Skip to main content
SpringerLink
Log in

Quantum Kibble–Zurek Mechanism

  • Chapter
  • First Online:
Equilibrium and Nonequilibrium Aspects of Phase Transitions in Quantum Physics

Part of the book series: Springer Theses ((Springer Theses))

  • 630 Accesses

Abstract

The Kibble–Zurek (KZ) mechanism, the paradigmatic theory addressing nonequilibrium dynamics involving continuous phase transitions, has played a key role throughout this thesis.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Recall that the upper critical dimension is defined as the spatial dimension of a system after which the mean field description becomes exact. For the nearest-neighbors Ising model \(d_U=4\).

References

  1. B. Damski, The simplest quantum model supporting the Kibble-Zurek mechanism of topological defect production: Landau-Zener transitions from a new perspective. Phys. Rev. Lett. 95, 035701 (2005). https://doi.org/10.1103/PhysRevLett.95.035701

    Article  ADS  Google Scholar 

  2. W.H. Zurek, U. Dorner, P. Zoller, Dynamics of a quantum phase transition. Phys. Rev. Lett. 95, 105701 (2005). https://doi.org/10.1103/PhysRevLett.95.105701

    Article  ADS  Google Scholar 

  3. J. Dziarmaga, Dynamics of a quantum phase transition: exact solution of the quantum Ising model. Phys. Rev. Lett. 95, 245701 (2005). https://doi.org/10.1103/PhysRevLett.95.245701

    Article  ADS  Google Scholar 

  4. A. Polkovnikov, Universal adiabatic dynamics in the vicinity of a quantum critical point. Phys. Rev. B 72, 161201 (2005). https://doi.org/10.1103/PhysRevB.72.161201

    Article  ADS  Google Scholar 

  5. M. Anquez, B.A. Robbins, H.M. Bharath, M. Boguslawski, T.M. Hoang, M.S. Chapman, Quantum Kibble-Zurek mechanism in a spin-1 Bose-Einstein condensate. Phys. Rev. Lett. 116, 155301 (2016). https://doi.org/10.1103/PhysRevLett.116.155301

    Article  ADS  Google Scholar 

  6. L.W. Clark, L. Feng, C. Chin, Universal space-time scaling symmetry in the dynamics of bosons across a quantum phase transition. Science 354, 606 (2016). https://doi.org/10.1126/science.aaf9657

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. J.-M. Cui, Y.-F. Huang, Z. Wang, D.-Y. Cao, J. Wang, W.-M. Lv, L. Luo, A. del Campo, Y.-J. Han, C.-F. Li, G.-C. Guo, Experimental trapped-ion quantum simulation of the Kibble-Zurek dynamics in momentum space. Sci. Rep. 6, 33381 (2016). https://doi.org/10.1038/srep33381

    Article  ADS  Google Scholar 

  8. X.-Y. Xu, Y.-J. Han, K. Sun, J.-S. Xu, J.-S. Tang, C.-F. Li, G.-C. Guo, Quantum simulation of Landau-Zener model dynamics supporting the Kibble-Zurek mechanism. Phys. Rev. Lett. 112, 035701 (2014). https://doi.org/10.1103/PhysRevLett.112.035701

    Article  ADS  Google Scholar 

  9. E. Ising, Beitrag zur Theorie des Ferromagnetismus. Z. Phys. 31, 253 (1925). https://doi.org/10.1007/BF02980577

    Article  ADS  Google Scholar 

  10. R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1989)

    MATH  Google Scholar 

  11. S. Sachdev, Quantum Phase Transitions, 2nd edn. (Cambridge University Press, Cambridge, UK, 2011)

    Book  Google Scholar 

  12. A. Dutta, G. Aeppli, B.K. Chakrabarti, U. Divakaran, T.F. Rosenbaum, D. Sen, Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information (Cambridge University Press, Cambridge, 2015)

    Book  Google Scholar 

  13. D. Porras, J.I. Cirac, Effective quantum spin systems with trapped ions. Phys. Rev. Lett. 92, 207901 (2004). https://doi.org/10.1103/PhysRevLett.92.207901

    Article  ADS  Google Scholar 

  14. A. Friedenauer, H. Schmitz, J.T. Glueckert, D. Porras, T. Schaetz, Simulating a quantum magnet with trapped ions. Nat. Phys. 4, 757 (2008). https://doi.org/10.1038/nphys1032

    Article  Google Scholar 

  15. K. Kim, M.-S. Chang, R. Islam, S. Korenblit, L.-M. Duan, C. Monroe, Entanglement and tunable spin-spin couplings between trapped ions using multiple transverse modes. Phys. Rev. Lett. 103, 120502 (2009). https://doi.org/10.1103/PhysRevLett.103.120502

    Article  ADS  Google Scholar 

  16. K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E.E. Edwards, J.K. Freericks, G.-D. Lin, L.-M. Duan, C. Monroe, Quantum simulation of frustrated Ising spins with trapped ions. Nature 465, 590 (2010). https://doi.org/10.1038/nature09071

    Article  ADS  Google Scholar 

  17. R. Islam, E.E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J.K. Freericks, C. Monroe, Onset of a quantum phase transition with a trapped ion quantum simulator. Nat. Commun. 2, 377 (2011). https://doi.org/10.1038/ncomms1374

    Article  Google Scholar 

  18. R. Islam, C. Senko, W.C. Campbell, S. Korenblit, J. Smith, A. Lee, E.E. Edwards, C.-C.J. Wang, J.K. Freericks, C. Monroe, Emergence and frustration of magnetism with variable-range interactions in a quantum simulator. Science 340, 583 (2013). https://doi.org/10.1126/science.1232296

    Article  ADS  Google Scholar 

  19. P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B.P. Lanyon, M. Heyl, R. Blatt, C.F. Roos, Direct observation of dynamical quantum phase transitions in an interacting many-body system. Phys. Rev. Lett. 119, 080501 (2017). https://doi.org/10.1103/PhysRevLett.119.080501

    Article  ADS  Google Scholar 

  20. J. Zhang, G. Pagano, P.W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A.V. Gorshkov, Z.-X. Gong, C. Monroe, Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature 551, 601 (2017). https://doi.org/10.1038/nature24654

    Article  ADS  Google Scholar 

  21. D. Jaschke, K. Maeda, J.D. Whalen, M.L. Wall, L.D. Carr, Critical phenomena and Kibble-Zurek scaling in the long-range quantum Ising chain. New J. Phys. 19, 033032 (2017), http://stacks.iop.org/1367-2630/19/i=3/a=033032

    Article  ADS  Google Scholar 

  22. U.  Schollwöck, The density-matrix renormalization group in the age of matrix product states. Ann. Phys. (N. Y.) 326, 96 (2011). https://doi.org/10.1016/j.aop.2010.09.012

    Article  ADS  MathSciNet  Google Scholar 

  23. H. Lipkin, N. Meshkov, A. Glick, Validity of many-body approximation methods for a solvable model. Nucl. Phys. 62, 188 (1965). https://doi.org/10.1016/0029-5582(65)90862-X

    Article  Google Scholar 

  24. F.J. Dyson, Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12, 91 (1969). https://doi.org/10.1007/BF01645907

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. J.M. Kosterlitz, D.J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181 (1973), http://stacks.iop.org/0022-3719/6/i=7/a=010

    Article  ADS  Google Scholar 

  26. J. Fröhlich, T. Spencer, The phase transition in the one-dimensional Ising model with \(1/r^{2}\) interaction energy. Comm. Math. Phys. 84, 87 (1982), https://projecteuclid.org:443/euclid.cmp/1103921047

  27. E. Luijten, H. Meßingfeld, Criticality in one dimension with inverse square-law potentials. Phys. Rev. Lett. 86, 5305 (2001). https://doi.org/10.1103/PhysRevLett.86.5305

    Article  ADS  Google Scholar 

  28. M.C. Angelini, G. Parisi, F. Ricci-Tersenghi, Relations between short-range and long-range Ising models. Phys. Rev. E 89, 062120 (2014). https://doi.org/10.1103/PhysRevE.89.062120

    Article  ADS  Google Scholar 

  29. P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A.V. Gorshkov, C. Monroe, Non-local propagation of correlations in quantum systems with long-range interactions. Nature 511, 198 (2014). https://doi.org/10.1038/nature13450

    Article  ADS  Google Scholar 

  30. J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P.W. Hess, P. Hauke, M. Heyl, D.A. Huse, C. Monroe, Many-body localization in a quantum simulator with programmable random disorder. Nat. Phys. 12, 907 (2016). https://doi.org/10.1038/nphys3783

    Article  Google Scholar 

  31. B.P. Lanyon, C. Maier, M. Holzäpfel, T. Baumgratz, C. Hempel, P. Jurcevic, I. Dhand, A.S. Buyskikh, A.J. Daley, M. Cramer, M.B. Plenio, R. Blatt, C.F. Roos, Efficient tomography of a quantum many-body system. Nat. Phys. EP (2017). https://doi.org/10.1038/nphys4244

    Article  Google Scholar 

  32. P. Jurcevic, B.P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, C.F. Roos, Quasiparticle engineering and entanglement propagation in a quantum many-body system. Nature 511, 202EP (2014). https://doi.org/10.1038/nature13461

    Article  ADS  Google Scholar 

  33. S. Olmschenk, K.C. Younge, D.L. Moehring, D.N. Matsukevich, P. Maunz, C. Monroe, Manipulation and detection of a trapped \({{\rm Yb}}^{+}\) hyperfine qubit. Phys. Rev. A 76, 052314 (2007). https://doi.org/10.1103/PhysRevA.76.052314

    Article  ADS  Google Scholar 

  34. D. James, Quantum dynamics of cold trapped ions with application to quantum computation. App. Phys. B 66, 181 (1998). https://doi.org/10.1007/s003400050373

    Article  ADS  Google Scholar 

  35. T. Koffel, M. Lewenstein, L. Tagliacozzo, Entanglement entropy for the long-range Ising chain in a transverse field. Phys. Rev. Lett. 109, 267203 (2012). https://doi.org/10.1103/PhysRevLett.109.267203

    Article  ADS  Google Scholar 

  36. P. Hauke, L. Tagliacozzo, Spread of correlations in long-range interacting quantum systems. Phys. Rev. Lett. 111, 207202 (2013). https://doi.org/10.1103/PhysRevLett.111.207202

    Article  ADS  Google Scholar 

  37. K. Binder, Finite size scaling analysis of Ising model block distribution functions. Z. Phys. B 43, 119 (1981). https://doi.org/10.1007/BF01293604

    Article  ADS  Google Scholar 

  38. M.E. Fisher, M.N. Barber, Scaling theory for finite-size effects in the critical region. Phys. Rev. Lett. 28, 1516 (1972). https://doi.org/10.1103/PhysRevLett.28.1516

    Article  ADS  Google Scholar 

  39. J.G. Brankov, Introduction to Finite-size Scaling (Leuven University Press, Leuven, 1996)

    Google Scholar 

  40. S. Mukherjee, A. Rajak, B.K. Chakrabarti, Classical-to-quantum crossover in the critical behavior of the transverse-field Sherrington-Kirkpatrick spin glass model. Phys. Rev. E 92, 042107 (2015). https://doi.org/10.1103/PhysRevE.92.042107

    Article  ADS  Google Scholar 

  41. E. Luijten, H.W.J. Blöte, Classical critical behavior of spin models with long-range interactions. Phys. Rev. B 56, 8945 (1997). https://doi.org/10.1103/PhysRevB.56.8945

    Article  ADS  Google Scholar 

  42. A.K. Chandra, A. Das, B.K.C. (Eds.), Quantum Quenching, Annealing and Computation (Springer, Berlin, 2010)

    MATH  Google Scholar 

  43. C. De Grandi, V. Gritsev, A. Polkovnikov, Quench dynamics near a quantum critical point: application to the sine-Gordon model. Phys. Rev. B 81, 224301 (2010). https://doi.org/10.1103/PhysRevB.81.224301

    Article  ADS  Google Scholar 

  44. M. Kac, C.J. Thompson, Critical behavior of several lattice models with long-range interaction. J. Math. Phys. 10, 1373 (1969). https://doi.org/10.1063/1.1664976

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. X.-L. Deng, D. Porras, J.I. Cirac, Effective spin quantum phases in systems of trapped ions. Phys. Rev. A 72, 063407 (2005). https://doi.org/10.1103/PhysRevA.72.063407

    Article  ADS  Google Scholar 

  46. M. Heyl, A. Polkovnikov, S. Kehrein, Dynamical quantum phase transitions in the transverse-field Ising model. Phys. Rev. Lett. 110, 135704 (2013). https://doi.org/10.1103/PhysRevLett.110.135704

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Theoretical Atomic, Molecular and Optical Physics, Queen’s University Belfast, Belfast, UK

    Ricardo Puebla

Authors
  1. Ricardo Puebla
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ricardo Puebla .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Puebla, R. (2018). Quantum Kibble–Zurek Mechanism. In: Equilibrium and Nonequilibrium Aspects of Phase Transitions in Quantum Physics. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-00653-2_6

Download citation

Publish with us

Policies and ethics

Search

Navigation