Discretization , z- averaging, thermodynamics, flow diagrams
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Discretization, z-averaging, thermodynamics, flow diagrams
Rok ŽitkoInstitute Jožef StefanLjubljana, Slovenia
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Numerical renormalization group (NRG)
L-n/2
Wilson, Rev. Mod. Phys. 47, 773 (1975)
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f0 – the first site of the Wilson chain
The f0 orbital is also the average state of all "representative states".
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Gram-Schmidt orthogonalization
http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process
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Lanczos algorithm
http://en.wikipedia.org/wiki/Lanczos_algorithm
The NRG Ljubljana contains a stand-alone tool nrgchain for tridiagonalization (in addition to the tridiagonalization code in initial.m and tridiag.h).
We take f0 instead of the random vector!Hband in truncated basis
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Logarithmic discretization
Good sampling of the states near the Fermi energy!
Wilson, Rev. Mod. Phys. 47, 773 (1975)
Correction:
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“z-averaging” or “interleaved method”Frota, Oliveira, Phys. Rev. B 33, 7871 (1986)Oliveira, Oliveira: Phys. Rev. B 49, 11986 (1994)
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2) Campo-Oliveira scheme1) Conventional scheme
Discretization schemes
Campo, Oliveira, PRB 72, 104432 (2005).
Chen, Jayaprakash, JPCM 7, L491 (1995); Ingersent, PRB 54, 11936 (1996); Bulla, Pruschke, Hewson, JPCM 9, 10463 (1997).
r(e) = density of states in the band
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With z-averaging it is possible to increase L quite significantly without introducing many artifacts in the results for thermodynamics (especially in the low-temperature limit).
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Can we obtain high-resolution spectral functions by using
1) many values of z and
2) narrow Gaussian broadening?
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Test case: resonant-level model
for a flat band,r(w)=const.
E
e
G=0E
e
G¹0
r(w)
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Higher-moment spectral sum rules
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7 percent discrepancy!
Spectral moments for the resonant-level model
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Origin of the band-edge artifacts
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for w near band-edges
Campo, Oliveira, PRB 72, 104432 (2005).
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NOTE: analytical results, not NRG.
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Can we do better?Yes! We demand
R. Žitko, Th. Pruschke, PRB 79, 085106 (2009)R. Žitko, Comput. Phys. Comm. 180, 1271 (2009)
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Main success: excellent convergence of various expectation values
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Höck, Schnack, PRB 2013
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Cancellation of L-periodic NRG discretization artifacts by the z-averaging
R. Žitko, M. Lee, R. Lopez, R. Aguado, M.-S. Choi, Phys. Rev. Lett. 105, 116803 (2010)
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Iterative diagonalization
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Bulla, Costi, Pruschke, RMP 2008
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Characteristic energy scale at the N-th step of the NRG iteration:
discretization=Y
discretization=C or Z
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RG, fixed points, operators
linear operator
relevant operator
irrelevant operator
marginal operator
See Krishnamurthy, Wilkins, Wilson,PRB 1980 for a very detailed analysisof the fixed points in the SIAM.
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Universality
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Fermi-liquid fixed points
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Thermodynamic quantities
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Computing the expectation values
Conventional approach: use HN as an approximation for full H at temperature scale TN.
Alternatively: use FDM for given T. Also works for thermodynamics:Merker, Weichselbaum, Costi, Phys. Rev. B 86, 075153 (2012)
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Local vs. global operators
• Local operators: impurity charge, impurity spin, d†
s . In general, any operator defined on the initial cluster.
• Global operators: total charge, total spin, etc.Example:
In itia lc luste r
Re m a ining Wilso nc ha in (a d d e d d uring
the NRG ite ra tio n)
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(Magnetic) susceptibility
If [H,Sz]=0:
General case:
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Kondo model
rJ=0.1
Zitko, Pruschke, NJP 2010
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Ground state energy (binding energy, correlation energy,
Kondo singlet formation energy)
Zitko, PRB 2009