- 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

Elastic quantum criticality in nematics and altermagnets via the elastocaloric effect

Charles R. W. Steward1, Grgur Palle1,2,3, Markus Garst4,5, Jörg Schmalian1,5, and Iksu Jang1

PDF

Phys. Rev. B 112, 064431 – Published 21 August, 2025

DOI: https://doi.org/10.1103/nlpj-1dt5

Abstract

The coupling between electronic nematic degrees of freedom and acoustic phonons is known to significantly alter the universality class of a nematic quantum critical point (QCP). While non-Fermi-liquid behavior emerges in the absence of lattice coupling, the inclusion of interactions with acoustic phonons results in observables such as heat capacity and single-particle scattering rate exhibiting only subleading nonanalytic corrections to dominant Fermi-liquid terms. In this work, we demonstrate that the elastocaloric effect (ECE), the adiabatic temperature change under varying strain, and the thermal expansion deviate from this pattern. Despite lattice coupling weakening the singularity of the ECE, it preserves a dominant temperature dependence that deviates from the prediction one would obtain from Fermi-liquid theory, an effect which we will refer to as an elastocaloric anomaly. By drawing analogies between nematic systems and field-tuned altermagnets, we further show that similar responses are expected for the ECE near altermagnetic QCPs. We classify the types of piezomagnetic couplings and analyze the regimes arising from field-tuned magnetoelastic interactions. Our findings are shown to be consistent with the scaling theory for elastic quantum criticality and they further emphasize the suitability of the ECE as a sensitive probe near QCPs.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (80)

  1. V. Oganesyan, S. A. Kivelson, and E. Fradkin, Quantum theory of a nematic fermi fluid, Phys. Rev. B 64, 195109 (2001).
  2. W. Metzner, D. Rohe, and S. Andergassen, Soft fermi surfaces and breakdown of fermi-liquid behavior, Phys. Rev. Lett. 91, 066402 (2003).
  3. H.-Y. Kee, E. H. Kim, and C.-H. Chung, Signatures of an electronic nematic phase at the isotropic-nematic phase transition, Phys. Rev. B 68, 245109 (2003).
  4. L. Dell'Anna and W. Metzner, Electrical resistivity near pomeranchuk instability in two dimensions, Phys. Rev. Lett. 98, 136402 (2007).
  5. M. Zacharias, P. Wölfle, and M. Garst, Multiscale quantum criticality: Pomeranchuk instability in isotropic metals, Phys. Rev. B 80, 165116 (2009).
  6. E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein, and A. P. Mackenzie, Nematic fermi fluids in condensed matter physics, Annu. Rev. Condens. Matter Phys. 1, 153 (2010).
  7. M. A. Metlitski and S. Sachdev, Quantum phase transitions of metals in two spatial dimensions. I.Ising-nematic order, Phys. Rev. B 82, 075127 (2010).
  8. R. Fernandes, A. Chubukov, and J. Schmalian, What drives nematic order in iron-based superconductors? Nat. Phys. 10, 97 (2014).
  9. L. Nie, G. Tarjus, and S. A. Kivelson, Quenched disorder and vestigial nematicity in the pseudogap regime of the cuprates, Proc. Natl. Acad. Sci. USA 111, 7980 (2014).
  10. A. Klein, A. V. Chubukov, Y. Schattner, and E. Berg, Normal state properties of quantum critical metals at finite temperature, Phys. Rev. X 10, 031053 (2020).
  11. O. Grossman, J. S. Hofmann, T. Holder, and E. Berg, Specific heat of a quantum critical metal, Phys. Rev. Lett. 127, 017601 (2021).
  12. C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, Phenomenology of the normal state of Cu-O high-temperature superconductors, Phys. Rev. Lett. 63, 1996 (1989).
  13. S. Lederer, Y. Schattner, E. Berg, and S. A. Kivelson, Enhancement of superconductivity near a nematic quantum critical point, Phys. Rev. Lett. 114, 097001 (2015).
  14. M. A. Metlitski, D. F. Mross, S. Sachdev, and T. Senthil, Cooper pairing in non-fermi liquids, Phys. Rev. B 91, 115111 (2015).
  15. S. Lederer, Y. Schattner, E. Berg, and S. A. Kivelson, Superconductivity and non-fermi liquid behavior near a nematic quantum critical point, Proc. Natl. Acad. Sci. USA 114, 4905 (2017).
  16. A. Klein and A. Chubukov, Superconductivity near a nematic quantum critical point: Interplay between hot and lukewarm regions, Phys. Rev. B 98, 220501(R) (2018).
  17. A. Klein, Y.-M. Wu, and A. V. Chubukov, Multiple intertwined pairing states and temperature-sensitive gap anisotropy for superconductivity at a nematic quantum-critical point, npj Quantum Mater. 4, 55 (2019).
  18. R. M. Fernandes, L. H. VanBebber, S. Bhattacharya, P. Chandra, V. Keppens, D. Mandrus, M. A. McGuire, B. C. Sales, A. S. Sefat, and J. Schmalian, Effects of nematic fluctuations on the elastic properties of iron arsenide superconductors, Phys. Rev. Lett. 105, 157003 (2010).
  19. A. E. Böhmer, P. Burger, F. Hardy, T. Wolf, P. Schweiss, R. Fromknecht, M. Reinecker, W. Schranz, and C. Meingast, Nematic susceptibility of hole-doped and electron-doped BaFe2As2 iron-based superconductors from shear modulus measurements, Phys. Rev. Lett. 112, 047001 (2014).
  20. T. Goto, R. Kurihara, K. Araki, K. Mitsumoto, M. Akatsu, Y. Nemoto, S. Tatematsu, and M. Sato, Quadrupole effects of layered iron pnictide superconductor Ba(Fe0.9Co0.1))2As2, J. Phys. Soc. Jpn. 80, 073702 (2011).
  21. U. Karahasanovic and J. Schmalian, Elastic coupling and spin-driven nematicity in iron-based superconductors, Phys. Rev. B 93, 064520 (2016).
  22. A. Levanyuk and A. Sobyanin, Second-order phase transitions without divergences in the second derivatives of the thermodynamic potential, Zh. Pisma Redaktsiiu 11, 540 (1970) [Second-order phase transitions without divergences in the second derivatives of the thermodynamic potential, J. Exp. Theor. Phys. Lett. 11, 371 (1970)].
  23. R. A. Cowley, Acoustic phonon instabilities and structural phase transitions, Phys. Rev. B 13, 4877 (1976).
  24. I. Paul and M. Garst, Lattice effects on nematic quantum criticality in metals, Phys. Rev. Lett. 118, 227601 (2017).
  25. M. S. Ikeda, J. A. Straquadine, A. T. Hristov, T. Worasaran, J. C. Palmstrom, M. Sorensen, P. Walmsley, and I. R. Fisher, AC elastocaloric effect as a probe for thermodynamic signatures of continuous phase transitions, Rev. Sci. Instrum. 90, 083902 (2019).
  26. M. S. Ikeda, T. Worasaran, E. W. Rosenberg, J. C. Palmstrom, S. A. Kivelson, and I. R. Fisher, Elastocaloric signature of nematic fluctuations, Proc. Natl. Acad. Sci. USA 118, e2105911118 (2021).
  27. C. R. W. Steward, I. Jang, M. Garst, and J. Schmalian (unpublished).
  28. E. W. Rosenberg, M. Ikeda, and I. R. Fisher, The nematic susceptibility of the ferroquadrupolar metal TmAg2 measured via the elastocaloric effect, npj Quantum Mater. 9, 46 (2024).
  29. Y.-S. Li, M. Garst, J. Schmalian, S. Ghosh, N. Kikugawa, D. A. Sokolov, C. W. Hicks, F. Jerzembeck, M. S. Ikeda, Z. Hu et al., Elastocaloric determination of the phase diagram of Sr2RuO4, Nature (London) 607, 276 (2022).
  30. G. Palle, C. Hicks, R. Valentí, Z. Hu, Y.-S. Li, A. Rost, M. Nicklas, A. P. Mackenzie, and J. Schmalian, Constraints on the superconducting state of Sr2RuO4 from elastocaloric measurements, Phys. Rev. B 108, 094516 (2023).
  31. S. Ghosh, M. S. Ikeda, A. R. Chakraborty, T. Worasaran, F. Theuss, L. B. Peralta, P. Lozano, J.-W. Kim, P. J. Ryan, L. Ye et al., Elastocaloric evidence for a multicomponent superconductor stabilized within the nematic state in Ba(Fe1−xCox)2As2, arXiv:2402.17945.
  32. L. Ye, Y. Sun, V. Sunko, J. F. Rodriguez-Nieva, M. S. Ikeda, T. Worasaran, M. E. Sorensen, M. D. Bachmann, J. Orenstein, and I. R. Fisher, Elastocaloric signatures of symmetric and antisymmetric strain-tuning of quadrupolar and magnetic phases in DyB2C2, Proc. Natl. Acad. Sci. USA 120, e2302800120 (2023).
  33. L. Ye, M. E. Sorensen, M. D. Bachmann, and I. R. Fisher, Measurement of the magnetic octupole susceptibility of PrV2Al20, Nat. Commun. 15, 7005 (2024).
  34. L. Zhu, M. Garst, A. Rosch, and Q. Si, Universally diverging grüneisen parameter and the magnetocaloric effect close to quantum critical points, Phys. Rev. Lett. 91, 066404 (2003).
  35. M. Zacharias, I. Paul, and M. Garst, Quantum critical elasticity, Phys. Rev. Lett. 115, 025703 (2015).
  36. I. Jang, RADAR4KIT, Data for “Elastic quantum criticality in nematics and altermagnets via the elasto-caloric effect” (2025), https://dx.doi.org/10.35097/qhcm524xjc27ct4y.
  37. I. I. Mazin, K. Koepernik, M. D. Johannes, R. González-Hernández, and L. Šmejkal, Prediction of unconventional magnetism in doped FeSb2, Proc. Natl. Acad. Sci. USA 118, e2108924118 (2021).
  38. L. Šmejkal, J. Sinova, and T. Jungwirth, Beyond conventional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry, Phys. Rev. X 12, 031042 (2022).
  39. L. Šmejkal, J. Sinova, and T. Jungwirth, Emerging research landscape of altermagnetism, Phys. Rev. X 12, 040501 (2022).
  40. I. Turek, Altermagnetism and magnetic groups with pseudoscalar electron spin, Phys. Rev. B 106, 094432 (2022).
  41. A. Urru and N. A. Spaldin, Magnetic octupole tensor decomposition and second-order magnetoelectric effect, Ann. Phys. 447, 168964 (2022).
  42. S. Bhowal and N. A. Spaldin, Magnetic octupoles as the order parameter for unconventional antiferromagnetism, Phys. Rev. X 14, 011019 (2024).
  43. L. Šmejkal, R. González-Hernández, T. Jungwirth, and J. Sinova, Crystal time-reversal symmetry breaking and spontaneous hall effect in collinear antiferromagnets, Sci. Adv. 6, eaaz8809 (2020).
  44. I. Mazin, Altermagnetism in MnTe: Origin, predicted manifestations, and routes to detwinning, Phys. Rev. B 107, L100418 (2023).
  45. Z. Feng, X. Zhou, L. Šmejkal, L. Wu, Z. Zhu, H. Guo, R. González-Hernández, X. Wang, H. Yan, P. Qin et al., An anomalous Hall effect in altermagnetic ruthenium dioxide, Nat. Electronics 5, 735 (2022).
  46. S. Voleti, D. D. Maharaj, B. D. Gaulin, G. Luke, and A. Paramekanti, Multipolar magnetism in d-orbital systems: Crystal field levels, octupolar order, and orbital loop currents, Phys. Rev. B 101, 155118 (2020).
  47. D. Fiore Mosca, L. V. Pourovskii, and C. Franchini, Modeling magnetic multipolar phases in density functional theory, Phys. Rev. B 106, 035127 (2022).
  48. R. G. Betancourt, J. Zubáč, R. Gonzalez-Hernandez, K. Geishendorf, Z. Šobáň, G. Springholz, K. Olejník, L. Šmejkal, J. Sinova, T. Jungwirth et al., Spontaneous anomalous hall effect arising from an unconventional compensated magnetic phase in a semiconductor, Phys. Rev. Lett. 130, 036702 (2023).
  49. R. Winkler and U. Zülicke, Theory of electric, magnetic, and toroidal polarizations in crystalline solids with applications to hexagonal lonsdaleite and cubic diamond, Phys. Rev. B 107, 155201 (2023).
  50. L.-D. Yuan, Z. Wang, J.-W. Luo, and A. Zunger, Prediction of low-z collinear and noncollinear antiferromagnetic compounds having momentum-dependent spin splitting even without spin-orbit coupling, Phys. Rev. Mater. 5, 014409 (2021).
  51. P. Liu, J. Li, J. Han, X. Wan, and Q. Liu, Spin-group symmetry in magnetic materials with negligible spin-orbit coupling, Phys. Rev. X 12, 021016 (2022).
  52. H. Bai, Y. Zhang, Y. Zhou, P. Chen, C. Wan, L. Han, W. Zhu, S. Liang, Y. Su, X. Han et al., Efficient spin-to-charge conversion via altermagnetic spin splitting effect in antiferromagnet RuO2, Phys. Rev. Lett. 130, 216701 (2023).
  53. L. Šmejkal, A. H. MacDonald, J. Sinova, S. Nakatsuji, and T. Jungwirth, Anomalous hall antiferromagnets, Nat. Rev. Mater. 7, 482 (2022).
  54. J. Yang, Z.-X. Liu, and C. Fang, Symmetry invariants and classes of quasi-particles in magnetically ordered systems having weak spin-orbit coupling, Nat. Commun. 15, 10203 (2024).
  55. Y. Jiang, Z. Song, T. Zhu, Z. Fang, H. Weng, Z.-X. Liu, J. Yang, and C. Fang, Enumeration of spin-space groups: Towards a complete description of symmetries of magnetic orders, Phys. Rev. X 14, 031039 (2024).
  56. P. A. McClarty and J. G. Rau, Landau theory of altermagnetism, Phys. Rev. Lett. 132, 176702 (2024).
  57. H. G. Giil and J. Linder, Superconductor-altermagnet memory functionality without stray fields, Phys. Rev. B 109, 134511 (2024).
  58. S.-W. Cheong and F.-T. Huang, Altermagnetism with non-collinear spins, npj Quantum Mater. 9, 13 (2024).
  59. E. W. Hodt and J. Linder, Spin pumping in an altermagnet/normal-metal bilayer, Phys. Rev. B 109, 174438 (2024).
  60. R. Jaeschke-Ubiergo, V. K. Bharadwaj, T. Jungwirth, L. Šmejkal, and J. Sinova, Supercell altermagnets, Phys. Rev. B 109, 094425 (2024).
  61. J. Krempaský, L. Šmejkal, S. D'Souza, M. Hajlaoui, G. Springholz, K. Uhlířová, F. Alarab, P. Constantinou, V. Strocov, D. Usanov et al., Altermagnetic lifting of kramers spin degeneracy, Nature (London) 626, 517 (2024).
  62. R. M. Fernandes, V. S. de Carvalho, T. Birol, and R. G. Pereira, Topological transition from nodal to nodeless Zeeman splitting in altermagnets, Phys. Rev. B 109, 024404 (2024).
  63. A. S. Patri, A. Sakai, S. Lee, A. Paramekanti, S. Nakatsuji, and Y. B. Kim, Unveiling hidden multipolar orders with magnetostriction, Nat. Commun. 10, 4092 (2019).
  64. T. Aoyama and K. Ohgushi, Piezomagnetic properties in altermagnetic mnte, Phys. Rev. Mater. 8, L041402 (2024).
  65. C. R. W. Steward, R. M. Fernandes, and J. Schmalian, Dynamic paramagnon-polarons in altermagnets, Phys. Rev. B 108, 144418 (2023).
  66. M. Garst and A. V. Chubukov, Electron self-energy near a nematic quantum critical point, Phys. Rev. B 81, 235105 (2010).
  67. A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 2012).
  68. H.-Y. Ma, M. Hu, N. Li, J. Liu, W. Yao, J.-F. Jia, and J. Liu, Multifunctional antiferromagnetic materials with giant piezomagnetism and noncollinear spin current, Nat. Commun. 12, 2846 (2021).
  69. F. Schwabl, Propagation of sound at continuous structural phase transitions, J. Stat. Phys. 39, 719 (1985).
  70. J. A. Hertz, Quantum critical phenomena, Phys. Rev. B 14, 1165 (1976).
  71. A. J. Millis, Effect of a nonzero temperature on quantum critical points in itinerant fermion systems, Phys. Rev. B 48, 7183 (1993).
  72. A. Abanov and A. Chubukov, Anomalous scaling at the quantum critical point in itinerant antiferromagnets, Phys. Rev. Lett. 93, 255702 (2004).
  73. S.-S. Lee, Low-energy effective theory of fermi surface coupled with U(1) gauge field in 2+1 dimensions, Phys. Rev. B 80, 165102 (2009).
  74. M. A. Metlitski and S. Sachdev, Quantum phase transitions of metals in two spatial dimensions. II. Spin density wave order, Phys. Rev. B 82, 075128 (2010).
  75. A. V. Chubukov and J. Schmalian, Superconductivity due to massless boson exchange in the strong-coupling limit, Phys. Rev. B 72, 174520 (2005).
  76. M. Yoshizawa, D. Kimura, T. Chiba, S. Simayi, Y. Nakanishi, K. Kihou, C.-H. Lee, A. Iyo, H. Eisaki, M. Nakajima et al., Structural quantum criticality and superconductivity in iron-based superconductor Ba(Fe1−xCox)2As2, J. Phys. Soc. Jpn. 81, 024604 (2012).
  77. C. Meingast, F. Hardy, R. Heid, P. Adelmann, A. Böhmer, P. Burger, D. Ernst, R. Fromknecht, P. Schweiss, and T. Wolf, Thermal expansion and grüneisen parameters of Ba(Fe1−xCox)2As2: A thermodynamic quest for quantum criticality, Phys. Rev. Lett. 108, 177004 (2012).
  78. H.-H. Kuo, J.-H. Chu, J. C. Palmstrom, S. A. Kivelson, and I. R. Fisher, Ubiquitous signatures of nematic quantum criticality in optimally doped Fe-based superconductors, Science 352, 958 (2016).
  79. S. Ghosh, A. Shekhter, F. Jerzembeck, N. Kikugawa, D. A. Sokolov, M. Brando, A. P. Mackenzie, C. W. Hicks, and B. J. Ramshaw, Thermodynamic evidence for a two-component superconducting order parameter in Sr2RuO4, Nat. Phys. 17, 199 (2021).
  80. M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Physics of Condensed Matter (Springer, New York, 2007).

Outline

Information

Phys. Rev. B 112, 064431– Published 21 August, 2025
  • Received 28 February 2025
  • Revised 29 July 2025
  • Accepted 30 July 2025
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