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Gapless chiral spin liquid from coupled chains on the kagome lattice
by Rodrigo G. Pereira, Samuel Bieri
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Samuel Bieri · Rodrigo Pereira 
Submission information  

Preprint Link:  https://arxiv.org/abs/1706.02421v2 (pdf) 
Date accepted:  20171223 
Date submitted:  20171117 01:00 
Submitted by:  Pereira, Rodrigo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Using a perturbative renormalization group approach, we show that the extended ($J_1$$J_2$$J_d$) Heisenberg model on the kagome lattice with a staggered chiral interaction ($J_\chi$) can exhibit a gapless chiral quantum spin liquid phase. Within a coupledchains construction, this phase can be understood as a chiral sliding Luttinger liquid with algebraic decay of spin correlations along the chain directions. We calculate the lowenergy properties of this gapless chiral spin liquid using the effective field theory and show that they are compatible with the predictions from parton meanfield theories with symmetryprotected line Fermi surfaces. These results may be relevant to the state observed in the kapellasite material.
Author comments upon resubmission
Second, we agree with the referee that our cosine potential differs significantly from a conventional sineGordon model. Nonetheless, we have rewritten the discussion around Eq. (50) to emphasize that the interactions are local in the sense that the operators at different positions (in two dimensions) commute when the distances are larger than the shortdistance cutoff. (Note this does not happen for more general cosine potentials with noninteger scaling dimensions.) We interpret this as a sign that the rightmoving bosons can be gapped out entirely, leaving only gapless leftmoving bosons. We cannot prove the latter statement rigorously, but we show that the lowenergy fixed point obtained within this assumption is stable and the theory is consistent. Clearly, the phase in which the coupling between chiral currents is the strongest interaction must be different from the magnetically ordered (cuboc) and valencebondcrystal phases. In our opinion, our proposal of a gapless chiral spin liquid is the simplest and most plausible picture that emerges from the RG analysis.
Concerning the parton construction, we clarify that the point here was to show that there are at least two meanfield ansatze which are able to reproduce the properties of the chiral spin liquid derived from the coupledchain approach. This is not a meaningless exercise because, once identified, the parton construction can be useful to analyze the properties of the chiral spin liquid in a regime where the starting point of weakly coupled chains is not reliable. The 1/r^2 decay of the correlation function indeed imposes a strong constraint on the theories that we can select, but it is not the only constraint. Our analysis is also guided by symmetry, as the meanfield ansatz must break time reversal and the reflection by lines perpendicular to the chain directions, but respect the symmetry of reflection by lines parallel to the chains. Out of large number of U(1) chiral spin liquids on the kagome lattice that have been classified in Ref. [45], we have narrowed the choice down to one. The other possible meanfield ansatz uses a Majorana fermion representation with a Z2 gauge structure. Selecting between the Z2 and U(1) parton constructions requires numerical methods to calculate the ground state energy (using e.g. variational Monte Carlo techniques) and it is beyond the scope of this paper. Most numerical studies of chiral spin liquids so far have focused on gapped phases (with the notable exception of Ref. [70]). We hope that our work will encourage the search for gapless chiral spin liquid phases.
List of changes
1. In the legend of Fig. 1, we now comment that the arrows indicate the positivex directions, which become the directions of rightmoving bosons for each q=1,2,3 in the continuum limit.
2. To address the referee’s concerns about the relevant cosine potential, we have rewritten the discussion around Eq. (50).
3. In section 6.4, we have included the energy magnetization correction in the expression for the thermal Hall conductivity, citing new references [79] and [80]. This correction does not change the conclusion that the thermal Hall response of the gapless chiral spin liquid vanishes by symmetry.
Published as SciPost Phys. 4, 004 (2018)
Reports on this Submission
Report #1 by Anonymous (Referee 2) on 20171214 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1706.02421v2, delivered 20171214, doi: 10.21468/SciPost.Report.294
Strengths
1 The authors approach the problem from different perspectives (coupled chains and parton mean fields), and find not inconsistent results.
2 The authors provide a detailed account of their calculation.
3 The topic is timely.
4 The approach is in principle a very good one to address questions such as "is it in principle possible to find this or that phase", as is the goal of the paper.
Weaknesses
1 The bosonization procedure is still unclear
Report
The authors have responded well to the comments raised in the first round. The changes in the manuscript in my view did increase the understandability of the paper, and I thus recommend the work for publication.
I have one final comment on Eq. (50). The authors state that the argument of the cosine does not commute with itself at different positions. In bosonization, that means that the field cannot order  I hope the authors agree on that.
The authors then show that the cosine as a whole commutes with itself at different positions and state that "the most plausible scenario" is that the fields are gapped. It is mathematically unclear to me why this "most plausible scenario" is realised, given that the fields themselves cannot be pinned. Is there a way to understand this, or do I have to accept this as, really a guess? In other words, I find the physics plausible, but currently feel like there is a leap of faith from writing down Eq. (50) to saying that the rightmoving fields are pinned. Sure, the phase where the rightmovers are gapped might be a stable fixed point, but it feels to me like the authors are writing down a Hamiltonian they cannot solve, and then state that there is a stable fixed point, of which they do not show under which conditions the model realises it, and how that is related to the chiral interaction. Is that true?
I thus ask the authors to reclarify this point.
Requested changes
1 Clear up the commutation of the argument of the cosine in Eq. (50).
Author: Rodrigo Pereira on 20180112 [id 200]
(in reply to Report 1 on 20171214)We thank the referee for his/her helpful comments. We have added some clarification to emphasize that the full gap in one chiral sector is a conjecture since we cannot pin the bosonic fields in the usual way. Although the gap scenario seems plausible and physically compelling, this particular step in our analysis is indeed an assumption which we are not able to rigorously prove. Nonetheless, our derivation of the field theory from the lattice model and the perturbative renormalization group analysis are completely unbiased. Similarly, the calculation of physical properties is well controlled once we make this assumption about the strong coupling regime. It is worth mentioning that the lack of analytical techniques to handle noncommuting bosonized cosine terms is not unique to our problem. See, for instance, D. Bulmash et al., Phys. Rev. B 96, 045134 (2017) for a recent discussion of other examples.
We stress that the theory relying on our conjecture is consistent as it describes a stable fixed point of the renormalization group flow. Moreover, we clearly state that the chiral spin liquid phase can appear for strong chiral interaction, while valencebond crystal and magnetic orders are expected to arise for small J_\chi. The validity of the conjecture could be tested by numerical simulations of the lattice model, by computing correlation functions that involve excitations of the gapped modes, such as dimerdimer or the staggered magnetization mentioned in section 6.1.