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Exact numerical methods for electron-phonon problems

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Abstract

In the last few years solid state physics has increasingly benefited from scientific computing and the significance of numerical techniques is likely to keep on growing quickly in this field. Because of the high complexity of solids, which are made of a huge number of interacting electrons and nuclei, a full understanding of their properties cannot be developed using analytical methods only. Numerical simulations do not only provide quantitative results for the properties of specific materials but are also widely used to test the validity of theories and analytical approaches.

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References

  1. For an overview on several important aspects of strongly correlated electron systems, see Science 288, Nr. 5465 (2000).

  2. Bishop A. R. and Swanson B. I., Novel Electronic Materials: the MX Family. Los Alamos Science, 21 (1993) 133.

    Google Scholar 

  3. Hubbard J., Proc. R. Soc. London, Ser. A, 276 (1963) 238.

    Article  ADS  Google Scholar 

  4. Holstein T., Ann. Phys. (N.Y.), 8 (1959) 325; 343.

    Article  Google Scholar 

  5. Bauml B., Wellein G. and Fehske H., Phys. Rev. B, 58 (1998) 3663.

    Article  ADS  Google Scholar 

  6. Wellein G., Röder H. and Fehske H., Phys. Rev. B, 53 (1996) 9666.

    Article  ADS  Google Scholar 

  7. Augier D. and Poilblanc D., Eur. Phys. J. B, 1 (1998) 19.

    Article  ADS  Google Scholar 

  8. Sykora S., Hübsch A., Becker K. W., Wellein G. and Fehske H., Phys. Rev. B, 71 (2005) 045112.

    Article  ADS  Google Scholar 

  9. Bonča J., Trugman S. A. and Batistić I., Phys. Rev. B, 60 (1999) 1633.

    Article  ADS  Google Scholar 

  10. Ku L. C., Trugman S. A. and Bonča J., Phys. Rev. B, 65 (2002) 174306.

    Article  ADS  Google Scholar 

  11. Cullum J. K. and Willoughby R. A., Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Volume I & II (Birkhäuser, Boston) 1985.

  12. Davidson E. R., J. Comput. Phys., 17 (1975) 87.

    Article  ADS  MathSciNet  Google Scholar 

  13. Barret R. et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (SIAM, Philadelphia) 1993.

    Google Scholar 

  14. Weisse A., Wellein G., Alvermann A. and Fehske H., The kernel polynomial method, to be published in Rev. Mod. Phys., 78 (2006), URL u]http://ar**v.org/abs/cond-mat/0504627.

  15. Silver R. N. and Röder H., Phys. Rev. E, 56 (1997) 4822.

    Article  ADS  Google Scholar 

  16. Sénéchal D., Perez D. and Pioro-Ladrière M., Phys. Rev. Lett., 84 (2000) 522.

    Article  ADS  Google Scholar 

  17. White S. R., Phys. Rev. Lett., 69 (1992) 2863; Phys. Rev. B, 48 (1993) 10345.

    Article  ADS  Google Scholar 

  18. For a recent review, see Schollwöck U., Rev. Mod. Phys., 77 (2005) 259.

  19. Noack R. M. and White S. R., The Density Matrix Renormalization Group,in Density-Matrix Renormalization, A New Numerical Method in Physics, edited by Peschel I., Wang X., Kaulke M. and Hallberg K. (Springer, Berlin) 1999, Chapt. 2.

  20. See http://alps.comp-phys.org/.

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

    Article  ADS  Google Scholar 

  22. Bray J. W. and Chui S. T., Phys. Rev. B, 19 (1979) 4876.

    Article  ADS  Google Scholar 

  23. White S. R. and Noack R. M., Phys. Rev. Lett., 68 (1992) 3487.

    Article  ADS  Google Scholar 

  24. Bursill R. J., Phys. Rev. B, 60 (1999) 1643.

    Article  ADS  Google Scholar 

  25. Caron L. G. and Moukouri S., Phys. Rev. Lett., 76 (1996) 4050; Phys. Rev. B, 56 (1997) R8471.

    Article  ADS  Google Scholar 

  26. Peschel I., Kaulke M. and Legeza Ö., Ann. Phys. (Leipzig), 8 (1999) 153; Peschel I. and Chung M.-C., J. Phys. A, 32 (1999) 8419; Chung M.-C. and Peschel I., Phys. Rev. B, 62 (2000) 4191; 64 (2001) 064412.

    Article  ADS  Google Scholar 

  27. Bonca J., Gubernatis J. E., Guerrero M., Jeckelmann E. and White S. R., Phys. Rev. B, 61 (2000) 3251.

    Article  ADS  Google Scholar 

  28. Jeckelmann E., Phys. Rev. Lett., 89 (2002) 236401.

    Article  ADS  Google Scholar 

  29. Jeckelmann E. and White S. R., Phys. Rev B, 57 (1998) 6376.

    Article  ADS  Google Scholar 

  30. Zhang C., Jeckelmann E. and White S. R., Phys. Rev. Lett., 80 (1998) 2661.

    Article  ADS  Google Scholar 

  31. Zhang C., Jeckelmann E. and White S. R., Phys. Rev. B, 60 (1999) 14092.

    Article  ADS  Google Scholar 

  32. Romero A. H., Brown D. W. and Lindenberg K., J. Chem. Phys., 109 (1998) 6540.

    Article  ADS  Google Scholar 

  33. Jeckelmann E., Zhang C. and White S. R., Phys. Rev B, 60 (1999) 7950.

    Article  ADS  Google Scholar 

  34. Weisse A., Fehske H., Wellein G. and Bishop A. R., Phys. Rev. B, 62 (2000) R747.

    Article  ADS  Google Scholar 

  35. Weisse A., Wellein G. and Fehske H., Density-Matrix Algorithm for Phonon Hilbert Space Reduction in the Numerical Diagonalization of Quantum Many-Body Systems, in High Performance Computing in Science and Engineering’01, edited by Krause E. and Jäger W. (Springer, Berlin) 2002, pp 131.

  36. Friedman B., Phys Rev. B, 61 (2000) 6701.

    Article  ADS  Google Scholar 

  37. Kuzmany H., Solid-State Spectroscopy (Springer, Berlin) 1998.

    Book  Google Scholar 

  38. Jeckelmann E., Gebhard F. and Essler F. H. L., Phys. Rev. Lett., 85 (2000) 3910.

    Article  ADS  Google Scholar 

  39. Jeckelmann E., Phys. Rev. B, 66 (2002) 045114.

    Article  ADS  Google Scholar 

  40. White S. R. and Huseuse D. A., Phys. Rev. B, 48 (1993) 3844.

    Article  ADS  Google Scholar 

  41. McCulloch I. P. and Gulácsi M., Europhys. Lett., 57 (2002) 852.

    Article  ADS  Google Scholar 

  42. Ramasesha S., Pati S. K., Krishnamurthy H. R., Shuai Z. and Brédas J. L., Phys. Rev. B, 54 (1996) 7598.

    Article  ADS  Google Scholar 

  43. Boman M. and Bursill R. J., Phys. Rev. B, 57 (1998) 15167.

    Article  ADS  Google Scholar 

  44. Ramasesha S., Pati S. K., Shuai Z. and Brédas J. L., in Advances in Quantum Chemistry, Vol. 38 (Academic Press) 2001, pp. 121–215.

    Article  ADS  Google Scholar 

  45. Daul S. and Noack R. M., Phys. Rev. B, 58 (1998) 2635.

    Article  ADS  Google Scholar 

  46. Hallberg K. A., Phys. Rev. B, 52 (1995) R9827.

    Article  ADS  Google Scholar 

  47. Zhang C., Numerical Study of the One-Dimensional Holstein Model, Ph.D. thesis, University of California at Irvine, 1999.

    Google Scholar 

  48. Pati S. K., Ramasesha S., Shuai Z. and Brédas J. L., Phys. Rev. B, 59 (1999) 14827; Ramasesha S., Pati S. K., Krishnamurthy H. R., Shuai Z. and Brédas J. L., Synth. Met., 85 (1997) 1019.

    Article  ADS  Google Scholar 

  49. Kühner T. D. and White S. R., Phys. Rev B, 60 (1999) 335.

    Article  ADS  Google Scholar 

  50. Essler F. H. L, Jeckelmann E. and Gebhard F., Phys. Rev. B, 64 (2001) 125119.

    Article  ADS  Google Scholar 

  51. Jeckelmann E., Phys. Rev. B, 67 (2003) 075106.

    Article  ADS  Google Scholar 

  52. Benthien H., Gebhard B. and Jeckelmann E., Phys. Rev. Lett., 92 (2004) 256401.

    Article  ADS  Google Scholar 

  53. Benthien H., Dynamical Properties of Quasi One-Dimensional Correlated Electron Systems (Ph.D. thesis, University of Marburg, Germany) 2005.

    Google Scholar 

  54. Nishimoto S. and Jeckelmann E., J. Phys. Condens. Matter, 16 (2004) 613.

    Article  ADS  Google Scholar 

  55. Hager G., Jeckelmann E., Fehske H. and Wellein G., Exact Numerical Treatment of Finite Quantum Systems using Leading-Edge Supercomputers,in Modelling, Simulation and Optimization of Complex Processes, edited by Bock H. G., Kostina E., Phu H.-X. and Rannacher R. (Springer, Heidelberg) 2005, pp. 165.

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Acknowledgments

We would like to thank B. Bäuml, H. Benthien, F. Essler, F. Gebhard, G. Hager. S. Nishimoto, G. Wellein, and A. Weisse for valuable discussions.

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Authors and Affiliations

  1. Institut für Theoretische Physik, Universität Hannover, Appelstraße 2, D-30167, Hannover, Germany

    E. Jeckelmann

  2. Institut für Physik, Ernst-Moritz-Arndt-Universitat Greifswald, D-17487, Greifswald, Germany

    H. Fehske

Authors
  1. E. Jeckelmann
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  2. H. Fehske
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Jeckelmann, E., Fehske, H. Exact numerical methods for electron-phonon problems. Riv. Nuovo Cim. 30, 259–292 (2007). https://doi.org/10.1393/ncr/i2007-10021-y

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