum
Meanwhile there are numerous studies of this model and there is
some consensus on the ground state phase diagram albeit the
discussion of the nature of the quantum paramagnetic phase is
The model may also serve as a test ground for new quantum many
body methods, such as the tensor network approach.
The focus of the present work is on the methodology of
differentiable symmetric tensor networks, where the J1-J2
model serves as a challenging problem of frustrated quantum
magnetism to demonstrate the potential of the method.
On the other hand, the main result, namely the dependence of the order Neel parameter on the frustrating coupling does not provide new
insight in the physics of the model, rather it confirms the
well-known second-order transition at J2/J2 ~ 0.45.
All in all, I felt that this paper provides
a valuable contribution to the field which deserves publication.
I have several concerns, however, that should be addressed before I
can recommend this paper for publication.
(1) Bearing in mind the long history of studies of the J1-J2 model
it is useful to mention and briefly discuss other (except tensor
network and related studies) successful approaches
such as high order series expansion, PRB 60, 7278 (1999) and
PRB 73, 184420 (2006);
coupled cluster approach, PRB 78, 214415 (2008), Eur.Phys.J.B 88, 2 (2015);
cluster mean-field theory, J. Phys Condens. Mat. 26, 115601 (2014).
In Fig.6 the authors compare their order parameter with QMC data at J2=0 and their previous results of Ref. 42.
It could be useful to compare also with order-parameter data
from DMRG, Ref 14 (PRL 113, 027201 (2014)), and coupled cluster
(3) For me it remains unclear why the authors do not discuss the
region J2/J1 > 0.5.
I believe it would be useful to add a brief discussion to what
extent the used method is appropriate to investigate the gap in the
quantum paramagnetic phase as well as the transition
quantum paramagnet and the semi-classical ordered striped phase
at J2/J1 ~ 0.6, which is most likely of 1st order.