Discretization z averaging thermodynamics flow diagrams
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Discretization , z- averaging, thermodynamics, flow diagrams. Rok Žitko Institute Jožef Stefan Ljubljana, Slovenia. L -n/2. Numerical renormalization group (NRG). Wilson, Rev. Mod. Phys. 47 , 773 (1975). f 0 – the first site of the Wilson chain.

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Discretization , z- averaging, thermodynamics, flow diagrams

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Discretization z averaging thermodynamics flow diagrams

Discretization, z-averaging, thermodynamics, flow diagrams

Rok Žitko

Institute Jožef Stefan

Ljubljana, Slovenia


Numerical renormalization group nrg

L-n/2

Numerical renormalization group (NRG)

Wilson, Rev. Mod. Phys. 47, 773 (1975)


F 0 the first site of the wilson chain

f0 – the first site of the Wilson chain

The f0 orbital is also the average state of all "representative states".


Gram schmidt orthogonalization

Gram-Schmidt orthogonalization

http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process


Lanczos algorithm

Lanczos algorithm

Hband in truncated basis

We take f0 instead of the random vector!

The NRG Ljubljana contains a stand-alone tool nrgchain for tridiagonalization (in addition to the tridiagonalization code in initial.m and tridiag.h).

http://en.wikipedia.org/wiki/Lanczos_algorithm


Logarithmic discretization

Logarithmic discretization

Good sampling of the states near the Fermi energy!

Correction:

Wilson, Rev. Mod. Phys. 47, 773 (1975)


Discretization z averaging thermodynamics flow diagrams

“z-averaging” or “interleaved method”

Frota, Oliveira, Phys. Rev. B 33, 7871 (1986)Oliveira, Oliveira: Phys. Rev. B 49, 11986 (1994)


Discretization schemes

Discretization schemes

1) Conventional scheme

2) Campo-Oliveira scheme

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


Discretization z averaging thermodynamics flow diagrams

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).


Discretization z averaging thermodynamics flow diagrams

Can we obtain high-resolution spectral functions by using 1)many values of zand 2) narrow Gaussian broadening?


Test case r esonant level model

Test case: resonant-level model

for a flat band,r(w)=const.


Higher moment spectral sum rules

Higher-moment spectral sum rules


Spectral moments for the resonant level model

Spectral moments for the resonant-level model

7 percent discrepancy!


Origin of the band edge art i facts

Origin of the band-edge artifacts


Discretization z averaging thermodynamics flow diagrams

Campo, Oliveira, PRB 72, 104432 (2005).

for w near band-edges


Discretization z averaging thermodynamics flow diagrams

NOTE: analytical results, not NRG.


Can we do better

Can we do better?

Yes! We demand

R. Žitko, Th. Pruschke, PRB 79, 085106 (2009)R. Žitko, Comput. Phys. Comm. 180, 1271 (2009)


Discretization z averaging thermodynamics flow diagrams

Main success: excellent convergence of various expectation values


Discretization z averaging thermodynamics flow diagrams

Höck, Schnack, PRB 2013


Discretization z averaging thermodynamics flow diagrams

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)


Iterative diagonalization

Iterative diagonalization


Discretization z averaging thermodynamics flow diagrams

Bulla, Costi, Pruschke, RMP 2008


Discretization z averaging thermodynamics flow diagrams

Characteristic energy scale at the N-thstep of the NRG iteration:

discretization=Y

discretization=C or Z


Rg fixed points operators

RG, fixed points, operators

linear operator

relevant operator

irrelevant operator

See Krishnamurthy, Wilkins, Wilson,

PRB 1980 for a very detailed analysis

of the fixed points in the SIAM.

marginal operator


Discretization z averaging thermodynamics flow diagrams

Universality


Fermi liquid fixed points

Fermi-liquid fixed points


Thermodynamic quantities

Thermodynamic quantities


Computing the expectation values

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)


Local vs global operators

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:


Magnetic susceptibility

(Magnetic) susceptibility

If [H,Sz]=0:

General case:


Discretization z averaging thermodynamics flow diagrams

Kondo model

rJ=0.1

Zitko, Pruschke, NJP 2010


Ground state energy binding energy correlation energy kondo singlet formation energy

Ground state energy (binding energy, correlation energy, Kondo singlet formation energy)

Zitko, PRB 2009


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