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5.32 Nils Blümer For a constant default model (within some finite frequency range), the entropic form (62) clearly favors smooth spectra. This is also true for a general smooth default model. It also enforces positivity of A and pushes the solution towards the (normalized) default model in absence of data. From (59), (60), and (61), the posterior probability can be read off as P(A| ¯ G,m,α) = e αS[A,m]−1 2 χ2 [ ¯ G,A] . (63) The balance between a tight match of data and a high entropy is calibrated by the Lagrange parameter α which may be chosen so that χ2 = Λ (historic MEM). Alternatively, one may use the value ofαwith the highest probabilityP(α| ¯ G,A,m) which can approximately be calculated within the method (classic MEM). Given the QMC data, a default model, a representation of the spectrum (i.e., a possibly inhomogeneous grid of ωj of frequencies for which A is going to be computed), and a starting guess for α, a simple MEM program thus both searches for the spectrum {A(ωj)} with maximum probability P(A| ¯ G,m,α) for given α using, e.g., the Newton-Raphson method and, in an outer loop, searches for the best value ofα. The former procedure can be stabilized using a singular value decomposition (SVD) of the kernel: K = VΣU T (64) Σ = diag(σ1,...,σs) whereσ1 > ... > σs > 0. Here U, V are orthogonal matrices. Typically, most of the singular values σi are equal to zero (within machine precision). The columns of U T , restricted to σi �= 0, then span the same space as the columns ofK. Projecting all related quantities to this new most smaller space (socalled singular space) a stable search can be performed. In practice, the width of the Newton- Raphson steps has to be restricted using a Levenberg-Marquardt-Parameter. An example for the application of this method is shown in Fig. 14. Alternatively to this deterministic approach, one may also use Markov chain Monte Carlo updates directly in the image space (i.e. on the level of the spectrum) with simulated annealing, as implemented, e.g., by Sandvik. This stochastic approach is however, quite sensitive to the choice of the frequency discretization, requires smoothing runs, and is computationally more costly.
Hirsch-Fye QMC 5.33 References [1] A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod. Phys. 68, 13 (1996) [2] J. E. Hirsch and R. M. Fye, Phys. Rev. Lett. 56, 2521 (1986) [3] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989) [4] E. Müller-Hartmann, Z. Phys. B 74, 507 (1989) [5] M. Jarrell, Phys. Rev. Lett. 69, 168 (1992) [6] A. Georges and G. Kotliar, Phys. Rev. B 45, 6479 (1992) [7] V. Janiˇs and D. Vollhardt, Int. J. Mod. Phys. B 6, 731 (1992) [8] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971) [9] Feliks Aleksandrovich Berezin, The method of second quantization (Academic Press, New York, 1966) [10] M. Caffarel and W. Krauth, Phys. Rev. Lett. 72, 1545 (1994) [11] H. Keiter and J. C. Kimball, Phys. Rev. Lett. 25, 672 (1970) [12] N. E. Bickers, D. L. Cox, and J. W. Wilkins, Phys. Rev. B 36, 2036 (1987) [13] T. Pruschke and N. Grewe, Z. Phys. B 74, 439 (1989) [14] T. Pruschke, D. L. Cox, and M. Jarrell, Phys. Rev. B 47, 3553 (1993) [15] N. E. Bickers, D. J. Scalapino, and S. R. White, Phys. Rev. Lett. 62, 961 (1989) [16] N. E. Bickers and D. J. Scalapino, Ann. Physik 193, 206 (1989) [17] N. E. Bickers and S. R. White, Phys. Rev. B 43, 8044 (1991) [18] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975) [19] H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, Phys. Rev. B 21, 1003 (1980) [20] T. A. Costi, A. C. Hewson, and V. Zlatić, J. Phys. Cond. Matter 6, 2519 (1994) [21] Ralf Bulla, Theo Costi, and Thomas Pruschke, Reviews of Modern Physics 80, 395 (2008), ISSN 0034-6861 [22] U. Schollwöck, Reviews of Modern Physics 77, 259 (2005), ISSN 0034-6861 [23] E. Gull, P. Werner, O. Parcollet, and M. Troyer, EPL (Europhysics Letters) 82, 57003 (2008), ISSN 0295-5075
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