Gapless chiral spin liquid from coupled chains on the kagome lattice

The referee has raised very good points. First, we would like to clarify the meaning of right-moving bosons for crossed chains. We label R and L modes in the limit of decoupled chains in such a way that R (L) refers to the modes that propagate in the positive-x (negative-x) direction of q-chains as indicated by the arrows in Fig. 1. Thus, the direction of propagation of R/L modes at a given point depends on the q index. This coordinate system with three different positive-x directions may seem confusing, but it was introduced by Gong et al. in Ref. [37] because it is convenient for analyzing the threefold rotational symmetry of the Hamiltonian.

Second, we agree with the referee that our cosine potential differs significantly from a conventional sine-Gordon 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 short-distance cutoff. (Note this does not happen for more general cosine potentials with non-integer scaling dimensions.) We interpret this as a sign that the right-moving bosons can be gapped out entirely, leaving only gapless left-moving bosons. We cannot prove the latter statement rigorously, but we show that the low-energy 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 valence-bond-crystal 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 mean-field ansatze which are able to reproduce the properties of the chiral spin liquid derived from the coupled-chain 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 mean-field 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 mean-field 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.

1. In the legend of Fig. 1, we now comment that the arrows indicate the positive-x directions, which become the directions of right-moving 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.