Extracting quantum-critical properties from directly evaluated enhanced perturbative continuous unitary transformations

At a quantum critical point, there is a gap that closes as |g-g_c|^z\mu as the tuning parameter g approaches critical value g_c. In this paper, the authors show that this exponent can be computed using a variant of CUT, referred to as deepCUT, in some two dimensional model magnets. As examples, they consider the transverse field Ising model (TFIM) on a square lattice (appendix C) and the triangular lattice for both signs of the exchange to access both 3D XY and 3D Ising universality. With these benchmarks they study the criticality in the bilayer triangular lattice TFIM with intra- and inter-layer exchange and layer dependent field. They provide evidence that the critical line between the clock state and the polarized phase is a 3D XY line. A key aspect to this paper is the method itself that can be adapted to other models. Underlying the successful extraction of exponents is a variant of finite size scaling where the perturbative expansion order plays the role of system size. An iterative scheme is devised that gives access to both the critical coupling and critical exponent. While most results are for the gap, an appendix describes attempts to extract exponent alpha.

This paper provides a technique to extract long distance physics from a truncated perturbative expansion. The results from the gap agree with known results in 2D quantum systems and the method is shown to be useful for a new case. I can imagine that this will find use in other instances in a field where few techniques are available. I recommend publication in Scipost after a few minor changes.

Optional Changes:

1. The review section is well referenced. But I found the description of the method in the paper itself not very pedagogical (both the main text and appendix B). One point that I found particularly opaque is central: how exactly the gap and its derivative $$\partial_J \Delta$$ are computed. As a suggestion for the authors: I can imagine other readers would appreciate some more discussion especially of this last point.

2. For the data in Fig. 2 I recommend including some measure of the goodness of fit to the computed points i.e. what are the errors on the exponent from this source? Errors are discussed at some points in the paper: for example coming from the choice of fitting window and in Fig. 3 and again in the lower panel of Fig. 5. Suggestion: could the authors bring together these contributions into a consolidated, quantitative discussion maybe at the end of the paper connecting to the remarks already in the discussion?

3. Fig. 5: the dashed blue line mentioned in the caption is not visible. If it overlaps with one of the other curves maybe change the plot style.

4. Fig. 8: I suggest including the known alpha exponents for 3D XY and Ising in the caption to this figure (or at least refer to table 1).

5. General questions: Is it possible to capture a first order phase transition using this method? What about a higher order continuous transition like the KT transition?

Publish (easily meets expectations and criteria for this Journal; among top 50%)

This manuscript applies the deepCUT method to determine quantum critical properties of transverse field Ising models in two spatial dimensions. Specifically, results are obtained for critical exponents and critical couplings for the ferromagnetic and antiferromegnetic transverse field Ising models on the single layer triangular lattice as well as for the bilayer antiferromagnetic one. A new iterative scheme is presented which allows the authors to simultaneously determine the critical couplings and the critical exponent nu. The single layer system is used as a benchmark to compare the obtained results with literature values, yielding good agreement. For the bilayer system the phase boundary between the polarized phase and the ordered phase is calculated in an intermediate region of coupling parameters, which has not been studied in previous works.

Overall, the manuscript represents a methodological advance in the detection of quantum critical points in two-dimensional spin models. The results are not only of relevance for the specific models investigated here but also open up the possibility of further applications. Furthermore, the numerical analysis seems to be thorough and solid as far as I can tell. Therefore, I recommend publication of this manuscript in SciPost.

I have a few points which the authors may want to address:

(1) The very technical discussion on the iterative procedure in the paragraph on page 11 starting with "One of the main sources of uncertainty..." was hard to follow for me. Since it seems that this iterative scheme is one of the main achievements in this work, I would like to motivate the authors to formulate it more clearly. The authors say that they start with nu=1 in their iterative scheme. How much do the results depend on this initial choice?

(2) The caption of Fig. 5 mentions a dashed blue line. However I cannot see such a line in the figure.

(3) On page 14 the authors say that their results in the decoupled bilayer limit do not agree with the single layer model. This sounds very strange since one would expect that the system is well behaved in the limit of small interlayer couplings. Can the authors comment on this in more detail or even eliminate this strange methodological artifact?

(4) The blue line from perturbation theory deviates from the deepCUT results in Fig. 5 (top). More discussion about the reasons would be helpful. Is it because of the low order perturbation theory of the blue line or due to errors in deepCUT?

Ask for minor revision

Report on "Extracting quantum-critical properties from directly evaluated enhanced perturbative continuous unitary transformations"

- Study using deepCUTs in order to extract critical points and critical exponents.
- Idea: Use deepCUT to obtain low energy model including single quasi-particle excitations and analyze parameter dependence of gap to find the QCP.
- Intuition: order of deepCUT corresponds to length-scale that is captured in the effective model -> connection to correlation length and thereby to critical exponents
- Obtain critical exponents as fits against order of the calculation
- Advantage of deepCUT: No need to have equidistant model in $H_0$
- Disadvantage: need to numerically evaluate flow equation for each value of expansion parameters
- Models: triangular Ising model with FM or AFM couplings, later bilayer model
- Results:
- very good fits for FM model
- slightly worse results for AFM model: possibly due to even/odd effect and larger critical$J_c$ due to frustration
- Proposal of iterative scheme: determine $J_c$ from first guess of gap closure, compute nu, get better result for$ J_c$ ...
- Works for single layer model in both cases: AFM and FM
- Check whether order n corresponds to xi by investigation of gap derivatives
- Application to frustrated Bilayer:
- Tune trough phase diagram for $J_\perp$ and symmetric h as function of $J_\parallel$
- Phase transition from polarized to clock-ordered phase with emergent O(2) symmetry in the 3d XY universality class.
- Determination of critical exponents trough the iterative scheme, consistent with 3d XY transition

Overall: Very solid numerical study and interesting proposal for an iterative scheme to determine quantum critical properties with CUT.

Some minor remarks that the authors may optionally address:

Abstract: "The data coincides with the perturbative series up to the order with respect to which the deepCUT is truncated." This sentence is a bit unclear. I propose "The data coincide with numerical evaluations of the truncated perturbative series and provides robust extrapolations beyond the perturbative regime."

Page 3: It makes sense to also cite Phys. Rev. B 87, 184406 (2013) here, as it deals with the 1D TFIM with deepCUTs. The correspondence between truncation order to the correlation length was also discussed in my thesis: https://cmt.physik.tu-dortmund.de/storages/cmt-physik/r/uhrig/master/master_Benedikt_Fauseweh_2012.pdf

Page 10: The ROD is converged to 10^-9, why not double precision? If 10^-9 is reached it should already converge exponentially. Btw. There is a nice connection between speed of convergence and order of the calculation at the QCP, see also my thesis.

Page 11: I like the idea of the iterative scheme very much. So far, the scheme works with J values already calculated from the flow equation. Can this be generalized by including the CUT in the scheme by computing the solution of the flow equation for a given estimate for J_c and then obtain nu and then iterate? This avoids the need to discretize J with 10^-3 close to J_c. At most an additional point is required to compute the finite difference for the derivative.

Page 12: It seems like the fit window is crucial for critical point and for the exponent AND they differ! This seems to me to be the most fine-tuned part of the method. Are there any ideas how to approach this problem systematically?

Fig 5: O2-7 and O2-5 seem to be more consistent but O3-7 not. Is this an even odd effect?

General: So far there are no error bars on the exponents. But from the fits, e.g. Fig 2 a) I would think the fitting algorithm gives you an estimate on the error. Is this smaller than 10^-3, so e.g. nu=0.687(..) ? Same question for J_c.

Page 16: "In particular, there is no sign of a sudden change of the nature of the transition which could have pointed towards a first-order transition otherwise." Would a perturbative CUT with a 1:n generator be able to capture a first order phase transition anyway?

Page 16: "Importantly, one would also like to be capable of quantifying the errors on the critical point and the critical exponents better." Yes, I think that is the most important point. It is discussed a bit in the conclusion, but I would think that at least the effect of different fit windows could be discussed already quantitatively with the data available, without the need to carry out any new CUT calculations. Not necessarily in the main text but as an appendix.

General: The y axis on various graphs read like "sec. der. GS. partial_j E_0 at J_c". I think the mathematical correct way of writing this would be much more reader friendly: $\partial_j E_0(J)|_{J={J_c}}$

Publish (meets expectations and criteria for this Journal)