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

We thank the referee for their effort in taking their time to carefully read and critique our manuscript. Ultimately, their remarks improve the manuscript for which we are grateful. Below, we want to answer all questions the referee raised and hope to do so in a satisfactory way. Furthermore, we have updated the arXiv preprint to v2 implementing several changes to the manuscript which address the requested changes. The replacement is scheduled to be announced at Wed, 26 Feb 2025 01:00:00 GMT.

Please also find the complete list of changes in the PDF file attached.

To begin with, I missed the motivation about why we should care about the transfer matrix spectral ratios. They are only said to be proportional to the effective DMRG Hamiltonian (although it is not very clear if this is a mathematical result or an empirical observation) and thus related to the low-energy excitations (of the original Hamiltonian?). I guess this correspondence is also the motivation to consider the TM spectrum an observable at the correlation length scale, or is there another argument for that?

The TM, is an object that has already attracted a lot of attention, since in the translationally invariant setup, is what dictates all correlations in space of the state. Here, furthermore, given that we are interested in the continuum limit of a relativistic invariant lattice model, such correlations are the ones that determine also the spectrum of the field theory in the continuum. As a result, they are the fingerprint of the field theory. Again, this is based on the emerging low-energy relativistic invariance, that given the linearity of the dispersion relation, ensures that correlations in space map to correlations in time up to a velocity factor. We guess this is exactly what the Referee has in mind in their question, and we hope that our added material at the beginning of Section 2.1 will further clarify this point.

The first part of Section 3 is quite nice. I am somewhat confused, nevertheless, by the notion of effective Hamiltonian after Eq. (14). Is it a Hamiltonian whose ground state is the finite b. d. MPS, as opposed to the actual ground state of (12).?

Yes, this is the idea. Given an MPS, it always has a parent Hamiltonian, of which it is the exact ground state. In general such Hamilton acts on several sites, but given that we are interested in the continuum limit we can study it from first principles. In other words, the effective Hamiltonian defines the effective field theory (EFT) which is only correct up to scales of the order of the correlation length and thus this length define the characteristic size of the effective system. In the case of the equation in question, the system is of finite size, and the characteristic length scale is the system size $L$.

What is the relationship between \hat{H} and \hat{H}^*?

The Hamiltonian $\hat{H}(L)$ is precisely the one describing the finite-size system while $\hat{H}^*$ describes the fixed point Hamiltonian of the renormalization group, that acts on an infinite system at the critical point, which is, by definition, the only point where scale invariance is actually realized for all scales.

I also wonder why the choice was made to use the non-symmetric tensors in this section (leading to a perturbation by sigma) when symmetric ones are used in the following section (leading to a perturbation by epsilon), although I actually don't mind it because that way we get to see both cases.

As implicit in the Referee comment, this is one of the possible choices. In particular, it is the one that gives access to the most relevant operator, and thus the one which is easier to characterize, since it deviates from the fixed point faster than any other choice. By using generic, non-preserving $\mathbb{Z}_2$ tensors, we have access to the relevant operator with the lowest scaling dimension, in this case $x_\sigma = 1/8$, as opposed to $x_\varepsilon = 1$, which would be the only accessible in a symmetric scenario. We hence require lower system sizes to observe the relevant effects of this perturbation, which is of practical advantage. Later, we also consider the explicit preservation of the $\mathbb{Z}_2$ symmetry for completeness’s sake.

Right at the beginning of Section 5, what does it mean that "this spectral ratio is beyond perturbative field theory"?

The approximative schemes of perturbation theory are always based on power series in terms of a small quantity, most often a physical coupling constant, and describe the full interaction of the many-body physics in terms of orders of interactions that are interpreted within the respective perturbative picture. These power series have some radius of convergence for which the perturbative scheme is applicable, and fail beyond that. The data of Fig 2, which is what "this spectral ratio" is referring to, is generated through iDMRG simulations on an iMPS. DMRG simulations do not necessarily access a perturbative regime, and can contain arbitrarily non-perturbative effects.

By looking at the right panel of Fig. 3 we can see that the TM spectrum for finite size system starts perturbatively deviating from its expected behaviour as the finite bond dimension effects start to become relevant, and then it settles into a completely different scale invariant spectrum, which is only dictated by the bond dimension, and it is not perturbatively connected to the original spectrum. This is why we say that the effects of finite bond dimension in an MPS, cannot be interpreted within a perturbative picture unless the bond dimension is increased polynomially with the system size.

As a result, we need to introduce a new effective lattice model explaining the finite bond dimension effects of MPS at a critical point.

In this Section [5], I understand the second argument given to set m=1 (namely that it matches the numerics that we want the model to reproduce), but the first one (33)-(35) is a bit cryptic, I do not know what assumptions are being made in order to be able to use those formulas.

A: We acknowledge the perhaps confusing way we expressed ourselves. We have expanded a bit on the explanation and the discrepancy between the correlation length and the single particle gap, which are inversely proportional by definition. However, the proportionality factor does not need be trivial, and it is this factor which makes up the entire argument for setting $m=1$ in the case of the Ising model (as opposed to $m=-1$ for example). We have included a Reference to analytical forms of the correlation functions and made the argument clearer that $\xi = |\delta|^{-1}$ in the paramagnetic case ($g>1$) and $\xi = \frac{1}{2} |\delta|^{-1}$ in the SSB phase ($g<1$).

Similarly, for the argument before 5.2. that says that the same correlation length "costs" twice the energy when produced in the direction of the paramagnet.

We have expanded the explanations further on why the DMRG simulations must introduce a finite correlation length, which is equivalent to a perturbation as per Section 3, and why this perturbation must always settle in the SSB side of the phase transition when considering the transverse-field Ising model.

Also, was it checked somewhere that the correlation length of the ground state of H(L) goes as L (i.e. that nu is 1 under Figure 8) or is it an assumption?

Figures 5 and 6 corroborate the standard critical exponent $\nu=1$ in our lattice model, i.e., the scaling behavior of the correlation length with the coupling $\delta$. There, we observe the first non-degenerate gap $\Delta_1 = 2|1-g|$ at exactly $g = 1 - 1/L$ for any $L$ and in the thermodynamic limit. This implies $\xi \sim 1/\Delta \sim \delta^{-\nu}$ with $\nu=1$, and $\delta = 1/L$ which is precisely the scaling of the perturbation $\delta$ such that the transverse field operator is rendered marginal in combination with its coupling.

Finally, in general I find the notation and the reasoning in Appendix A a bit confusing. We assume that there is a perturbation that generates a correlation length, and then we incur in an error from truncating to finite bond dimension, but if the perturbation that induces the correlation length is the finite bond dimension itself, then there should not be an error due to truncation since psi_0 is already the ground state of the perturbed system, with the finite bond dimension in place, should it?

In this appendix, we only recite the work from Refs. [15] and [16]. The main idea is to assume the existence of a finite correlation length. This by itself implies that the energy is different from the ground state energy that would require an infinite correlation length, call it $\delta E(\xi)$. On the other hand one can take the critical ground state and truncate its Schmidt spectrum to a given bond dimension. This approximation also produce an error in the ground state energy, call it $\delta E(D)$. By asking which bond dimension would induce the same error as the one induced by the finite correlation length, namely asking $\delta E(\xi)= \delta E(D)$ then one arrives to an expression of the correlation length as a function of the bond dimension.

More details can be found in the original works.

1. On page 3, the sentence "This has been the subject of several studies" is not followed by a reference.

A: We have taken out that sentence as we discuss these references in the paragraph just below.

1. Beginning of page 5, is "ratio of the gap ratio" the correct expression?

A: Certainly not, thank you for pointing out the typo.

1. For the unfamiliar reader, it could help to refresh what the scaling hypothesis is. E.g. page 6, "by virtue of the scaling hypothesis, all universal quantities only depend on the correlation length". Is it all there is to the scaling hypothesis, or is it a consequence of a more fundamental assumption?

This is indeed a concise recitation of the scaling hypothesis. We have added a clarifying sentence and a reference.

1. In Fig. 2, the non-symmetry enforcing case is called symmetry broken instead. Is this intentional? Does the Hamiltonian actually have a degeneracy?

We have changed the label of Fig 2 to consistently call the non-symmetry preserving spectrum non-conserving. Indeed, the Ising model has a two-fold degenerate ground state in the ferromagnetic phase (g < 1), which is spontaneously broken. We have added a reference when introducing the spontaneous symmetry-breaking (SSB) acronym on page 18.

1. The text after Eq (13) is a bit misleading: this equation represents the difference between the puMPS and the finite size energy, while the plot shows the difference between the puMPS value and the infinite size energy.

A: Thank you for pointing that out, we adapted the phrasing in the main text.

1. On page 13, "transitionally" should read "translationally".

A: Thanks for spotting this malicious autocorrection typo.

1. Typo in the caption of Fig 6, 1-1/xi instead of 1-1/L.

A: Thanks for spotting this typo.

1. On page 21, "the theory is confined to the region", I think I know what is meant: in the spatial region where dS blows up, fluctuations of any field are suppressed and thus effectively log xi is a boundary, i.e. an interface with the vacuum. I would verbalize it a bit more, though.

A: We added the following clarifying sentence below E. (40): Since the perturbation $\hat{\Phi}_g$ is relevant, the RG dimension $(2-\Delta_g)$ is positive. Since the contribution to the partition function is weighted by $e^{-S}$ in the path-integral, the configuration with larger $\delta S$ has a smaller contribution.

1. On page 22,"effectively halting it as the relevant operator was marginal Section 3" is missing an "if" and brackets?

A: Yes, this required some rephrasing. We have changed the wording of the sentence by adding a subclause specifying the marginal nature of the relevant operator when combining it with its scale dependent coupling

1. Right before A.11, should ket(psi) be ket(psi_0) everywhere? Also e_0 and e seem to be the same from (A.6),

A: Indeed, to both! Thank you.

1. In appendix B, the model is only KW invariant at g=1 (also the transformation would more accurately be represented as sigma_Z -> sigma_X sigma_X and vice versa). What is the value of p_c?

A: Correctly pointed out that the original Kramers--Wanier transformations do not apply here. We have corrected the duality transformation and copied the original ones from Alcaraz in Ref. [59]. Furthermore, Alcaraz tested the model within $0 \leq p \leq 1.5$ and showed that it is in the Ising universality class approximately until $p \approx 1.5$, cf. Fig 1 in Ref [59] and text page 5. We are unaware of a numerical study estimating $p_c$ more rigorously.

Attachment:

list-of-changes_Q7jkuUM.pdf

We thank the referee for their effort and taking their time to carefully read and critique our manuscript. As their critique is ultimately improving the quality of the manuscript, we are grateful for their comments. Regarding the points raised, we have the following replies:

(a) The vertical axis of Fig.3 (a) is E_0(L,x)-e_0L, and it takes a couple of time to identify the meaning of e_0, since different form is used in the main text.

We have included a more prominent definition of $\epsilon_0$ in the caption, and in the main text which discusses Figure 3 below Eq. (13).

(b) The abbreviation "SSB" is used many times. It is relatively odd that the corresponding term is symmetry-broken phase.

We have updated the first time we used the abbreviation "SSB" to refer to "spontaneous symmetry-broken" phase, on page 17.

The updates are uploaded to the arXiv and are rendered visible to the public within the next publication cycle from the time of this post. Please find attached a document with the complete list of changes, which are also in response to the second referee's critique.

Kind regards The authors.

Attachment:

list-of-changes.pdf