Rainbow chains and numerical renormalisation group for accurate chiral conformal spectra

I thank the referee for their positive assessment of the paper and their helpful comments, especially for highlighting clearer ways to present the inhomogeneous Hamiltonians used.

• Section 2 should be written more clearly. The author says "we propose discretizing the continuum CFT to sites [...]". What does this mean in practice? How does on go from a CFT to a spin chain Hamiltonian concretely?
• The author keeps referring to Eqs. (3)-(4)-(6) as defining the equations defining the models. But how do these couplings enter the Hamiltonian exactly? Given the Hamiltonian of a critical spin chain, how should we modify it to obtain the models investigated numerically in this paper? This is rather obscure in the present version. (Maybe the author can take inspiration from papers on inhomogeneous spin chains such as [Katsura, J. Phys. A 45 115003, 2012] or [Dubail, Stephan, Calabrese, SciPost Phys. 3, 019, 2017] where this was explained more clearly in my opinion)

Reply: I thank the referee for highlighting these issues and suggesting the above references. The Dubail et al. reference in particular has also provided a very neat argument for interpreting space-dependent coupling strengths as deformation of the space the low-energy CFT is defined on. I have largely rewritten Sec. 2 relying on these arguments. I also introduced an explicit scaling function $f(x)$ in the definition of all Hamiltonians, so it is clearer what the definition of rainbow chains etc. [now Eqs. (7-9) and (12)] refer to. While the fact that the same scaling functions are used in several different Hamiltonians is necessarily a little confusing, I believe the current presentation is significantly clearer.

• Again about Eqs. (3)-(4)-(6) : are Eqs. (3) and (6) different? Or are they the same?

Reply: They are formally similar, but semantically different. Eq. (3) → (7) refers to a chain, so the variable n runs from 0 at the edges of the chain to (L/2-1) in the middle. Eq. (6) → (12) refers to a ring, so the equivalent variable runs from -(L/4-1/2) to +(L/4-1/2), resulting in reflection symmetry. To make this and similar distinctions clearer, these ranges are now listed in all equations where they are relevant.

• Fig. 2 is not clear, both the figure and the caption should be improved.

Reply: Given the improved discussion of the relationship between inhomogeneous spin chains and CFTs in Sec. 2, the TFIM → Majorana chain argument is no longer needed to fix the magnitude of single-site Hamiltonian terms. Accordingly, Fig. 2 has been removed.

• In the discussion of the numerical procedure and of the numerical results in Sections 3.1, 3.2, and 4.1, the Hamiltonian should be written explicitly, so that the reader knows exactly what Hamiltonian is used to obtain the numerical results.

Reply: The Hamiltonian was written explicitly already [Eqs. (14), (15), (24) in the current version]. As mentioned above, the inhomogeneity of Hamiltonian terms is now parameterised in terms of a function $f$, which is written explicitly for all geometries in Sec. 2.

• finally, there is a comment in the conclusion that could also be clarified: "computing entanglement spectra only requires wave functions, not necessarily Hamiltonians, at the critical point". The fact that the entanglement spectrum is a property of the wavefunction alone is obvious and has been known since the very first papers on this topic. But of course, to obtain the wavefunction numerically, usually one has to look for the ground state of a Hamiltonian. So what does the author mean exactly here?

Reply: Wave functions consistent with conformal criticality can be obtained by other means than diagonalising a Hamiltonian. For example, Refs. [52,53] explain how to obtain ansatz wave functions for $SU(n)_k$ WZW models from parton constructions. This might be useful to study the edges of topological phases, because those are often described in terms of partons or other wave function constructions and not parent Hamiltonians. This was already explained in the rest of the final paragraph, but I made it more explicit in the new version.

I thank the referee for their constructive comments, which are addressed below:

1- Sometimes the author introduces notation which might be standard in the field, but is not known to most readers. For a non-expert in the Potts model, the "Q=1 sector" is non-standard notation. Please revise the paper with this idea in mind.

Reply: Q=1 is indeed nonstandard notation - I meant the eigenvalue under the $Z_3$ subgroup of the full symmetry group of the Potts model. I have now clarified this on page 16.

2- I am intrigued by the cases in which the author does not find the expected behavior. While I appreciate the honesty and candor associated, I miss an explanation, even if incomplete. It might even go into the conclusion, as further work. Let me provide a list:

• Page 10: why the 1/16 chiral conformal block is not found? Maybe it is a numerical problem?

Reply: In the ring geometry, both effective boundaries follow from entanglement cuts (see Fig. 1d), i.e., they correspond to free boundary conditions. The ground-state entanglement spectrum then consists of the 0 + 1/2 blocks. There may be tricks similar to the ones discussed for the rainbow chains to obtain the 1/16 block as well. I have not found a suitable one, and whether there is one is tangential to the main thrust of the paper, especially as the NRG does not generalise to the ring geometry.

• Page 12: why do we observe the $L_{eff}^{-2}$ scaling? It is not expected as a correction to the dispersion relation.

Reply: The $L_{eff}^{-2}$ scaling is expected. Entanglement energies scale as $L_{eff}^{-1} + L_{eff}^{-3}$ (expansion of a sine dispersion), but the quantity of interest is their ratio, which scales as ${\rm const.} + L_{eff}^{-2}$. This is now spelled out in more detail on Page 12-13.

• Page 17: why do we observe a $\nu\sim16/3$? As the author says, it is "vastly different" from the theoretical value, $\nu=5/6$.

Reply: This is a very important and interesting point. I have studied this in more detail, which is discussed on Page 19-20 and Appendix C. The exact value of 16/3 is not perfectly universal, but a universal value around 5-5.5 seems to underlie it. For the TFIM, I also did not find the expected scaling exponent $\nu=1$, but rather an exponentially long correlation length ($\nu=\infty$). I could not find an explanation for this behaviour, but I have highlighted it as an important item for future work.

3- I would appreciate more detail referred to the "unzipping" algorithm. I find it novel and intriguing.

Reply: I thank the referee for their interest in this algorithm. I have expanded its description substantially on Page 15.

4- I find the sentence "since the entanglement entropy represented by the MPS is maximised if all Schmidt vectors are equal" (page 18) extremely confusing.

Reply: I thank the referee for highlighting this - I have meant "Schmidt values" rather than "Schmidt vectors". I have fixed this and expanded the sentence to clarify it further.

5- I miss some further exploration regarding the maximal $\chi_{NRG}$ required in order to capture the state faithfully. The author discusses low values and then fixes $\chi_{NRG}=400$ or 800, both cases are relevant. But, how should $\chi_{NRG}$ scale in order to be safe, for a given $L_{eff}$? Page 19 mentions some discrete jumps which spoil the scaling behavior, perhaps this is the reason?

Reply: I think there is some confusion between $\chi_{NRG}$, the number of low-energy states kept in the NRG, and $\chi$, the MPS bond dimension used for the unzipping. The necessary $\chi_{NRG}$ does not depend on system size - the folded MPS tensors converge to a fixed point, which can be repeated indefinitely. The unzipping bond dimension does depend on system size, since the unzipped MPS has to represent a growing entanglement entropy. The rule of thumb is that the Cardy-Calabrese entanglement entropy must be below $\log\chi$, which limits how much entanglement is represented by the MPS. This latter is now spelled out explicitly in Eq. (27).

6- The NRG approach might find a precedent in the analysis of the SPT-like phase that the rainbow chain presents when the number of sites is odd, check Samos et al arxiv:1812.04869. There they provide an MPS structure in the folded regime, much in the line of the NRG approach used by the author.

Reply: I thank the referee for pointing out this paper. I have highlighted rainbow chains of odd length as a possible route to generating all conformal blocks of a CFT in the conclusion.