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Simulating thermal density operators with cluster expansions and tensor networks
by Bram Vanhecke,David Devoogdt, Frank Verstraete, Laurens Vanderstraeten
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Laurens Vanderstraeten · Bram Vanhecke |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202210_00027v2 (pdf) |
| Date accepted: | 2023-02-13 |
| Date submitted: | 2022-10-14 15:24 |
| Submitted by: | Vanhecke, Bram |
| Submitted to: | SciPost Physics |
| 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.
Author comments upon resubmission
We have tried to add in all the missing details one might need to decide if this is the appropriate method for the case one might be interested in. This includes a mention of the numerical complexity in the case of 2D, as well as a more in depth comparison to the alternative tensor network approach. We also clarified that the method works wonderfully for beta/t below or around unity, yet falters above that.
List of changes
added a subsection 'discussion' the seciton 'benchmarks'
elaborated on the numerical complexity just above eq. 50
added comparison with PEPS method under eq. 54
included possibilities for low temperature above 'outlook'
Published as SciPost Phys. 14, 085 (2023)
Reports on this Submission
Report #2 by Didier Poilblanc (Referee 3) on 2022-12-22 (Invited Report)
Strengths
This work introduces a powerful tensor network method to deal with real/imaginary time evolution, a very timely topic.
Weaknesses
I did not find any real weakness.
Report
The field of out-of-equilibrium dynamics/thermodynamics is evolving fast. This work introduces a very efficient method to compute time evolution using tensor network, overcoming some of the limitations of previous work. I find it very promising for future applications besides the usual 1D/2D traverse fields Ising models.
Requested changes
Suggestion: may the other write down the 1D models displayed in Eqs. (40), (41) (46) explicitly as 1D model, using sum_i instead of \sum_<ij> since the whole section is devoted to 1D and besides, the 2D version of the models are introduced again in a subsequent section IV.
Author: Bram Vanhecke on 2023-01-17 [id 3244]
(in reply to Report 2 by Didier Poilblanc on 2022-12-22)We thank Dr. Poilblanc for reviewing our paper.
We will make the desired changes to the Hamiltonian definitions.
We also agree that the old definitions could benefit from a clearer form.
Attachment:
ClusterPepo_1.pdf