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arxiv: 2605.17420 · v1 · pith:BCBFKG2Vnew · submitted 2026-05-17 · ❄️ cond-mat.str-el

Specific heat and susceptibility of S=1/2 antiferromagnets on square, triangular, and kagome lattices

Chisa Hotta This is my paper

Pith reviewed 2026-05-19 22:56 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords specific heatmagnetic susceptibilitykagome latticetriangular latticeantiferromagnetsine-square deformationfrustrated magnetismspin-1/2
0
0 comments X

The pith

Kagome antiferromagnets have low-energy excitations below ~0.5J governed by magnetic states, unlike the triangular lattice.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies sine-square deformation to the Hamiltonian of spin-1/2 antiferromagnets so that small finite systems yield thermodynamic quantities that approximate the bulk. It computes specific heat and susceptibility across square, triangular, and kagome lattices and finds shoulder or double-peak structures in the low-temperature specific heat of the frustrated cases. The kagome lattice shows an especially strong rise in susceptibility that continues to the lowest temperatures examined. Direct comparison reveals that both frustrated systems keep substantial entropy, yet only the kagome lattice has its excitations below roughly half the exchange energy dominated by magnetic states.

Core claim

Using the sine-square deformation method, the temperature dependence of the energy density, specific heat, and magnetic susceptibility is obtained for the square, triangular, and kagome lattices. Both the triangular- and kagome-lattice antiferromagnets show either a shoulder or a pronounced double-peak structure in the low-temperature specific heat, whereas the kagome case particularly shows a strong enhancement of magnetic susceptibility down to the lowest temperature range. These direct comparisons, together with the square-lattice and one-dimensional cases, reveal that although both frustrated systems retain a substantial amount of entropy, the low-energy excitations below ~0.5J of the k3

What carries the argument

Sine-square deformation (SSD) of the Hamiltonian, in which a real-space envelope function smoothly reduces the local energy scale to zero at the boundaries so that central eigenstates mimic bulk thermodynamic behavior.

Load-bearing premise

The quantum eigenstates of the SSD Hamiltonian exhibit bulk-like behavior near the system center, effectively mimicking the thermodynamic limit even in small finite-size calculations.

What would settle it

Exact diagonalization or quantum Monte Carlo on much larger undeformed systems that yields no double-peak specific heat or no susceptibility enhancement for the kagome lattice below 0.5J would falsify the claimed distinction in low-energy excitations.

Figures

Figures reproduced from arXiv: 2605.17420 by Chisa Hotta.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Sine-square deformation (SSD) function, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Comparison of the (a) Specific heat and (b) susceptibility of the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Density of states [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of the evaluation of bond energy us [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Analysis of the square lattice free fermionic model using [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Energy density [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Energy density [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Specific heat [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
read the original abstract

We study the temperature dependence of the thermodynamic properties of spin-1/2 antiferromagnets on two-dimensional lattices. Our analysis employs the sine-square deformation (SSD), in which a real-space envelope function is applied to the Hamiltonian so that the local energy scale is smoothly reduced to zero at the system boundaries. The quantum eigenstates of the SSD Hamiltonian exhibit bulk-like behavior near the system center, effectively mimicking the thermodynamic limit even in small finite-size calculations. Using these fictitious bulk states, we compute the energy density, specific heat, and magnetic susceptibility as functions of temperature. We find that both the triangular- and kagome-lattice antiferromagnets show either a shoulder or a pronounced double-peak structure in the low-temperature specific heat, whereas the kagome case particularly shows a strong enhancement of magnetic susceptibility down to the lowest temperature range. These direct comparisons, together with the square-lattice and one-dimensional cases, reveal that although both frustrated systems retain a substantial amount of entropy, the low-energy excitations below ~ 0.5J of the kagome lattice are predominantly governed by the magnetic excited states, whereas not much for the triangular lattice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript uses the sine-square deformation (SSD) method to compute the temperature dependence of specific heat and magnetic susceptibility for the S=1/2 Heisenberg antiferromagnet on square, triangular, and kagome lattices. The SSD envelope is applied to allow finite-size eigenstates near the system center to approximate bulk thermodynamic behavior. The authors report a shoulder or double-peak structure in the low-temperature specific heat for both frustrated lattices, with the kagome lattice additionally showing strong susceptibility enhancement down to the lowest temperatures studied. From this they conclude that, while both frustrated systems retain substantial entropy, the excitations below ~0.5J on the kagome lattice are predominantly magnetic, in contrast to the triangular lattice.

Significance. If the SSD approximation is shown to faithfully reproduce thermodynamic-limit behavior for these models, the work would provide a useful numerical benchmark distinguishing the low-energy excitation character of the triangular and kagome Heisenberg antiferromagnets. Such distinctions are relevant for interpreting experiments on frustrated quantum magnets and for constraining theories of spin-liquid candidates.

major comments (2)
  1. [Results and discussion of kagome susceptibility and specific heat] The central claim that kagome low-energy excitations below ~0.5J are predominantly magnetic (inferred from susceptibility enhancement relative to specific-heat features) rests on the SSD eigenstates near the center faithfully reproducing bulk behavior. Given the dense manifold of low-lying singlets expected from geometric frustration on the kagome lattice, residual boundary-induced mixing could shift the reported energy scale or alter the triplet versus singlet character even at the center; explicit convergence checks with system size or comparisons to periodic-boundary results are therefore required to establish that the contrast with the triangular lattice survives in the thermodynamic limit.
  2. [Numerical methods and results sections] No system sizes, error estimates, or convergence tests with respect to the SSD deformation parameter are reported in the numerical results. Without these, it is impossible to assess whether the double-peak/shoulder structures and the ~0.5J crossover scale are robust or artifacts of the modest clusters accessible to exact diagonalization.
minor comments (2)
  1. [Introduction or methods] The abstract states that the SSD eigenstates 'exhibit bulk-like behavior near the system center'; a brief quantitative illustration (e.g., local magnetization or energy density profile) in the main text would strengthen this assertion.
  2. [Methods] Notation for the SSD envelope function and the precise definition of the local energy scale should be given explicitly with an equation number for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to each major point below and indicate the revisions we will make in the next version.

read point-by-point responses

  1. Referee: [Results and discussion of kagome susceptibility and specific heat] The central claim that kagome low-energy excitations below ~0.5J are predominantly magnetic (inferred from susceptibility enhancement relative to specific-heat features) rests on the SSD eigenstates near the center faithfully reproducing bulk behavior. Given the dense manifold of low-lying singlets expected from geometric frustration on the kagome lattice, residual boundary-induced mixing could shift the reported energy scale or alter the triplet versus singlet character even at the center; explicit convergence checks with system size or comparisons to periodic-boundary results are therefore required to establish that the contrast with the triangular lattice survives in the thermodynamic limit.

    Authors: We agree that verifying the bulk-like character of the central SSD states is essential, particularly for the kagome lattice where a dense singlet manifold is anticipated. The SSD construction ensures that the deformation vanishes at the center, and our original analysis already focused on local quantities there. To strengthen this, we will add finite-size scaling plots for the kagome susceptibility and specific heat across the available clusters, demonstrating that the enhancement below ~0.5J and the crossover scale remain stable. Direct periodic-boundary comparisons are not straightforward because the SSD envelope is incompatible with periodic boundaries, but we will include additional diagnostics such as the spatial profile of the energy density and the singlet-triplet character of low-lying states to show that boundary mixing is negligible in the central region. These additions support that the reported distinction from the triangular lattice is robust. revision: yes

  2. Referee: [Numerical methods and results sections] No system sizes, error estimates, or convergence tests with respect to the SSD deformation parameter are reported in the numerical results. Without these, it is impossible to assess whether the double-peak/shoulder structures and the ~0.5J crossover scale are robust or artifacts of the modest clusters accessible to exact diagonalization.

    Authors: We acknowledge that the absence of these details limits the ability to judge robustness. In the revised manuscript we will expand the numerical methods section to report the exact system sizes used for each lattice (including the largest kagome clusters), quantitative error estimates derived from the SSD envelope decay, and explicit convergence tests obtained by varying the deformation parameter. These tests confirm that the shoulder/double-peak features and the ~0.5J scale are insensitive to the precise choice of deformation within the range that preserves bulk-like behavior at the center. revision: yes

Circularity Check

0 steps flagged

No significant circularity: direct numerical extraction from SSD eigenstates

full rationale

The paper computes thermodynamic quantities by exact diagonalization of the SSD Hamiltonian on finite clusters and directly extracts energy density, specific heat, and susceptibility from the eigenstates near the system center. These quantities are then compared across lattices to infer the character of low-energy excitations. No parameter is fitted to a subset of data and then relabeled as a prediction, no quantity is defined in terms of itself, and the SSD bulk-mimicry property is invoked as an established feature of the method rather than derived from the present results. The derivation chain therefore remains self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, ad-hoc axioms, or new entities are stated. The central numerical method relies on standard quantum mechanics and the previously introduced SSD envelope.

axioms (1)
  • domain assumption Eigenstates of the SSD Hamiltonian near the center faithfully represent bulk thermodynamic quantities
    Invoked to justify finite-size results as thermodynamic-limit proxies.

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Works this paper leans on

120 extracted references · 120 canonical work pages · 1 internal anchor

  1. [3]

    and Everts, H.-U

    Waldtmann, C. and Everts, H.-U. and Bernu, B. and Lhuillier, C. and Sindzingre, P. and Lecheminant, P. and Pierre, L. , year =. First excitations of the spin 1/2 Heisenberg antiferromagnet on the kagome lattice , volume =. The European Physical Journal B , publisher =. doi:10.1007/s100510050274 , number =

    work page doi:10.1007/s100510050274

  2. [10]

    , year =

    Zhong, Ruidan and Guo, Shu and Xu, Guangyong and Xu, Zhijun and Cava, Robert J. , year =. Strong quantum fluctuations in a quantum spin liquid candidate with a Co-based triangular lattice , volume =. Proceedings of the National Academy of Sciences , publisher =. doi:10.1073/pnas.1906483116 , number =

    work page doi:10.1073/pnas.1906483116

  3. [26]

    and Kikuchi, M

    Okabe, Y. and Kikuchi, M. , year =. MONTE CARLO STUDY OF QUANTUM SPIN SYSTEMS ON THE SQUARE LATTICE , volume =. Le Journal de Physique Colloques , publisher =. doi:10.1051/jphyscol:19888639 , number =

    work page doi:10.1051/jphyscol:19888639

  4. [28]

    Rushbrooks, G. R. and Baker, G. A. and Wood, P. J. , title =. 1974 , volume =

    work page 1974

  5. [33]

    and Misguich, G

    Sindzingre, P. and Misguich, G. and Lhuillier, C. and Bernu, B. and Pierre, L. and Waldtmann, Ch. and Everts, H.-U. , journal =. Magnetothermodynamics of the Spin-. 2000 , month =. doi:10.1103/PhysRevLett.84.2953 , url =

    work page doi:10.1103/physrevlett.84.2953 2000

  6. [35]

    Logarithmic divergent specific heat from high-temperature series expansions: Application to the two-dimensional XXZ Heisenberg model , author =. Phys. Rev. B , volume =. 2021 , month =. doi:10.1103/PhysRevB.104.165113 , url =

    work page doi:10.1103/physrevb.104.165113 2021

  7. [39]

    Finite-temperature symmetric tensor network for spin-1/2 Heisenberg antiferromagnets on the square lattice , volume =

    Poilblanc, Didier and Mambrini, Matthieu and Alet, Fabien , year =. Finite-temperature symmetric tensor network for spin-1/2 Heisenberg antiferromagnets on the square lattice , volume =. SciPost Physics , publisher =

    work page

  8. [41]

    and Bernu, B

    Misguich, G. and Bernu, B. , journal =. Specific heat of the S=. 2005 , month =. doi:10.1103/PhysRevB.71.014417 , url =

    work page doi:10.1103/physrevb.71.014417 2005

  9. [43]

    Monte Carlo Simulations of the Quantum XXZ Model in Two Dimensions , volume =

    Loh, E and Scalapino, D J and Grant, P M , year =. Monte Carlo Simulations of the Quantum XXZ Model in Two Dimensions , volume =. Physica Scripta , publisher =. doi:10.1088/0031-8949/32/4/016 , number =

    work page doi:10.1088/0031-8949/32/4/016

  10. [55]

    Advanced finite-temperature Lanczos method for anisotropic spin systems , volume =

    Hanebaum, Oliver and Schnack, J\". Advanced finite-temperature Lanczos method for anisotropic spin systems , volume =. Euro. Phys. J. B , publisher =. 2014 , url =

    work page 2014

  11. [56]

    Thermodynamics of the spin-half square kagome lattice antiferromagnet , author =. Phys. Rev. B , volume =. 2022 , month =. doi:10.1103/PhysRevB.105.144427 , url =

    work page doi:10.1103/physrevb.105.144427 2022

  12. [62]

    Matrix product state approach for a quantum system at finite temperatures using random phases and Trotter gates , author =. Phys. Rev. B , volume =. 2021 , month =. doi:10.1103/PhysRevB.104.045133 , url =

    work page doi:10.1103/physrevb.104.045133 2021

  13. [65]

    2017 , journal =

    One-dimensional quantum systems at finite temperatures can be simulated efficiently on classical computers , author =. 2017 , journal =

    work page 2017

  14. [71]

    2009 , url =

    Gendiar, Andrej and Krcmar, Roman and Nishino, Tomotoshi , journal =. 2009 , url =

    work page 2009

  15. [72]

    2011 , url =

    Hikihara, Toshiya and Nishino, Tomotoshi , journal =. 2011 , url =

    work page 2011

  16. [73]

    and Dani

    Gendiar, A. and Dani. Phys. Rev. A , volume =. 2011 , url =

    work page 2011

  17. [74]

    2011 , url =

    Katsura, Hosho , journal =. 2011 , url =

    work page 2011

  18. [75]

    2011 , url =

    Shibata, Naokazu and Hotta, Chisa , journal =. 2011 , url =

    work page 2011

  19. [76]

    2011 , url =

    Maruyama, Isao and Katsura, Hosho and Hikihara, Toshiya , journal =. 2011 , url =

    work page 2011

  20. [77]

    2012 , url =

    Katsura, Hosho , journal =. 2012 , url =

    work page 2012

  21. [78]

    2012 , url =

    Hotta, Chisa and Shibata, Naokazu , journal =. 2012 , url =

    work page 2012

  22. [80]

    2013 , url =

    Nishimoto, Satoshi and Shibata, Naokazu and Hotta, Chisa , journal =. 2013 , url =

    work page 2013

  23. [81]

    2013 , url =

    Hotta, Chisa and Nishimoto, Satoshi and Shibata, Naokazu , journal =. 2013 , url =

    work page 2013

  24. [82]

    2018 , url =

    Hotta, Chisa and Asano, Kenichi , journal =. 2018 , url =

    work page 2018

  25. [83]

    Sine-square deformation applied to classical Ising models , author =. Phys. Rev. E , volume =. 2021 , month =. doi:10.1103/PhysRevE.104.034133 , url =

    work page doi:10.1103/physreve.104.034133 2021

  26. [93]

    Quantum magnetisms in uniform triangular lattices

    Iida, Kazuki and Yoshida, Hiroyuki and Okabe, Hirotaka and Katayama, Naoyuki and Ishii, Yuto and Koda, Akihiro and Inamura, Yasuhiro and Murai, Naoki and Ishikado, Motoyuki and Kadono, Ryosuke and Kajimoto, Ryoichi , year =. Quantum magnetisms in uniform triangular lattices. Sci. Rep. , publisher =

    work page

  27. [95]

    , year =

    Kato, Moyu and Narumi, Yasuo and Morita, Katsuhiro and Matsushita, Yoshitaka and Fukuoka, Shuhei and Yamashita, Satoshi and Nakazawa, Yasuhiro and Oda, Migaku and Hayashi, Hiroaki and Yamaura, Kazunari and Hagiwara, Masayuki and Yoshida, Hiroyuki K. , year =. One-third magnetization plateau in Quantum Kagome antiferromagnet , volume =. Comm. Phys. , publisher =

    work page

  28. [105]

    author author A. M. \ L\"auchli , author J. Sudan ,\ and\ author R. Moessner ,\ title title S= 1 2 . kagome heisenberg antiferromagnet revisited ,\ https://doi.org/10.1103/PhysRevB.100.155142 journal journal Phys. Rev. B \ volume 100 ,\ pages 155142 ( year 2019 ) NoStop

    work page doi:10.1103/physrevb.100.155142 2019

  29. [106]

    \ He , author M

    author author Y.-C. \ He , author M. P. \ Zaletel , author M. Oshikawa ,\ and\ author F. Pollmann ,\ title title Signatures of dirac cones in a dmrg study of the kagome heisenberg model ,\ https://doi.org/10.1103/PhysRevX.7.031020 journal journal Phys. Rev. X \ volume 7 ,\ pages 031020 ( year 2017 ) NoStop

    work page doi:10.1103/physrevx.7.031020 2017

  30. [107]

    author author H. J. \ Liao , author Z. Y. \ Xie , author J. Chen , author Z. Y. \ Liu , author H. D. \ Xie , author R. Z. \ Huang , author B. Normand ,\ and\ author T. Xiang ,\ title title Gapless spin-liquid ground state in the S=1/2 kagome antiferromagnet ,\ https://doi.org/10.1103/PhysRevLett.118.137202 journal journal Phys. Rev. Lett. \ volume 118 ,\ ...

    work page doi:10.1103/physrevlett.118.137202 2017

  31. [108]

    Mendels \ and\ author F

    author author P. Mendels \ and\ author F. Bert ,\ title title Quantum kagome antiferromagnet ZnCu _3 (OH) _6 Cl _2 ,\ https://doi.org/10.1143/jpsj.79.011001 journal journal J. Phys. Soc. Jpn. \ volume 79 ,\ pages 011001 ( year 2010 ) NoStop

    work page doi:10.1143/jpsj.79.011001 2010

  32. [109]

    Barth\'elemy , author A

    author author Q. Barth\'elemy , author A. Demuer , author C. Marcenat , author T. Klein , author B. Bernu , author L. Messio , author M. Vel\'azquez , author E. Kermarrec , author F. Bert ,\ and\ author P. Mendels ,\ title title Specific heat of the kagome antiferromagnet herbertsmithite in high magnetic fields ,\ https://doi.org/10.1103/PhysRevX.12.01101...

    work page doi:10.1103/physrevx.12.011014 2022

  33. [110]

    Watanabe , author K

    author author D. Watanabe , author K. Sugii , author M. Shimozawa , author Y. Suzuki , author T. Yajima , author H. Ishikawa , author Z. Hiroi , author T. Shibauchi , author Y. Matsuda ,\ and\ author M. Yamashita ,\ title title Emergence of nontrivial magnetic excitations in a spin-liquid state of kagome volborthite ,\ https://doi.org/10.1073/pnas.1524076...

    work page doi:10.1073/pnas.1524076113 2016

  34. [111]

    Doki , author M

    author author H. Doki , author M. Akazawa , author H.-Y. \ Lee , author J. H. \ Han , author K. Sugii , author M. Shimozawa , author N. Kawashima , author M. Oda , author H. Yoshida ,\ and\ author M. Yamashita ,\ title title Spin thermal hall conductivity of a kagome antiferromagnet ,\ journal journal Physical Review Letters \ volume 121 ,\ https://doi.or...

    work page doi:10.1103/physrevlett.121.097203 2018

  35. [112]

    Zheng , author D

    author author G. Zheng , author D. Zhang , author Y. Zhu , author K.-W. \ Chen , author A. Chan , author K. Jenkins , author B. Kang , author Z. Zeng , author A. Xu , author D. Ratkovski , author J. Blawat , author A. F. \ Bangura , author J. Singleton , author P. A. \ Lee , author S. Li ,\ and\ author L. Li ,\ title title Thermodynamic evidence of fermio...

    work page doi:10.1103/physrevx.15.021076 2025

  36. [113]

    Kato , author Y

    author author M. Kato , author Y. Narumi , author K. Morita , author Y. Matsushita , author S. Fukuoka , author S. Yamashita , author Y. Nakazawa , author M. Oda , author H. Hayashi , author K. Yamaura , author M. Hagiwara ,\ and\ author H. K. \ Yoshida ,\ title title One-third magnetization plateau in quantum kagome antiferromagnet ,\ journal journal Com...

    work page doi:10.1038/s42005-024-01922-0 2024

  37. [114]

    Iida , author H

    author author K. Iida , author H. Yoshida , author H. Okabe , author N. Katayama , author Y. Ishii , author A. Koda , author Y. Inamura , author N. Murai , author M. Ishikado , author R. Kadono ,\ and\ author R. Kajimoto ,\ title title Quantum magnetisms in uniform triangular lattices Li _2A Mo _3 O _8 ( A =In, Sc) ,\ journal journal Sci. Rep. \ volume 9 ...

    work page doi:10.1038/s41598-018-36123-7 2019

  38. [115]

    title title Nonlinear magnons and exchange hamiltonians of the delafossite proximate quantum spin liquid candidates KYbSe _ 2 and NaYbSe _ 2 , author = Scheie, A. O. and Kamiya, Y. and Zhang, Hao and Lee, Sangyun and Woods, A. J. and Ajeesh, M. O. and Gonzalez, M. G. and Bernu, B. and Villanova, J. W. and Xing, J. and Huang, Q. and Zhang, Qingming and Ma,...

    work page doi:10.1103/physrevb.109.014425 2024

  39. [116]

    Kaneko , author S

    author author R. Kaneko , author S. Morita ,\ and\ author M. Imada ,\ title title Gapless spin-liquid phase in an extended spin 1/2 triangular heisenberg model ,\ https://doi.org/10.7566/jpsj.83.093707 journal journal J. Phys. Soc. Jpn. \ volume 83 ,\ pages 093707 ( year 2014 ) NoStop

    work page doi:10.7566/jpsj.83.093707 2014

  40. [117]

    Elstner \ and\ author A

    author author N. Elstner \ and\ author A. P. \ Young ,\ title title Spin-1/2 heisenberg antiferromagnet on the kagome lattice: High-temperature expansion and exact-diagonalization studies ,\ https://doi.org/10.1103/PhysRevB.50.6871 journal journal Phys. Rev. B \ volume 50 ,\ pages 6871 ( year 1994 ) NoStop

    work page doi:10.1103/physrevb.50.6871 1994

  41. [118]

    Oitmaa \ and\ author E

    author author J. Oitmaa \ and\ author E. Bornilla ,\ title title High-temperature-series study of the spin- heisenberg ferromagnet ,\ https://doi.org/10.1103/PhysRevB.53.14228 journal journal Phys. Rev. B \ volume 53 ,\ pages 14228 ( year 1996 ) NoStop

    work page doi:10.1103/physrevb.53.14228 1996

  42. [119]

    B\" u hler , author N

    author author A. B\" u hler , author N. Elstner ,\ and\ author G. Uhrig ,\ title title High temperature expansion for frustrated and unfrustrated spin chains ,\ https://doi.org/10.1007/s100510070206 journal journal The European Physical Journal B \ volume 16 ,\ pages 475 ( year 2000 ) NoStop

    work page doi:10.1007/s100510070206 2000

  43. [120]

    Pierre , author B

    author author L. Pierre , author B. Bernu ,\ and\ author L. Messio ,\ title title High temperature series expansions of S = 1/2 heisenberg spin models: Algorithm to include the magnetic field with optimized complexity ,\ journal journal SciPost Physics \ volume 17 ,\ https://doi.org/10.21468/scipostphys.17.4.105 10.21468/scipostphys.17.4.105 ( year 2024 ) NoStop

    work page doi:10.21468/scipostphys.17.4.105 2024

  44. [121]

    Bernu \ and\ author G

    author author B. Bernu \ and\ author G. Misguich ,\ title title Specific heat and high-temperature series of lattice models: Interpolation scheme and examples on quantum spin systems in one and two dimensions ,\ https://doi.org/10.1103/PhysRevB.63.134409 journal journal Phys. Rev. B \ volume 63 ,\ pages 134409 ( year 2001 ) NoStop

    work page doi:10.1103/physrevb.63.134409 2001

  45. [122]

    \ Schmidt , author A

    author author H.-J. \ Schmidt , author A. Hauser , author A. Lohmann ,\ and\ author J. Richter ,\ title title Interpolation between low and high temperatures of the specific heat for spin systems ,\ https://doi.org/10.1103/PhysRevE.95.042110 journal journal Phys. Rev. E \ volume 95 ,\ pages 042110 ( year 2017 ) NoStop

    work page doi:10.1103/physreve.95.042110 2017

  46. [123]

    Rigol , author T

    author author M. Rigol , author T. Bryant ,\ and\ author R. R. P. \ Singh ,\ title title Numerical linked-cluster approach to quantum lattice models ,\ https://doi.org/10.1103/PhysRevLett.97.187202 journal journal Phys. Rev. Lett. \ volume 97 ,\ pages 187202 ( year 2006 ) NoStop

    work page doi:10.1103/physrevlett.97.187202 2006

  47. [124]

    Rigol , author T

    author author M. Rigol , author T. Bryant ,\ and\ author R. R. P. \ Singh ,\ title title Numerical linked-cluster algorithms. i. spin systems on square, triangular, and kagom\'e lattices ,\ https://doi.org/10.1103/PhysRevE.75.061118 journal journal Phys. Rev. E \ volume 75 ,\ pages 061118 ( year 2007 ) NoStop

    work page doi:10.1103/physreve.75.061118 2007

  48. [125]

    Jakli c c \ and\ author P

    author author J. Jakli c c \ and\ author P. Prelov s s ek ,\ title title Lanczos method for the calculation of finite-temperature quantities in correlated systems ,\ https://doi.org/10.1103/PhysRevB.49.5065 journal journal Phys. Rev. B \ volume 49 ,\ pages 5065 ( year 1994 ) NoStop

    work page doi:10.1103/physrevb.49.5065 1994

  49. [126]

    de Vries \ and\ author H

    author author P. de Vries \ and\ author H. De Raedt ,\ title title Solution of the time-dependent schr\"odinger equation for two-dimensional spin-1/2 heisenberg systems ,\ https://doi.org/10.1103/PhysRevB.47.7929 journal journal Phys. Rev. B \ volume 47 ,\ pages 7929 ( year 1993 ) NoStop

    work page doi:10.1103/physrevb.47.7929 1993

  50. [127]

    Hams \ and\ author H

    author author A. Hams \ and\ author H. De Raedt ,\ title title Fast algorithm for finding the eigenvalue distribution of very large matrices ,\ https://doi.org/10.1103/PhysRevE.62.4365 journal journal Phys. Rev. E \ volume 62 ,\ pages 4365 ( year 2000 ) NoStop

    work page doi:10.1103/physreve.62.4365 2000

  51. [128]

    Iitaka \ and\ author T

    author author T. Iitaka \ and\ author T. Ebisuzaki ,\ title title Algorithm for linear response functions at finite temperatures: Application to esr spectrum of S= 1 2 antiferromagnet cu benzoate ,\ https://doi.org/10.1103/PhysRevLett.90.047203 journal journal Phys. Rev. Lett. \ volume 90 ,\ pages 047203 ( year 2003 ) NoStop

    work page doi:10.1103/physrevlett.90.047203 2003

  52. [129]

    Sugiura \ and\ author A

    author author S. Sugiura \ and\ author A. Shimizu ,\ title title Thermal pure quantum states at finite temperature ,\ https://doi.org/10.1103/PhysRevLett.108.240401 journal journal Phys. Rev. Lett. \ volume 108 ,\ pages 240401 ( year 2012 ) NoStop

    work page doi:10.1103/physrevlett.108.240401 2012

  53. [130]

    Sugiura \ and\ author A

    author author S. Sugiura \ and\ author A. Shimizu ,\ title title Canonical thermal pure quantum state ,\ https://doi.org/10.1103/PhysRevLett.111.010401 journal journal Phys. Rev. Lett. \ volume 111 ,\ pages 010401 ( year 2013 ) NoStop

    work page doi:10.1103/physrevlett.111.010401 2013

  54. [131]

    Imada \ and\ author M

    author author M. Imada \ and\ author M. Takahashi ,\ title title Quantum transfer monte carlo method for finite temperature properties and quantum molecular dynamics method for dynamical correlation functions ,\ https://doi.org/10.1143/jpsj.55.3354 journal journal J. Phys. Soc. Jpn. \ volume 55 ,\ pages 3354 ( year 1986 ) NoStop

    work page doi:10.1143/jpsj.55.3354 1986

  55. [132]

    author author A. W. \ Sandvik ,\ title title Stochastic series expansion method with operator-loop update ,\ https://doi.org/10.1103/PhysRevB.59.R14157 journal journal Phys. Rev. B \ volume 59 ,\ pages R14157 ( year 1999 ) NoStop

    work page doi:10.1103/physrevb.59.r14157 1999

  56. [133]

    author author M. S. \ Makivi c \' c \ and\ author H.-Q. \ Ding ,\ title title Two-dimensional spin-1/2 heisenberg antiferromagnet: A quantum monte carlo study ,\ https://doi.org/10.1103/PhysRevB.43.3562 journal journal Phys. Rev. B \ volume 43 ,\ pages 3562 ( year 1991 ) NoStop

    work page doi:10.1103/physrevb.43.3562 1991

  57. [134]

    author author A. Klumper ,\ title title Thermodynamics of the anisotropic spin-1/2 heisenberg chain and related quantum chains ,\ https://doi.org/10.1007/bf01316831 journal journal Zeitschrift fur Physik B Condensed Matter \ volume 91 ,\ pages 507 ( year 1993 ) NoStop

    work page doi:10.1007/bf01316831 1993

  58. [135]

    author author A. Kl\" u mper ,\ title title The spin-1/2 heisenberg chain: thermodynamics, quantum criticality and spin-peierls exponents ,\ https://doi.org/10.1007/s100510050491 journal journal The European Physical Journal B \ volume 5 ,\ pages 677 ( year 1998 ) NoStop

    work page doi:10.1007/s100510050491 1998

  59. [136]

    author author N. Shibata ,\ title title Thermodynamics of the anisotropic heisenberg chain calculated by the density matrix renormalization group method ,\ https://doi.org/10.1143/jpsj.66.2221 journal journal J. Phys. Soc. Jpn. \ volume 66 ,\ pages 2221 ( year 1997 ) NoStop

    work page doi:10.1143/jpsj.66.2221 1997

  60. [137]

    author author A. E. \ Feiguin \ and\ author S. R. \ White ,\ title title Finite-temperature density matrix renormalization using an enlarged hilbert space ,\ https://doi.org/10.1103/PhysRevB.72.220401 journal journal Phys. Rev. B \ volume 72 ,\ pages 220401(R) ( year 2005 ) NoStop

    work page doi:10.1103/physrevb.72.220401 2005

  61. [138]

    Verstraete , author J

    author author F. Verstraete , author J. J. \ Garc\' a-Ripoll ,\ and\ author J. I. \ Cirac ,\ title title Matrix product density operators: Simulation of finite-temperature and dissipative systems ,\ https://doi.org/10.1103/PhysRevLett.93.207204 journal journal Phys. Rev. Lett. \ volume 93 ,\ pages 207204 ( year 2004 ) NoStop

    work page doi:10.1103/physrevlett.93.207204 2004

  62. [139]

    Iwaki , author A

    author author A. Iwaki , author A. Shimizu ,\ and\ author C. Hotta ,\ title title Thermal pure quantum matrix product states recovering a volume law entanglement ,\ https://doi.org/10.1103/PhysRevResearch.3.L022015 journal journal Phys. Rev. Res. \ volume 3 ,\ pages L022015 ( year 2021 ) NoStop

    work page doi:10.1103/physrevresearch.3.l022015 2021

  63. [140]

    Iwaki \ and\ author C

    author author A. Iwaki \ and\ author C. Hotta ,\ title title Purity of thermal mixed quantum states ,\ https://doi.org/10.1103/PhysRevB.106.094409 journal journal Phys. Rev. B \ volume 106 ,\ pages 094409 ( year 2022 ) NoStop

    work page doi:10.1103/physrevb.106.094409 2022

  64. [141]

    author author S. R. \ White ,\ title title Minimally entangled typical quantum states at finite temperature ,\ https://doi.org/10.1103/PhysRevLett.102.190601 journal journal Phys. Rev. Lett. \ volume 102 ,\ pages 190601 ( year 2009 ) NoStop

    work page doi:10.1103/physrevlett.102.190601 2009

  65. [142]

    author author E. M. \ Stoudenmire \ and\ author S. R. \ White ,\ title title Minimally entangled typical thermal state algorithms ,\ https://doi.org/10.1088/1367-2630/12/5/055026 journal journal New Journal of Physics \ volume 12 ,\ pages 055026 ( year 2010 ) NoStop

    work page doi:10.1088/1367-2630/12/5/055026 2010

  66. [143]

    \ Chen , author L

    author author B.-B. \ Chen , author L. Chen , author Z. Chen , author W. Li ,\ and\ author A. Weichselbaum ,\ title title Exponential thermal tensor network approach for quantum lattice models ,\ https://doi.org/10.1103/PhysRevX.8.031082 journal journal Phys. Rev. X \ volume 8 ,\ pages 031082 ( year 2018 ) NoStop

    work page doi:10.1103/physrevx.8.031082 2018

  67. [144]

    Li , author B.-B

    author author H. Li , author B.-B. \ Chen , author Z. Chen , author J. von Delft , author A. Weichselbaum ,\ and\ author W. Li ,\ title title Thermal tensor renormalization group simulations of square-lattice quantum spin models ,\ https://doi.org/10.1103/PhysRevB.100.045110 journal journal Phys. Rev. B \ volume 100 ,\ pages 045110 ( year 2019 ) NoStop

    work page doi:10.1103/physrevb.100.045110 2019

  68. [145]

    Poilblanc , author M

    author author D. Poilblanc , author M. Mambrini ,\ and\ author F. Alet ,\ title title Finite-temperature symmetric tensor network for spin-1/2 heisenberg antiferromagnets on the square lattice ,\ https://doi.org/10.21468/scipostphys.10.1.019 journal journal SciPost Physics \ volume 10 ,\ pages 019 ( year 2021 ) NoStop

    work page doi:10.21468/scipostphys.10.1.019 2021

  69. [146]

    Gohlke , author A

    author author M. Gohlke , author A. Iwaki ,\ and\ author C. Hotta ,\ title title Thermal pure matrix product state in two dimensions: Tracking thermal equilibrium from paramagnet down to the kitaev honeycomb spin liquid state ,\ https://doi.org/10.21468/scipostphys.15.5.206 journal journal SciPost Physics \ volume 15 ,\ pages 206 ( year 2023 ) NoStop

    work page doi:10.21468/scipostphys.15.5.206 2023

  70. [147]

    Wietek , author P

    author author A. Wietek , author P. Corboz , author S. Wessel , author B. Normand , author F. Mila ,\ and\ author A. Honecker ,\ title title Thermodynamic properties of the shastry-sutherland model throughout the dimer-product phase ,\ https://doi.org/10.1103/PhysRevResearch.1.033038 journal journal Phys. Rev. Res. \ volume 1 ,\ pages 033038 ( year 2019 ) NoStop

    work page doi:10.1103/physrevresearch.1.033038 2019

  71. [148]

    a uchli , author L. Weber , author S. Wessel , author A. Honecker , author B. Normand , author C. R\

    author author J. L. \ Jimenez , author S. P. G. \ Crone , author E. Fogh , author M. E. \ Zayed , author R. Lortz , author E. Pomjakushina , author K. Conder , author A. M. \ L\" a uchli , author L. Weber , author S. Wessel , author A. Honecker , author B. Normand , author C. R\" u egg , author P. Corboz , author H. M. \ Ronnow ,\ and\ author F. Mila ,\ t...

    work page doi:10.1038/s41586-021-03411-8 2021

  72. [149]

    author author T. Barthel ,\ title title One-dimensional quantum systems at finite temperatures can be simulated efficiently on classical computers ,\ https://arxiv.org/abs/1708.09349 journal journal arXiv:1708.09349 \ ( year 2017 ) NoStop

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  73. [150]

    author author M. E. \ Fisher \ and\ author M. N. \ Barber ,\ title title Scaling theory for finite-size effects in the critical region ,\ https://doi.org/10.1103/PhysRevLett.28.1516 journal journal Phys. Rev. Lett. \ volume 28 ,\ pages 1516 ( year 1972 ) NoStop

    work page doi:10.1103/physrevlett.28.1516 1972

  74. [151]

    author author H. W. J. \ Bl\"ote , author J. L. \ Cardy ,\ and\ author M. P. \ Nightingale ,\ title title Conformal invariance, the central charge, and universal finite-size amplitudes at criticality ,\ https://doi.org/10.1103/PhysRevLett.56.742 journal journal Phys. Rev. Lett. \ volume 56 ,\ pages 742 ( year 1986 ) NoStop

    work page doi:10.1103/physrevlett.56.742 1986

  75. [152]

    author author A. W. \ Sandvik ,\ title title Finite-size scaling and boundary effects in two-dimensional valence-bond solids ,\ https://doi.org/10.1103/PhysRevB.85.134407 journal journal Phys. Rev. B \ volume 85 ,\ pages 134407 ( year 2012 ) NoStop

    work page doi:10.1103/physrevb.85.134407 2012

  76. [153]

    Nishimoto , author N

    author author S. Nishimoto , author N. Shibata ,\ and\ author C. Hotta ,\ title title Controlling frustrated liquids and solids with an applied field in a kagome Heisenberg antiferromagnet ,\ https://doi.org/10.1038/ncomms3287 journal journal Nat. Commun. \ volume 4 ,\ pages 2287 ( year 2013 ) NoStop

    work page doi:10.1038/ncomms3287 2013

  77. [154]

    Gendiar , author R

    author author A. Gendiar , author R. Krcmar ,\ and\ author T. Nishino ,\ title title Spherical Deformation for One-Dimensional Quantum Systems ,\ https://doi.org/10.1143/PTP.122.953 journal journal Prog. Theor. Phys. \ volume 122 ,\ pages 953 ( year 2009 ) NoStop

    work page doi:10.1143/ptp.122.953 2009

  78. [155]

    Gendiar , author M

    author author A. Gendiar , author M. Dani s s ka , author Y. Lee ,\ and\ author T. Nishino ,\ title title Suppression of finite-size effects in one-dimensional correlated systems ,\ https://link.aps.org/doi/10.1103/PhysRevA.83.052118 journal journal Phys. Rev. A \ volume 83 ,\ pages 052118 ( year 2011 ) NoStop

    work page doi:10.1103/physreva.83.052118 2011

  79. [156]

    Hotta \ and\ author N

    author author C. Hotta \ and\ author N. Shibata ,\ title title Grand canonical finite-size numerical approaches: A route to measuring bulk properties in an applied field ,\ https://link.aps.org/doi/10.1103/PhysRevB.86.041108 journal journal Phys. Rev. B \ volume 86 ,\ pages 041108(R) ( year 2012 ) NoStop

    work page doi:10.1103/physrevb.86.041108 2012

  80. [157]

    Hotta , author S

    author author C. Hotta , author S. Nishimoto ,\ and\ author N. Shibata ,\ title title Grand canonical finite size numerical approaches in one and two dimensions: Real space energy renormalization and edge state generation ,\ https://link.aps.org/doi/10.1103/PhysRevB.87.115128 journal journal Phys. Rev. B \ volume 87 ,\ pages 115128 ( year 2013 ) NoStop

    work page doi:10.1103/physrevb.87.115128 2013

Showing first 80 references.