Abstract
We examine the framework of relativistic spin hydrodynamics in the context of the Dirac quasiparticle fluid in graphene. We develop a spin hydrodynamic model for a (2 + 1)-dimensional system of fermions under the condition of small spin polarization. Our analysis confirms that thermal vorticity, which satisfies the global equilibrium condition, is also a solution to the spin hydrodynamic equations. Additionally, we calculate the magnetization of the system in global equilibrium and introduce a novel phenomenon — thermovortical magnetization — resulting from thermal vorticity, which can be experimentally observed in graphene.
References
[1] M. J. H. Ku et al., Imaging viscous flow of the Dirac fluid in graphene, Nature (London) 583 (7817) (2020) 537–541. doi:10.1038/s41586-020-2507-2. DOI: https://doi.org/10.1038/s41586-020-2507-2
[2] G. Varnavides, A. S. Jermyn, P. Anikeeva, C. Felser, P. Narang, Electron hydrodynamics in anisotropic materials, Nat. Commun. 11 (1) (2020) 4710. doi:10.1038/s41467-020-18553-y. DOI: https://doi.org/10.1038/s41467-020-18553-y
[3] E. H. Hasdeo, J. Ekström, E. G. Idrisov, T. L. Schmidt, Electron hydrodynamics of twodimensional anomalous Hall materials, Phys. Rev. B 103 (2021) 125106. doi:10.1103/PhysRevB. 103.125106. DOI: https://doi.org/10.1103/PhysRevB.103.125106
[4] A. Hui, V. Oganesyan, E.-A. Kim, Beyond Ohm’s law: Bernoulli effect and streaming in electron hydrodynamics, Phys. Rev. B 103 (2021) 235152. doi:10.1103/PhysRevB.103.235152. DOI: https://doi.org/10.1103/PhysRevB.103.235152
[5] X. Huang, A. Lucas, Electron-phonon hydrodynamics, Phys. Rev. B 103 (2021) 155128. doi: 10.1103/PhysRevB.103.155128. DOI: https://doi.org/10.1103/PhysRevB.103.155128
[6] O. Tavakol, Y. B. Kim, Artificial electric field and electron hydrodynamics, Phys. Rev. Res. 3 (2021) 013290. doi:10.1103/PhysRevResearch.3.013290. DOI: https://doi.org/10.1103/PhysRevResearch.3.013290
[7] D. Di Sante, J. Erdmenger, M. Greiter, I. Matthaiakakis, R. Meyer, D. Rodríguez Fernández, R. Thomale, E. van Loon, T. Wehling, Turbulent hydrodynamics in strongly correlated Kagome metals, Nat. Commun. 11 (1) (2020) 3997. doi:10.1038/s41467-020-17663-x. DOI: https://doi.org/10.1038/s41467-020-17663-x
[8] B. N. Narozhny, I. V. Gornyi, M. Titov, Hydrodynamic collective modes in graphene, Phys. Rev. B 103 (2021) 115402. doi:10.1103/PhysRevB.103.115402. DOI: https://doi.org/10.1103/PhysRevB.103.115402
[9] J. A. Sulpizio, L. Ella, A. Rozen, J. Birkbeck, D. J. Perello, D. Dutta, M. Ben-Shalom, T. Taniguchi, K. Watanabe, T. Holder, et al., Visualizing Poiseuille flow of hydrodynamic electrons, Nature (London) 576 (7785) (2019) 75–79. doi:10.1038/s41586-019-1788-9. DOI: https://doi.org/10.1038/s41586-019-1788-9
[10] A. I. Berdyugin, S. G. Xu, F. M. D. Pellegrino, R. K. Kumar, A. Principi, I. Torre, M. B. Shalom, T. Taniguchi, K. Watanabe, I. V. Grigorieva, M. Polini, A. K. Geim, D. A. Bandurin, Measuring Hall viscosity of graphene’s electron fluid, Science 364 (6436) (2019) 162–165. doi:10.1126/science.aau0685. DOI: https://doi.org/10.1126/science.aau0685
[11] L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, et al., Simultaneous voltage and current density imaging of flowing electrons in two dimensions, Nat. Nanotechnol. 14 (5) (2019) 480–487. doi:10.1038/s41565-019-0398-x. DOI: https://doi.org/10.1038/s41565-019-0398-x
[12] D. A. Bandurin, A. V. Shytov, L. S. Levitov, R. K. Kumar, A. I. Berdyugin, M. Ben Shalom, I. V. Grigorieva, A. K. Geim, G. Falkovich, Fluidity onset in graphene, Nat. Commun. 9 (1) (2018) 4533. doi:10.1038/s41467-018-07004-4. DOI: https://doi.org/10.1038/s41467-018-07004-4
[13] D. A. Bandurin, I. Torre, R. K. Kumar, M. B. Shalom, A. Tomadin, A. Principi, G. H. Auton, E. Khestanova, K. S. Novoselov, I. V. Grigorieva, L. A. Ponomarenko, A. K. Geim, M. Polini, Negative local resistance caused by viscous electron backflow in graphene, Science 351 (6277) (2016) 1055–1058. doi:10.1126/science.aad0201. DOI: https://doi.org/10.1126/science.aad0201
[14] C. W. Aung, T. Z. Win, G. Khandal, S. Ghosh, Shear viscosity expression for a graphene system in relaxation time approximation, Phys. Rev. B 108 (23) (2023) 235172. arXiv:2306.14747, doi:10.1103/PhysRevB.108.235172. DOI: https://doi.org/10.1103/PhysRevB.108.235172
[15] E. G. Idrisov, E. H. Hasdeo, B. N. Radhakrishnan, T. L. Schmidt, Hydrodynamic Navier–Stokes equations in two-dimensional systems with Rashba spin–orbit coupling, Low Temp. Phys. 49 (12) (2023) 1385–1397. doi:10.1063/10.0022364. DOI: https://doi.org/10.1063/10.0022364
[16] A. Lucas, K. C. Fong, Hydrodynamics of electrons in graphene, J. Phys.: Condens. Matter 30 (5) (2018) 053001. doi:10.1088/1361-648X/aaa274. DOI: https://doi.org/10.1088/1361-648X/aaa274
[17] B. N. Narozhny, Electronic hydrodynamics in graphene, Ann. Phys. 411 (2019) 167979. doi: https://doi.org/10.1016/j.aop.2019.167979. DOI: https://doi.org/10.1016/j.aop.2019.167979
[18] B. N. Narozhny, Hydrodynamic approach to two-dimensional electron systems, Riv. Nuovo Cim. 45 (10) (2022) 661–736. doi:10.1007/s40766-022-00036-z. DOI: https://doi.org/10.1007/s40766-022-00036-z
[19] P. R. Wallace, The band theory of graphite, Phys. Rev. 71 (9) (1947) 622. doi:10.1103/PhysRev. 71.622. DOI: https://doi.org/10.1103/PhysRev.71.622
[20] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, A. A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature 438 (2005) 197. arXiv:cond-mat/0509330, doi:10.1038/nature04233. DOI: https://doi.org/10.1038/nature04233
[21] Y. Zhang, Y.-W. Tan, H. L. Stormer, P. Kim, Experimental observation of the quantum Hall effect and Berry’s phase in graphene, Nature 438 (2005) 201–204. arXiv:cond-mat/0509355, doi:10.1038/nature04235. DOI: https://doi.org/10.1038/nature04235
[22] U. Heinz, R. Snellings, Collective flow and viscosity in relativistic heavy-ion collisions, Annu. Rev. Nucl. Part. Sci. 63 (2013) 123–151. arXiv:1301.2826, doi:10.1146/annurev-nucl-102212-170540. DOI: https://doi.org/10.1146/annurev-nucl-102212-170540
[23] W. Florkowski, B. Friman, A. Jaiswal, E. Speranza, Relativistic fluid dynamics with spin, Phys. Rev. C 97 (4) (2018) 041901. arXiv:1705.00587, doi:10.1103/PhysRevC.97.041901. DOI: https://doi.org/10.1103/PhysRevC.97.041901
[24] W. Florkowski, A. Kumar, R. Ryblewski, Relativistic hydrodynamics for spin-polarized fluids, Prog. Part. Nucl. Phys. 108 (2019) 103709. arXiv:1811.04409, doi:10.1016/j.ppnp.2019.07.001. DOI: https://doi.org/10.1016/j.ppnp.2019.07.001
[25] S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, R. Ryblewski, Dissipative spin dynamics in relativistic matter, Phys. Rev. D 103 (1) (2021) 014030. arXiv:2008.10976, doi:10.1103/PhysRevD.103.014030. DOI: https://doi.org/10.1103/PhysRevD.103.014030
[26] S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, R. Ryblewski, Relativistic dissipative spin dynamics in the relaxation time approximation, Phys. Lett. B 814 (2021) 136096. arXiv:2002. 03937, doi:10.1016/j.physletb.2021.136096. DOI: https://doi.org/10.1016/j.physletb.2021.136096
[27] F. Becattini, M. A. Lisa, Polarization and vorticity in the quark–gluon plasma, Annu. Rev. Nucl. Part. Sci. 70 (2020) 395–423. arXiv:2003.03640, doi:10.1146/annurev-nucl-021920-095245. DOI: https://doi.org/10.1146/annurev-nucl-021920-095245
[28] J. Hu, Relativistic first-order spin hydrodynamics via the Chapman–Enskog expansion, Phys. Rev. D 105 (7) (2022) 076009. arXiv:2111.03571, doi:10.1103/PhysRevD.105.076009. DOI: https://doi.org/10.1103/PhysRevD.105.076009
[29] S. Shi, C. Gale, S. Jeon, From chiral kinetic theory to relativistic viscous spin hydrodynamics, Phys. Rev. C 103 (4) (2021) 044906. arXiv:2008.08618, doi:10.1103/PhysRevC.103.044906. DOI: https://doi.org/10.1103/PhysRevC.103.044906
[30] B. Fu, K. Xu, X.-G. Huang, H. Song, Hydrodynamic study of hyperon spin polarization in relativistic heavy ion collisions, Phys. Rev. C 103 (2) (2021) 024903. arXiv:2011.03740, doi: 10.1103/PhysRevC.103.024903. DOI: https://doi.org/10.1103/PhysRevC.103.024903
[31] E. Speranza, N. Weickgenannt, Spin tensor and pseudo-gauges: From nuclear collisions to gravitational physics, Eur. Phys. J. A 57 (5) (2021) 155. arXiv:2007.00138, doi:10.1140/epja/s10050-021-00455-2. DOI: https://doi.org/10.1140/epja/s10050-021-00455-2
[32] E. Speranza, F. S. Bemfica, M. M. Disconzi, J. Noronha, Challenges in solving chiral hydrodynamics, Phys. Rev. D 107 (5) (2023) 054029. arXiv:2104.02110, doi:10.1103/PhysRevD.107.054029. DOI: https://doi.org/10.1103/PhysRevD.107.054029
[33] D. She, A. Huang, D. Hou, J. Liao, Relativistic viscous hydrodynamics with angular momentum, Sci. Bull. 67 (2022) 2265–2268. arXiv:2105.04060, doi:10.1016/j.scib.2022.10.020. DOI: https://doi.org/10.1016/j.scib.2022.10.020
[34] H.-H. Peng, J.-J. Zhang, X.-L. Sheng, Q. Wang, Ideal spin hydrodynamics from the Wigner function approach, Chin. Phys. Lett. 38 (11) (2021) 116701. arXiv:2107.00448, doi:10.1088/0256-307X/38/11/116701. DOI: https://doi.org/10.1088/0256-307X/38/11/116701
[35] D.-L. Wang, S. Fang, S. Pu, Analytic solutions of relativistic dissipative spin hydrodynamics with Bjorken expansion, Phys. Rev. D 104 (11) (2021) 114043. arXiv:2107.11726, doi:10.1103/PhysRevD.104.114043. DOI: https://doi.org/10.1103/PhysRevD.104.114043
[36] C. Yi, S. Pu, J.-H. Gao, D.-L. Yang, Hydrodynamic helicity polarization in relativistic heavy ion collisions, Phys. Rev. C 105 (4) (2022) 044911. arXiv:2112.15531, doi:10.1103/PhysRevC.105.044911. DOI: https://doi.org/10.1103/PhysRevC.105.044911
[37] D.-L. Wang, X.-Q. Xie, S. Fang, S. Pu, Analytic solutions of relativistic dissipative spin hydrodynamics with radial expansion in Gubser flow, Phys. Rev. D 105 (11) (2022) 114050. arXiv:2112.15535, doi:10.1103/PhysRevD.105.114050. DOI: https://doi.org/10.1103/PhysRevD.105.114050
[38] W. Florkowski, A. Kumar, R. Ryblewski, R. Singh, Spin polarization evolution in a boost invariant hydrodynamical background, Phys. Rev. C 99 (4) (2019) 044910. arXiv:1901.09655, doi:10.1103/PhysRevC.99.044910. DOI: https://doi.org/10.1103/PhysRevC.99.044910
[39] R. Singh, G. Sophys, R. Ryblewski, Spin polarization dynamics in the Gubser-expanding background, Phys. Rev. D 103 (7) (2021) 074024. arXiv:2011.14907, doi:10.1103/PhysRevD.103.074024. DOI: https://doi.org/10.1103/PhysRevD.103.074024
[40] R. Singh, M. Shokri, R. Ryblewski, Spin polarization dynamics in the Bjorken-expanding resistive MHD background, Phys. Rev. D 103 (9) (2021) 094034. arXiv:2103.02592, doi:10.1103/PhysRevD.103.094034. DOI: https://doi.org/10.1103/PhysRevD.103.094034
[41] L. Tinti, A. Jaiswal, R. Ryblewski, Quasiparticle second-order viscous hydrodynamics from kinetic theory, Phys. Rev. D 95 (5) (2017) 054007. arXiv:1612.07329, doi:10.1103/PhysRevD.95.054007. DOI: https://doi.org/10.1103/PhysRevD.95.054007
[42] A. Jaiswal, R. S. Bhalerao, S. Pal, New relativistic dissipative fluid dynamics from kinetic theory, Phys. Lett. B 720 (2013) 347–351. arXiv:1204.3779, doi:10.1016/j.physletb.2013.02.025. DOI: https://doi.org/10.1016/j.physletb.2013.02.025
[43] W. Florkowski, R. Ryblewski, R. Singh, G. Sophys, Spin polarization dynamics in the nonboost-invariant background, Phys. Rev. D 105 (5) (2022) 054007. arXiv:2112.01856, doi:10.1103/PhysRevD.105.054007. DOI: https://doi.org/10.1103/PhysRevD.105.054007
[44] A. Das, W. Florkowski, A. Kumar, R. Ryblewski, R. Singh, Semi-classical kinetic theory for massive spin-half fermions with leading-order spin effects, Acta Phys. Polon. B 54 (8) (2023) 8–A4. arXiv:2203.15562, doi:10.5506/APhysPolB.54.8-A4. DOI: https://doi.org/10.5506/APhysPolB.54.8-A4
[45] D. Montenegro, L. Tinti, G. Torrieri, Ideal relativistic fluid limit for a medium with polarization, Phys. Rev. D 96 (5) (2017) 056012 [Addendum: Phys. Rev. D 96 (2017) 079901]. arXiv: 1701.08263, doi:10.1103/PhysRevD.96.056012. DOI: https://doi.org/10.1103/PhysRevD.96.079901
[46] D. Montenegro, L. Tinti, G. Torrieri, Sound waves and vortices in a polarized relativistic fluid, Phys. Rev. D 96 (7) (2017) 076016. arXiv:1703.03079, doi:10.1103/PhysRevD.96.076016. DOI: https://doi.org/10.1103/PhysRevD.96.076016
[47] D. Montenegro, G. Torrieri, Causality and dissipation in relativistic polarizable fluids, Phys. Rev. D 100 (5) (2019) 056011. arXiv:1807.02796, doi:10.1103/PhysRevD.100.056011. DOI: https://doi.org/10.1103/PhysRevD.100.056011
[48] D. Montenegro, G. Torrieri, Linear response theory and effective action of relativistic hydrodynamics with spin, Phys. Rev. D 102 (3) (2020) 036007. arXiv:2004.10195, doi:10.1103/PhysRevD.102.036007. DOI: https://doi.org/10.1103/PhysRevD.102.036007
[49] A. D. Gallegos, U. G¨ursoy, A. Yarom, Hydrodynamics of spin currents, SciPost Phys. 11 (2021) 041. arXiv:2101.04759, doi:10.21468/SciPostPhys.11.2.041. DOI: https://doi.org/10.21468/SciPostPhys.11.2.041
[50] M. Hongo, X.-G. Huang, M. Kaminski, M. Stephanov, H.-U. Yee, Relativistic spin hydrodynamics with torsion and linear response theory for spin relaxation, JHEP 11 (2021) 150. arXiv:2107.14231, doi:10.1007/JHEP11(2021)150. DOI: https://doi.org/10.1007/JHEP11(2021)150
[51] C. Cartwright, M. G. Amano, M. Kaminski, J. Noronha, E. Speranza, Convergence of hydrodynamics in a rotating strongly coupled plasma, Phys. Rev. D 108 (4) (2023) 046014. arXiv:2112.10781, doi:10.1103/PhysRevD.108.046014. DOI: https://doi.org/10.1103/PhysRevD.108.046014
[52] A. D. Gallegos, U. Gursoy, A. Yarom, Hydrodynamics, spin currents and torsion, JHEP 05 (2023) 139. arXiv:2203.05044, doi:10.1007/JHEP05(2023)139. DOI: https://doi.org/10.1007/JHEP05(2023)139
[53] F. Becattini, L. Tinti, The ideal relativistic rotating gas as a perfect fluid with spin, Ann. Phys. 325 (2010) 1566–1594. arXiv:0911.0864, doi:10.1016/j.aop.2010.03.007. DOI: https://doi.org/10.1016/j.aop.2010.03.007
[54] F. Becattini, V. Chandra, L. Del Zanna, E. Grossi, Relativistic distribution function for particles with spin at local thermodynamical equilibrium, Ann. Phys. 338 (2013) 32–49. arXiv:1303.3431, doi:10.1016/j.aop.2013.07.004. DOI: https://doi.org/10.1016/j.aop.2013.07.004
[55] F. Becattini, I. Karpenko, M. Lisa, I. Upsal, S. Voloshin, Global hyperon polarization at local thermodynamic equilibrium with vorticity, magnetic field and feed-down, Phys. Rev. C 95 (5) (2017) 054902. arXiv:1610.02506, doi:10.1103/PhysRevC.95.054902. DOI: https://doi.org/10.1103/PhysRevC.95.054902
[56] I. Karpenko, F. Becattini, Study of Λ polarization in relativistic nuclear collisions at √sNN = 7.7−200 GeV, Eur. Phys. J. C 77 (4) (2017) 213. arXiv:1610.04717, doi:10.1140/epjc/s10052-017-4765-1. DOI: https://doi.org/10.1140/epjc/s10052-017-4765-1
[57] H. Li, L.-G. Pang, Q. Wang, X.-L. Xia, Global Λ polarization in heavy-ion collisions from a transport model, Phys. Rev. C 96 (5) (2017) 054908. arXiv:1704.01507, doi:10.1103/PhysRevC.96.054908. DOI: https://doi.org/10.1103/PhysRevC.96.054908
[58] F. Becattini, I. Karpenko, Collective longitudinal polarization in relativistic heavy-ion collisions at very high energy, Phys. Rev. Lett. 120 (1) (2018) 012302. arXiv:1707.07984, doi:10.1103/PhysRevLett.120.012302. DOI: https://doi.org/10.1103/PhysRevLett.120.012302
[59] W. Florkowski, A. Kumar, R. Ryblewski, A. Mazeliauskas, Longitudinal spin polarization in a thermal model, Phys. Rev. C 100 (5) (2019) 054907. arXiv:1904.00002, doi:10.1103/PhysRevC.100.054907. DOI: https://doi.org/10.1103/PhysRevC.100.054907
[60] B. Fu, S. Y. F. Liu, L. Pang, H. Song, Y. Yin, Shear-induced spin polarization in heavy-ion collisions, Phys. Rev. Lett. 127 (14) (2021) 142301. arXiv:2103.10403, doi:10.1103/PhysRevLett.127.142301. DOI: https://doi.org/10.1103/PhysRevLett.127.142301
[61] F. Becattini, M. Buzzegoli, A. Palermo, Spin-thermal shear coupling in a relativistic fluid, Phys.Lett. B 820 (2021) 136519. arXiv:2103.10917, doi:10.1016/j.physletb.2021.136519. DOI: https://doi.org/10.1016/j.physletb.2021.136519
[62] F. Becattini, M. Buzzegoli, G. Inghirami, I. Karpenko, A. Palermo, Local polarization and isothermal local equilibrium in relativistic heavy ion collisions, Phys. Rev. Lett. 127 (27) (2021) 272302. arXiv:2103.14621, doi:10.1103/PhysRevLett.127.272302. DOI: https://doi.org/10.1103/PhysRevLett.127.272302
[63] W. Florkowski, A. Kumar, A. Mazeliauskas, R. Ryblewski, Effect of thermal shear on longitudinal spin polarization in a thermal model, Phys. Rev. C 105 (6) (2022) 064901. arXiv:2112.02799, doi:10.1103/PhysRevC.105.064901. DOI: https://doi.org/10.1103/PhysRevC.105.064901
[64] J. I. Kapusta, E. Rrapaj, S. Rudaz, Relaxation time for strange quark spin in rotating quark– gluon plasma, Phys. Rev. C 101 (2) (2020) 024907. arXiv:1907.10750, doi:10.1103/PhysRevC.101.024907. DOI: https://doi.org/10.1103/PhysRevC.101.024907
[65] A. Ayala, D. de la Cruz, L. A. Hernández, J. Salinas, Relaxation time for the alignment between the spin of a finite-mass quark or antiquark and the thermal vorticity in relativistic heavy-ion collisions, Phys. Rev. D 102 (5) (2020) 056019. arXiv:2003.06545, doi:10.1103/PhysRevD.102.056019. DOI: https://doi.org/10.1103/PhysRevD.102.056019
[66] A. Kumar, B. Müller, D.-L. Yang, Spin alignment of vector mesons by glasma fields, Phys. Rev. D 108 (1) (2023) 016020. arXiv:2304.04181, doi:10.1103/PhysRevD.108.016020. DOI: https://doi.org/10.1103/PhysRevD.108.016020
[67] Y. Hidaka, M. Hongo, M. A. Stephanov, H.-U. Yee, Spin relaxation rate for baryons in a thermal pion gas, Phys. Rev. C 109 (5) (2024) 054909. arXiv:2312.08266, doi:10.1103/PhysRevC.109.054909. DOI: https://doi.org/10.1103/PhysRevC.109.054909
[68] D. Wagner, M. Shokri, D. H. Rischke, Damping of spin waves, Phys. Rev. Res. 6 (4) (2024) 043103. arXiv:2405.00533, doi:10.1103/PhysRevResearch.6.043103. DOI: https://doi.org/10.1103/PhysRevResearch.6.043103
[69] S. Banerjee, S. Bhadury, W. Florkowski, A. Jaiswal, R. Ryblewski, Longitudinal spin polarization in a thermal model with dissipative corrections, Phys. Rev. C 111 (6) (2025) 064912. arXiv: 2405.05089, doi:10.1103/923l-yxkc. DOI: https://doi.org/10.1103/923l-yxkc
[70] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (2009) 109–162. doi:10.1103/RevModPhys.81.109. DOI: https://doi.org/10.1103/RevModPhys.81.109
[71] J. Crossno, J. K. Shi, K. Wang, X. Liu, A. Harzheim, A. Lucas, S. Sachdev, P. Kim, T. Taniguchi, K. Watanabe, T. A. Ohki, K. C. Fong, Observation of the Dirac fluid and the breakdown of the Wiedemann–Franz law in graphene, Science 351 (6277) (2016) 1058–1061. doi:10.1126/science.aad0343. DOI: https://doi.org/10.1126/science.aad0343
[72] A. Lucas, J. Crossno, K. C. Fong, P. Kim, S. Sachdev, Transport in inhomogeneous quantum critical fluids and in the Dirac fluid in graphene, Phys. Rev. B 93 (2016) 075426. doi:10.1103/PhysRevB.93.075426. DOI: https://doi.org/10.1103/PhysRevB.93.075426
[73] P. Gallagher, C.-S. Yang, T. Lyu, F. Tian, R. Kou, H. Zhang, K. Watanabe, T. Taniguchi, F. Wang, Quantum-critical conductivity of the Dirac fluid in graphene, Science 364 (6436) (2019) 158–162. doi:10.1126/science.aat8687. DOI: https://doi.org/10.1126/science.aat8687
[74] A. Lucas, An exotic quantum fluid in graphene, Science 364 (6436) (2019) 125. doi:10.1126/science.aaw9869. DOI: https://doi.org/10.1126/science.aaw9869
[75] D. Kochan, M. Gmitra, J. Fabian, Spin relaxation mechanism in graphene: Resonant scattering by magnetic impurities, Phys. Rev. Lett. 112 (2014) 116602. doi:10.1103/PhysRevLett.112.116602. DOI: https://doi.org/10.1103/PhysRevLett.112.116602
[76] W. Han, K. McCreary, K. Pi, W. Wang, Y. Li, H. Wen, J. Chen, R. Kawakami, Spin transport and relaxation in graphene, J. Magn. Magn. Mater. 324 (4) (2012) 369–381. doi:https://doi.org/10.1016/j.jmmm.2011.08.001. DOI: https://doi.org/10.1016/j.jmmm.2011.08.001
[77] D. Van Tuan, F. Ortmann, A. W. Cummings, D. Soriano, S. Roche, Spin dynamics and relaxation in graphene dictated by electron–hole puddles, Sci. Rep. 6 (1) (2016) 21046. doi:10.1038/srep21046. DOI: https://doi.org/10.1038/srep21046
[78] F. Becattini, L. Bucciantini, E. Grossi, L. Tinti, Local thermodynamical equilibrium and the beta frame for a quantum relativistic fluid, Eur. Phys. J. C 75 (5) (2015) 191. arXiv:1403.6265, doi:10.1140/epjc/s10052-015-3384-y. DOI: https://doi.org/10.1140/epjc/s10052-015-3384-y
[79] F. Becattini, W. Florkowski, E. Speranza, Spin tensor and its role in non-equilibrium thermodynamics, Phys. Lett. B 789 (2019) 419 425. arXiv:1807.10994, doi:10.1016/j.physletb.2018.12.016. DOI: https://doi.org/10.1016/j.physletb.2018.12.016
[80] F. W. Hehl, On the energy tensor of spinning massive matter in classical field theory and general relativity, Rept. Math. Phys. 9 (1976) 55–82. doi:10.1016/0034-4877(76)90016-1. DOI: https://doi.org/10.1016/0034-4877(76)90016-1
[81] F. Becattini, L. Tinti, Thermodynamical inequivalence of quantum stress-energy and spin tensors, Phys. Rev. D 84 (2011) 025013. arXiv:1101.5251, doi:10.1103/PhysRevD.84.025013. DOI: https://doi.org/10.1103/PhysRevD.84.025013
[82] F. Becattini, C. Hoyos, Pseudo-gauge invariant non-equilibrium density operator (July 2025). arXiv:2507.09249. DOI: https://doi.org/10.1103/sk54-wd8z
[83] S. R. De Groot, Relativistic kinetic theory. Principles and applications, North Holland Publ. Comp., Amsterdam, 1980.
[84] L. Suttorp, S. De Groot, Covariant equations of motion, for a charged particle with a magnetic dipole moment, Il Nuovo Cimento A (1965–1970) 65 (1) (1970) 245–274. DOI: https://doi.org/10.1007/BF02752917
[85] C. G. van Weert, On the relativistic kinetic theory of particles with a magnetic dipole moment in an external electromagnetic field (PhD thesis), University of Amsterdam, 1970.
[86] S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, R. Ryblewski, Relativistic spin magnetohydrodynamics, Phys. Rev. Lett. 129 (19) (2022) 192301. arXiv:2204.01357, doi:10.1103/PhysRevLett.129.192301. DOI: https://doi.org/10.1103/PhysRevLett.129.192301
[87] N. Weickgenannt, X.-L. Sheng, E. Speranza, Q. Wang, D. H. Rischke, Kinetic theory for massive spin-1/2 particles from the Wigner-function formalism, Phys. Rev. D 100 (5) (2019) 056018. arXiv:1902.06513, doi:10.1103/PhysRevD.100.056018. DOI: https://doi.org/10.1103/PhysRevD.100.056018
[88] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87 (2015) 1213–1260. doi:10.1103/RevModPhys.87.1213. DOI: https://doi.org/10.1103/RevModPhys.87.1213
[89] C. L. Kane, E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett. 95 (2005) 226801. doi:10.1103/PhysRevLett.95.226801. DOI: https://doi.org/10.1103/PhysRevLett.95.226801
[90] S. J. Barnett, Gyromagnetic and electron-inertia effects, Rev. Mod. Phys. 7 (2) (1935) 129. doi:10.1103/RevModPhys.7.129. DOI: https://doi.org/10.1103/RevModPhys.7.129
[91] A. Yamaguchi, S. Nasu, H. Tanigawa, T. Ono, K. Miyake, K. Mibu, T. Shinjo, Effect of Joule heating in current-driven domain wall motion, Appl. Phys. Lett. 86 (1) (2004) 012511. doi:10.1063/1.1847714. DOI: https://doi.org/10.1063/1.1847714
[92] G. Meier, M. Bolte, R. Eiselt, B. Krüger, D.-H. Kim, P. Fischer, Direct imaging of stochastic domain-wall motion driven by nanosecond current pulses, Phys. Rev. Lett. 98 (2007) 187202. doi:10.1103/PhysRevLett.98.187202. DOI: https://doi.org/10.1103/PhysRevLett.98.187202
[93] H. Fangohr, D. S. Chernyshenko, M. Franchin, T. Fischbacher, G. Meier, Joule heating in nanowires, Phys. Rev. B 84 (2011) 054437. doi:10.1103/PhysRevB.84.054437. DOI: https://doi.org/10.1103/PhysRevB.84.054437
[94] K. Ravi Chandran, Transient Joule heating of graphene, nanowires and filaments: Analytical model for current-induced temperature evolution including substrate and end effects, Int. J. Heat Mass Transfer 88 (2015) 14–19. doi:https://doi.org/10.1016/j.ijheatmasstransfer. 2015.04.014. DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2015.04.014
[95] K. Wang, M. Shi, X. Li, N. Wang, Z. Qu, H. Du, Compact dynamic cantilever magnetometry, Rev. Sci. Instrum. 96 (4) (2025) 045104. doi:10.1063/5.0259803. DOI: https://doi.org/10.1063/5.0259803
[96] C. H. Lee, D. R. Kim, X. Zheng, Fabricating nanowire devices on diverse substrates by simple transfer-printing methods, Proc. Natl. Acad. Sci. U.S.A. 107 (22) (2010) 9950–9955. doi:10.1073/pnas.0914031107. DOI: https://doi.org/10.1073/pnas.0914031107
[97] Z. Xu, M. J. Buehler, Heat dissipation at a graphene–substrate interface, J. Phys.: Condens. Matter 24 (47) (2012) 475305. doi:10.1088/0953-8984/24/47/475305. DOI: https://doi.org/10.1088/0953-8984/24/47/475305
[98] K. Hattori, M. Hongo, X.-G. Huang, M. Matsuo, H. Taya, Fate of spin polarization in a relativistic fluid: An entropy-current analysis, Phys. Lett. B 795 (2019) 100–106. arXiv:1901.06615, doi:10.1016/j.physletb.2019.05.040. DOI: https://doi.org/10.1016/j.physletb.2019.05.040
[99] P. Hosur, X. Qi, Recent developments in transport phenomena in Weyl semimetals, Compt. Rend. Phys. 14 (2013) 857–870. arXiv:1309.4464, doi:10.1016/j.crhy.2013.10.010. DOI: https://doi.org/10.1016/j.crhy.2013.10.010
[100] O. Vafek, A. Vishwanath, Dirac fermions in solids: From high-Tc cuprates and graphene to topological insulators and Weyl semimetals, Annu. Rev. Condens. Matter Phys. 5 (2014) 83–112. arXiv:1306.2272, doi:10.1146/annurev-conmatphys-031113-133841. DOI: https://doi.org/10.1146/annurev-conmatphys-031113-133841
[101] B. Yan, C. Felser, Topological materials: Weyl semimetals, Annu. Rev. Condens. Matter Phys. 8 (2017) 337–354. arXiv:1611.04182, doi:10.1146/annurev-conmatphys-031016-025458. DOI: https://doi.org/10.1146/annurev-conmatphys-031016-025458
[102] A. A. Burkov, Weyl metals, Annu. Rev. Condens. Matter Phys. 9 (2018) 359–378. arXiv: 1704.06660, doi:10.1146/annurev-conmatphys-033117-054129. DOI: https://doi.org/10.1146/annurev-conmatphys-033117-054129
[103] N. P. Armitage, E. J. Mele, A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90 (1) (2018) 015001. arXiv:1705.01111, doi:10.1103/RevModPhys.90.015001. DOI: https://doi.org/10.1103/RevModPhys.90.015001
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