Interacting fermions in rotation: chiral symmetry restoration, moment of inertia and thermodynamics


We study rotating fermionic matter at finite temperature in the framework of the Nambu-Jona-Lasinio model. In order to respect causality the rigidly rotating system must be bound by a cylindrical boundary with appropriate boundary conditions that confine the fermions inside the cylinder. We show the finite geometry with the MIT boundary conditions affects strongly the phase structure of the model leading to three distinct regions characterized by explicitly broken (gapped), partially restored (nearly gapless) and spontaneously broken (gapped) phases at, respectively, small, moderate and large radius of the cylinder. The presence of the boundary leads to specific steplike irregularities of the chiral condensate as functions of coupling constant, temperature and angular frequency. These steplike features have the same nature as the Shubnikov-de Haas oscillations with the crucial difference that they occur in the absence of both external magnetic field and Fermi surface. At finite temperature the rotation leads to restoration of spontaneously broken chiral symmetry while the vacuum at zero temperature is insensitive to rotation (“cold vacuum cannot rotate”). As the temperature increases the critical angular frequency decreases and the transition becomes softer. A phase diagram in angular frequency-temperature plane is presented. We also show that at fixed temperature the fermion matter in the chirally restored (gapless) phase has a higher moment of inertia compared to the one in the chirally broken (gapped) phase.

A preprint version of the article is available at ArXiv.


  1. [1]

    L.P. Csernai, V.K. Magas and D.J. Wang, Flow Vorticity in Peripheral High Energy Heavy Ion Collisions, Phys. Rev. C 87 (2013) 034906 [arXiv:1302.5310] [INSPIRE].

    ADS  Google Scholar 

  2. [2]

    F. Becattini et al., A study of vorticity formation in high energy nuclear collisions, Eur. Phys. J. C 75 (2015) 406 [arXiv:1501.04468] [INSPIRE].

    ADS  Article  Google Scholar 

  3. [3]

    Y. Jiang, Z.-W. Lin and J. Liao, Rotating quark-gluon plasma in relativistic heavy ion collisions, Phys. Rev. C 94 (2016) 044910 [arXiv:1602.06580] [INSPIRE].

    ADS  Google Scholar 

  4. [4]

    W.-T. Deng and X.-G. Huang, Vorticity in Heavy-Ion Collisions, Phys. Rev. C 93 (2016) 064907 [arXiv:1603.06117] [INSPIRE].

    ADS  Google Scholar 

  5. [5]

    D.T. Son and A.R. Zhitnitsky, Quantum anomalies in dense matter, Phys. Rev. D 70 (2004) 074018 [hep-ph/0405216] [INSPIRE].

  6. [6]

    D.T. Son and P. Surowka, Hydrodynamics with Triangle Anomalies, Phys. Rev. Lett. 103 (2009) 191601 [arXiv:0906.5044] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  7. [7]

    A. Vilenkin, Parity Violating Currents in Thermal Radiation, Phys. Lett. 80B (1978) 150.

    ADS  Article  Google Scholar 

  8. [8]

    A. Vilenkin, Macroscopic Parity Violating Effects: Neutrino Fluxes From Rotating Black Holes And In Rotating Thermal Radiation, Phys. Rev. D 20 (1979) 1807 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  9. [9]

    A. Vilenkin, Quantum Field Theory At Finite Temperature In A Rotating System, Phys. Rev. D 21 (1980) 2260 [INSPIRE].

    ADS  Google Scholar 

  10. [10]

    M. Kaminski, C.F. Uhlemann, M. Bleicher and J. Schaffner-Bielich, Anomalous hydrodynamics kicks neutron stars, Phys. Lett. B 760 (2016) 170 [arXiv:1410.3833] [INSPIRE].

    ADS  Article  Google Scholar 

  11. [11]

    N. Yamamoto, Chiral transport of neutrinos in supernovae: Neutrino-induced fluid helicity and helical plasma instability, Phys. Rev. D 93 (2016) 065017 [arXiv:1511.00933] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  12. [12]

    E. Shaverin and A. Yarom, An anomalous propulsion mechanism, arXiv:1411.5581 [INSPIRE].

  13. [13]

    G. Basar, D.E. Kharzeev and H.-U. Yee, Triangle anomaly in Weyl semimetals, Phys. Rev. B 89 (2014) 035142 [arXiv:1305.6338] [INSPIRE].

    ADS  Article  Google Scholar 

  14. [14]

    K. Landsteiner, Anomalous transport of Weyl fermions in Weyl semimetals, Phys. Rev. B 89 (2014) 075124 [arXiv:1306.4932] [INSPIRE].

    ADS  Article  Google Scholar 

  15. [15]

    M.N. Chernodub, A. Cortijo, A.G. Grushin, K. Landsteiner and M.A.H. Vozmediano, Condensed matter realization of the axial magnetic effect, Phys. Rev. B 89 (2014) 081407 [arXiv:1311.0878] [INSPIRE].

    ADS  Article  Google Scholar 

  16. [16]

    V.E. Ambrus and E. Winstanley, Rotating quantum states, Phys. Lett. B 734 (2014) 296 [arXiv:1401.6388] [INSPIRE].

    ADS  Article  Google Scholar 

  17. [17]

    V.E. Ambrus and E. Winstanley, Rotating fermions inside a cylindrical boundary, Phys. Rev. D 93 (2016) 104014 [arXiv:1512.05239] [INSPIRE].

    ADS  Google Scholar 

  18. [18]

    B.R. Iyer, Dirac field theory in rotating coordinates, Phys. Rev. D 26 (1982) 1900 [INSPIRE].

    ADS  Google Scholar 

  19. [19]

    A. Manning, Fermions in Rotating Reference Frames, arXiv:1512.00579 [INSPIRE].

  20. [20]

    F. Becattini and F. Piccinini, The Ideal relativistic spinning gas: Polarization and spectra, Annals Phys. 323 (2008) 2452 [arXiv:0710.5694] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  21. [21]

    F. Becattini and L. Tinti, Thermodynamical inequivalence of quantum stress-energy and spin tensors, Phys. Rev. D 84 (2011) 025013 [arXiv:1101.5251] [INSPIRE].

    ADS  Google Scholar 

  22. [22]

    E.R. Bezerra de Mello, V.B. Bezerra, A.A. Saharian and A.S. Tarloyan, Fermionic vacuum polarization by a cylindrical boundary in the cosmic string spacetime, Phys. Rev. D 78 (2008) 105007 [arXiv:0809.0844] [INSPIRE].

    ADS  Google Scholar 

  23. [23]

    H.-L. Chen, K. Fukushima, X.-G. Huang and K. Mameda, Analogy between rotation and density for Dirac fermions in a magnetic field, Phys. Rev. D 93 (2016) 104052 [arXiv:1512.08974] [INSPIRE].

    ADS  Google Scholar 

  24. [24]

    Y. Jiang and J. Liao, Pairing Phase Transitions of Matter under Rotation, Phys. Rev. Lett. 117 (2016) 192302 [arXiv:1606.03808] [INSPIRE].

    ADS  Article  Google Scholar 

  25. [25]

    S. Ebihara, K. Fukushima and K. Mameda, Boundary effects and gapped dispersion in rotating fermionic matter, Phys. Lett. B 764 (2017) 94 [arXiv:1608.00336] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  26. [26]

    B. McInnes, Angular Momentum in QGP Holography, Nucl. Phys. B 887 (2014) 246 [arXiv:1403.3258] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  27. [27]

    B. McInnes, Inverse Magnetic/Shear Catalysis, Nucl. Phys. B 906 (2016) 40 [arXiv:1511.05293] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. [28]

    B. McInnes, A rotation/magnetism analogy for the quark-gluon plasma, Nucl. Phys. B 911 (2016) 173 [arXiv:1604.03669] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  29. [29]

    A. Yamamoto and Y. Hirono, Lattice QCD in rotating frames, Phys. Rev. Lett. 111 (2013) 081601 [arXiv:1303.6292] [INSPIRE].

    ADS  Article  Google Scholar 

  30. [30]

    Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. II, Phys. Rev. 124 (1961) 246 [INSPIRE].

  31. [31]

    Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. 1., Phys. Rev. 122 (1961) 345 [INSPIRE].

  32. [32]

    O. Levin, Y. Peleg and A. Peres, Unruh effect for circular motion in a cavity, J. Phys. A 26 (1993) 3001.

    ADS  MathSciNet  Google Scholar 

  33. [33]

    P.C.W. Davies, T. Dray and C.A. Manogue, The Rotating quantum vacuum, Phys. Rev. D 53 (1996) 4382 [gr-qc/9601034] [INSPIRE].

  34. [34]

    G. Duffy and A.C. Ottewill, The Rotating quantum thermal distribution, Phys. Rev. D 67 (2003) 044002 [hep-th/0211096] [INSPIRE].

    ADS  Google Scholar 

  35. [35]

    V.A. Miransky and I.A. Shovkovy, Quantum field theory in a magnetic field: From quantum chromodynamics to graphene and Dirac semimetals, Phys. Rept. 576 (2015) 1 [arXiv:1503.00732] [INSPIRE].

    ADS  Article  Google Scholar 

  36. [36]

    E.V. Gorbar, V.A. Miransky and I.A. Shovkovy, Normal ground state of dense relativistic matter in a magnetic field, Phys. Rev. D 83 (2011) 085003 [arXiv:1101.4954] [INSPIRE].

    ADS  MATH  Google Scholar 

  37. [37]

    B.K. Ridley, Quantum Processes in Semiconductors, 4th edition, Oxford University Press, Oxford U.K. (2000).

  38. [38]

    L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics. Vol. 5: Statistical Physics, Part 1, Butterworth-Heinemann, Oxword U.K. (1980).

  39. [39]

    S. Stringari, Moment of Inertia and Superfluidity of a Trapped Bose Gas, Phys. Rev. Lett. 76 (1996) 1405.

    ADS  Article  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Author information



Corresponding author

Correspondence to M. N. Chernodub.

Additional information

ArXiv ePrint: 1611.02598

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chernodub, M.N., Gongyo, S. Interacting fermions in rotation: chiral symmetry restoration, moment of inertia and thermodynamics. J. High Energ. Phys. 2017, 136 (2017).

Download citation


  • Chiral Lagrangians
  • Global Symmetries