Flow Equations for Hamiltonians

Some papers with similar ideas

There are at least two lines of similar ideas which preceeded the flow equations

• The idea to diagonalize a quantum Hamiltonian by considering it at different discretely chosen energy and length scales and thus performing a renormalization group analysis dates back to the seventies. Two main streams were
  • on the one hand side the solution of the Kondo problem by
    • K.G. Wilson, The Renormalization Group: Critical Phenomena and the Kondo Problem, Rev. Mod. Phys. 47 (1975) 773
  • and on the other hand side the investigation of the Anderson localization by
    • D.C. Licciardello and D.J. Thouless, Constancy of Minimum Metallic Conductivity in Two Dimensions, Phys. Rev. Lett. 35 (1975) 1475,
    • F.J. Wegner, Electrons in Disordered Systems. Scaling near the Mobility Edge, Z. Phys. B25 (1976) 327,
    • P.A. Lee, Real-Space Scaling Studies of Localization, Phys. Rev. Lett. 42 (1979) 1492,
      Scaling Studies of Localization, J. Noncr. Solids 35+36 (1980) 21,
    • and E. Domany and S. Sarker, Renormalization Group Study of Anderson Localization, Phys. Rev B20 (1979) 945,
      A Scaling Theory of Anderson Localization, J. Phys. C13 (1980) L273.
• A second line came from two Russian groups, which eventually informed us about their work: They performed continuous transformations for spin systems with a constant generator of the unitary transformation,
  • one around D.A. Garanin and V.S. Lutinov,
    • High-temperature spin wave dynamics of the uniaxial antiferromagnets, Solid State Communications 44 (1982) 1359;
    • Normal modes and relaxation processes in magnetically ordered materials with single-site anisotropy, Teor. mat. Fiz. 60 (1984) 133;
    • Spin waves in the easy-plane ferromagnets, Physica A 126 (1984) 133;
    • and with L.V. Panina, The magnon-magnon interactions in easy-plane antiferromagnets, Physica A 184 (1992) 523-557.
  • The other one is around V.L. Safonov,
    • Method of the Spin Hamiltonian Diagonalization, Phys. Lett. A 97 (1983) 164;
    • with A.V. Andrienko, V.I. Ozhogin, A. Yu. Yakubovskii, Study of mechanisms of nuclear-spin-wave relaxation in the weakly anisotropic antiferromagnet CsMnF₃, Zh. Eksp. Teor. Fiz. 84 (1983) 1158 [Sov. Phys. JETP 57 (1983)673];
    • with R.M. Farzetdinova, On a possibility of magnon pumping in crystals with two-level defects, J. Magn. Magn. Mater. 98 (1991) L235;
    • Theory of superconductivity for quasiparticles with parastatistics, Phys. Stat. Solidi (b) 174 (1992) 223;
    • with Q. Shi, M. Mino, H. Yamazaki, Unitary transformations in weakly nonideal Bose gases, Phys. Lett. A238 (1998) 258;
    • Continuous Unitary Transformations, quant-ph/0202095.
  • If this second approach is applied to the elimination of the electron-phonon interaction, then it yields the singular result by Fröhlich and not the smooth effective interaction as obtained with the flow equations, where the generator is varied with the corresponding effective Hamiltonian.

November 2002