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Finite-temperature critical behavior of long-range quantum Ising models

by E. Gonzalez-Lazo, M. Heyl, M. Dalmonte, A. Angelone

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Submission summary

Authors (as registered SciPost users): Marcello Dalmonte · Eduardo Gonzalez Lazo
Submission information
Preprint Link:  (pdf)
Date accepted: 2021-09-29
Date submitted: 2021-08-11 14:22
Submitted by: Gonzalez Lazo, Eduardo
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Computational
  • Statistical and Soft Matter Physics
Approach: Computational


We study the phase diagram and critical properties of quantum Ising chains with long-range ferromagnetic interactions decaying in a power-law fashion with exponent $\alpha$, in regimes of direct interest for current trapped ion experiments. Using large-scale path integral Monte Carlo simulations, we investigate both the ground-state and the nonzero-temperature regimes. We identify the phase boundary of the ferromagnetic phase and obtain accurate estimates for the ferromagnetic-paramagnetic transition temperatures. We further determine the critical exponents of the respective transitions. Our results are in agreement with existing predictions for interaction exponents $\alpha > 1$ up to small deviations in some critical exponents. We also address the elusive regime $\alpha < 1$, where we find that the universality class of both the ground-state and nonzero-temperature transition is consistent with the mean-field limit at $\alpha = 0$. Our work not only contributes to the understanding of the equilibrium properties of long-range interacting quantum Ising models, but can also be important for addressing fundamental dynamical aspects, such as issues concerning the open question of thermalization in such models.

List of changes

1. Acronyms for paramagnetic (PM) and ferromagnetic (FM) were introduced.

2. A citation to [J. Phys. A: Math. Theor. 53 013001]( have been added at line 55.

3. The paragraph
>"Similarly, $p$-wave superconductors with long-range pairing[] have been shown to display exotic critical behavior, even if, due to the presence of Jordan-Wigner strings, those models do in general differ from Ising chains with similarly decaying interactions"
have been added at line 63.

4. We have changed
>"While this picture has been essentially confirmed for the case \yellow{$\alpha = 3$}"
>"While this picture has been essentially confirmed for the case \yellow{$\alpha = 2$}"
at line 72.

5. We have changed
>" be of the mean-field type"
>"belong to the Gaussian UC"
at line 135.

6. References to the exponent $\sigma$ were changed to the corresponding $\alpha$ depending value in lines 205 and 265.

7. We have changed
>"errorbar(s) "
>"error bar(s)"

8. We have changed
>"while the susceptibility one displays deviations at most up to the order"
>"while the susceptibility exponent displays deviations at most up to the order"
at line 368.

9. The paragraph
>"For future works, it would be interesting to verify if some of these findings also apply to long-range $p$-wave superconductors
[], that, while described by free theories, could still display some of the phenomenology we discuss."
have been added at line 371.

10. The style of Figure 1 has been changed.

11. A citation to [Phys. Rev. B 96, 104436]( have been added at line 379.

12. A citation to [Phys. Rev. B 95, 024302]( have been added at line 381.

Published as SciPost Phys. 11, 076 (2021)

Reports on this Submission

Anonymous Report 2 on 2021-9-17 (Invited Report)


I have read the revised version of the manusscript which I found very convincing. I therefore suggest to publish the article in SciPost.

Requested changes

The following article is already piblished:

J. A. Koziol, A. Langheld, S. C. Kapfer, K. P. Schmidt
Quantum-critical properties of the long-range transverse-field Ising model from quantum Monte Carlo simulations
Physical Review B 103, 245135 (2021)

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

Anonymous Report 1 on 2021-8-12 (Invited Report)


See report


See report


The authors have appropriately taken care of the suggestions of both the referees. I therefore recommend that the present version of the paper
should be accepted for publication.

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

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