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Extended Coulomb liquid of paired hardcore boson model on a pyrochlore lattice

by Chun-Jiong Huang, Changle Liu, Ziyang Meng, Yue Yu, Youjin Deng, and Gang Chen

Submission summary

As Contributors: Gang Chen
Preprint link: scipost_201908_00004v1
Date submitted: 2019-08-04
Submitted by: Chen, Gang
Submitted to: SciPost Physics
Domain(s): Theor. & Comp.
Subject area: Condensed Matter Physics - Theory


There is a growing interest in the U(1) Coulomb liquid in both quantum materials in pyrochlore ice and cluster Mott insulators and cold atom systems. We explore a paired hardcore boson model on a pyrochlore lattice. This model is equivalent to the XYZ spin model that was proposed for rare-earth pyrochlores with “dipole-octupole” doublets. Since our model has no sign problem for quantum Monte Carlo (QMC) simulations in a large parameter regime, we carry out both analytical and QMC calculations. We find that the U (1) Coulomb liquid is quite stable and spans a rather large portion of the phase diagram with boson pairing. Moreover, we numerically find a thermodynamic evidence that the boson pairing could induce a possible Z2 liquid in the vicinity of the phase boundary between Coulomb liquid and Z2 symmetry-broken phase. Besides the materials’ relevance with quantum spin ice, we point to quantum simulation with cold atoms on optical lattices.

Current status:
Editor-in-charge assigned

Reports on this Submission

Anonymous Report 2 on 2019-9-30 Invited Report


1. Interesting problem
2. Complementary analytical and numerical approach.
3. Realistic models for real quantum magnets that are proposed candidates for the novel quantum spin liquid / ice phases.
4. Very nice entanglement entropy and specific heat data from QMC simulations.
5. Powerful extension of QMC algorithm to study pair hopping.


1. No discussion of finite size effects - data presented for only one system size.
2. No discussion of finite temperature scaling to estimate ground state properties.
3. Lack of strong evidence for Z2 spin liquid phase.
4. No order parameter studied for determining phase boundaries.


The authors have studied hard core bosons with nearest neighbor interactions and an additional pair hopping term on the pyrochlore lattice. Using complementary mean field theory of spinon-gauge decomposition of the bosonic operators and quantum Monte Carlo simulation of the parent model, the authors propose multiple ground state phases with unconventional order as the relative strength between the single (t1) and pair hopping (t2) amplitudes is varied. These include a Coulomb liquid at small values of t1 and t2, an ordered phase (superfluid?) at large values of the hopping amplitudes and an intermediate Z2 spin liquid intervening the ordered phase and the Coulomb liquid at small values of the pair hopping.

The model is certainly interesting, especially since it is relevant to existing quantum magnets that are quantum spin liquid / spin ice candidates. The results are also interesting. However, while the entanglement entropy and specific heat data are beautiful, the same cannot be said for the correlation functions. I’d like the authors to address the following points before I can recommend the manuscript for publication:

1. Does the pair hopping process arise from the asymmetry of the XX and YY terms in the XYZ model? If so, can the authors please specify that?
2. If that is indeed the case, then is there any estimate of the location of the experimental systems on the t1-t2 parameter space?
3. Can the authors show the actual lattice for which the simulations were done? The “supplementary material” only shows 2D Kagome lattice.
4. It is better to rename the Supplementary Material as Appendix – but that is probably a stylistic convention fixed by the journal. Either way, it is better to refer to it as “SM” instead of Ref.[49].
5. There is a difference of one order of magnitude between the critical values of the hopping parameters obtained from mean field theory and QMC simulations – can the authors comment on that?
6. Are there any longer range interactions beyond nearest neighbor?
7. Page 2, col. 2, Table 1: Is the ordered phase a superfluid (as they describe it in the t2=0 limit)? If so, why is it gapped? Isn’t a SF phase gapless?
8. Page 2, col. 2, 3rd para: Since the model being studied is a hard core boson Hamiltonian, it is better to adopt a bosonic terminology after having discussed the correspondence between the spin and boson models in the introduction. Accordingly, the ground state with \langle b_i + b^\dagger_i\rangle \neq 0 should be termed a SF phase – it is less confusing.
9. Page 3, col 2, Eq. (6): Does the free energy equal total energy at these temperatures?
10. All the data are presented for only 1 system size – the authors need to discuss finite size effects, even if qualitatively for only a few system sizes.
11. Similarly, the ground state properties are estimated from simulations at only one temperature. Have the authors conducted any finite temperature scaling and confirmed that this is adequate to estimate ground state properties?
12. Can the authors calculate the order parameter for the SF phase to study the phase transition, to complement their analysis using the derivative of the energy w.r.t. hopping parameters? It is not entirely clear why the order of transition should be reflected in dE/dt1.
13. Page 4, col 1: Can the authors plot the Cv-T data on a log-log scale to confirm the T^3 dependence at large t2? Similarly, can they plot the data on a log-linear scale to confirm exponential decay at small t2?
14. A rough estimate of the magnitude of the gap can be obtained from imaginary time dependence of the single particle Green’s function (assuming single mode approximation) – since the authors are already calculating the Green’s function, it should be possible to confirm their estimate from specific heat data with this approach.
15. The evidence for a Z2 spin liquid is very weak. Additionally, it is confined to a very narrow range of parameters. One needs extensive finite size scaling analysis to rule out finite size effects. Can the authors provide some more evidence for this phase? Is there any signal in the entanglement entropy?

The manuscript addresses an interesting (and difficult to study) problem. The results are interesting and deserve to be published, but the authors need to address the above queries satisfactorily before that.

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

Anonymous Report 1 on 2019-9-13 Invited Report


1. The model simulated is a non-trivial extension of previous research on Coulomb spin liquids. It could be relevant for some pyrochlore compounds and possibly amenable to cold-atom physics emulation. It is also interesting on its own, with a possibility of an intermediate phase (even though I have doubts on its existence in the thermodynamic limit).

2. The physics is well exposed, and the tone of the paper is quite pedagogical.

3. Even though it is not that surprising that the Coulomb phase survives to a non-zero t2, it is important to show it through an explicit computation


1. The simulations are well-performed, but not outstanding. There is no discussion of finite-size effects.

2. Evidence for the Z_2 liquid are not convincing.


This manuscript reports the ground-state phase diagram of a hard-core Boson (equivalent to XYZ) model on the pyrochlore lattice in presence of a pair hopping term t2, through mean-field and quantum Monte Carlo computations. The Coulomb phase (known to exist for small hopping term and t2=0) is found to persist in a sizeable part of the phase diagram. Possible evidence for an intermediate Z_2 liquid phase is also discussed.

The mean-field theory for these type of frustrated models is well established in previous literature. The simulations for low or intermediate temperature for this type of model in 3d are known to be difficult, even though there is no Monte Carlo sign problem.

The existence of the Coulomb phase is quite clear from the entropy plateaus. The evidence from the correlators are slightly less convincing, but acceptable.

The evidence for the Z_2 liquid on the other hand is not compelling in my opinion (fit of the specific heat and of correlators). There are other possible smoking guns of Z_2 liquid (e.g. from entanglement entropy) which are possible in QMC (albeit difficult). From the phase diagram of Fig. 1, it seems that the Z_2 liquid would persist up to t_2=0, which was not seen in previous simulations. Finally, the discussion on the CDW in the appendix is pretty much useless: it mentions that a CDW is possible and could be detected from density correlations, but the calculations is not done!

Actually there is no study of finite size effects in this manuscript: all results are provided for 8^3 lattices. Even though I mentioned simulations are difficult, this number is not really impressive, especially with respect to what was done more than 10 years ago in Ref. 43, using similar techniques. It is important to have at least a qualitative impression of where finite-size scaling is leading. This is particularly true for the phase boundary of the putative Z_2 liquid, which might as well disappear. I strongly suggest to show data at other sizes (the authors must certainly have results at L=6 at least) to see the trends (on the size of the entropy plateau, discontinuity in energy etc).

A few technical remarks:
* The phase diagram Fig1b is really too small, it's almost impossible to see that there is an intermediate phase (orange region)
* The values of (t1,t2) for each point 1...6 should be explicitly given somewhere.
* The lattice should be shown somewhere, this would be very useful in particular with respect to the perturbation theory notations.
* The use of the 'our model' every other line is exaggerated.
* Typos : Solid and dashed should be interchanged in Fig4; Fig 7 should refer to Fig 2 and not 1; Procedure (1) and (2) in the QMC appendix are not defined; Typo in equation (10) (no dagger);
* The manuscript would benefit from proofreading by someone with professional proficiency in English

Requested changes

1. Discuss finite size effects. Show data at other L to have at least an opinion on how finite-size effects could change the phase diagram.

2. Improve presentation according to the technical remarks above.

  • validity: good
  • significance: ok
  • originality: good
  • clarity: high
  • formatting: reasonable
  • grammar: acceptable

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