Self-congruent point in critical matrix product states: An effective field theory for finite-entanglement scaling

We thank the referee for their positive evaluation of our previous feedback and for the careful critique of our revised second version. Below, we intend to respond satisfactorily to all points raised by the referee.

The explanation of why we are interested in the spectrum of the transfer matrix makes sense. I would make a bit more emphasis on the fact that the correspondence between the spectra of the TM and the effective Hamiltonian is a numerical observation (unless there is a proof of it in some reference). I would suggest also adding is a comment about the fact whether the transfer matrix is Hermitian, since it feels like it need not be, yet its spectrum is real anyway.

A: We have added the clarification of the numerical observation of the mapping of the TM spectrum to the effective Hamiltonian spectrum. On the second point, indeed the TM spectrum does not need to be real as the transfer matrix is not Hermitian in general. In the Ising model, everything is real-valued because the Ising model has not just a Hermitian Hamiltonian but indeed a symmetric real Hamiltonian. The ground state of the Ising model is thus real-valued, and therefore so is the transfer matrix. We added a note stating that this is generally not true.

In Fig. 3(a), the reader is not given an explanation for the appearance of $\nu c$ in the cyan dotted line, the expansion of the $E_\mathrm{exact}(L)$ only gives $\pi/6L$

A: On page 11, in the main text discussing Fig. 3(a), we cite Refs. [53, 54], the relevant references which give the definition of the Casimir energy in the context of a CFT given the features of the model, i.e., speed of sound $v=2$, and conformal charge $c=1/2$. Note that the referee mistakenly read the cursive Latin letter $v$ as the cursive Greek letter $\nu$, possibly mistaking it for the critical exponent $\nu=1$. We have added a footnote to warn the reader. We have made the formula explicit in the main text in the new version, explaining the contributing factors in the Ising model. Essentially, $v \cdot c = 1$ as $v=2$ and $c=1/2$ in the Ising model, which hopefully should resolve the confusion.

At the end of the first paragraph of page 14, it reads "Their distance from the unstable CFT fixed point increases logarithmically as we increase the system size." What distance measure are we talking about?

The logarithmic distance originates from the RG flow equations which are always Ordinary Differential Equations (ODEs) in the logarithm of the system size, as in Eq. (15). Thus, the "flow" in the RG sense happens in a space of couplings with a notion of distance (or time if you want) given by the logarithm of the ratios of two system sizes, $\log(L_2/L_1)$, which are given by the initial and final point of the flow, $L_1$ and $L_2$, respectively. The maximum distance RG can flow in a physical system with system size $L$ is $\log(L/1)$, that is, from the microscopic definition of a single site (or unit cell), $L_1=1$, up to the notion of the complete system, $L_2=L$. The distance of the initial coupling flowing due to RG to its final point hence grows logarithmically with the size of the system and approaches the fixed point in the thermodynamic limit.

Above Eq. 25, I would use a different letter to count the number of fermionic modes, since m already means something else.

A: Thank you for noticing the double notation. We have adapted the label for the number of fermion modes.

If I have now understood the Eqs. (33)-(35) argument correctly, the statement is that we can compute the correlation length exactly at both sides of the phase transition, thanks to (33) (which, I'd say, is a nontrivial result, as opposed to the author's reply saying that correlation length and single particle gap are inversely proportional "by definition"), and we pick the value of $m$ so that $\xi=L/2$. Is this the case?

A: We have restructured the arguments slightly and taken out the previous Eq. (33), which only served to give the speed of sound (or speed of light) $v$ the entrance as a necessary proportionality constant into the well-known inverse proportionality relation of the correlation length to the gap, for dimensional reasons. In addition to the dimensional argument, i.e., $\xi \propto v |\Delta_1|^{-1}$, there is a dimensionless factor of 2 arising from the domain-wall nature of the quasi-particle excitations. Furthermore, we have argued above Eq. (23), the definition of the lattice model, that we expect a factor of 2 between the correlation length and the system size, since the correlation of the effective Hamiltonian is mediated through the finite bond dimension tensors on both sides. As the induced spatial dimension of the finite bond dimension is given by the correlation length, this gives rise to a notion of the total system size given by twice the correlation length, $L = 2\xi$. We furthermore numerically verify our thusly constructed model, giving rise to the correct behavior. This consequently fixes $|m| = 1$ according to Eq. (37). The only ambiguity left is the sign of $m$, which is equivalent to which side of the phase transition does $\delta$ settle, i.e., in the SSB or the PM phase. We show that statement with the same arguments as before, which also prompted us to include the graphic showing the approximate symmetry of the variational ground state error w.r.t. the coupling $\delta$. This additionally yields the reference to the claim in the footnote the referee was asking for in the next question just below. We will resubmit this manuscript with the reference to this comment and its attachment and acknowledge Referee 2 for their constructive, clarifying, and helpful critique.

On the sixth line of page 19, I would advise against using footnotes that can be mistook for exponents. Also, is there a reference for the information in the footnote (the bias of the variational energy) or is it an observation of the authors?

A: Thank you for suggesting this clearer typographic choice. We have swapped the footnote marker to symbols. We do not have a direct source for the bias of the variational energy. We computed it numerically exactly through the free fermion framework and the numerical packages of Ref. [64]. We have attached a plot to this comment, which shows the variational ground state energy error as a function of the perturbative coupling $\delta$. We hope this plot clears things up.

Towards the end of 5.2.2. there is a reference to Fig. 8(c) which does not exist.

A: Thank you for spotting that typo. The label is now corrected to Fig. 8(a).

I find the explanation for the influence of the choice of symmetric vs. nonsymmetric tensors on the spectrum of the TM is a bit rushed and sandwiched in the discussion of a different spectrum (the entanglement Hamiltonian spectrum, last paragraphs before Section 6), making it easy to miss.

A: We appreciate the remark and acknowledge the brevity with which we go over this specialized detail. We have expanded the explanations and added a discussion of the matter in Section 2 and 3. Furthermore, at the very end of Section 2, when we previously talked about shedding more light onto the puzzle of the different spectra of the TM when enforcing the $\mathbb{Z}_2$-symmetry or not, we have added the cross-reference to the section (Section 5.2.3) where we discuss this in more detail.

It would be nice to give it some more entity and discuss whether the relevant boundary conditions can be implemented in the toy model of the paper.

A: On the implementation of the boundary conditions in the toy model of our paper: Indeed, it remains open to further investigations to incorporate other boundary conditions in our toy model which could correspond to different spectra.

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MPS-TFM-attachement-plot.pdf