- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Stacked quantum Ising systems and quantum Ashkin-Teller model

Davide Rossini1 and Ettore Vicari2

PDF

Phys. Rev. B 113, 155143 – Published 21 April, 2026

DOI: https://doi.org/10.1103/y9n7-d6vr

Abstract

We analyze the quantum states of an isolated composite system consisting of two stacked quantum Ising (SQI) subsystems, coupled by a local Hamiltonian term that preserves the Z2 symmetry of each subsystem. The coupling strength is controlled by an intercoupling parameter w, with w=0 corresponding to decoupled quantum Ising subsystems. We focus on the quantum correlations of one of the two SQI subsystems, S, in the ground state of the global system and study their dependence on both the state of the weakly coupled complementary part E and the intercoupling strength. We concentrate on regimes in which S develops critical long-range correlations. The most interesting physical scenario arises when both SQI subsystems are critical. In particular, for identical SQI subsystems, the global system is equivalent to the quantum Ashkin-Teller model, characterized by an additional Z2 interchange symmetry between the two subsystem operators. In this limit, one-dimensional SQI systems exhibit a peculiar critical line along which the length-scale critical exponent ν varies continuously with w, while two-dimensional systems develop quantum multicritical behaviors characterized by an effective enlargement of the symmetry of the critical modes, from the actual discrete Z2⊕Z2 symmetry to an effective continuous O(2) symmetry.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (125)

  1. W. H. Zurek, Environment-induced superselection rules, Phys. Rev. D 26, 1862 (1982).
  2. W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75, 715 (2003).
  3. M. Gaudin, Diagonalisation d'une classe d'Hamiltoniens de spin, J. Phys. France 37, 1087 (1976).
  4. N. V. Prokof'ev and P. C. E. Stamp, Theory of the spin bath, Rep. Prog. Phys. 63, 669 (2000).
  5. F. M. Cucchietti, J. P. Paz, and W. H. Zurek, Decoherence from spin environments, Phys. Rev. A 72, 052113 (2005).
  6. H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Decay of Loschmidt echo enhanced by quantum criticality, Phys. Rev. Lett. 96, 140604 (2006).
  7. D. Rossini, T. Calarco, V. Giovannetti, S. Montangero, and R. Fazio, Decoherence induced by interacting quantum spin baths, Phys. Rev. A 75, 032333 (2007).
  8. F. M. Cucchietti, S. Fernandez-Vidal, and J. P. Paz, Universal decoherence induced by an environmental quantum phase transition, Phys. Rev. A 75, 032337 (2007).
  9. Z.-G. Yuan, P. Zhang, and S.-S. Li, Loschmidt echo and Berry phase of a quantum system coupled to an XY spin chain: Proximity to a quantum phase transition, Phys. Rev. A 75, 012102 (2007).
  10. C. Cormick and J. P. Paz, Decoherence induced by a dynamic spin environment: The universal regime, Phys. Rev. A 77, 022317 (2008).
  11. W. H. Zurek, Quantum Darwinism, Nat. Phys. 5, 181 (2009).
  12. M. Bortz, S. Eggert, C. Schneider, R. Stübner, and J. Stolze, Dynamics and decoherence in the central spin model using exact methods, Phys. Rev. B 82, 161308(R) (2010).
  13. B. Damski, H. T. Quan, and W. H. Zurek, Critical dynamics of decoherence, Phys. Rev. A 83, 062104 (2011).
  14. B.-B. Wei and R.-B. Liu, Lee-Yang zeros and critical times in decoherence of a probe spin coupled to a bath, Phys. Rev. Lett. 109, 185701 (2012).
  15. T. Nag, U. Divakaran, and A. Dutta, Scaling of the decoherence factor of a qubit coupled to a spin chain driven across quantum critical points, Phys. Rev. B 86, 020401(R) (2012).
  16. S. Suzuki, T. Nag, and A. Dutta, Dynamics of decoherence: Universal scaling of the decoherence factor, Phys. Rev. A 93, 012112 (2016).
  17. R. Jafari and H. Johannesson, Decoherence from spin environments: Loschmidt echo and quasiparticle excitations, Phys. Rev. B 96, 224302 (2017).
  18. E. Vicari, Decoherence dynamics of qubits coupled to systems at quantum transitions, Phys. Rev. A 98, 052127 (2018).
  19. E. Fiorelli, A. Cuccoli, and P. Verrucchi, Critical slowing down and entanglement protection, Phys. Rev. A 100, 032123 (2019).
  20. D. Rossini and E. Vicari, Scaling of decoherence and energy flow in interacting quantum spin systems, Phys. Rev. A 99, 052113 (2019).
  21. P. Haikka, J. Goold, S. McEndoo, F. Plastina, and S. Maniscalco, Non-Markovianity, Loschmidt echo, and criticality: A unified picture, Phys. Rev. A 85, 060101(R) (2012).
  22. F. Liu, X. Zhou, and Z.-W. Zhou, Memory effect and non-Markovian dynamics in an open quantum system, Phys. Rev. A 99, 052119 (2019).
  23. J.-X. Liu, H.-L. Shi, Y.-H. Shi, X.-H. Wang, and W.-L. Yang, Entanglement and work extraction in the central-spin quantum battery, Phys. Rev. B 104, 245418 (2021).
  24. H.-Y. Yang, K. Zhang, X.-H. Wang, and H.-L. Shi, Optimal energy storage and collective charging speedup in the central-spin quantum battery, Phys. Rev. B 111, 085410 (2025).
  25. C.-Y. Lai, J.-T. Hung, C.-Y. Mou, and P. Chen, Induced decoherence and entanglement by interacting quantum spin baths, Phys. Rev. B 77, 205419 (2008).
  26. R. Vasseur, S. A. Parameswaran, and J. E. Moore, Quantum revivals and many-body localization, Phys. Rev. B 91, 140202(R) (2015).
  27. C.-Z. Yao and W.-M. Zhang, Probing topological states through the exact non-Markovian decoherence dynamics of a spin coupled to a spin bath in the real-time domain, Phys. Rev. B 102, 035133 (2020).
  28. A. Franchi, D. Rossini, and E. Vicari, Quantum many-body spin rings coupled to ancillary spins: The sunburst quantum Ising model, Phys. Rev. E 105, 054111 (2022).
  29. A. Franchi, D. Rossini, and E. Vicari, Decoherence and energy flow in the sunburst quantum Ising model, J. Stat. Mech. (2022) 083103.
  30. A. Mitra and S. C. L. Srivastava, Sunburst quantum Ising model under interaction quench: Entanglement and role of initial state coherence, Phys. Rev. E 108, 054114 (2023).
  31. A. Franchi and F. Tarantelli, Liouvillian gap and out-of-equilibrium dynamics of a sunburst Kitaev ring: From local to uniform dissipation, Phys. Rev. B 108, 094114 (2023).
  32. A. Mitra and S. C. L. Srivastava, Sunburst quantum Ising battery, Phys. Rev. A 110, 012227 (2024).
  33. S. Barbarino, J. Yu, P. Zoller, and J. C. Budich, Preparing atomic topological quantum matter by adiabatic nonunitary dynamics, Phys. Rev. Lett. 124, 010401 (2020).
  34. S. Bhattacharjee, S. Bandyopadhyay, D. Sen, and A. Dutta, Bilayer Haldane system: Topological characterization and adiabatic passages connecting Chern phases, Phys. Rev. B 103, 224304 (2021).
  35. M. Mannaï, J.-N. Fuchs, F. Piéchon, and S. Haddad, Stacking-induced Chern insulator, Phys. Rev. B 107, 045117 (2023).
  36. A. Franchi, A. Pelissetto, and E. Vicari, Quantum critical behaviors and decoherence of weakly coupled quantum Ising models within an isolated global system, Phys. Rev. E 107, 014113 (2023).
  37. S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, Continuous quantum phase transitions, Rev. Mod. Phys. 69, 315 (1997).
  38. S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, Cambridge, UK, 2011).
  39. D. Rossini and E. Vicari, Coherent and dissipative dynamics at quantum phase transitions, Phys. Rep. 936, 1 (2021).
  40. U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005).
  41. M. Kohmoto, M. den Nijs, and L. P. Kadanoff, Hamiltonian studies of the d=2 Ashkin-Teller model, Phys. Rev. B 24, 5229 (1981).
  42. F. Igloi and J. Solyom, Phase diagram and critical properties of the (1 + 1)-dimensional Ashkin-Teller model, J. Phys. A: Math. Gen. 17, 1531 (1984).
  43. F. C. Alcaraz and J. R. Drugowich de Felicio, Finite size studies of the Ashkin-Teller model, J. Phys. A: Math. Gen. 17, L651 (1984).
  44. E. Fradkin, N-Color Ashkin-Teller model in two dimensions: Solution in the large-N limit, Phys. Rev. Lett. 53, 1967 (1984).
  45. R. Shankar, Ashkin-Teller and Gross-Neveu models: New relations and results, Phys. Rev. Lett. 55, 453 (1985).
  46. G. von Gehlen and V. Rittenberg, The Ashkin-Teller quantum chain and conformal invariance, J. Phys. A: Math. Gen. 20, 227 (1987).
  47. F. C. Alcaraz, M. N. Barber, and M. T. Batchelor, Conformal invariance, the XXZ chain and the operator content of two-dimensional critical systems, Ann. Phys. 182, 280 (1988).
  48. J. Ashkin and E. Teller, Statistics of two-dimensional lattices with four components, Phys. Rev. 64, 178 (1943).
  49. F. Y. Wu and K. Y. Lin, Two phase transitions in the Ashkin-Teller model, J. Phys. C: Solid State Phys. 7, L181 (1974).
  50. A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, and D. Sen, Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information (Cambridge University Press, Cambridge, UK, 2015).
  51. P. Pfeuty, The one-dimensional Ising model with a transverse field, Ann. Phys. 57, 79 (1970).
  52. M. Campostrini, A. Pelissetto, and E. Vicari, Quantum Ising chains with boundary fields, J. Stat. Mech. (2015) P11015.
  53. M. Campostrini, A. Pelissetto, and E. Vicari, Finite-size scaling at quantum transitions, Phys. Rev. B 89, 094516 (2014).
  54. A. Pelissetto and E. Vicari, Critical phenomena and renormalization group theory, Phys. Rep. 368, 549 (2002).
  55. M. Caselle, M. Hasenbusch, A. Pelissetto, and E. Vicari, Irrelevant operators in the two-dimensional Ising model, J. Phys. A: Math. Gen. 35, 4861 (2002).
  56. M. Baake, G. von Gehlen, and V. Rittenberg, Operator content of the Ashkin-Teller quantum chain, superconformal and Zamolodchikov-Fateev invariance: I. Free boundary conditions, J. Phys. A: Math. Gen. 20, L479 (1987); Operator content of the Ashkin-Teller quantum chain, superconformal and Zamolodchikov-Fateev invariance. II. Boundary conditions compatible with the torus, 20, L487 (1987).
  57. S.-K. Yang, Modular invariant partition function of the Ashkin-Teller model on the critical line and N = 2 superconformal invariance, Nucl. Phys. B 285, 183 (1987).
  58. S.-K. Yang and H. B. Zheng, Superconformal invariance in the two-dimensional Ashkin-Teller model, Nucl. Phys. B 285, 410 (1987).
  59. M. Yamanaka, Y. Hatsugai, and M. Kohmoto, Phase diagram of the Ashkin-Teller quantum spin chain, Phys. Rev. B 50, 559 (1994).
  60. M. Yamanaka and M. Kohmoto, Line of continuously varying criticality in the Ashkin-Teller quantum chain, Phys. Rev. B 52, 1138 (1995).
  61. J. C. Bridgeman, A. O'Brien, S. D. Barlett, and A. C. Doherty, Multiscale entanglement renormalization ansatz for spin chains with continuously varying criticality, Phys. Rev. B 91, 165129 (2015).
  62. A O'Brien, S. D. Barlett, A. C. Doherty, and S. T. Flammia, Symmetry-respecting real-space renormalization for the quantum Ashkin-Teller model, Phys. Rev. E 92, 042163 (2015).
  63. B. E. Lüscher, F. Mila, and N. Chepiga, Critical properties of the quantum Ashkin-Teller chain with chiral perturbations, Phys. Rev. B 108, 184425 (2023).
  64. Y. Liu, N. Chepiga, Y. Fukusumi, and M. Oshikawa, Boundary critical phenomena in the quantum Ashkin-Teller model, arXiv:2601.16951.
  65. F. Y. Wu, Ashkin-Teller model as a vertex problem, J. Math. Phys. 18, 611 (1977).
  66. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, New York, 1982).
  67. E. Domany and E. K. Riedel, Two-dimensional anisotropic N-vector models, Phys. Rev. B 19, 5817 (1979).
  68. J. R. Drugowich de Felício and R. Köberle, Critical exponents of the Ashkin-Teller model, Phys. Rev. B 25, 511(R) (1982).
  69. S. Wiseman and E. Domany, Cluster method for the Ashkin–Teller model, Phys. Rev. E 48, 4080 (1993).
  70. G. Kamieniarz, P. Kozłowski, and R. Dekeyser, Critical Ising lines of the d=2 Ashkin-Teller model, Phys. Rev. E 55, 3724 (1997).
  71. G. Delfino and P. Grinza, Universal ratios along a line of critical points. The Ashkin-Teller model, Nucl. Phys. B 682, 521 (2004).
  72. A. Giuliani and V. Mastropietro, Anomalous universality in the anisotropic Ashkin-Teller model, Commun. Math. Phys. 256, 681 (2005).
  73. I. Mukherjee and P. K. Mohanty, Hidden superuniversality in systems with continuous variation of critical exponents, Phys. Rev. B 108, 174417 (2023).
  74. I. Keçoğlu and A. N. Berker, Global Ashkin-Teller phase diagrams in two and three dimensions: Multicritical bifurcation versus double tricriticality-endpoint, Physica A 630, 129248 (2023).
  75. Y. Aoun, M. Dober, and A. Glazman, Phase diagram of the Ashkin-Teller model, Commun. Math. Phys. 405, 37 (2024).
  76. M. Dober, On antiferromagnetic regimes in the Ashkin-Teller model, Electron. J. Probab. 30, 1 (2025).
  77. C. Itzykson and J.-M. Drouffe, Statistical Field Theory (Cambridge University Press, Cambridge, UK, 1989).
  78. P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer Verlag, New York, 1997).
  79. N. Khan, P. Sarkar, A. Midya, P. Mandal, and P. K. Mohanty, Continuously varying critical exponents beyond weak universality, Sci. Rep. 7, 45004 (2017).
  80. L. P. Kadanoff, Connections between the critical behavior of the planar model and that of the eight-vertex model, Phys. Rev. Lett. 39, 903 (1977); Multicritical behavior at the Kosterlitz-Thouless critical point, Ann. Phys. 120, 39 (1979).
  81. L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80, 517 (2008).
  82. S.-J. Gu, Fidelity approach to quantum phase transitions, Int. J. Mod. Phys. B 24, 4371 (2010).
  83. D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirandola, Quantum-enhanced measurements without entanglement, Rev. Mod. Phys. 90, 035006 (2018).
  84. D. Rossini and E. Vicari, Ground-state fidelity at first-order quantum transitions, Phys. Rev. E 98, 062137 (2018).
  85. R. Guida and J. Zinn-Justin, Critical exponents of the N-vector model, J. Phys. A: Math. Gen. 31, 8103 (1998).
  86. M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, 25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple cubic lattice, Phys. Rev. E 65, 066127 (2002).
  87. M. Hasenbusch, Finite size scaling study of lattice models in the three-dimensional Ising universality class, Phys. Rev. B 82, 174433 (2010).
  88. M. V. Kompaniets and E. Panzer, Minimally subtracted six-loop renormalization of O(n)-symmetric ϕ4 theory and critical exponents, Phys. Rev. D 96, 036016 (2017).
  89. A. M. Ferrenberg, J. Xu, and D. P. Landau, Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model, Phys. Rev. E 97, 043301 (2018).
  90. M. Hasenbusch, Restoring isotropy in a three-dimensional lattice model: The Ising universality class, Phys. Rev. B 104, 014426 (2021).
  91. C.-H. Chang, V. Dommes, R. S. Erramilli, A. Homrich, P. Kravchuk, A. Liu, M. S. Mitchell, D. Poland, and D. Simmons-Duffin, Bootstrapping the 3d Ising stress tensor, J. High Energy Phys. 03 (2025) 136.
  92. K.-S. Liu and M. E. Fisher, Quantum lattice gas and the existence of a supersolid, J. Low Temp. Phys. 10, 655 (1973).
  93. M. E. Fisher and D. R. Nelson, Spin flop, supersolids, and bicritical and tetracritical points, Phys. Rev. Lett. 32, 1350 (1974).
  94. D. R. Nelson, J. M. Kosterlitz, and M. E. Fisher, Renormalization-group analysis of bicritical and tetracritical points, Phys. Rev. Lett. 33, 813 (1974); J. M. Kosterlitz, D. R. Nelson, and M. E. Fisher, Bicritical and tetracritical points in anisotropic antiferromagnetic systems, Phys. Rev. B 13, 412 (1976).
  95. P. Calabrese, A. Pelissetto, and E. Vicari, Multicritical phenomena in O(n1)⊕O(n2)-symmetric systems, Phys. Rev. B 67, 054505 (2003).
  96. M. Hasenbusch and E. Vicari, Anisotropic perturbations in three-dimensional O(N)-symmetric vector models, Phys. Rev. B 84, 125136 (2011).
  97. C. Bonati, A. Pelissetto, and E. Vicari, Multicritical point of the three-dimensional Z2 gauge Higgs model, Phys. Rev. B 105, 165138 (2022).
  98. M. Campostrini, M. Hasenbusch, A. Pelissetto, and E. Vicari, Theoretical estimates of the critical exponents of the superfluid transition in He4 by lattice methods, Phys. Rev. B 74, 144506 (2006).
  99. S. M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su, and A. Vichi, Carving out OPE space and precise O(2) model critical exponents, J. High Energy Phys. 06 (2020) 142.
  100. C. Bonati, A. Pelissetto, and E. Vicari, Three-dimensional Abelian and non-Abelian gauge Higgs theories, Phys. Rep. 1133, 1 (2025).
  101. M. Hasenbusch, Eliminating leading and subleading corrections to scaling in the three-dimensional XY universality class, Phys. Rev. B 112, 184512 (2025).
  102. M. Hasenbusch, Precision estimates of large charge RG exponents Yq in the 3D XY universality class, Phys. Rev. D 113, 014507 (2026).
  103. J. M. Carmona, A. Pelissetto, and E. Vicari, N-component Ginzburg-Landau Hamiltonian with cubic anisotropy: A six-loop study, Phys. Rev. B 61, 15136 (2000).
  104. U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 2012).
  105. A. O. Caldeira and A. J. Leggett, Quantum tunnelling in a dissipative system, Ann. Phys. 149, 374 (1983).
  106. A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Dynamics of the dissipative two-state system, Rev. Mod. Phys. 59, 1 (1987); 67, 725(E) (1995).
  107. P. Werner, M. Troyer, and S. Sachdev, Quantum spin chains with site dissipation, J. Phys. Soc. Jpn. 74, 67 (2005).
  108. S. Sachdev, P. Werner, and M. Troyer, Universal conductance of nanowires near the superconductor-metal quantum transition, Phys. Rev. Lett. 92, 237003 (2004).
  109. P. Werner, K. Völker, M. Troyer, and S. Chakravarty, Phase diagram and critical exponents of a dissipative Ising spin chain in a transverse magnetic field, Phys. Rev. Lett. 94, 047201 (2005).
  110. S. Yin, P. Mai, and F. Zhong, Nonequilibrium quantum criticality in open systems: The dissipation rate as an additional indispensable scaling variable, Phys. Rev. B 89, 094108 (2014).
  111. O. Alberton, J. Ruhman, E. Berg, and E. Altman, Fate of the one dimensional Ising quantum critical point coupled to a gapless boson, Phys. Rev. B 95, 075132 (2017).
  112. M. Keck, S. Montangero, G. E. Santoro, R. Fazio, and D. Rossini, Dissipation in adiabatic quantum computers: Lessons from an exactly solvable model, New J. Phys. 19, 113029 (2017).
  113. H. Weisbrich, C. Saussol, W. Belzig, and G. Rastelli, Decoherence in the quantum Ising model with transverse dissipative interaction in the strong coupling regime, Phys. Rev. A 98, 052109 (2018).
  114. D. Nigro, D. Rossini, and E. Vicari, Competing coherent and dissipative dynamics close to quantum criticality, Phys. Rev. A 100, 052108 (2019); D. Rossini and E. Vicari, Scaling behavior of stationary states arising from dissipation at continuous quantum transitions, Phys. Rev. B 100, 174303 (2019).
  115. D. Rossini and E. Vicari, Dynamic Kibble-Zurek scaling framework for open dissipative many-body systems crossing quantum transitions, Phys. Rev. Res. 2, 023211 (2020).
  116. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, 2002).
  117. A. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer Briefs in Phyiscs (Springer, Berlin, 2012).
  118. Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018).
  119. Y. Li, X. Chen, and M. P. A. Fisher, Measurement-driven entanglement transition in hybrid quantum circuits, Phys. Rev. B 100, 134306 (2019).
  120. X. Cao, A. Tilloy, and A. De Luca, Entanglement in a fermion chain under continuous monitoring, SciPost Phys. 7, 024 (2019).
  121. B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9, 031009 (2019).
  122. D. Rossini and E. Vicari, Measurement-induced dynamics of many-body systems at quantum criticality, Phys. Rev. B 102, 035119 (2020).
  123. M. Tsitsishvili, D. Poletti, M. Dalmonte, and G. Chiriacò, Measurement induced transitions in non-Markovian free fermion ladders, SciPost Phys. Core 7, 011 (2024).
  124. C. Muzzi, M. Tsitsishvili, and G. Chiriacò, Entanglement enhancement induced by noise in inhomogeneously monitored systems, Phys. Rev. B 111, 014312 (2025).
  125. M. Campostrini and E. Vicari, Trap-size scaling in confined-particle systems at quantum transitions, Phys. Rev. A 81, 023606 (2010).

Outline

Information

Phys. Rev. B 113, 155143– Published 21 April, 2026
  • Received 31 January 2026
  • Accepted 2 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation