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Finite-Size Scaling Theory of Rényi Thermal Entropy for Second-Order, First-Order, and Weakly First-Order Quantum Phase Transitions

Zhe Wang1,2,*, Yanzhang Zhu1,2,3, Chunhao Guo1,2,†, Bin-Bin Chen4, Yi-Ming Ding1,2,3, Zenan Liu5, and Zheng Yan1,2,‡

  • 1Department of Physics, School of Science and Research Center for Industries of the Future, Westlake University, Hangzhou 310030, China
  • 2Institute of Natural Sciences, Westlake Institute for Advanced Study, Hangzhou 310024, China
  • 3State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China
  • 4Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China
  • 5Department of Physics, South China University of Technology, Guangzhou 510640, China

  • *Contact author: wangzhe90@westlake.edu.cn
  • †Contact author: guochunhao@westlake.edu.cn
  • ‡Contact author: zhengyan@westlake.edu.cn
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Phys. Rev. X 16, 041031 – Published 3 November, 2026

DOI: https://doi.org/10.1103/42kr-81xw

Abstract

Establishing the nature of a quantum phase transition in finite-size simulations—whether continuous, first order, or weakly first order—is a fundamental challenge in quantum many-body computation. The weakly first-order transition is particularly elusive: its anomalously large correlation length invariably mimics a continuous critical point at accessible system sizes, and no unified finite-size scaling theory has existed to distinguish these scenarios under realistic computational constraints. In this work, we address this problem by introducing a finite-size scaling framework based on the Rényi thermal entropy (RTE) and its derivative (DRTE). We derive the finite-size scaling forms for the RTE and DRTE at continuous, first-order, and weakly first-order quantum phase transitions and show that the DRTE isolates the universal contribution of the free energy and strengthens the characteristics of various phase transitions in finite sizes. Using quantum Monte Carlo simulations, we demonstrate accurate data collapse and extraction of critical exponents at (2+1)-dimensional O(N) critical points. Most importantly, the DRTE provides a definitive signature of weakly first-order transitions at visitable sizes—a zero crossing and a double-peak structure—which we observe in the debated deconfined quantum criticality candidates such as the J−Q and SU(8) J1−J2 models. Our approach offers a general, order-parameter-agnostic, and numerically efficient diagnostic tool, resolving long-standing ambiguities between continuous and weakly first-order scenarios in the finite-size simulations.

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Determining whether a quantum phase transition is continuous, first order, or weakly first order in finite-size simulations remains a major challenge. We address this problem by introducing a finite-size scaling framework based on the Rényi thermal entropy and its derivative. By comparing a quantum system at two different temperatures, our approach suppresses the dominant analytic background contributions to isolate the universal free energy. We derive distinct scaling forms for each transition type and uncover a zero-crossing double-peak signature that reveals weakly first-order transitions even in modest simulation sizes. Our order-parameter-agnostic method provides a reliable diagnostic tool to resolve long-standing ambiguities in quantum criticality.

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References (101)

  1. S. Sachdev, Quantum phase transitions, Phys. World 12, 33 (1999).
  2. G. H. Wannier, Statistical Physics (Courier Corporation, New York, 1987).
  3. S. M. Girvin and K. Yang, Modern Condensed Matter Physics (Cambridge University Press, Cambridge, England, 2019).
  4. T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. Fisher, Deconfined quantum critical points, Science 303, 1490 (2004).
  5. A. W. Sandvik, Evidence for deconfined quantum criticality in a two-dimensional Heisenberg model with four-spin interactions, Phys. Rev. Lett. 98, 227202 (2007).
  6. A. W. Sandvik, Continuous quantum phase transition between an antiferromagnet and a valence-bond solid in two dimensions: Evidence for logarithmic corrections to scaling, Phys. Rev. Lett. 104, 177201 (2010).
  7. A. Nahum, P. Serna, J. T. Chalker, M. Ortuño, and A. M. Somoza, Emergent SO(5) symmetry at the néel to valence-bond-solid transition, Phys. Rev. Lett. 115, 267203 (2015).
  8. N. Ma, G.-Y. Sun, Y.-Z. You, C. Xu, A. Vishwanath, A. W. Sandvik, and Z. Y. Meng, Dynamical signature of fractionalization at a deconfined quantum critical point, Phys. Rev. B 98, 174421 (2018).
  9. H. Shao, W. Guo, and A. W. Sandvik, Quantum criticality with two length scales, Science 352, 213 (2016).
  10. C. Wang, A. Nahum, M. A. Metlitski, C. Xu, and T. Senthil, Deconfined quantum critical points: Symmetries and dualities, Phys. Rev. X 7, 031051 (2017).
  11. T. Senthil, Deconfined quantum critical points: A review, 50 Years of the Renormalization Group: Dedicated to the Memory of Michael E Fisher edited by A. Aharony, O. Entin-Wohlman, D. A. Huse, and L. Radzihovsky (World Scientific, Singapore, 2024), p. 169, 10.1142/9789811282386_0014.
  12. B.-B. Mao, Z. Wang, B.-B. Chen, and Z. Yan, Detecting the emergent continuous symmetry of criticality via a subsystem’s entanglement spectrum, Phys. Rev. Lett. 136, 046401 (2026).
  13. S. Yang, F. Xu, D.-C. Lu, Y.-Z. You, H.-Q. Lin, and X.-J. Yu, Deconfined criticality as intrinsically gapless topological state in one dimension, Phys. Rev. B 113, L201105 (2026).
  14. Y. Liu, T. Sato, D. Hou, Z. Wang, W. Guo, and F. F. Assaad, Edge modes of topological Mott insulators and deconfined quantum critical points, arXiv:2508.04455.
  15. R. Ma and C. Wang, Theory of deconfined pseudocriticality, Phys. Rev. B 102, 020407(R) (2020).
  16. D.-C. Lu, Nonlinear sigma model description of deconfined quantum criticality in arbitrary dimensions, SciPost Phys. Core 6, 047 (2023).
  17. Z. Wang, M. P. Zaletel, Roger S. K. Mong, and F. F. Assaad, Phases of the (2+1) dimensional so(5) nonlinear sigma model with topological term, Phys. Rev. Lett. 126, 045701 (2021).
  18. Z. Zhou, L. Hu, W. Zhu, and Y.-C. He, The SO(5) deconfined phase transition under the fuzzy sphere microscope: Approximate conformal symmetry, pseudo-criticality, and operator spectrum, Phys. Rev. X 14, 021044 (2024).
  19. J. Zhao, Y.-C. Wang, Z. Yan, M. Cheng, and Z. Y. Meng, Scaling of entanglement entropy at deconfined quantum criticality, Phys. Rev. Lett. 128, 010601 (2022).
  20. J. D’Emidio and A. W. Sandvik, Entanglement entropy and deconfined criticality: emergent SO(5) symmetry and proper lattice bipartition, Phys. Rev. Lett. 133, 166702 (2024).
  21. Z. Deng, L. Liu, W. Guo, and H.-Q. Lin, Diagnosing quantum phase transition order and deconfined criticality via entanglement entropy, Phys. Rev. Lett. 133, 100402 (2024).
  22. J. Lou, A. W. Sandvik, and N. Kawashima, Antiferromagnetic to valence-bond-solid transitions in two-dimensional su (n) Heisenberg models with multispin interactions, Phys. Rev. B 80, 180414(R) (2009).
  23. Z. Wang, Z. Deng, Z. Liu, Z. Wang, Y.-M. Ding, L. Zhang, W. Guo, and Z. Yan, Universal behavior in entanglement entropy reveals quantum criticality and underlying symmetry breaking, Chin. Phys. Lett. 42, 110712 (2025).
  24. Y. Liu, Z. Wang, T. Sato, M. Hohenadler, C. Wang, W. Guo, and F. F. Assaad, Superconductivity from the condensation of topological defects in a quantum spin-hall insulator, Nat. Commun. 10, 2658 (2019).
  25. Y. D. Liao, X. Y. Xu, Z. Y. Meng, and Y. Qi, Dirac fermions with Plaquette interactions. I. SU(2) phase diagram with gross-neveu and deconfined quantum criticalities, Phys. Rev. B 106, 075111 (2022).
  26. X.-F. Zhang, Y.-C. He, S. Eggert, R. Moessner, and F. Pollmann, Continuous easy-plane deconfined phase transition on the kagome lattice, Phys. Rev. Lett. 120, 115702 (2018).
  27. D.-X. Liu, Z. Xiong, Y. Xu, and X.-F. Zhang, Deconfined quantum phase transition on the kagome lattice: Distinct velocities of spinon and string excitations, Phys. Rev. B 109, L140404 (2024).
  28. V. Gorbenko, S. Rychkov, and B. Zan, Walking, weak first-order transitions, and complex CFTs, J. High Energy Phys. 10 (2018) 108.
  29. V. Gorbenko, S. Rychkov, and B. Zan, Walking, weak first-order transitions, and complex CFTs II. Two-dimensional Potts model at q>4, SciPost Phys. 5, 050 (2018).
  30. Y. Tang, Q. Liu, Q. Tang, and W. Zhu, Boundary criticality of complex conformal field theory: A case study in the non-hermitian 5-state Potts model, SciPost Phys. 19, 164 (2025).
  31. J. Takahashi, H. Shao, B. Zhao, W. Guo, and A. W. Sandvik, SO(5) multicriticality in two-dimensional quantum magnets, arXiv:2405.06607.
  32. B.-B. Chen, X. Zhang, Y. Wang, K. Sun, and Z. Y. Meng, Phases of (2+1) d so (5) nonlinear sigma model with a topological term on a sphere: Multicritical point and disorder phase, Phys. Rev. Lett. 132, 246503 (2024).
  33. W.-Y. Liu, J. Hasik, S.-S. Gong, D. Poilblanc, W.-Q. Chen, and Z.-C. Gu, Emergence of gapless quantum spin liquid from deconfined quantum critical point, Phys. Rev. X 12, 031039 (2022).
  34. Y. Zhu, Z. Liu, Z. Wang, Y.-C. Wang, and Z. Yan, Bipartite entanglement and surface criticality: The extra contribution of the nonordinary edge in entanglement, Phys. Rev. Lett. 136, 046501 (2026).
  35. Z. Wang, C. Guo, B.-B. Mao, and Z. Yan, Universal and non-universal contributions of entanglement under different bipartitions, arXiv:2601.12365.
  36. Y. Zhu, Z. Wang, M. Cheng, and Z. Yan, Criticality on Rényi defects at (2+1) d o (3) quantum critical points, Phys. Rev. Lett. 137, 156501 (2026).
  37. M. Song, J. Zhao, M. Cheng, C. Xu, M. Scherer, L. Janssen, and Z. Y. Meng, Evolution of entanglement entropy at su (n) deconfined quantum critical points, Sci. Adv. 11, eadr0634 (2025).
  38. M. E. Fisher and M. N. Barber, Scaling theory for finite-size effects in the critical region, Phys. Rev. Lett. 28, 1516 (1972).
  39. A. W. Sandvik, Computational studies of quantum spin systems, AIP Conf. Proc. 1297, 135 (2010).
  40. K. G. Wilson, Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture, Phys. Rev. B 4, 3174 (1971).
  41. H. B. G. Casimir, On the attraction between two perfectly conducting plates, Proc. K. Ned. Akad. Wet. 51, 793 (1948).
  42. G. Plunien, B. Müller, and W. Greiner, The Casimir effect, Phys. Rep. 134, 87 (1986).
  43. K. A. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy (World Scientific, Singapore, 2001).
  44. M. E. Fisher and P.-G. de Gennes, Phénomènes aux parois dans un mélange binaire critique, C. R. Acad. Sci. Paris B 287, 207 (1978).
  45. M. Krech, The Casimir Effect in Critical Systems (World Scientific, Singapore, 1994).
  46. J. L. Cardy and I. Peschel, Finite-size dependence of the free energy in two-dimensional critical systems, Nucl. Phys. B300, 377 (1988).
  47. S. Ma, H. Miao, Y. Xiang, and S. Zhang, Enhanced dynamic Casimir effect in temporally and spatially modulated Josephson transmission line, Laser Photonics Rev. 13, 1900164 (2019).
  48. A. Rényi, On measures of entropy and information, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics (University of California Press, Berkeley, California, 1961), Vol. 1, pp. 547–561.
  49. M. Ozawa and N. Javerzat, Perspective on physical interpretations of Rényi entropy in statistical mechanics, Europhys. Lett. 147, 11001 (2024).
  50. J. C. Baez, Rényi entropy and free energy, Entropy 24, 706 (2022).
  51. L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80, 517 (2008).
  52. J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010).
  53. N. Laflorencie, Quantum entanglement in condensed matter systems, Phys. Rep. 646, 1 (2016).
  54. B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phases of Many-Body Systems, Quantum Science and Technology (Springer, New York, 2019), 10.1007/978-1-4939-9084-9.
  55. Z. Wang, Z. Wang, Y.-M. Ding, B.-B. Mao, and Z. Yan, Bipartite reweight-annealing algorithm to extract large-scale data of entanglement entropy and its derivative in high precision, Nat. Commun. 16, 5880 (2025).
  56. Y.-M. Ding, Y. Tang, Z. Wang, Z. Wang, B.-B. Mao, and Z. Yan, Tracking the variation of entanglement rényi negativity: A quantum Monte Carlo study, Phys. Rev. B 111, L241108 (2025).
  57. Z. Wang, Z. Liu, B.-B. Mao, Z. Wang, and Z. Yan, Addressing general measurements in quantum Monte Carlo, Nat. Commun. 17, 679 (2026).
  58. A. W. Sandvik, Stochastic series expansion method with operator-loop update, Phys. Rev. B 59, R14157 (1999).
  59. A. W. Sandvik, Stochastic series expansion methods, arXiv:1909.10591.
  60. O. F. Syljuåsen and A. W. Sandvik, Quantum Monte Carlo with directed loops, Phys. Rev. E 66, 046701 (2002).
  61. Z. Yan, Y. Wu, C. Liu, O. F. Syljuåsen, J. Lou, and Y. Chen, Sweeping cluster algorithm for quantum spin systems with strong geometric restrictions, Phys. Rev. B 99, 165135 (2019).
  62. Z. Yan, Global scheme of sweeping cluster algorithm to sample among topological sectors, Phys. Rev. B 105, 184432 (2022).
  63. K.-H. Wu, T.-C. Lu, C.-M. Chung, Y.-J. Kao, and T. Grover, Entanglement Renyi negativity across a finite temperature transition: A Monte Carlo study, Phys. Rev. Lett. 125, 140603 (2020).
  64. Y.-M. Ding, J.-S. Sun, N. Ma, G. Pan, C. Cheng, and Z. Yan, Reweight-annealing method for evaluating the partition function via quantum Monte Carlo calculations, Phys. Rev. B 110, 165152 (2024).
  65. Z. Dai and X. Y. Xu, Residual entropy from the temperature incremental Monte Carlo method, Phys. Rev. B 111, L081108 (2025).
  66. N. Ma, J.-S. Sun, G. Pan, C. Cheng, and Z. Yan, Defining a universal sign to strictly probe a phase transition, Phys. Rev. B 110, 125141 (2024).
  67. S. Wu, X. Ran, B. Yin, Q.-F. Li, B.-B. Mao, Y.-C. Wang, and Z. Yan, Classical model emerges in quantum entanglement: Quantum Monte Carlo study for an Ising-Heisenberg bilayer, Phys. Rev. B 107, 155121 (2023).
  68. R. Guida and J. Zinn-Justin, Critical exponents of the N-vector model, J. Phys. A 31, 8103 (1998).
  69. Y. Deng and Henk W. J. Blöte, Simultaneous analysis of several models in the three-dimensional Ising universality class, Phys. Rev. E 68, 036125 (2003).
  70. A. Pelissetto and E. Vicari, Critical phenomena and renormalization-group theory, Phys. Rep. 368, 549 (2002).
  71. M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Critical behavior of the three-dimensional XY universality class, Phys. Rev. B 63, 214503 (2001).
  72. M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Critical exponents and equation of state of the three-dimensional Heisenberg universality class, Phys. Rev. B 65, 144520 (2002).
  73. Y. Deng, Henk W. J. Blöte, and M. P. Nightingale, Surface and bulk transitions in three-dimensional O(n) models, Phys. Rev. E 72, 016128 (2005).
  74. M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Ground-state phase diagram of quantum Heisenberg antiferromagnets on the anisotropic dimerized square lattice, Phys. Rev. B 65, 014407 (2001).
  75. C. Ding, L. Zhang, and W. Guo, Engineering surface critical behavior of (2+1)-dimensional o (3) quantum critical points, Phys. Rev. Lett. 120, 235701 (2018).
  76. W. Zhu, C. Ding, L. Zhang, and W. Guo, Exotic surface behaviors induced by geometrical settings of the two-dimensional dimerized quantum XXZ model, arXiv:2111.12336.
  77. Z. Deng, L. Liu, W. Guo, and H. Q. Lin, Improved scaling of the entanglement entropy of quantum antiferromagnetic Heisenberg systems, Phys. Rev. B 108, 125144 (2023).
  78. B.-B. Mao, Y.-M. Ding, Z. Wang, S. Hu, and Z. Yan, Sampling reduced density matrix to extract fine levels of entanglement spectrum and restore entanglement Hamiltonian, Nat. Commun. 16, 2880 (2025).
  79. K. Binder and D. P. Landau, Finite-size scaling at first-order phase transitions, Phys. Rev. B 30, 1477 (1984).
  80. B. Zhao, P. Weinberg, and A. W. Sandvik, Symmetry-enhanced discontinuous phase transition in a two-dimensional quantum magnet, Nat. Phys. 15, 678+ (2019).
  81. T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and Matthew P. A. Fisher, Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm, Phys. Rev. B 70, 144407 (2004).
  82. R. G. Melko and R. K. Kaul, Scaling in the fan of an unconventional quantum critical point, Phys. Rev. Lett. 100, 017203 (2008).
  83. R. K. Kaul, Quantum criticality in SU(3) and SU(4) antiferromagnets, Phys. Rev. B 84, 054407 (2011).
  84. A. Nahum, J. T. Chalker, P. Serna, M. Ortuño, and A. M. Somoza, Deconfined quantum criticality, scaling violations, and classical loop models, Phys. Rev. X 5, 041048 (2015).
  85. Y. Nakayama and T. Ohtsuki, Necessary condition for emergent symmetry from the conformal bootstrap, Phys. Rev. Lett. 117, 131601 (2016).
  86. B. Zhao, J. Takahashi, and A. W. Sandvik, Multicritical deconfined quantum criticality and Lifshitz point of a helical valence-bond phase, Phys. Rev. Lett. 125, 257204 (2020).
  87. D.-C. Lu, C. Xu, and Y.-Z. You, Self-duality protected multicriticality in deconfined quantum phase transitions, Phys. Rev. B 104, 205142 (2021).
  88. A. B. Kuklov, M. Matsumoto, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, Deconfined criticality: Generic first-order transition in the SU(2) symmetry case, Phys. Rev. Lett. 101, 050405 (2008).
  89. K. Chen, Y. Huang, Y. Deng, A. B. Kuklov, N. V. Prokof’ev, and B. V. Svistunov, Deconfined criticality flow in the Heisenberg model with ring-exchange interactions, Phys. Rev. Lett. 110, 185701 (2013).
  90. Y. D. Liao, G. Pan, W. Jiang, Y. Qi, and Z. Y. Meng, The teaching from entanglement: 2d SU(2) antiferromagnet to valence bond solid deconfined quantum critical points are not conformal, arXiv:2302.11742.
  91. M. Song, J. Zhao, Z. Y. Meng, C. Xu, and M. Cheng, Extracting subleading corrections in entanglement entropy at quantum phase transitions, SciPost Phys. 17, 010 (2024).
  92. Q. Li, Y. Gao, Y.-Y. He, Y. Qi, B.-B. Chen, and W. Li, Tangent space approach for thermal tensor network simulations of the 2D Hubbard model, Phys. Rev. Lett. 130, 226502 (2023).
  93. Y.-C. Wang, N. Ma, M. Cheng, and Z. Y. Meng, Scaling of the disorder operator at deconfined quantum criticality, SciPost Phys. 13, 123 (2022).
  94. B.-B. Chen, L. Chen, Z. Chen, W. Li, and A. Weichselbaum, Exponential thermal tensor network approach for quantum lattice models, Phys. Rev. X 8, 031082 (2018).
  95. R. M. Neal, Annealed importance sampling, Stat. Comput. 11, 125 (2001).

  96. How to measure the energy in SSE has been carefully explained in Professor Sandvik’s tutorial https://physics.bu.edu/~sandvik/programs/ssebasic/ssebasic.html.

  97. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, England, 2011).
  98. M. Campostrini, A. Pelissetto, and E. Vicari, Finite-size scaling at quantum transitions, Phys. Rev. B 89, 094516 (2014).
  99. P. W. Anderson, An approximate quantum theory of the antiferromagnetic ground state, Phys. Rev. 86, 694 (1952).
  100. H. Watanabe, Counting rules of Nambu–Goldstone modes, Annu. Rev. Condens. Matter Phys. 11, 169 (2020).
  101. R. K. Kaul and A. W. Sandvik, Lattice model for the SU(n) Néel to valence-bond solid quantum phase transition at large n, Phys. Rev. Lett. 108, 137201 (2012).

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Phys. Rev. X 16, 041031– Published 3 November, 2026
  • Received 3 February 2026
  • Revised 22 July 2026
  • Accepted 6 October 2026
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