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Simulating thermal density operators with cluster expansions and tensor networks
by Bram Vanhecke,David Devoogdt, Frank Verstraete, Laurens Vanderstraeten
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Submission summary
| Authors (as registered SciPost users): | Laurens Vanderstraeten · Bram Vanhecke |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202210_00027v1 (pdf) |
| Date submitted: | 2022-10-03 17:19 |
| Submitted by: | Vanhecke, Bram |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
Abstract
We provide an efficient approximation for the exponential of a local operator in quantum spin systems using tensor-network representations of a cluster expansion. We benchmark this cluster ten- sor network operator (cluster TNO) for one-dimensional systems, and show that the approximation works well for large real- or imaginary-time steps. We use this formalism for representing the ther- mal density operator of a two-dimensional quantum spin system at a certain temperature as a single cluster TNO, which we can then contract by standard contraction methods for two-dimensional ten- sor networks. We apply this approach to the thermal phase transition of the transverse-field Ising model on the square lattice, and we find through a scaling analysis that the cluster-TNO approx- imation gives rise to a continuous phase transition in the correct universality class; by increasing the order of the cluster expansion we find good values of the critical point up to surprisingly low temperatures.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 3) on 2022-10-11 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202210_00027v1, delivered 2022-10-11, doi: 10.21468/SciPost.Report.5868
Strengths
The manuscript proposes a highly original approach to thermal states of quantum Hamiltonians. In some applications it may surpass other methods based on tensor networks.
Weaknesses
What I miss in the manuscript is a more quantitative discussion how the numerical cost of performing the algorithm depends on the cluster size or the correlation length (or whatever else is more relevant). Could the main bottleneck be identified? It would help a reader to figure out where this is the method of choice and where it is not.
To be more specific, for g=2.5 the critical temperature is in agreement with a PEPS [25] and a quantum Monte Carlo [23] but references [23,25,27] also consider g=2.9 where the critical temperature is much lower but the PEPS still agrees with the QMC. What precisely prevents comparison also in this case? The same question could also be applied to the bottom panel in Fig. 7 where g_c is close g=2.9 and something prevents the same accuracy as in the top panel. What is it?
Report
This certainly is a publishable work but, given its novelty, I would like the authors to elaborate more on what sets the limits to the proposed method.