Thermodynamic integration, fermion sign problem, and real-space renormalization

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No change requested as I do not think that simply revising the paper will make it suitable for publication.

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This paper is inspired by a chapter in Wilson's famous Reviews of Modern Physics (1975). That paper is famous because of the numerical solution of the Kondo model. It's less known that in another chapter of the same paper Wilson sets up a numerical renormalization group analysis for 2D Ising model with 217 coupling constants. The results of that analysis are promising but poorly documented and as far I know they have never been reproduced. It's an interesting open problem to reproduce them and to see how much they can be improved with the present-day computers.

Wilson's clever algorithm had several steps. 1. Hamiltonian was encoded by a sequence of coupling constants. 2. The Hamiltonian, and the coupling constants, were transformed from original $\pm 1$ spin variables to lattice gas $0,1$ variables. 3. Working in the lattice gas variables, renormalized partition function was evaluated for many configurations of block spins. This computation was done approximately, using the fact that the system for fixed block spins is short-range correlated. 4. Using the results of step 3, renormalized couplings were evaluated, first in lattice gas variables, and then in block spin variables. As a result of approximated procedure, some couplings breaking $\mathbb{Z}_2$ invariance were generated. The size of these generated couplings was used to monitor accuracy. Steps 1-4 define the RG map. There was a separate algorithm how to look for fixed points of this RG map. As I said Wilson used 217 coupling constants.

The paper under review is kind of narrow in scope as it focuses only on one aspect of Wilson's algorithm - step 3. They propose an alternative method to evaluate the renormalized partition function, which is via Monte Carlo integration instead of Wilson's direct, approximate, summation. To do this they have to set up various technical tools such as thermodynamic integration, and overcome a sign problem appearing as a result of a non-positive RG kernel.

The paper reads as a highly technical report of their exploration of this one aspect of the problem. It appears that a lot of work still remains, if the goal is to reproduce Wilson's results in their framework. The jury is still out if their method is going to be superior to Wilson's method, for 2D Ising. They operate at a toy level of 2 couplings or 14 couplings, way below Wilson's 217 couplings. And they do not demonstrate their ability to evaluate renormalized couplings. In fact some parts of their discussion, where they discuss correlated sampling, sound to me like they don't intend to evaluate the renormalized couplings at all, which would imply another major change to Wilson's algorithm.

However, if this work does succeed in the future, it could have one clear advantage over Wilson's approach of direct approximate summation. Namely, Wilson's approach, by his own admission, appeared impossible to extend to 3D because of exponential wall in the number of configurations to sum over, even in approximate summation. But their approach, based on Monte Carlo, may be able to avoid this wall.

This paper contains therefore an interesting idea, and I would like to encourage further work in this direction, so I would like to see it eventually published. SciPost Physics Core has lower criteria of acceptance than SciPost Physics. Still, I believe that to be published, this paper has to be made more self-contained. Right now it is basically unaccessible for anyone who has not already studied Wilson's RMP section. A longer introduction needs to be included describing the main steps of Wilson's algorithm and which parts of the algorithm they are focussing on. I would like to see a more explicit discussion about whether they plan to evaluate the renormalized couplings or not. If yes, how, and what is the expected error? If not, why do they think they can get away without evaluating them?

The authors should also connect with real-space RG state of the art, by including some perspective on other real-space RG methods which came to the fore in the 50 years of work after Wilson. Notable established real-space RG methods include tensor network RG (see e.g. the review in App.C of https://arxiv.org/abs/2107.11464) and Swendsen's Monte Carlo RG. The latter uses Monte Carlo and does not evaluate the couplings but only variations. Is their method by any chance exactly what Monte Carlo RG does already. It's a natural question anyone familiar with Monte Carlo RG would ask?

There follows a list of further remarks: p.2 "Despite its conceptual appeal as adirect application of scale invariance, real-space renormalization" - this remark concerns only block-spin real-space RG, not all real-space renormalization. Indeed tensor network RG and Monte Carlo RG are both real-space RG methods which are well-established.

p.2 "the result as the same hamiltonian" => "the result as a hamiltonian of the same kind"

p.3 "interaction $K_T$" This would normally be called "coupling", "coupling constant" or "interaction strength"

p.4 "this is inconsistent." what is inconsistent? Isn't Kadanoff transformation well defined?

p.5 $\rho$ parameter makes it appearance but nothing is said about the theory saying that it should take the value $2^{1/16}$ to have an exact fixed point. Without this, the later assignments $\rho=1.04$ are incomprehensible.

p.6 Section 3.2 title "Fermion sign problem" That's confusing, since there are no fermions. Why not call it simply "sign problem", here and everywhere else?

Further remarks: The authors took a conscious decision to modify Wilson's evaluation method of the partition function, but they keep the RG kernel same as Wilson. Once they start to modify things, why not modify the RG kernel? They could have chosen a different kernel, like a majority rule, which would be positive, and so no sign problem would present itself. This needs to be commented upon. Relevant references for the work using different kernels include Kennedy https://arxiv.org/abs/0905.2601.

The authors don't stress the importance of the short-range correlations in the Hamiltonian conditioned on block spins. This short-rangedness was important for Wilson, and it also must be important for them, for a different reason--it helps avoiding critical slowdown. A good recent reference discussing this point is https://arxiv.org/abs/2504.01264, Section X.B, with references to prior work by Kennedy and Ould-Lemrabot.

Because of number of requested corrections is significant, I'm qualifying the requested revision as a major one.

Ask for major revision

The authors carry out the real-space RG program proposed by K.G. Wilson a long time ago. In this program, a Hamiltonian with a finite number of interaction terms is considered. The corresponding coupling constants are computed by the fixed-point criterion that the renormalized system and the original system are characterized by the same set of coupling constants. This condition is boiled down to the computation of the thermodynamic weights of various renormalized spin configurations. The RG map is characterized by the parameter referred to as \rho in the manuscript, which controls the proximity between the renormalized and the original spins at the same spatial position. They took the two-dimensional Ising model as an example, and considered the simplest case with only two coupling terms (the nearest neighbor pair and the next-nearest neighbor pair interactions). When the value for the parameter \rho suggested in Wilson's original paper was used, they found that the optimal values of the two coupling constants turned out to be close to the ones obtained in Wilson's paper.

Though I read through the manuscript with a lot of interest, I had to ask myself about the value of the method viewed from the present standard. I do not agree with the authors statement in the introduction, "Despite its conceptual appeal as a direct application of scale invariance, real-space renormalization has mainly been presented in a pedagogical context [3], with only a few quantitative applications [4-9]." In fact, there are a large and still increasing number of publications on the real-space renormalization based on tensor networks, many of which produce estimates of various quantities with the accuracy comparable to or even higher than any other numerical methods. In this regard, the authors need stronger justification for publication of the present manuscript, which, as the authors write, only aims at the proof of concept.

As for the technical aspect, I wonder why they did not estimate the value of \rho from the same fixed-point condition instead of simply taking the value from the old paper, though this is of a secondary importance.

Reject