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Quantum Monte Carlo simulations in the trimer basis: first-order transitions and thermal critical points in frustrated trilayer magnets

by L. Weber, A. Honecker, B. Normand, P. Corboz, F. Mila, S. Wessel

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Submission summary

Authors (as registered SciPost users): Philippe Corboz · Andreas Honecker · Lukas Weber
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-08-17 09:44
Submitted by: Weber, Lukas
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


The phase diagrams of highly frustrated quantum spin systems can exhibit first-order quantum phase transitions and thermal critical points even in the absence of any long-ranged magnetic order. However, all unbiased numerical techniques for investigating frustrated quantum magnets face significant challenges, and for generic quantum Monte Carlo methods the challenge is the sign problem. Here we report on a general quantum Monte Carlo approach with a loop-update scheme that operates in any basis, and we show that, with an appropriate choice of basis, it allows us to study a frustrated model of coupled spin-1/2 trimers: simulations of the trilayer Heisenberg antiferromagnet in the spin-trimer basis are sign-problem-free when the intertrimer couplings are fully frustrated. This model features a first-order quantum phase transition, from which a line of first-order transitions emerges at finite temperatures and terminates in a thermal critical point. The trimer unit cell hosts an internal degree of freedom that can be controlled to induce an extensive entropy jump at the quantum transition, which alters the shape of the first-order line. We explore the consequences for the thermal properties in the vicinity of the critical point, which include profound changes in the lines of maxima defined by the specific heat. Our findings reveal trimer quantum magnets as fundamental systems capturing in full the complex thermal physics of the strongly frustrated regime.

Author comments upon resubmission

Dear Editor,

Thank you very much for forwarding us the first referee’s report on our
manuscript “Quantum Monte Carlo simulations in the trimer basis: first-order
transitions and thermal critical points in frustrated trilayer magnets.”
Following your advice, we have improved the manuscript based on the comments
of this referee and we resubmit the revised version for your further action.

We provide a full response to the points raised by the referee and a
summary of changes made in the revision process. This response is accompanied
by two figures.

Best regards,

Lukas Weber, Andreas Honecker, Bruce Normand, Philippe Corboz, Frédéric Mila
and Stefan Wessel

List of changes

-- three new paragraphs added in the introduction to Sec. 3 to make clear
its embedding as an integral part of the study (critique (2) of the
-- modified and additional sentences included in Secs. 1, 2, 4, and the abstract to assist
with the embedding of Sec. 3.
-- additional panel in Fig. 8 and additional sentences in the accompanying
text (Sec. 4.3) concerning how criticality is reflected in the data
shown (from critique (1) of the referee).
-- add subscript $I$ to the Ising specific heat, $C_I$ for clarity.
-- subscript “c” corrected in the y-axis labels of Fig. 7.
-- missing square added in the denominator of Eq. (39).
-- missing index $i$ added in the definition of the Ising magnetization.
-- citation added to a recent reference concerning the sign problem.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2021-10-14 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2105.05271v3, delivered 2021-10-14, doi: 10.21468/SciPost.Report.3671


1. A new QMC algorithm that overcomes the sign problem is certain subclass of frustrated magnets.
2. Explain the behaviour of specific heat in some frustrated spin systems.


1. No real quantum magnet cited that correspond to the model studied.


The authors have studied thermal phase transition in a model of coupled trimers. By switching to a larger trimer basis with an extended Hilbert space, they are able to overcome the sign problem is a wide range of interactions parameters. This is an extension of their earlier work on frustrated bilayer systems.

The authors have also developed a new approach to constructing directed loop updates within the SSE QMC to work in the new basis.

Using the new algorithm, the authors have studied ground state and thermal phase transitions in a system of coupled trimers. An important finding of the study is that by tuning the intra-trimer degree of frustration, the nature of the 1st order thermal transition can be varied over a wide range. This is an interesting result and expands our understanding of the effects of geometric frustration of physical properties of quantum magnets.

The use of expanded local Hilbert space to overcome the sign problem is not new, but any development in this direction is useful as it allows the unbiased simulation of additional class of frustrated systems. The work is valuable for the algorithmic developments as well as the findings related to the specific heat are important to people working in this area. The numerics are sound, the paper is very well written and in the opinion of the present referee, the manuscript deserves to be published. I have only two optional suggestions that the authors might want to address:
1. Can the authors cite some real quantum magnet to which their model is applicable?
2. Can the authors elaborate a little more on how their choice of actions differ from what one would naively expect based on the local Hamiltonian operators by giving specific examples for the case studied?

  • validity: top
  • significance: good
  • originality: ok
  • clarity: top
  • formatting: excellent
  • grammar: perfect

Author:  Lukas Weber  on 2021-10-25  [id 1876]

(in reply to Report 2 on 2021-10-14)

We would like to thank the referee for a careful reading of our manuscript and for the helpful suggestions regarding (1) the connection to real quantum magnets and (2) the direct comparison of the abstract loop actions with local Hamiltonian operators.

(1) The results of our study are expected to serve as a reference for the fate of a first-order transition at finite temperatures when one of the phases has a degenerate or quasi-degenerate ground state, in the same way as the fully frustrated bilayer served as a reference for the non-degenerate case. While a materials realisation of the fully frustrated bilayer is known only for a system with non-Heisenberg spin interactions, it turned out that the same physics is realised almost exactly in the compound SrCu$_2$(BO$_3$)$_2$. We are in the same situation here: although it seems difficult to find a realistic compound with identical inter-trimer bonds in the trilayer structure (or to engineer one using atomically thin magnetic layers), quantum magnets based on triangular motifs with frustrated coupling certainly do exist, and some of these may show part of the same phenomenology covered by varying our $J_2$ parameter when driven to their phase transitions. We have summarised this reasoning and listed some triangle-based materials in Sec. 1 of the revised manuscript. However, we are not currently aware of a triangle-based compound undergoing this type of first-order transition under pressure.

(2) Because the abstract actions act on the local basis states in the same way as simple ket-bra operations, we have added a comparison between a ket-bra operator and the equivalent combination of local spin-1/2 operators. These operators are significantly more complicated than the spin-sum and -difference operators used in the dimer basis and thus they illustrate the increasing complexity of the “physical-operator” picture in higher-dimensional cluster bases. The suggestion of the referee also led us to improve our presentation of the xor actions, clarifying the case where the basis dimension is not a power of two.

Anonymous Report 1 on 2021-10-13 (Invited Report)


The authors have satisfyingly answered the minor points I raised in the first report. This paper is ready for publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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