Skip to main content
SpringerLink
Log in

Fermions on a torus knot

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

In this work, we investigate the effects of a nontrivial topology (and geometry) of a system considering interacting and noninteracting particle modes, which are restricted to follow a closed path over the torus surface. In order to present a prominent thermodynamical investigation of this system configuration, we carry out a detailed analysis using statistical mechanics within the grand canonical ensemble approach to deal with noninteracting fermions. In an analytical manner, we study the following thermodynamic functions in such context: the Helmholtz free energy, the mean energy, the magnetization and the susceptibility. Further, we take into account the behavior of Fermi energy of the thermodynamic system. Finally, we briefly outline how to proceed in case of interacting fermions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Notes

  1. Note that the procedure used here to carried out the factorization is not a general result. It suits only for the results presented in this paper.

  2. It should be noted that we have considered a magnetic field which, in technical terms, is referred as the solenoidal magnetic field (uniform magnetic field pointing in the \(\varvec{\hat{z}}\) direction). Nevertheless, for a particle on a torus knot, two other forms of magnetic field, referred to as toroidal (along the \(\varvec{\hat{\phi }}\) direction) and poloidal (along the \(\varvec{\hat{\theta }}\) direction) forms are also relevant. If one is interested in more details ascribed to them, please see Ref. [33].

References

  1. D.R. Gaskell, Introduction to the Thermodynamics of Materials (CRC Press, Boca Raton, 2012)

    Book  Google Scholar 

  2. B. Mühlschlegel, D. Scalapino, R. Denton, Thermodynamic properties of small superconducting particles. Phys. Rev. B 6(5), 1767 (1972)

    Article  ADS  Google Scholar 

  3. R. DeHoff, Thermodynamics in Materials Science (CRC Press, Boca Raton, 2006)

    Book  Google Scholar 

  4. R. Jones, Thermodynamics and its Applications: An Overview (Mintek, Randburg, 1974)

    Google Scholar 

  5. J. Davidovits, Geopolymers: inorganic polymeric new materials. J. Therm. Anal. Calorim. 37(8), 1633–1656 (1991)

    Article  Google Scholar 

  6. S. Safran, Statistical Thermodynamics of Surfaces, Interfaces, and Membranes (CRC Press, Boca Raton, 2018)

    Book  MATH  Google Scholar 

  7. F.S. Bates, G.H. Fredrickson, Block copolymer thermodynamics: theory and experiment. Annu. Rev. Phys. Chem. 41(1), 525–557 (1990)

    Article  ADS  Google Scholar 

  8. A.A. Araújo-Filho, F.L. Silva, A. Righi, M.B. da Silva, B.P. Silva, E.W. Caetano, V.N. Freire, Structural, electronic and optical properties of monoclinic Na\(_2\)Ti\(_3\)O\(_7\) from density functional theory calculations: a comparison with xrd and optical absorption measurements. J. Solid State Chem. 250, 68–74 (2017)

    Article  ADS  Google Scholar 

  9. F.L.R. Silva, A.A. Araújo Filho, M.B. da Silva, K. Balzuweit, J.-L. Bantignies, E.W.S. Caetano, R.L. Moreira, V.N. Freire, A. Righi, Polarized raman, ftir, and dft study of Na\(_2\)Ti\(_3\)O\(_7\) microcrystals. J. Raman Spectrosc. 49(3), 538–548 (2018)

  10. M.E. Casida, C. Jamorski, K.C. Casida, D.R. Salahub, Molecular excitation energies to high-lying bound states from time-dependent density-functional response theory: characterization and correction of the time-dependent local density approximation ionization threshold. J. Chem. Phys. 108(11), 4439–4449 (1998)

    Article  ADS  Google Scholar 

  11. C. Lee, W. Yang, R.G. Parr, Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 37(2), 785 (1988)

    Article  ADS  Google Scholar 

  12. M. Segall, P.J. Lindan, M. Probert, C.J. Pickard, P.J. Hasnip, S. Clark, M. Payne, First-principles simulation: ideas, illustrations and the CASTEP code. J. Phys. Condens. Matter 14(11), 2717 (2002)

    Article  ADS  Google Scholar 

  13. J.P. Perdew, A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 23(10), 5048 (1981)

    Article  ADS  Google Scholar 

  14. J.P. Perdew, Density-functional approximation for the correlation energy of the inhomogeneous electron gas. Phys. Rev. B 33(12), 8822 (1986)

    Article  ADS  Google Scholar 

  15. J.P. Perdew, Y. Wang, Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B 45(23), 13244 (1992)

    Article  ADS  Google Scholar 

  16. J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple. Phys. Rev. Lett. 77(18), 3865 (1996)

    Article  ADS  Google Scholar 

  17. W.-S. Dai, M. Xie, Quantum statistics of ideal gases in confined space. Phys. Lett. A 311(4–5), 340–346 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. W.-S. Dai, M. Xie, Geometry effects in confined space. Phys. Rev. E 70(1), 016103 (2004)

    Article  ADS  Google Scholar 

  19. H. Potempa, L. Schweitzer, Dependence of critical level statistics on the sample shape. J. Phys. Condens. Matter 10(25), L431 (1998)

    Article  ADS  Google Scholar 

  20. L. Angelani, L. Casetti, M. Pettini, G. Ruocco, F. Zamponi, Topology and phase transitions: from an exactly solvable model to a relation between topology and thermodynamics. Phys. Rev. E 71(3), 036152 (2005)

    Article  ADS  Google Scholar 

  21. M.P. Bendsoe, O. Sigmund, Topology Optimization: Theory, Methods, and Applications (Springer, Berlin, 2013)

    MATH  Google Scholar 

  22. R. Oliveira, A.A. Araùjo Filho, R. Maluf, C. Almeida, The relativistic Aharonov–Bohm–Coulomb system with position-dependent mass. J. Phys. A Math. Theor. 53(4), 045304 (2020)

  23. P. Narang, C.A. Garcia, C. Felser, The topology of electronic band structures. Nat. Mater. 20(3), 293–300 (2021)

    Article  ADS  Google Scholar 

  24. L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics (Elsevier, Amsterdam, 2013)

    Google Scholar 

  25. L.E. Reichl, A Modern Course in Statistical Physics (Wiley, New York, 1999)

    MATH  Google Scholar 

  26. K. Huang, Introduction to Statistical Physics (CRC Press, Boca Raton, 2009)

    Book  Google Scholar 

  27. N. Zettili, Quantum Mechanics: Concepts and Applications (Jacksonville State University, Jacksonville, 2003)

    Google Scholar 

  28. E.M. Purcell, E.M. Purcell, E.M. Purcell, E.M. Purcell, Electricity and Magnetism, vol. 2 (McGraw-Hill, New York, 1965)

    MATH  Google Scholar 

  29. P.A. Tipler, R. Llewellyn, Modern Physics (Macmillan, New York, 2003)

    Google Scholar 

  30. H. Weyl, Gesammelte Abhandlungen: Band 1 bis 4, vol. 4 (Springer, Berlin, 1968)

    MATH  Google Scholar 

  31. V. Sreedhar, The classical and quantum mechanics of a particle on a knot. Ann. Phys. 359, 20–30 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. S. Ghosh, Particle on a torus knot: Anholonomy and Hannay angle. Int. J. Geom. Methods Mod. Phys. 15(06), 1850097 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. P. Das, S. Pramanik, S. Ghosh, Particle on a torus knot: constrained dynamics and semi-classical quantization in a magnetic field. Ann. Phys. 374, 67–83 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. D. Biswas, S. Ghosh, Quantum mechanics of a particle on a torus knot: curvature and torsion effects. EPL 132(1), 10004 (2020)

    Article  ADS  Google Scholar 

  35. R. Da Costa, Quantum mechanics of a constrained particle. Phys. Rev. A 23(4), 1982 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  36. Y.-L. Wang, M.-Y. Lai, F. Wang, H.-S. Zong, Y.-F. Chen, Geometric effects resulting from square and circular confinements for a particle constrained to a space curve. Phys. Rev. A 97(4), 042108 (2018)

    Article  ADS  Google Scholar 

  37. C. Ortix, Quantum mechanics of a spin-orbit coupled electron constrained to a space curve. Phys. Rev. B 91(24), 245412 (2015)

    Article  ADS  Google Scholar 

  38. L.C. da Silva, C.C. Bastos, F.G. Ribeiro, Quantum mechanics of a constrained particle and the problem of prescribed geometry-induced potential. Ann. Phys. 379, 13–33 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. G. Ferrari, G. Cuoghi, Schrödinger equation for a particle on a curved surface in an electric and magnetic field. Phys. Rev. Lett. 100(23), 230403 (2008)

    Article  ADS  Google Scholar 

  40. L.W. Tu, Manifolds. An Introduction to Manifolds (Springer, Berlin, 2011), pp. 47–83

    Google Scholar 

  41. L.W. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes, vol. 275 (Springer, Berlin, 2017)

    MATH  Google Scholar 

  42. L. Liu, C. Jayanthi, S. Wu, Structural and electronic properties of a carbon nanotorus: effects of delocalized and localized deformations. Phys. Rev. B 64(3), 033412 (2001)

    Article  ADS  Google Scholar 

  43. O.V. Kharissova, M.G. Castañón, B.I. Kharisov, Inorganic nanorings and nanotori: state of the art. J. Mater. Res. 34(24), 3998–4010 (2019)

    Article  ADS  Google Scholar 

  44. N. Chen, M.T. Lusk, A.C. Van Duin, W.A. Goddard III., Mechanical properties of connected carbon nanorings via molecular dynamics simulation. Phys. Rev. B 72(8), 085416 (2005)

    Article  ADS  Google Scholar 

  45. S. Madani, A.R. Ashrafi, The energies of (3, 6)-fullerenes and nanotori. Appl. Math. Lett. 25(12), 2365–2368 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  46. Z.A. Lewicka, Y. Li, A. Bohloul, W.Y. William, V.L. Colvin, Nanorings and nanocrescents formed via shaped nanosphere lithography: a route toward large areas of infrared metamaterials. Nanotechnology 24(11), 115303 (2013)

    Article  ADS  Google Scholar 

  47. H.Y. Feng, F. Luo, R. Kekesi, D. Granados, D. Meneses-Rodríguez, J.M. Garcia, A. García-Martín, G. Armelles, A. Cebollada, Magnetoplasmonic nanorings as novel architectures with tunable magneto-optical activity in wide wavelength ranges. Adv. Opt. Mater. 2(7), 612–617 (2014)

    Article  Google Scholar 

  48. L. Liu, G. Guo, C. Jayanthi, S. Wu, Colossal paramagnetic moments in metallic carbon nanotori. Phys. Rev. Lett. 88(21), 217206 (2002)

    Article  ADS  Google Scholar 

  49. S. Vojkovic, A.S. Nunez, D. Altbir, V.L. Carvalho-Santos, Magnetization ground state and reversal modes of magnetic nanotori. J. Appl. Phys. 120(3), 033901 (2016)

    Article  ADS  Google Scholar 

  50. J. Rodríguez-Manzo, F. Lopez-Urias, M. Terrones, H. Terrones, Magnetism in corrugated carbon nanotori: the importance of symmetry, defects, and negative curvature. Nano Lett. 4(11), 2179–2183 (2004)

    Article  ADS  Google Scholar 

  51. H. Du, W. Zhang, Y. Li, Silicon nitride nanorings: synthesis and optical properties. Chem. Lett. 43(8), 1360–1362 (2014)

    Article  Google Scholar 

  52. Y. Chan, B.J. Cox, J.M. Hill, Carbon nanotori as traps for atoms and ions. Physica B 407(17), 3479–3483 (2012)

    Article  ADS  Google Scholar 

  53. L. Peña-Parás, D. Maldonado-Cortés, O.V. Kharissova, K.I. Saldívar, L. Contreras, P. Arquieta, B. Castaños, Novel carbon nanotori additives for lubricants with superior anti-wear and extreme pressure properties. Tribol. Int. 131, 488–495 (2019)

    Article  Google Scholar 

  54. A.A. Araùjo Filho, J.A.A.S. Reis, Thermal aspects of interacting quantum gases in Lorentz-violating scenarios. Eur. Phys. J. Plus 136(3), 1–30 (2021)

  55. J. Reis et al., How does geometry affect quantum gases? arXiv preprint arXiv:2012.13613 (2020)

  56. R.R. Oliveira, A.A. Araújo Filho, F.C. Lima, R.V. Maluf, C.A. Almeida, Thermodynamic properties of an Aharonov–Bohm quantum ring. Eur. Phys. J. Plus 134(10), 495 (2019)

    Article  Google Scholar 

  57. R. Oliveira, A. Araújo Filho, Thermodynamic properties of neutral Dirac particles in the presence of an electromagnetic field. Eur. Phys. J. Plus 135(1), 99 (2020)

    Article  Google Scholar 

  58. A. Araújo Filho, R. Maluf, Thermodynamic properties in higher-derivative electrodynamics. Braz. J. Phys. 51(3), 820–830 (2021)

    Article  ADS  Google Scholar 

  59. A. Araújo Filho, Lorentz-violating scenarios in a thermal reservoir. Eur. Phys. J. Plus 136(4), 1–14 (2021)

    Google Scholar 

  60. A. Araújo Filho, A.Y. Petrov, Higher-derivative Lorentz-breaking dispersion relations: a thermal description. Eur. Phys. J. C 81(9), 1–16 (2021)

    Article  ADS  Google Scholar 

  61. A.A. Araújo Filho, A.Y. Petrov, Bouncing universe in a heat bath. Int. J. Mod. Phys. A 36(34 & 35), 2150242 (2021). https://doi.org/10.1142/S0217751X21502420

    Article  ADS  MathSciNet  Google Scholar 

  62. A.A. Araújo Filho, Thermodynamics of massless particles in curved spacetime. arXiv preprint arXiv:2201.00066 (2021)

Download references

Acknowledgements

Particularly, A. A. Araújo Filho acknowledges the Facultad de Física-Universitat de València and Gonzalo J. Olmo for the kind hospitality when part of this work was made. Moreover, this work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)-142412/2018-0, Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)-Finance Code 001, and CAPES-PRINT (PRINT - PROGRAMA INSTITUCIONAL DE INTERNACIONALIZAÇÃO) - 88887.508184/2020-00.

Author information

Authors and Affiliations

  1. Departamento de Física, Universidade Federal do Ceará (UFC), Campus do Pici, C.P. 6030, Fortaleza, CE, 60455-760, Brazil

    A. A. Araújo Filho

  2. Departamento de Física, Universidade Federal do Maranhão (UFMA), Campus Universitário do Bacanga, São Luís, MA, 65080-805, Brazil

    J. A. A. S. Reis

  3. Departamento de Física, Universidade Estadual do Maranhão (UEMA), Cidade Universitária Paulo VI, São Luís, MA, 65055-310, Brazil

    J. A. A. S. Reis

  4. Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T.Road, Kolkata, 700108, India

    Subir Ghosh

Authors
  1. A. A. Araújo Filho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. A. A. S. Reis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Subir Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Araújo Filho.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Araújo Filho, A.A., Reis, J.A.A.S. & Ghosh, S. Fermions on a torus knot. Eur. Phys. J. Plus 137, 614 (2022). https://doi.org/10.1140/epjp/s13360-022-02828-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-022-02828-y

Search

Navigation