Skip to main content
SpringerLink
Log in

Local Transformations and Long-Range Entanglement

  • Chapter
  • First Online:
Quantum Information Meets Quantum Matter

Part of the book series: Quantum Science and Technology ((QST))

  • 4340 Accesses

Abstract

To understand the origin of the topological phenomena discussed in the previous chapters, we need a microscopic theory for topological order. It was realized that the key microscopic feature of topologically ordered systems is the existence of long-range many-body entanglement in the ground-state wave function. Useful tools from quantum information theory to characterize many-body entanglement are local transformations, including local unitary (LU) transformations and stochastic local (SL) transformations. In this chapter, we apply these tools to the study of gapped quantum phases and phase transitions and establish the connection between topological order and long/short-range entanglement. This allows us to obtain a general theory to study topological order and symmetry-breaking order within the same framework. This leads to a basic understanding of the structure of the full quantum phase diagram.

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 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
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.

    We note that the symmetric local unitary transformation in the form \( \mathscr {T} \Big ( e^{-i \int _0^1 d g\; \tilde{H}(g)} \Big )\) always connect to the identity transformation continuously. This may not be the case for the transformation in the form \(U^M_{circ}\). To rule out that possibility, we define symmetric local unitary transformations as those that connect to the identity transformation continuously.

References

  1. Bennett, C. H., Popescu, S., Rohrlich, D., Smolin, J. A., & Thapliyal, A. V. (2000, December). Exact and asymptotic measures of multipartite pure-state entanglement. Physical Review A, 63, 012307.

    Google Scholar 

  2. Bernevig, B. A., & Zhang, S. C. (2006, March). Quantum spin hall effect. Physical Review Letter, 96, 106802.

    Google Scholar 

  3. Bravyi, S., Hastings, M. B., & Verstraete, F. (2006, July). Lieb-robinson bounds and the generation of correlations and topological quantum order. Physical Review Letter, 97, 050401.

    Google Scholar 

  4. Bravyi, S., Hastings, M. B., & Michalakis, S. (2010). Topological quantum order: Stability under local perturbations. Journal of Mathematical Physics, 51(9), 093512.

    Article  ADS  MathSciNet  Google Scholar 

  5. Bravyi, S., Hastings, M. B., & Michalakis, S. (2010). Topological quantum order: Stability under local perturbations. Journal of Mathematical Physics, 51(9), 093512.

    Article  ADS  MathSciNet  Google Scholar 

  6. Chen, X., Zheng-Cheng, G., & Wen, X.-G. (2010, October). Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. Physical Review B, 82(15), 155138.

    Google Scholar 

  7. Duncan, F., & Haldane, M. (1983). Continuum dynamics of the 1-D heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model. Physics Letters A, 93, 464.

    Article  ADS  MathSciNet  Google Scholar 

  8. Furukawa, S., Misguich, G., & Oshikawa, M. (2006). Systematic derivation of order parameters through reduced density matrices. Physical Review Letters, 96(4), 047211.

    Article  ADS  Google Scholar 

  9. Furukawa, S., Misguich, G., & Oshikawa, M. (2007). Reduced density matrices and topological order in a quantum dimer model. Journal of Physics: Condensed Matter, 19(14), 145212.

    ADS  Google Scholar 

  10. Haah, J. (2011, April). Local stabilizer codes in three dimensions without string logical operators. Physical Review A, 83, 042330.

    Google Scholar 

  11. Haah, J. (2014). Bifurcation in entanglement renormalization group flow of a gapped spin model. Physical Review B, 89(7), 075119.

    Article  ADS  Google Scholar 

  12. Hastings, M. B., & Wen, X.-G. (2005, July). Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance. Physical Review B, 72, 045141.

    Google Scholar 

  13. Kalmeyer, V., & Laughlin, R. B. (1987). Equivalence of the resonating-valence-bond and fractional quantum hall states. Physical Review Letter, 59, 2095–2098.

    Article  ADS  Google Scholar 

  14. Kane, C. L., & Mele, E. J. (2005, November). Quantum spin hall effect in graphene. Physical Review Letter, 95, 226801.

    Google Scholar 

  15. Kane, C. L., & Mele, E. J. (2005, September). \({Z}_{2}\) topological order and the quantum spin hall effect. Physical Review Letter, 95, 146802.

    Google Scholar 

  16. Keski-Vakkuri, E., & Wen, X.-G. (1993). The ground state structure and modular transformations of fractional quantum hall states on a torus. International Journal of Modern Physics B, 7(25), 4227–4259.

    Article  ADS  Google Scholar 

  17. Kitaev, A. Y. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2–30.

    Article  ADS  MathSciNet  Google Scholar 

  18. Ko, W.-H., Lee, P. A., & Wen, X.-G. (2009, June). Doped kagome system as exotic superconductor. Physical Review B, 79(21), 214502.

    Google Scholar 

  19. Kong, L., & Wen, X-G. (2014). Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions. arXiv:1405.5858.

  20. Kou, S.-P., & Wen, X.-G. (2009, December). Translation-symmetry-protected topological orders in quantum spin systems. Physical Review B, 80(22), 224406.

    Google Scholar 

  21. Kou, S.-P., Levin, M., & Wen, X.-G. (2008, October). Mutual chern-simons theory for \(z_{2}\) topological order. Physical Review B, 78(15), 155134.

    Google Scholar 

  22. Lan, T., & Wen, X.-G. (2013). Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models.

    Google Scholar 

  23. Levin, M. A., & Wen, X.-G. (2005, January). String-net condensation: A physical mechanism for topological phases. Physical Review B, 71(4), 045110.

    Google Scholar 

  24. Levin, M., & Wen, X.-G. (2006, March). Detecting topological order in a ground state wave function. Physical Review Letter, 96(11), 110405.

    Google Scholar 

  25. Liang, F., Kane, C. L., & Mele, E. J. (2007). Topological insulators in three dimensions. Physical Review Letter, 98, 106803.

    Article  ADS  Google Scholar 

  26. Lieb, E. H., & Robinson, D. W. (1972). The finite group velocity of quantum spin systems. Communications in Mathematical Physics, 28, 251–257. https://doi.org/10.1007/BF01645779.

    Article  ADS  MathSciNet  Google Scholar 

  27. Lloyd, S. (1996). Universal quantum simulators. Science, 273(5278), 1073–1078.

    Article  ADS  MathSciNet  Google Scholar 

  28. Maciejko, J., Qi, X.-L., Karch, A., & Zhang, S.-C. (2010, December). Fractional topological insulators in three dimensions. Physical Review Letter, 105, 246809.

    Google Scholar 

  29. Moessner, R., & Sondhi, S. L. (2001, February). Resonating valence bond phase in the triangular lattice quantum dimer model. Physical Review Letter, 86, 1881–1884.

    Google Scholar 

  30. Moore, J. E., & Balents, L. (2007, March). Topological invariants of time-reversal-invariant band structures. Physical Review B, 75, 121306.

    Google Scholar 

  31. Moore, G., & Read, N. (1991). Nonabelions in the fractional quantum hall effect. Nuclear Physics B, 360(2), 362–396.

    Article  ADS  MathSciNet  Google Scholar 

  32. Nussinov, Z., & Ortiz, G. (2009). A symmetry principle for topological quantum order. Annals of Physics, 324(5), 977–1057.

    Article  ADS  MathSciNet  Google Scholar 

  33. Orús, R., Wei, T.-C., Buerschaper, O., & Van den Nest, M. (2014). Geometric entanglement in topologically ordered states. New Journal of Physics, 16(1), 013015.

    Article  ADS  Google Scholar 

  34. Owari, M., Braunstein, S. L., Nemoto, K., & Murao, M. (2008). \(\varepsilon \)-convertibility of entangled states and extension of schmidt rank in infinite-dimensional systems. Quantum Information & Computation, 8(1), 30–52.

    MathSciNet  MATH  Google Scholar 

  35. Pesin, D., & Balents, L. (2010, May). Mott physics and band topology in materials with strong spin-orbit interaction. Nature Physics, 6(5), 376–381.

    Google Scholar 

  36. Pollmann, F., Berg, E., Turner, A. M., & Oshikawa, M. (2012, February). Symmetry protection of topological phases in one-dimensional quantum spin systems. Physical Review B, 85, 075125.

    Google Scholar 

  37. Qi, X.-L., Hughes, T. L., & Zhang, S.-C. (2008). Topological field theory of time-reversal invariant insulators. Physical Review B, 78(19), 195424.

    Article  ADS  Google Scholar 

  38. Raghu, S., Qi, X.-L., Honerkamp, C., & Zhang, S.-C. (2008, April). Topological mott insulators. Physical Review Letter, 100, 156401.

    Google Scholar 

  39. Read, N., & Green, D. (2000, April). Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum hall effect. Physical Review B, 61, 10267–10297.

    Google Scholar 

  40. Read, N., & Sachdev, S. (1991, April). Large- N expansion for frustrated quantum antiferromagnets. Physical Review Letter, 66, 1773–1776.

    Google Scholar 

  41. Schuch, N., Perez-Garcia, D., & Cirac, I. (2011, October). Classifying quantum phases using matrix product states and projected entangled pair states. Physical Review B, 84, 165139.

    Google Scholar 

  42. Schuch, N., Cirac, I., & Pérez-García, D. (2010). Peps as ground states: Degeneracy and topology. Annals of Physics, 325(10), 2153–2192.

    Article  ADS  MathSciNet  Google Scholar 

  43. Swingle, B., & McGreevy, J. (2014). Renormalization group constructions of topological quantum liquids and beyond. arXiv:1407.8203.

  44. Swingle, B., Barkeshli, M., McGreevy, J., & Senthil, T. (2011, May). Correlated topological insulators and the fractional magnetoelectric effect. Physical Review B, 83(19), 195139-+.

    Google Scholar 

  45. Verstraete, F., Cirac, J. I., Latorre, J. I., Rico, E., & Wolf, M. M. (2005, April). Renormalization-group transformations on quantum states. Physical Review Letter, 94, 140601.

    Google Scholar 

  46. Vidal, G. (2007, November). Entanglement renormalization. Physical Review Letter, 99, 220405.

    Google Scholar 

  47. Wei, T.-C., Das, D., Mukhopadyay, S., Vishveshwara, S., & Goldbart, P. M. (2005). Global entanglement and quantum criticality in spin chains. Physical Review A, 71(6), 060305.

    Article  ADS  Google Scholar 

  48. Wen, X. G. (1989, October). Vacuum degeneracy of chiral spin states in compactified space. Physical Review B, 40, 7387–7390.

    Google Scholar 

  49. Wen, X.-G. (1991, August). Mean-field theory of spin-liquid states with finite energy gap and topological orders. Physical Review B, 44(6), 2664–2672.

    Google Scholar 

  50. Wen, X. G. (1991, February). Non-abelian statistics in the fractional quantum hall states. Physical Review Letter, 66, 802–805.

    Google Scholar 

  51. Wen, X.-G. (2002, April). Quantum orders and symmetric spin liquids. Physical Review B, 65(16), 165113.

    Google Scholar 

  52. Wen, X.-G., & Niu, Q. (1990, May). Ground-state degeneracy of the fractional quantum hall states in the presence of a random potential and on high-genus riemann surfaces. Physical Review B, 41(13), 9377–9396.

    Google Scholar 

  53. Wen, X. G., & Niu, Q. (1990, May). Ground-state degeneracy of the fractional quantum hall states in the presence of a random potential and on high-genus riemann surfaces. Physical Review B, 41, 9377–9396.

    Google Scholar 

  54. Wen, X.-G., Wilczek, F., & Zee, A. (1989, June). Chiral spin states and superconductivity. Physical Review B, 39(16), 11413–11423.

    Google Scholar 

  55. Wen, X.-G. (1990). Topological orders in rigid states. International Journal of Modern Physics B, 4(02), 239–271.

    Article  ADS  MathSciNet  Google Scholar 

  56. Yang, B.-J., & Kim, Y. B. (2010, August). Topological insulators and metal-insulator transition in the pyrochlore iridates. Physical Review B, 82, 085111.

    Google Scholar 

  57. Yoshida, B. (2011, January). Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes. Annals of Physics, 326(1), 15–95.

    Google Scholar 

  58. Zeng, B., & Wen, X.-G. (2015). Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity. Physical Review B, 91(12), 125121.

    Article  ADS  Google Scholar 

  59. Zhang, Y., Ran, Y., & Vishwanath, A. (2009, June). Topological insulators in three dimensions from spontaneous symmetry breaking. Physical Review B, 79, 245331.

    Google Scholar 

  60. Zheng-Cheng, G., & Wen, X.-G. (2009, October). Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order. Physical Review B, 80(15), 155131.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Guelph, Guelph, ON, Canada

    Bei Zeng

  2. Physics, California Institute of Technology, Pasadena, CA, USA

    Xie Chen

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

    Duan-Lu Zhou

  4. Massachusetts Institute of Technology, Cambridge, MA, USA

    Xiao-Gang Wen

Authors
  1. Bei Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xie Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Duan-Lu Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiao-Gang Wen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bei Zeng .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Science+Business Media, LLC, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Zeng, B., Chen, X., Zhou, DL., Wen, XG. (2019). Local Transformations and Long-Range Entanglement. In: Quantum Information Meets Quantum Matter. Quantum Science and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9084-9_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-4939-9084-9_7

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4939-9082-5

  • Online ISBN: 978-1-4939-9084-9

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation