Quantum Monte Carlo Simulation of the 3D Ising Transition on the Fuzzy Sphere

Requested changes:

1. We thank the reviewer for this suggestion and have added a discussion of the N^4 scaling under equation 16 in the QMC simulations section of Results.

2. At the Ising critical point, the finite-size effect can be significantly reduced by tuning away the irrelevant operator epsilon’ (with scaling dimension ~3.83). In the present work, we tune the interaction parameters to reach this goal, i.e. g1=0.4 and g0=1.3 from the exact diagonalization. By changing these numbers to the conventions of QMC parameters, it gives the repeating decimals V0=0.61818181... and V1=0.1111111... We then scaled them so that V1=0.1, giving the repeating decimal V0=0.55636363... In principle, the Ising transition and its typical critical exponents should not depend on the specific parameters in the model setup in the UV limit. But the RG flow distances to the IR fixed point are distinct, if choosing the different parameters in the UV limit. So when we tune away from the "sweet point" as shown above, the finite-size effect will become significant and one needs much heavier computation on larger system sizes to reach the critical phenomena.

We have attached an updated version of the manuscript.

Attachment:

RevisedFuzzySphereQMC.pdf

We would like to thank the referee for taking the time to read our manuscript and for providing a report that helped us to improve this paper. We are happy to hear that the referee finds our work interesting and for rating the originality of this study as 'High'.

The target of the present work is to introduce a new method, namely the fuzzy sphere QMC, and the feasibility of this approach is demonstrated via the benchmark on the Ising model. We believe this approach will advance the coming studies on more challenging problems, which we leave for the future work. We do think, as a paper introducing new techniques, it should not influence the quality of the current paper.

First, we agree that high-precision results for critical exponents have already been obtained for the Ising universality class by various methods such as the conformal bootstrap, and QMC for spin lattice models. Instead of competing with these already very efficient methods for the Ising universality class, the fuzzy sphere setup allows us to examine the proposed emergent conformal symmetry at the critical point, and further make use of the conformal symmetry to compute universal properties such as critical exponents.

Second, in a previous study introducing the fuzzy sphere setup, we have used both exact diagonalization and DMRG to identify the Ising critical point and analyze the Hamiltonian spectrum at criticality to verify the emergent conformal symmetry. However, both methods struggle with increasing fermion degrees of freedom, due to the exponential compute cost. Here, we introduce a polynomial algorithm to overcome these limitations and benchmark the new method using the Ising criticality.

Finally, the polynomial scaling allows us to study models with larger local Hilbert spaces and the plethora of interesting quantum critical phenomena therein. We agree with the referee that it would be rewarding to study critical Dirac fermions, for example. However, this is left for future work.

Hence, we strongly believe that the presented research is already an important result and meets the acceptance criteria of SciPost. A detailed response to the list of requested changes can be found below. With these changes we believe the manuscript is ready for publication.

Requested changes:

1. We agree that it is useful to clarify the projection. As given in the manuscript, we have H = H_{kin} + H_{int}, with H_{kin} giving rise to Landau levels and H_{int} the Hamiltonian of physical interest. The level spacing of the Landau levels is controlled by the cyclotron frequency $\omega_c$, and we consider a half-filled lowest Landau level (LLL). In general, the interaction scale could induce excitations to higher Landau levels, however, this can be avoided by taking the limit $\omega_c \rightarrow \infty$. Hence, the projection of H_int, encoding the physics of interest, to the LLL is not posing any constraint. We have added a couple of sentences above equation (3) to clarify this.

2.,3. To reiterate the argument from above, the purpose of this work is not to produce high-precision critical exponents for the Ising universality class. Instead, we are introducing a sign-free QMC algorithm with a polynomial computational effort. While in principle, one can produce relatively precise exponents, we would like to highlight that this setup also enables us to analyze the existence of an emergent conformal symmetry at criticality, especially for models with larger local Hilbert spaces. However, applying the new algorithm to such models to study, e.g., critical gauge theories, relevant for deconfined quantum critical points, is left for future studies. Here, we have shown that the algorithm actually works by benchmarking it using the Ising model. To address the referee's comments, we have added a discussion of the numerical complexity below Eq. (16), and discuss future directions in the conclusion.

4.,5.,6. We have added the definition of the chi^2 to the paper, see new Eq. (20). While there are seven parameters, we also have thirty-two data points in our fit, so overfitting should not be a big concern. Note that we are not trying to extract high-precision critical exponents from the scaling collapse, which are known to be prone to finite size scaling corrections, but we are rather giving evidence that the critical point of the fuzzy sphere model belongs to the Ising universality class. We believe the current system sizes are sufficiently large for this statement as shown by the quality of the scaling collapse in Fig. 1(c).

7., We thank the referee for this interesting comment and agree that this would be an interesting future direction. However, it is still an open nontrivial question how to properly describe Dirac fermions on a fuzzy sphere. A more directly accessible follow-up is to study critical gauge theories, as already mentioned in the conclusion. However, we believe that this would be a sufficiently different and new result to be left for a future work.

We have attached a revised version of the manuscript for the reviewer to examine.

Attachment:

RevisedFuzzySphereQMC_BUzhntM.pdf

We thank the referee for taking the time to read our manuscript and provide a useful report. As we detail below, we have addressed the referee's comments appropriately and feel confident that the modified manuscript can now be published by SciPost.

1. We have added a discussion of the numerical limitation, i.e., the scaling of compute time with system size, below Eq. [16]. Due to the rank of the operators that couple to the auxiliary fields, the computational effort scales as N^4 instead of the usual N^3, which can be achieved for conventional lattice models. However, we stress that this method still scales polynomially with size, in contrast to exact diagonalization and DMRG. Hence, the QMC remains the method of choice when many fermionic degrees of freedom are studied, e.g., for studying critical gauge theories with multiple fermion flavors.

2. We have added a more detailed discussion of the guiding design principle for the additional flavor in the last paragraph of Sec. 2.3. It might indeed be interesting to study how generically one can use this trick to study universal aspects of critical phenomena. However, the purpose of this paper is not to show that introducing an additional flavor can always provide a sign-free model to study any universality class of interest. Instead, we introduce the possibility of using the QMC method on the fuzzy sphere, which has never been done before in a way that exploits the unique features of it. For example, the state-operator correspondence on the fuzzy sphere gives rise to unambiguous fingerprints of conformal symmetry in the spectrum (Fig. 3). Note that this cannot be achieved on a tight-binding Hamiltonian on a 2D lattice with periodic boundary conditions, nor by conformal bootstrap as the conformal symmetry that assumes conformal symmetry from the beginning. Similarly, the conformal symmetry highly constrains the functional form of (equal-time) correlation functions which allow a fit-free extraction of critical exponents/operator dimensions (Fig. 2c) We agree with the referee that this paper mostly focuses on the Ising universality class to benchmark the method. In response to their comments, we have also added a discussion regarding the "generality of the method" to the conclusion. For the Ising model, we needed an additional flavor degree of freedom to avoid the sign problem. However, many interesting universality classes can be studied using sign-free model Hamiltonians without the need to introduce artificial flavor degrees of freedom! Examples are the critical gauge theories that have been proposed to describe deconfined quantum critical points.

Attachment:

RevisedFuzzySphereQMC_hQfbJu0.pdf