Effect of Fluctuation in the Coupling Strength on Critical Dynamics of 1D Transverse Field Quantum Ising Model

Abstract

The effect of uniformly distributed fluctuation in the coupling strength of a 100 spins transverse field quantum Ising chain is studied using the Density Matrix Renormalization Group on the Matrix Product States formalism. Uniform noise with mean zero and amplitude η is introduced to uniform nearest neighbour coupling of value unity. Disordered averages of thermodynamic quantities are calculated from 100 disordered realizations for each transverse field reading. We show that the system exhibits distinct behaviour of ordered and disordered state separated at η~1.0. For η ≲ 1.0, thermodynamic behaviour of the system gradually deviates from pure quantum Ising model. For η > 1.0 system exhibits highly fluctuating thermodynamic behaviour. Edward-Anderson order parameters are calculated and shown to be much less fluctuating and suitable as an order parameter for η > 1.0. Qualitative behaviour of the system is not affected by finite-size effect and replacing the fluctuation with normally distributed noise. Finally, for noise level between η = 1.0 and 1.5, the system exhibits a faster phase transition and enhances transverse magnetization before the critical point. For certain ranges of noise amplitude before the critical point, the quantum fluctuations are amplified. This suggests a potential improvement of quantum annealing within that regime.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. 1.

    Inoue, J.I.: Application of the quantum spin glass theory to image restoration. Phys. Rev. E. 63(4), 046114 (2001)

    ADS  Article  Google Scholar 

  2. 2.

    Venturelli, D., Mandra, S., Knysh, S., O’Gorman, B., Biswas, R., Smelyanskiy, V.: Quantum optimization of fully connected spin glasses. Physical Review X. 5(3), 031040 (2015)

    ADS  Article  Google Scholar 

  3. 3.

    McMahon, P.L., Marandi, A., Haribara, Y., Hamerly, R., Langrock, C., Tamate, S., et al.: A fully programmable 100-spin coherent Ising machine with all-to-all connections. Science. 354(6312), 614–617 (2016)

    ADS  Article  Google Scholar 

  4. 4.

    Blinc, R.: On the isotopic effects in the ferroelectric behaviour of crystals with short hydrogen bonds. J. Phys. Chem. Solids. 13(3–4), 204–211 (1960)

    ADS  Article  Google Scholar 

  5. 5.

    Sengupta, K., Powell, S., Sachdev, S.: Quench dynamics across quantum critical points. Phys. Rev. A. 69, 053616 (2004)

    ADS  Article  Google Scholar 

  6. 6.

    Calabrese, P., Cardy, J.: Time dependence of correlation functions following a quantum quench. Phys. Rev. Lett. 96, 136801 (2006)

    ADS  Article  Google Scholar 

  7. 7.

    Rossini, D., Silva, A., Mussardo, G., Santoro, G.E.: Effective thermal dynamics following a quantum quench in a spin chain. Phys. Rev. Lett. 102, 127204 (2009)

    ADS  Article  Google Scholar 

  8. 8.

    Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse Ising model. Phys. Rev. E. 58, 5355–5363 (1998)

    ADS  Article  Google Scholar 

  9. 9.

    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution. arXiv:quant-ph/0001106 (2000)

  10. 10.

    Jouzdani, P., Novais, E., Tupitsyn, I.S., Mucciolo, E.R.: Fidelity threshold of the surface code beyond single-qubit error models. Phys. Rev. A. 90(4), 042315 (2014)

    ADS  Article  Google Scholar 

  11. 11.

    de Falco, D., Tamascelli, D.: An introduction to quantum annealing. RAIRO-Theoretical Informatics and Applications. 45(1), 99–116 (2011)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Kryzhanovsky, B., Malsagov, M., Karandashev, I.: Investigation of finite-size 2D Ising model with a noisy matrix of spin-spin interactions. Entropy. 20(8), 585 (2018)

    ADS  Article  Google Scholar 

  13. 13.

    Sondhi, S.L., Girvin, S.M., Carini, J.P., Shahar, D.: Continuous quantum phase transitions. Rev. Mod. Phys. 69(1), 315 (1997)

    ADS  Article  Google Scholar 

  14. 14.

    Chakrabarti, B.K.: Critical behaviour of the Ising spin-glass models in a transverse field. Phys. Rev. B. 24(7), 4062 (1981)

    ADS  Article  Google Scholar 

  15. 15.

    Dos Santos, R.R., dos Santos, R.Z., Kischinhevsky, M.: Transverse Ising spin-glass model. Phys. Rev. B. 31(7), 4694 (1985)

    ADS  Article  Google Scholar 

  16. 16.

    Eisert, J., Cramer, M., Plenio, M.B.: Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys. 82(1), 277 (2010)

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Pfeuty, P.: The one-dimensional Ising model with a transverse field. Ann. Phys. 57(1), 79–90 (1970)

    ADS  Article  Google Scholar 

  18. 18.

    Sachdev, S.: Quantum phase transitions. Cambridge University Press, New York (2011)

    Google Scholar 

  19. 19.

    Elliott, R.J., Pfeuty, P., Wood, C.: Ising model with a transverse field. Phys. Rev. Lett. 25(7), 443 (1970)

    ADS  Article  Google Scholar 

  20. 20.

    Park, S.B., Cha, M.C.: Matrix product state approach to the finite-size scaling properties of the one-dimensional critical quantum Ising model. J. Korean Phys. Soc. 67(9), 1619–1623 (2015)

    ADS  Article  Google Scholar 

  21. 21.

    Schollwöck, U.: The density-matrix renormalization group. Rev. Mod. Phys. 77(1), 259 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  22. 22.

    Schollwöck, U.: The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326(1), 96–192 (2011)

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    White, S.R.: Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69(19), 2863 (1992)

    ADS  Article  Google Scholar 

  24. 24.

    White, S.R.: Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B. 48(14), 10345 (1993)

    ADS  Article  Google Scholar 

  25. 25.

    Kole, A. H.: Density Matrix Renormalization Group calculations for the Ising Model with a Transverse Field (Bachelor's thesis) (2018)

  26. 26.

    ITensor C++ Library, available at itensor.org

  27. 27.

    Young, A. P.: Simulations of Spin Glass Systems. In Finite-size scaling and numerical simulation of statistical systems (pp. 466–488) (1990)

    Google Scholar 

Download references

Acknowledgements

The authors thank the Malaysian Ministry of Higher Education (MOHE) for FRGS grant: FP031-2017A and University of Malaya Frontier Research Grant: FG032-17AFR. S.Y. Pang thanks Dr. Miles Stoudenmire for his kind assistance in technical matters related to the ITensor Library. S.Y. Pang is supported by Skim Biasiswa MyBrainSc Scholarship under the Malaysian Ministry of Higher Education (MOHE).

Author information

Affiliations

Authors

Corresponding author

Correspondence to S. Y. Pang.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pang, S.Y., Muniandy, S.V. & Kamali, M.Z.M. Effect of Fluctuation in the Coupling Strength on Critical Dynamics of 1D Transverse Field Quantum Ising Model. Int J Theor Phys 59, 250–260 (2020). https://doi.org/10.1007/s10773-019-04320-3

Download citation

Keywords

  • Quantum Ising model
  • Random/Noisy Coupling
  • Matrix Product States
  • Critical Dynamics