Dear Dr. Sun,
Thank you for submitting your manuscript entitled "Global quantum discord in transverse-field Ising models on N×N lattices" (Original Paper, No. pssb.201800666) to Physica Status Solidi B: Basic Solid State Physics. The reviewer report and comments are included at the end of this e-mail.
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REVIEWER REPORT:
Reviewer #1:
The work investigates multipartite correlations as measured by the global quantum discord (GQD) in the 2D transverse-field Ising model. The analysis is performed either at zero or finite temperatures. For the case of zero-temperature, a quantum phase transition (QPT) is successfully characterized by the GQD measure. For finite temperatures, robustness against thermal fluctuations and its relationship with the transverse field is discussed.
In my opinion, the work seems correct and contains relevant results. Since multipartite correlations are expected to play an important role for the properties of many-body systems, I consider it is indeed interesting to see clear characterizations of QPTs and finite-temperature behaviors through GQD. So I recommend the publication. In order to promote some improvements in the readability of the paper, I would like to suggest the Authors to address the following items:
i) In the introduction, the Author mentions that quantum discord exhibits remarkable properties concerning long-range decay. This is true, but it has been firstly pointed out in Ref. J. Maziero et al., Phys. Rev. A 82, 012106 (2010). I suggest to include this reference in addition to the current Ref. [16] quoted.
Response: We thank the referee for bring us to the paper "Quantum and classical thermal correlations in the XY spin-1/2
chain, Phys. Rev. A 82, 012106 (2010)" by J. Maziero. We have cited this paper in the revised manuscript.
ii) In Section 2.1, the Author mentions that "… because of its operational interpretation, GQD can be used in quantum communication.". I suggest to add a reference to make this point stronger.
We have added two papers into the sentence. The first one is
"No-Local-Broadcasting Theorem for Multipartite Quantum Correlations, Phys. Rev. Lett. 100, 119903 (2008)" by Marco Piani. In addition, we find this paper when we read Liu's paper "Anna. Phys. 348 (2014) 256–269". Thereby we have also cited both papers in the revised manuscript.
iii) How is the 2D transverse-field Ising model solved in the infinite-size limit? The Author should improve the discussion in Section 3.2. It is clear the finite-size solution, but the limitations (if any) involved in the infinite case should be better discussed.
The finite-size lattices are solved by the finite-size projected-entangled-pair-state (PEPS) algorithm. The infinite-size lattices are solved by the infinite-size PEPS (iPEPS) algorithm. In both situations, we use the imaginary-time evolution method to figure out the ground state. The limitation of the iPEPS method is also added in Page 4 (add some sentences about the precision of the algorithm). According to the referee's comment, we have added some sentences in Sec. 3.1 about some details of the method.
iv) In Figure 3, after the QPT critical point, all the curves (for the various N) approximately coincide with each other. The Author provides a brief explanation in terms of a little size effect. However, is there a more intuitive reason for the approximate size independence after the QPT?
v) I agree with the Author that the robustness of GQD against temperature for high fields is indeed remarkable, as shown in Figs. 6c and 7c. However, curiously, since large strong fields tend to completely destroy the quantum correlations, how come do these large fields show more robustness as function of T? I think this point could be better discussed (and perhaps emphasized) in the text.
Indeed, the strong fields tend to completely destroy the quantum correlations, as illustrated in the sub-lattice discord in Fig. 3 and the entire-lattice discord in Fig. 4. The strength of the quantum correlations weakened by the magnetic field, and would completely vanish in the strong field limit.
The robustness of quantum correlation against thermal fluctuation is determined by energy gap in the low-lying energy states. Suppose a system is gapless in its low-lying energy spectrum, a slight increase of the temperature will bring excited state into the thermal state , thus affects the quantum correlations. On the other hand, if a system is gapped in its low-lying energy spectrum, when the temperature increases slightly, the weights of the excited states would be quite small. In such a situation, quantum correlations would behave robust against thermal fluctuation.
In summary, increasing the magnetic field would (1) weaken the strength of quantum correlations, (2) increase the energy gap (see Fig. 8), and consequently enhance the thermal robustness of quantum correlations.
Reviewer #2:
Reviewer report on the manuscript pssb.201800666: "Global quantum discord in transverse-field Ising models on N×N lattices" by Zhao-Yu Sun, Mu Zhou, Mei Wang, Jin Fu
The present manuscript is devoted to a theoretical study of the global quantum discord in the quantum Ising model on N×N square lattice in a transverse magnetic field. The results presented in the manuscript seem to be scientifically sound and the investigated topic is quite interesting. However, the manuscript has several drawbacks, which should be removed before the paper can be accepted for publication
- A description of the method used for theoretical calculations is totally absent and the authors just simply refer to their previous two works (Refs. 21 and 22) at the end of Sect. 2.1 for all details. From this perspective, the paper is not self-contained and it disables reader to get a full insight into the investigated topic. Of course, it is not necessary to repeat all technical details of the calculation procedure, but instead of referring to Refs. 21 and 22 the authors should specifically mention at least basic steps of the calculation procedure when it is applied to the quantum Ising model on N×N square lattice in a transverse magnetic field (topic of the present study).
We thank the referee for the comment. In order to help the reader to get a full insight into the investigated topic, we have added a subsection in the end of Sec. 2 in the revised manuscript. Thereby, Sec. 2 is arranged as follows. In Sec.2.1, we introduce the concept of global discord. In Sec.2.2, we introduce the two-dimensional tensor network states. Finally, in Sec. 2.3, we introduce the basic steps of calculating global discord in the framework of tensor networks.
For such a purpose, first of all, we will re-write the original formula in Eq. (3) in to Campbell's formula. Please see Eq. (4) in the revised manuscript. Campbell's formula contains two main terms.
- The first term is a numerical optimization, with a large operator.
- The second term is just the von Neumann entropy for the quantum state of the models. At zero temperature, the ground state of the model concerned in this manuscript is unique, thus we simply have . At finite temperatures, since we only study very small lattices, we exactly figure out , and consequently , of the models.
- The Hamiltonian of the investigated system is not comprehensively described. I am lacking in the manuscript information that the coupling constant is set to unity and the subscripts defining lattice positions are quite unusual and not clearly defined.
We are sorry about our poor expression about the Hamiltonian of the system. In Sec. 3.1, we have carefully rephrased the first paragraph. We use a pair of interger to label the sites, with . In addition, we state explicitly that is the coupling constant, which we use as the unit in the calculations.
- I am missing throughout the whole paper information that the investigated spin model has a geometric topology of square lattice. This missing information should be supplemented on several places of the manuscript (including title, abstract, introduction, … and specifically, in Sect. 3.1 where the authors mention critical field \lambda_c=3.04), because the magnetic behavior of the quantum Ising model might basically depend on a lattice geometry (e.g. the size of quantum critical fields is lattice dependent and more importantly, there is a fundamental difference between the behavior of loose-packed and close-packed lattices due to a spin-frustration effect).
We would like to express our gratitude to the reviewers' comment. In the manuscript we deal with square lattices, but this information is missed in the manuscript. According to the referee's comment, we have added the word "square lattice" in the title, abstract, introduction, and in the first paragraph of Sec. 3.1. Especially, we use a pair of interger (x,y) with x,y=1,...,N to label the sites. We hope the geometric structure of the model becomes clear in the revised manuscript.
- From my point of view, the most interesting result presented in the manuscript is a thermal enhancement of quantum correlations exemplified by a nonmonotonous temperature dependence of the qlobal quantum discord and specifically its thorough finite-size analysis. Please know, however, that a similar thermal enhancement of the quantum-entanglement measures (e.g. concurrence) has been recently reported for several low-dimensional quantum spin chains (e.g. Ising-Heisenberg two-leg ladder, tetrahedral chain, diamond chains, and many others). Hence, the mechanism of strengthening quantum correlations through thermal excitations has been already suggested in several previously published papers, which should be specifically mentioned in the list of references. The main novelty of the present study lies in a finite-size analysis of this effect, which has not been reported yet (at least to my knowledge). The authors should accordingly put their work into the proper context when
this note should be mentioned in the manuscript together with several references dealing with thermal enhancement of the quantum correlations.
According to referee's comment, we have add some sentences in Page 5, in which we have cited the papers about the thermal enhancement of two-site quantum entanglement, for instance, the Ising-Heisenberg two-leg ladder with alternating Ising and Heisenberg inter-leg couplings, the Ising-Heisenberg tetrahedral chain, the Ising-XYZ diamond chain, the spin trimerized model. We thank referee for the comment, and we hope the revised manuscript may offer a more wide perspective for readers who are interested in thermal quantum correlations.
- The paper needs some polishing of English, e.g. a few sentences like "Quantum entanglement may be the most famous method to characterize quantum correlations" should be rephrased (quantum entanglement is not method).
To summarize, the investigated topic is quite interesting, the results seem to be scientifically sound, but the submitted manuscript has several shortcomings that has to be removed before it can be accepted for publication in Physica status solidi (b) journal.