Abstract
We show the existence of supersymmetry and degeneracy for an arbitrary \((\)even or odd\()\) number of Majorana fermions without invoking any symmetry of the Hamiltonian. We analyze supersymmetry at a finite temperature using the thermofield dynamics formalism. Such an analysis emerges from the construction of a thermal formulation for Majorana fermions in the thermofield dynamics scenario. Furthermore, we derive thermal braiding operators via Bogoliubov transformations and find its action on a thermal Bell state. Based on quantum fidelity, we measure the distance between the thermal states and the corresponding pure states. In the limit as the temperature tends to zero, we recover the pure states and the fidelity approaches unity. This analysis makes it possible to analyze the influence of temperature on the evolution of the system.
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References
E. Majorana, “Teoria simmetrica dell’elettrone e del positrone,” Nuovo Cimento, 14, 171–184 (1937).
A. Gando, Y. Gando, T. Hachiya et al. [KamLAND-Zen Collab.], “Search for Majorana neutrinos near the inverted mass hierarchy region with KamLAND-Zen,” Phys. Rev. Lett., 117, 082503, 6 pp. (2016).
A. Yu. Kitaev, “Unpaired Majorana fermions in quantum wires,” Phys. Usp., 44, 131–136 (2001).
J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, “Non-Abelian statistics and topological quantum information processing in 1D wire networks,” Nature Phys., 7, 412–417 (2011).
Z. H. Wang, Topological Quantum Computation (Conference Board of the Mathematical Sciences. Regional Conference Series in Mathematics, Vol. 112), AMS, Providence, RI (2008).
L.-W. Yu and M.-L. Ge, “More about the doubling degeneracy operators associated with Majorana fermions and Yang–Baxter equation,” Sci. Rep., 5, 8102, 7 pp. (2015).
X.-M. Zhao, J. Yu, J. He, Q.-B. Chen, Y. Liang, and S.-P. Kou, “The simulation of non-Abelian statistics of Majorana fermions in Ising chain with Z2 symmetry,” Modern Phys. Lett. B, 31, 1750123, 10 pp. (2017); arXiv: 1602.07444.
L. Long and Y. Yue, “An anyon model in a toric honeycomb lattice,” Commun. Theor. Phys., 55, 80–84 (2011); arXiv: 1006.0804.
A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons,” Ann. Phys., 303, 2–30 (2003).
A. Yu. Kitaev, “Anyons in an exactly solved model and beyond,” Ann. Phys., 321, 2–111 (2006); arXiv: cond-mat/0506438.
C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Modern Phys., 80, 1083–1159 (2008).
J. Lee and F. Wilczek, “Algebra of Majorana doubling,” Phys. Rev. Lett., 111, 226402, 4 pp. (2013); arXiv: 1307.3245.
X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang, “Time-reversal-invariant topological superconductors and superfluids in two and three dimensions,” Phys. Rev. Lett., 102, 187001, 4 pp. (2009); arXiv: 0803.3614.
T. H. Hsieh, G. B. Halász, and T. Grover, “All Majorana models with translation symmetry are supersymmetric,” Phys. Rev. Lett., 117, 166802, 6 pp. (2016); arXiv: 1604.08591.
R. U. Haq and L. H. Kauffman, “\(Z_2\) topological order and topological protection of Majorana fermion qubits,” Condens. Matter, 6, 11, 22 pp. (2021).
L. Van Hove, “Supersymmetry and positive temperature for simple systems,” Nucl. Phys. B, 207, 15–28 (1982).
R. Parthasarathy and R. Sridhar, “Supersymmetry in thermofield dynamics,” Phys. Lett. A, 279, 17–22 (2001).
F. Khanna, A. P. C. Malbouisson, J. M. C. Malbouisson, and A. E. Santana, Thermal Quantum Field Theory: Algebraic Aspects and Applications, World Sci., Singapore (2009).
V. G. Kac, “Lie superalgebras,” Adv. Math., 26, 8–96 (1977).
M. R. de Traubenberg, “Clifford algebras in physics,” Adv. Appl. Clifford Alg., 19, 869–908 (2009).
M. Takesaki, Tomita’s Theory of Modular Hilbert Algebras and its Applications (Lecture Notes in Mathematics, Vol. 128), Springer, Berlin, Heidelberg (1970).
I. Ojima, “Gauge fields at finite temperatures – ‘Thermo field dynamics’ and the KMS condition and their extension to gauge theories,” Ann. Phys., 137, 1–32 (1981).
O. Bratelli and D. W. Robinson, Operators Algebras and Quantum Statistical Mechanics: Equilibrium States Models in Quantum Statistical Mechanics, Vols. 1, 2 (Texts and Monographs in Physics) Springer, Berlin (1997).
A. E. Santana, A. M. Neto, J. D. M. Vianna, and F. C. Khanna, “w\(^{*}\)-Algebra, Poincare group, and quantum kinetic theory,” Internat. J. Theor. Phys., 38, 641–651 (1999).
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, Cambridge (2000).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 209, pp. 502–514 https://doi.org/10.4213/tmf10106.
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Trindade, M.A.S., Floquet, S. Majorana fermions, supersymmetry, and thermofield dynamics. Theor Math Phys 209, 1747–1757 (2021). https://doi.org/10.1134/S0040577921120072
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DOI: https://doi.org/10.1134/S0040577921120072