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Boosting Determinant Quantum Monte Carlo with Submatrix Updates: Unveiling the Phase Diagram of the 3D Hubbard Model
by Fanjie Sun and Xiao Yan Xu
Submission summary
| Authors (as registered SciPost users): | Fanjie Sun |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202409_00036v1 (pdf) |
| Date submitted: | 2024-09-30 11:25 |
| Submitted by: | Sun, Fanjie |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
Abstract
The study of strongly correlated fermionic systems, crucial for understanding condensed matter physics, has been significantly advanced by numerical computational methods. Among these, the Determinant Quantum Monte Carlo (DQMC) method stands out for its ability to provide exact numerical solutions. However, the computational complexity of DQMC, particularly in dealing with large system sizes and the notorious sign problem, limits its applicability. We introduce an innovative approach to enhance DQMC efficiency through the implementation of submatrix updates. Building upon the foundational work of conventional fast updates and delay updates, our method leverages a generalized submatrix update algorithm to address challenges in simulating strongly correlated fermionic systems with both onsite and extended interactions at both finite and zero temperatures. We demonstrate the method's superiority by comparing it with previous update methods in terms of computational complexity and efficiency. Specifically, our submatrix update method significantly reduces the computational overhead, enabling the simulation of system sizes up to 8,000 sites without pushing hard. This advancement allows for a more accurate determination of the finite temperature phase diagram of the 3D Hubbard model at half-filling. Our findings not only shed light on the phase transitions within these complex systems but also pave the way for more effective simulations of strongly correlated electrons, potentially guiding experimental efforts in cold atom simulations of the 3D Hubbard model.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report
The manuscript “Boosting Determinant Quantum Monte Carlo with Submatrix Updates: Unveiling the Phase Diagram of the 3D Hubbard Model” presents recent improvements to the Determinant Quantum Monte Carlo (DQMC) method. These improvements make it possible to simulate larger systems. In terms of physics results, the manuscript presents data for the half-filled 3D Hubbard model (when the sign problem is absent). The phase diagram of the half-filled Hubbard model has been extensively studied and the present paper does not shed new light on it. The main contribution is to simulate larger systems, enabling the authors to give a more accurate determination of the Néel temperature.
The core of the paper is fairly technical, aimed at specialists who want to know the details of these recent improvements. The article deals with three different update schemes: fast updates, delayed updates and submatrix updates. One of the main points of the manuscript is the implementation of submatrix updates
Although the article is very technical, providing such technical details is important for any researcher who wants to write their own DQMC code. Furthermore, understanding the physics of the Hubbard model is an important challenge in condensed-matter physics. It has become very clear in recent times that such an understanding goes hand in hand with technical developments / improvements in numerical techniques. That's why I am recommending publication. But before fully accepting the manuscript, I would like the following points to be addressed:
-The authors call their “innovative approach” a “generalized submatrix updating algorithm” because it can handle extended interactions and zero temperature. However, for the standard finite-temperature Hubbard model, which is the main model studied here, the introduction does not clearly explain, in my opinion, what has been done before and what is new. What's the difference with previous submatrix updating schemes for the finite-T half-filled Hubbard model (that couldn't go up to such large lattice sizes)? This is not clear from the current version of the manuscript.
-On page 3, the authors write “As this additional overhead is implemented with Level 1 BLAS, its time cost can surpass the time cost for the update of Green’s function in practical calculation if one increase nd, as shown in Fig. 4.” But for the values of $n_d$ shown in Fig. 4, this doesn't seem to be the case. That's why I found this sentence somewhat confusing.
- It seems that the lattice sizes that were reached prior to this work were large enough to obtain a good estimate of the Néel temperature (using finite size scaling by crossing analysis). It is mentioned that previously one could go up to $N=1000$, or $L=10$. It's not very clear from the figures that going significantly beyond L=10 implies a significant improvement in the calculation of T_c. Could the authors indicate how much moving to larger lattice sites has decreased the error bar on $T_c$, which is what matters in the end. A related question: are there other quantities for which the effect of finite size is more important?
-The paper focuses on a static property of the Hubbard model, in order to determine the Néel temperature. What about dynamical properties of the system? Is it expected that one could also go up to $L=20$?
Recommendation
Ask for minor revision