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Quantum Monte Carlo detection of SU(2) symmetry breaking in the participation entropies of line subsystems
by David J. Luitz, Nicolas Laflorencie
This is not the current version.
|As Contributors:||David J. Luitz|
|Arxiv Link:||https://arxiv.org/abs/1612.06338v1 (pdf)|
|Date submitted:||2016-12-23 01:00|
|Submitted by:||Luitz, David J.|
|Submitted to:||SciPost Physics|
Using quantum Monte Carlo simulations, we compute the participation (Shannon-R\'enyi) entropies for groundstate wave functions of Heisenberg antiferromagnets for one-dimensional (line) subsystems of length $L$ embedded in two-dimensional ($L\times L$) square lattices. We also study the line entropy at finite temperature, i.e. of the diagonal elements of the density matrix, for three-dimensional ($L\times L\times L$) cubic lattices. The breaking of SU(2) symmetry is clearly captured by a universal logarithmic scaling term $l_q\ln L$ in the R\'enyi entropies, in good agreement with the recent field-theory results of Misguish, Pasquier and Oshikawa [arXiv:1607.02465]. We also study the dependence of the log prefactor $l_q$ on the R\'enyi index $q$ for which a phase transition is detected at $q_c\simeq 1$.
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Anonymous Report 1 on 2017-1-30 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1612.06338v1, delivered 2017-01-30, doi: 10.21468/SciPost.Report.71
(1) Timely contribution to a fundamental property of quantum many-body systems in terms of entanglement.
(2) Useful contribution from a numerical method in good accord with recent field theory predictions.
(3) Clear presentation of the paper's overall strategy and methodology.
(4) Interesting observation of the transition in the q-dependece of the log-prefactor, calls for further work.
(1) Only minor issues, to be detailed below.
The authors use a quantum Monte Carlo approach to calculate the participation entropy in 2D (in the groundstate) and 3D (finite T) Heisenberg antiferromagnets for an embedded 1D line subsystem. They identify a logarithmic scaling term in accord with previous field-theory predicions, which quantifies the spontaneous symmetry-breaking contribution from Goldstone modes. Furthermore, they identify an interesting transition in the dependence of the log-prefactor on the Renyi index, which they associate with a possible thermal transition in the entanglement Hamiltonian.
The numerical results are sound and have been carefully analyzed, including an error analyis in the fitting procedure to the scaling laws. The employed analysis, focusing on the sublattice particpation is original and in good overall accord with recent field-theory predictions with respect to the Goldstone-mode contribution to the log-corrections, which is known to be difficult to extract in e.g. a direct analysis of the entanglement entropy. The obervation of a transition in the Renyi-index dependence and its linkage to the entanglement entropy points towards an interesting perspective for future analysis.
The paper is well written and explains very clearly the overall idea and strategy behind the taken approach. Nevertheless, I think that the presentation should be improved with respect to the points given below:
(1) In addition to Ref. 1, several other general reviews of the topic exist, which should also be cited.
(2) In addition to Ref. 2, the authors should cite also O. A. Castro-Alvaredo and B. Doyon, Phys. Rev. Lett. 108, 120401 (2012), with regards to theory work on the general Goldstone-mode contribution.
(3) At the beginning of 3.1.1, the actual estimator for the limit q->\infnty should be stated (similar to eq (6)) for clarity.
(4) An explicit algorithm should be cited after "Here, we study the system at zero temperatures..." in the second paragraph of 3.1.1.
(5) For the inset of Fig. 2, only two values of J2 are considered, while in the main panel of that figure, more values of J2 are shown. A reason should be given for this selection of J2 values for the inset (it appears that for these two values of J2, data on many more small systems was collected). Why is the most simple case of J2=0 not considered at all?
(6) Why is the convergence in the inset of Fig. 2 from below? Can this be understood?
(7) It appears more appropriate from the present analysis to write that "a transition is detected at q_c", instead of "phase transition", as the later has a defined meaning, and it is unclear, if this really applies here.