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Different Bell Inequalities as Probes to Detect Quantum Phase Transitions

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Abstract

We introduce a method which based on Bell inequalities, to study quantum phase transitions. By using the non-linear programming, we compare two different kinds of Bell inequalities, the original Bell inequality and Clauser-Horne-Shimony-Holt (CHSH) inequality. And we find that the original Bell inequality is more accurate in detecting the Bell non-locality. By defining the maximal violation of Bell inequalities, we calculate two kinds of transitions, the one is magnetic transition in the spin-\(\frac {1}{2}\) XX model and the other is topological transition in the Kitaev honeycomb model. The critical points are detected successfully. Compared with traditional methods, our method requires no prior knowledge of order parameters and it is base-free.

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References

  1. Sachdev, S.: Quantum Phase Transitions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  2. Sondhi, S.L., Girvin, S.M., Carini, J.P., Shahar, D.: Continuous quantum phase transitions. Rev. Mod. Phys. 69, 315 (1997)

    Article  ADS  Google Scholar 

  3. Sachdev, S., Keimer, B.: Quantum criticality. Phys. Today 64, 29 (2011)

    Article  Google Scholar 

  4. Tsui, D.C., Stormer, H.L., Gossard, A.C.: Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559 (1982)

    Article  ADS  Google Scholar 

  5. Kitaev, A., Preskill, J.: Topological entanglement entropy. Phys. Rev. Lett. 96, 110404 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  6. Pollmann, F., Turner, A.M., Berg, E., Oshikawa, M.: Entanglement spectrum of a topological phase in one dimension. Phys. Rev. B 81, 064439 (2010)

    Article  ADS  Google Scholar 

  7. Franchini, F., Cui, J., Amico, L., Fan, H., Gu, M., Korepin, V., Kwek, L.C., Vedral, V.: Local convertibility and the quantum simulation of edge states in Many-Body systems. Phys. Rev. X 4, 041028 (2014)

    Google Scholar 

  8. Hamma, A., Zhang, W., Haas, S., Lidar, D.A.: Entanglement, fidelity, and topological entropy in a quantum phase transition to topological order. Phys. Rev. B 77, 155111 (2008)

    Article  ADS  Google Scholar 

  9. Cui, J., Cao, J.P., Fan, H.: Quantum-Information Approach to the quantum phase transition in the kitaev honeycomb model. Phys. Rev. A 82, 022319 (2010)

    Article  ADS  Google Scholar 

  10. Mandel, S.S.: Generalization of laughlin’s theory for the fractional quantum hall effect. J. Phys.-Condens. Mat. 30, 405605 (2018)

    Article  Google Scholar 

  11. Parameswaran, S.A., Feldman, B.E.: Quantum hall valley nematics. J. Phys.-Condens. Mat. 31, 273001 (2019)

    Article  ADS  Google Scholar 

  12. Allen, M., Cui, Y.T., Yue, M.E., Mogi, M., Kawamura, M., Fulga, I.C., Goldhaber-Gordon, D., Tokura, Y., Shen, Z.X.: Visualization of an axion insulating state at the transition between 2 chiral quantum anomalous hall states. Proc. Natl. Acad. Sci. USA 116, 14511 (2019)

    Article  ADS  Google Scholar 

  13. Prosniak, O.A.: On the size of boundary effects in the ising chain. Phys. Scripta. 94, 085201 (2019)

    Article  ADS  Google Scholar 

  14. Zhang, Y.Q., Sun, Y.T., He, Q.L.: Einstein-Podolsky-Rosen Steering and quantum phase transition in spin chains. Int. J. Theor. Phys. 57, 2978 (2018)

    Article  MATH  Google Scholar 

  15. Zhao, B.W., Weinberg, P., Sandvik, A.W.: Symmetry-enhanced discontinuous phase transition in a two-dimensional quantum magnet. Nat. Phys. 15, 678 (2019)

    Article  Google Scholar 

  16. Bishop, R.F., Li, P.H.Y., Gotze, O., Richter, J.: Frustrated Spin-1/2 heisenberg magnet on a square-lattice bilayer: High-order study of the quantum critical behavior of the J(1)-J(2)-J(1)(Perpendicular to) Model. Phys. Rev. B 100, 024401 (2019)

    Article  ADS  Google Scholar 

  17. Strecka, J., Pojas, O., de Souza, S.M.: Absence of a spontaneous Long-Range order in a mixed spin-(1/2,3/2) ising model on a decorated square lattice due to anomalous spin frustration driven by a magnetoelastic coupling. Phys. Lett. A 383, 2451 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  18. Xiao, X.B., Liu, Z.F., Wu, Q.P., Li, Y., Li, F., Du, Y.: Anisotropic magnetoelectronic structures and Magneto-Transport properties of topological dirac semimetal nanowires. J. Magn. Magn. Mater. 484, 373 (2019)

    Article  ADS  Google Scholar 

  19. Chen, X.H., Wang, X.J.: Topological orders and quantum phase transitions in a One-Dimensional extended quantum compass model. Acta. Phys. Sin. 67, 190301 (2018)

    Article  Google Scholar 

  20. Dana, I., Kubo, K.: Floquet systems with Hall effect: Topological properties and phase transitions. Phys. Rev. B 100, 045107 (2019)

    Article  ADS  Google Scholar 

  21. Longhi, S.: Topological phase transition in Non-Hermitian quasicrystals. Phys. Rev. Lett. 122, 237601 (2019)

    Article  ADS  Google Scholar 

  22. Chen, Q., Zhang, G.Q., Chen, J.Q., Xu, J.B.: Topological quantum phase transitions in the 2-D kitaev honeycomb model. Quantum Inf. Process. 18, 1 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Landau, L.D., Lifschitz, E.M.: Statistical Physics: Course of theoretical physics, vol. 5. Pergamon, London (1958)

    Google Scholar 

  24. Amico, L., Patanè, D.: Entanglement crossover close to a quantum critical point. Europhys. Lett. 77, 17001 (2006)

    Article  ADS  Google Scholar 

  25. Amico, L., Fazio, R., Osterloh, A., Vedral, V.: Entanglement in Many-Body systems. Rev. Mod. Phys. 80, 517 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Osborne, T.J., Nielsen, M.A.: Entanglement in a simple quantum phase transition. Phys. Rev. A 66, 032110 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  27. Vidal, G., Latorre, J.I., Rico, E., Kitaev, A.: Entanglement in quantum critical phenomena. Phys. Rev. Lett. 90, 227902 (2003)

    Article  ADS  Google Scholar 

  28. Osterloh, A., Amico, L., Falci, G., Fazio, R.: Scaling of entanglement close to a quantum phase transition. Nature (London) 416, 608 (2002)

    Article  ADS  Google Scholar 

  29. Lieb, E., Schultz, T., Mattis, D.: Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407 (1961)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Vedral, V.: Quantum entanglement. Nat. Phys. 10, 256 (2014)

    Article  Google Scholar 

  31. Ollivier, H., Zurek, W.H.: Quantum discord: A measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)

    Article  ADS  MATH  Google Scholar 

  32. Henderson, L., Vedral, V.: Classical, quantum and total correlations. J. Phys. A 34, 6899 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Modi, K., Brodutch, A., Cable, H., Paterek, T., Vedral, V.: The Classical-Quantum Boundary for correlations: Discord and Related Measures. Rev. Mod. Phys. 84, 1655 (2012)

    Article  ADS  Google Scholar 

  34. Sarandy, M.S.: Classical correlation and quantum discord in critical systems. Phys. Rev. A 80, 022108 (2009)

    Article  ADS  Google Scholar 

  35. Werlang, T., Trippe, C., Ribeiro, G.A.P., Rigolin, G.: Quantum correlations in spin chains at finite temperatures and quantum phase transitions. Phys. Rev. Lett. 105, 095702 (2010)

    Article  ADS  Google Scholar 

  36. Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014)

    Article  ADS  Google Scholar 

  37. Streltsov, A., Adesso, G., Plenio, M.B.: Colloquium: Quantum coherence as a resource. Rev. Mod. Phys. 89, 041003 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  38. Shi, H.L., Liu, S.Y., Wang, X.H., Yang, W.L., Yang, Z.Y., Fan, H.: Coherence depletion in the grover quantum search algorithm. Phys. Rev. A 95, 032307 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  39. Hou, J.X., Liu, S.Y., Wang, X.H., Yang, W.L.: Role of coherence during classical and quantum decoherence. Phys. Rev. A 96, 042324 (2017)

    Article  ADS  Google Scholar 

  40. Chen, J.J., Cui, J., Zhang, Y.R., Fan, H.: Coherence susceptibility as a probe of quantum phase transitions. Phys. Rev. A 94, 022112 (2016)

    Article  ADS  Google Scholar 

  41. Girolami, D., Adesso, G: Quantum discord for general two-qubit states: Analytical progress. Phys. Rev. A 83, 052108 (2011)

    Article  ADS  Google Scholar 

  42. Luo, S.L.: Quantum Discord for Two-qubit Systems. Phys. Rev. A 77, 042303 (2008)

    Article  ADS  Google Scholar 

  43. Werner, R.F.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277 (1989)

    Article  ADS  MATH  Google Scholar 

  44. Wiseman, H.M., Jones, S.J., Doherty, A.C.: Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox. Phys. Rev. Lett. 98, 140402 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Einstein, A., Podolsky, B., Rosen, N.: Can Quantum-Mechanical description of physical reality be considered complete?. Phys. Rev. 47, 777 (1935)

    Article  ADS  MATH  Google Scholar 

  46. Bell, J.S.: On the einstein podolsky rosen paradox. Phys. 1, 195 (1964)

    Article  MathSciNet  Google Scholar 

  47. Braunstein, S.L., Mann, A., Revzen, M.: Maximal violation of bell inequalities for mixed states. Phys. Rev. Lett. 68, 3259 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Batle, J., Casas, M.: Nonlocality and Entanglement in the XY model. Phys. Rev. A 82, 062101 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  49. Hirsch, F., Quintino, M.T., Vértesi, T., Pusey, M.F., Brunner, N.: Algorithmic construction of local hidden variable models for entangled quantum states. Phys. Rev. Lett. 117, 190402 (2016)

    Article  ADS  Google Scholar 

  50. Vandenberghe, L., Boyd, S.: Semidefinite programming. Soc. Ind. Appl. Math. 38, 49 (1996)

    MathSciNet  MATH  Google Scholar 

  51. Cavalcanti, D., Guerini, L., Rabelo, R., Skrzypczyk, P.: General method for constructing local hidden variable models for entangled quantum states. Phys. Rev. Lett. 117, 190401 (2016)

    Article  ADS  Google Scholar 

  52. Quan, Q., Zhu, H.J., Liu, S.Y., Fei, S.M., Fan, H., Yang, W.L.: Steering Bell-Diagonal states. Sci. Rep. 6, 22025 (2016)

    Article  ADS  Google Scholar 

  53. De Pasquale, A., Costantini, G., Facchi, P., Florio, G., Pascazio, S., Yuasa, K.: XX Model on the circle. Eur. Phys. J. Spec. Top. 160, 127 (2008)

    Article  Google Scholar 

  54. Horodecki, R., Horodecki, P., Horodecki, M.: Violating Bell Inequality by Mixed Spin-1/2 states: Necessary and Sufficient Condition. Phys. Lett. A 200, 340 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Son, W., Amico, L., Plastina, F., Vedral, V.: Quantum instability and edge entanglement in the Quasi-Long-Range order. Phys. Rev. A 79, 022302 (2009)

    Article  ADS  Google Scholar 

  56. Kitaev, A.: Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We acknowledge Jin-Jun Chen and Jin-Xing Hou for their valuable discussions. This work was supported by the National Natural Science Foundation of China under Grant Nos. 11775177, 11775178, 11647057 and 11705146, the Peng Huaiwu Center for Fundamental Theory under Grant No.12047502 the Basic Research Plan of Natural Science in Shaanxi Province under Grant No. 2018JQ1014, the Major Basic Research Program of Natural Science of Shaanxi Province under Grant No. 2017ZDJC-32, the Key Innovative Research Team of Quantum Many-Body Theory and Quantum Control in Shaanxi Province under Grant No. 2017KCT-12, the Double First-Class University Construction Project of Northwest University.

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Authors and Affiliations

  1. Institute of Modern Physics, Northwest University, Xi’an, 710127, China

    Feng-Lin Wu, Si-Yuan Liu, Wen-Li Yang & Heng Fan

  2. Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an, 710127, China

    Feng-Lin Wu, Si-Yuan Liu & Wen-Li Yang

  3. Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China

    Feng-Lin Wu, Si-Yuan Liu & Heng Fan

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  1. Feng-Lin Wu
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Correspondence to Si-Yuan Liu.

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Wu, FL., Liu, SY., Yang, WL. et al. Different Bell Inequalities as Probes to Detect Quantum Phase Transitions. Int J Theor Phys 60, 1611–1623 (2021). https://doi.org/10.1007/s10773-021-04784-2

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