Skip to main content
SpringerLink
Log in

The Density Matrix Renormalization Group

  • Chapter
  • First Online:
Strongly Correlated Systems

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 176))

Abstract

Since its creation in 1992, the density matrix renormalization group (DMRG) method has evolved and mutated. From its original formulation in a condensed matter context, it has been adapted to study problems in various fields, such as nuclear physics and quantum chemistry, becoming one of the dominant numerical techniques to simulate strongly correlated systems. In this chapter, we shall cover many technical aspects of the DMRG, from a “traditional”, or “conventional” perspective, describing the theoretical fundamentation, as well as the details of the algorithm.

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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

Similar content being viewed by others

References

  1. F.D.M. Haldane, Phys. Lett. 80A, 281 (1980)

    MathSciNet  ADS  Google Scholar 

  2. T.C. Choy, Phys. Lett. 80A, 49 (1980)

    Google Scholar 

  3. See also S.R. Manmana et al., Phys. Rev. A 84, 043601 (2011)

    Google Scholar 

  4. R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983)

    Article  ADS  Google Scholar 

  5. R. Schrieffer, Theory of Superconductivity, Advanced Books Classics (Perseus, Chicago, 1999)

    Google Scholar 

  6. S.R. White, Phys. Rev. Lett. 69, 2863 (1992)

    Article  ADS  Google Scholar 

  7. S.R. White, Phys. Rev. B 48, 10345 (1993)

    Article  ADS  Google Scholar 

  8. K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975)

    Article  ADS  Google Scholar 

  9. R. Bulla, T.A. Costi, T. Pruschke, Rev. Mod. Phys. 80, 395 (2008)

    Article  ADS  Google Scholar 

  10. S. Ostlund, S. Rommer, Phys. Rev. Lett. 75, 3537 (1995)

    Article  ADS  Google Scholar 

  11. S. Rommer, S. Ostlund, Phys. Rev. B 55, 2164 (1997)

    Article  ADS  Google Scholar 

  12. H.A. Kramers, G.H. Wannier, Phys. Rev. 60, 263 (1941)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. R.J. Baxter, J. Math. Phys. 9, 650 (1968)

    Article  ADS  Google Scholar 

  14. I. Peschel, X. Wang, M. Kaulke, K. Hallberg (eds.), Density-Matrix Renormalization—A New Numerical Method in Physics: Lectures of a Seminar and Workshop held at the Max-Planck-Institut für Physik, Lecture Notes in Physics (Springer, Berlin, 1999)

    Google Scholar 

  15. U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005)

    Article  ADS  Google Scholar 

  16. K. Hallberg, Density matrix renormalization: a review of the method and its applications, in Theoretical Methods for Strongly Correlated Electrons, CRM Series in Mathematical Physics, ed. by D. Senechal, A.-M. Tremblay, C. Bourbonnais (Springer, New York, 2003)

    Google Scholar 

  17. K. Hallberg, Adv. Phys. 55, 477 (2006)

    Article  ADS  Google Scholar 

  18. R.M. Noack, S.R. Manmana, Proceedings of the “IX. Training course in the physics of correlated electron systems and high-Tc superconductors”, Vietri sul Mare (Salerno, Italy, October 2004). AIP Conf. Proc. 789, 93 (2005)

    Google Scholar 

  19. G. De Chiara, M. Rizzi, D. Rossini, S. Montangero, J. Comput. Theor. Nanosci. 5, 1277 (2008)

    Article  Google Scholar 

  20. A.E. Feiguin, Proceedings of the “XV. Training course in the physics of correlated electron systems”, Vietri sul Mare (Salerno, Italy, October 2010). AIP Conf. Proc. 1419, 5 (2011)

    Google Scholar 

  21. A.C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge, 1997)

    Google Scholar 

  22. C. Lanczos, J. Res. Natl. Bur. Stand. 45, 255 (1950)

    Google Scholar 

  23. D.G. Pettifor, D.L. Weaire (eds.), The Recursion Method and Its Applications, Springer Series in Solid State Sciences, vol. 58 (Springer, Berlin, 1985)

    Google Scholar 

  24. S.R. White, Phys. Rev. Lett. 77, 3633 (1996)

    Article  ADS  Google Scholar 

  25. R.M. Noack, S.R. White, D.J. Scalapino, The density-matrix renormalization group for fermion systems, in Computer Simulations in Condensed Matter Physics, vol. VII, ed. by D.P. Landau, K.K. Mon, H.B. Schüttler (Spinger, Heidelberg, 1994), UCI-CMTHE-94-03

    Google Scholar 

  26. S. Liang, H. Pang, Europhys. Lett. 32, 173 (1995)

    Article  ADS  Google Scholar 

  27. M.S.L. du Croo de Jongh, J.M.J. van Leeuwen, Phys. Rev. B 57, 8494 (1998)

    Google Scholar 

  28. T. Xiang, J.Z. Lou, Z.B. Zu, Phys. Rev. B 64, 104414 (2001)

    Article  ADS  Google Scholar 

  29. G. Vidal, Phys. Rev. Lett. 91, 147902 (2003)

    Article  ADS  Google Scholar 

  30. G. Vidal, Phys. Rev. Lett. 93, 040502 (2004)

    Article  ADS  Google Scholar 

  31. M.B. Plenio, S. Virmani, Quantum Inf. Comput. 7, 1 (2007)

    MathSciNet  MATH  Google Scholar 

  32. L. Amico, R. Fazio, A. Osterloh, V. Vedral, Rev. Mod. Phys. 80, 517 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Med. Phys. 81, 865 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. F. Verstraete, J.I. Cirac, Phys. Rev. B 73, 094423 (2006)

    Article  ADS  Google Scholar 

  35. J. Eisert, M. Cramer, M.B. Plenio, Rev. Mod. Phys. 82, 277 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. M.B. Hastings, J. Stat. Mech.: Theory Exp. 2007, P08024 (2007)

    Google Scholar 

  37. P. Calabrese, J. Cardy, J. Stat. Mech.: Theory Exp. 2004, P06002 (2004)

    Google Scholar 

  38. P. Calabrese, J. Cardy, Int. J. Quantum Inf. 4, 429 (2006)

    Google Scholar 

  39. M.M. Wolf, Phys. Rev. Lett. 96, 010404 (2006)

    Article  ADS  Google Scholar 

  40. D. Gioev, I. Klich, Phys. Rev Lett. 96, 100503 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  41. W. Li, L. Ding, R. Yu, T. Roscilde, S. Haas, Phys. Rev. B 74, 073103 (2006)

    Article  ADS  Google Scholar 

  42. T. Barthel, M.-C. Chung, U. Schollwöck, Phys. Rev. A 74, 022329 (2006)

    Article  ADS  Google Scholar 

  43. F. Verstraete, M.M. Wolf, D. Perez-Garcia, J.I. Cirac, Phys. Rev. Lett. 96, 220601 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  44. U. Schollwöck, Ann. Phys. 326, 96 (2011)

    Article  ADS  MATH  Google Scholar 

  45. F. Verstraete, D. Porras, J.I. Cirac, Phys. Rev. Lett. 93, 227205 (2004)

    Article  ADS  Google Scholar 

  46. See also P. Pippan, S.R. White, H.G. Evertz, Phys. Rev. B 81, 081103 (2010)

    Google Scholar 

  47. I. McCulloch, J. Stat. Mech. 2007, P10014 (2007)

    Google Scholar 

  48. B. Bauer et al. (ALPS collaboration), J. Stat. Mech. 2011, P05001 (2011)

    Google Scholar 

  49. F. Albuquerque et al. (ALPS collaboration), J. Magnet. Magnet. Mater. 310, 1187 (2007)

    Google Scholar 

  50. F. Alet et al. (ALPS collaboration), J. Phys. Soc. Jpn. Suppl. 74, 30 (2005)

    Google Scholar 

  51. R. Orus, G. Vidal, Phys. Rev. B 78, 155117 (2008)

    Article  ADS  Google Scholar 

  52. I.P. McCulloch, arXiv:0804.2509

    Google Scholar 

  53. B. Pirvu, F. Verstraete, G. Vidal, Phys. Rev. B 83, 125104 (2011)

    Article  ADS  Google Scholar 

  54. J. Jordan, R. Orus, G. Vidal, F. Verstraete, J.I. Cirac, Phys. Rev. Lett. 101, 250602 (2008)

    Article  ADS  Google Scholar 

  55. M.C. Bañuls, M.B. Hastings, F. Verstraete, J.I. Cirac, Phys. Rev. Lett. 102, 240603 (2009)

    Article  ADS  Google Scholar 

  56. I. Pizorn, L. Wang, F. Verstraete, Phys. Rev. A 83, 052321 (2011)

    Article  ADS  Google Scholar 

  57. M.B. Hastings, J. Math. Phys. 50, 095207 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  58. N. Schuch, M.M. Wolf, F. Verstraete, J.I. Cirac, Phys. Rev. Lett. 100, 030504 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  59. N. Schuch, M.M. Wolf, F. Verstraete, J. Ignacio Cirac, Phys. Rev. Lett. 98, 140506 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  60. N. Schuch, I. Cirac, F. Verstraete, Phys. Rev. Lett. 100, 250501 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  61. F. Verstraete, J.I. Cirac, Phys. Rev. Lett. 104, 190405 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  62. G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)

    Article  ADS  Google Scholar 

  63. G. Evenbly, G. Vidal, Phys. Rev. B 79, 144108 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  64. G. Evenbly, G. Vidal, Phys. Rev. Lett. 102, 180406 (2009)

    Article  ADS  Google Scholar 

  65. G. Vidal, in Understanding Quantum Phase Transitions, ed. by L.D. Carr (Taylor & Francis, Boca Raton, 2010)

    Google Scholar 

  66. P. Corboz, S.R. White, G. Vidal, M. Troyer, arXiv:1104.5463

    Google Scholar 

  67. S. Yan, D.A. Huse, S.R. White, Science 332, 1173 (2011)

    Article  ADS  Google Scholar 

  68. Gu Zheng-Cheng, Michael Levin, Xiao-Gang Wen, Phys. Rev. B 78, 205116 (2008)

    Article  ADS  Google Scholar 

  69. Z.Y. Xie, H.C. Jiang, Q.N. Chen, Z.Y. Weng, T. Xiang, Phys. Rev. Lett. 103, 160601 (2009)

    Article  ADS  Google Scholar 

  70. H.H. Zhao, Z.Y. Xie, Q.N. Chen, Z.C. Wei, J.W. Cai, T. Xiang, Phys. Rev. B 81, 174411 (2010)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Northeastern University, Boston, MA, 02115, USA

    Adrian E. Feiguin

Authors
  1. Adrian E. Feiguin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adrian E. Feiguin .

Editor information

Editors and Affiliations

  1. Dipartimento di Fisica "E.R. Caianiello", Università degli Studi di Salerno, Via Ponte don Melillo, Fisciano (SA), 84084, Italy

    Adolfo Avella

  2. Dipartimento di Fisica "E.R. Caianiello", Università degli Studi di Salerno, Via Ponte don Melillo, Fisciano (SA), 84084, Italy

    Ferdinando Mancini

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Feiguin, A.E. (2013). The Density Matrix Renormalization Group. In: Avella, A., Mancini, F. (eds) Strongly Correlated Systems. Springer Series in Solid-State Sciences, vol 176. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35106-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-35106-8_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-35105-1

  • Online ISBN: 978-3-642-35106-8

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

Publish with us

Policies and ethics

Search

Navigation