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Thermal pure matrix product state in two dimensions: tracking thermal equilibrium from paramagnet down to the Kitaev honeycomb spin liquid state

by Matthias Gohlke, Atsushi Iwaki, Chisa Hotta

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Submission summary

Authors (as registered SciPost users): Matthias Gohlke
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-11-10
Date submitted: 2023-10-20 06:33
Submitted by: Gohlke, Matthias
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Computational
Approach: Computational


We present the first successful application of the matrix product state (MPS) representing a thermal quantum pure state (TPQ) in equilibrium in two spatial dimensions over almost the entire temperature range. We use the Kitaev honeycomb model as a prominent example hosting a quantum spin liquid (QSL) ground state to target the two specific-heat peaks previously solved nearly exactly using the free Majorana fermionic description. Starting from the high-temperature random state, our TPQ-MPS framework on a cylinder precisely reproduces these peaks, showing that the quantum many-body description based on spins can still capture the emergent itinerant Majorana fermions in a ${\mathbb Z}_2$ gauge field. The truncation process efficiently discards the high-energy states, eventually reaching the long-range entangled topological state approaching the exact ground state for a given finite size cluster. An advantage of TPQ-MPS over exact diagonalization or purification-based methods is its lowered numerical cost coming from a reduced effective Hilbert space even at finite temperature.

Author comments upon resubmission

Dear Editor, dear Referees,

We like to thank you for carefully reading our draft and providing useful comments on how to improve it.
Both referees judge our work to be significant and original. We have carefully read their critique and provide point-by-point discussion within separate replies to each report. We have made an effort to improve the presentation of our result based on the issues raised by the referees.

Best regards,
Matthias Gohlke
Atsushi Iwaki
Chisa Hotta

List of changes

We have modified our draft according to the points raised by the referees.

We first list the changes in correspondence to the first report:
- in Fig.1(a) caption, we have added a brief explanation about the puritication approach.
- added XTRG data to Fig.~2(b) and extended the comparison with TPQ-MPS in the paragraph starting from ``A recent XTRG calculation ... " of Section 3. For better comparison with the XTRG data, we are including an inset to Fig. 2(b) showing the specific heat vs. T in a linear scale.
- added a brief footnote 3 in that using $N_{aux}$ spin is equivalent to a single site with Hilbert space dimension $d^{N_{aux}}$ and the it is mostly for the benefit of implementation and intuition.
- moved the footnote on page 5 to the last part of the Appendix A, where we have added Fig.4(b) and an explanation regarding the preparation of the initial random state.
- in Appendix A, we have added the second paragraph containing an explanation about the differences and similarities of TPQ-MPS and METTS.

In the following we list the changes in correspondence to the second report:
- revised the first and the last sentence of the abstract to summarize limitations and advantages of our work.
- modified the last paragraph of Sec. 4 providing an explanation on why we expect TPQ-MPS to be fine for non-frustrated systems in contrast to TPQ. We have also added two related remarks in the conclusion section.
- added references for TPQ on frustrated magnetism:
[Phys. Rev. B 98, 140405 (2018), Phys. Rev. Lett. 121, 220601 (2018), Phys. Rev. B 93, 174425 (2016), Phys. Rev. B 100, 045117 (2019), Phys. Rev. B 105, 144427 (2022), Phys. Rev. B 107, 245115 (2023)]
- fixed the mentioned typos, added the definitions of some inline equations by words, and replaced the part "from the front door" by a small explanation similar to the reply to the second referee.

Moreover, we have added TPQ-MPS data for both geometries at a larger bond dimension, $\chi=1024$, to Fig. 2, 3, and 4 further strengthening our interpretation.

Published as SciPost Phys. 15, 206 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-11-3 (Invited Report)


The authors have addressed all the points raised by the referees in their revised version and I can thus recommend this manuscript for publication in SciPost Physics.

  • validity: good
  • significance: high
  • originality: high
  • clarity: high
  • formatting: good
  • grammar: good

Anonymous Report 1 on 2023-10-30 (Invited Report)


The Authors have addressed all points raised by the referees, have improved the presentation, and made some of their claims stronger. I am happy to recommend the article for publication in this form.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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