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Dual applications of Chebyshev polynomials method: Efficiently finding thousands of central eigenvalues for many-spin systems
by Haoyu Guan, Wenxian Zhang
This is not the current version.
|As Contributors:||Haoyu Guan|
|Date submitted:||2021-06-28 14:20|
|Submitted by:||Guan, Haoyu|
|Submitted to:||SciPost Physics|
Computation of a large group of interior eigenvalues at the middle spectrum is an important problem for quantum many-body systems, where the level statistics provide characteristic signatures of quantum chaos. We propose an exact numerical method, dual applications of Chebyshev polynomials (DACP), to simultaneously find thousands of central eigenvalues, which are exponentially close to each other in terms of the system size. To cope with the near-degenerate problem, we employ twice the Chebyshev polynomials, to construct an exponential semicircle filter as a preconditioning step and to generate a large set of proper basis states in the desired subspace. Numerical experiments on Ising spin chain and spin glass shards confirm the correctness and efficiency of DACP. As numerical results demonstrate, DACP is $30$ times faster than the state-of-the-art shift-invert method for the Ising spin chain while $8$ times faster for the spin glass shards. In contrast to the shift-invert method, the computation time of DACP is only weakly influenced by the required number of eigenvalues, which renders it a powerful tool for large scale eigenvalues computations. Moreover, the consumed memory also remains a small constant (5.6 GB) for spin-$1/2$ systems consisting of up to $20$ spins, making it desirable for parallel computing.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 2021-7-27 (Invited Report)
- interesting topic
- well written & easy to understand
- design choices for the algorithm not clearly motivated
- no mathematical error analysis
The authors propose a new numerical method to calculate interior
eigenvalues of large sparse matrices, in particular eigenvalues of the
Hamiltonians of two disordered interacting spin systems. This problem
is relevant, for instance, to study the level-spacing distribution of
the eigenvalues which characterizes quantum chaos and integrability.
The method is based on a two-fold application of Chebyshev expansion:
1) as a spectral filter, and 2) to construct a basis of a restricted
Hilbert space. Within this space eigenvalues are calculated using standard libraries.
Overall, the approach is plausible and seems to outperform other
methods. However, I would prefer a more detailed explanation of the
chosen design. Chebyshev expansion can be used for many other types of
filters. Why did the authors pick the exp-semicircle filter? What is
the particular advantage of this filter?
Figure 2 shows errors of the computed eigenvalues and attributes the
particular shape of the error curve to the shape of the filter. Is
there any theory for this observation, or mathematical error
estimates? The authors also speculate that densely clustered
eigenvalues reduce precision and cause the spikes in Figure 2(a).
Is there any theory to describe this observation, or ideas to improve
The study of level-spacing statistics is the main motivation for
calculating interior eigenvalues. The authors should provide examples
of level-spacing distributions obtained with their method. Is the
number of $10^3$ to $10^4$ eigenvalues calculated in one run sufficient
for good statistics, or are multiple runs required (e.g., with
different window position or different disorder sample)?
On the whole, the manuscript is well written with only few typos
(e.g. 'midlle' on page 4). If the above remarks are taken into account
it could be suitable for publication.
Concerning supplemental material: I was able to compile the C code
provided by the authors, and it seemed to work. However, if readers
are to benefit from the code it needs more comments, better structure
and some simplification. Today such algorithmic examples can be
presented very well with interactive Jupyter notebooks written in some
free language like Python or Julia.
See report above. In short:
- Explain the motivation for the exp-semicircle filter. What is the particular advantage of this filter?
- Provide error estimates and try to explain the error data in Figure 2 with some math.
- Show some data for level-spacing statistics calculated with DACP.
- Fix typos.
- If the C source code is to be attached to the paper, please add more comments and simplify. Maybe a Jupyter Notebook with descriptions and simple Julia or Python code is more instructive for the readers.
Anonymous Report 1 on 2021-7-21 (Invited Report)
- The three-step method is simple and seems effective (runtime and memory)
- A particularly nice part is the use of an exponential semicircle filter, which is probably optimal computationally (though this is not proven)
- It is claimed that the creation of the non-orthogonal basis T_k(H)|psi_E> is more stable than the Lanczos scheme
- There is insufficient context given in the intro. The field of filtered interior eigensolvers is quite large. A comparison to other approaches like FEAST, EVSL or FILTLAN would be necessary.
- What advantage does the proposed method bring over EVSL in particular?
- It is unclear to me how a potentially ill-conditioned overlap matrix S is handled.
This work could potentially meet the acceptance criteria ("breakthrough computational method"). This essentially hinges on whether the DACP method has some tangible advantages over more established methods.
Unfortunately this has not been shown convincingly in the paper. Only a comparison to shift-and-invert methods are mentioned. However there is a vast body of work on filtered interior eigensolvers in the literature. Two examples are FEAST (with a different approach) and EVSL (with a rather similar approach). I would recommend focusing the intro on a comparison to these methods, giving actual figures for the computational advantages of DACP (runtime, memory and/or stability).
My guess is that the strongest feature of the DACP approach compared to others is the filtering step, which is probably better (optimal?) compared to rational and polynomial filters, although this is not demonstrated. The second step (building the basis) is claimed to be better than Lanczos, but at first sight you pay the price of a potentially ill-conditioned overlap matrix S. How is this problem remedied? How precisely does an iteration with Chebyshev polynomials instead of H^k help in building the basis?
- Devote a full section to comparing the DACP method to (a) simple Kernel Polynomial Method (eigenvalues from DOS), (b) EVSL and (c) FEAST approaches. Probably (a) is quite bad, but it is the most well known technique. No need for a detailed dissection/discussion of these methods, just runtime comparisons for the same problem (which will be undoubtedly in the interest of both the authors and the readers)
- Discuss in a little more detail why the basis construction with T_k(H) is better than Lanczos. In particular, add to Appendix D a plot of the final eigenvalue errors and runtime using T_k(H) and H^k with and without reorthogonalization.