Dynamical mean-field theory and the NRG as the impurity solver
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Dynamical mean-field theory andthe NRG as the impurity solver
Rok ŽitkoInstitute Jožef StefanLjubljana, Slovenia
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DMFT
• Goal: study and explain lattice systems of strongly correlated electron systems.
• Approximation: local self-energy S• Dynamical: on-site quantum fluctuations treated
exactly• Pro: tractable (reduction to an effective quantum
impurity problem subject to self-consistency condition)
• Con: non-local correlations treated at the static mean-field level
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Infinite-D limit
See, for example, D. Vollhardt in The LDA+DMFT approach to strongly correlated materials, Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (Eds.), 2011
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Infinite-D limit
Derivation of the DMFT equations by cavity method:• write the partition function as an integral over Grassman
variables• integrate out all fermions except those at the chosen single
site• split the effective action• take the D to infinity limit to simplify the expression• result: self-consistency equation
Georges et al., RMP 1996
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Hubbard SIAM⇒
hybridization plays therole of the Weiss fieldG(w)=-Im[ D(w+id) ]
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Bethe lattice
Im
Re
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Hypercubic lattice
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3D cubic lattice
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2D cubic lattice
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1D cubic lattice (i.e., chain)
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Flat band
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Achieving convergence: self-consistency constraint viewed as a system of equations
DMFT step:
Self-consistency:
Linear mixing (parameter α in [0:1]):
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Broyden method
Newton-Raphson method:
Broyden update:
F(m)=F[ V(m) ]
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R. Žitko, PRB 80, 125125 (2009).Note: can also be used to control the chemical potential infixed occupancy calculations.
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Recent improvements in NRG
• DM-NRG, CFS, FDM algorithms more ⇒reliable spectral functions, even at finite T
• discretization scheme with reduced artifacts ⇒possibility for improved energy resolution
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NRG vs. CT-QMC• NRG: extremely fast for single-impurity problems• NRG: access to arbitrarily low temperature scales, but decent
results even at high T (despite claims to the contrary!)• NRG: spectral functions directly on the real-frequency axis, no
analytic continuation necessary• NRG: any local Hamiltonian can be used, no minus sign
problem• NRG: efficient use of symmetries• CT-QMC: (numerically) exact• CT-QMC: can handle multi-orbital problems (even 7-orbital f-
level electrons)
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Mott-Hubbard phase transition
m
U»0 non-correlated metalW
W W Mott insulator
U
U>>W
U W»
U/W
DMFT prediction
Kotliar, Vollhardt
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Bulla, 1999, Bulla et al. 2001.
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k-resolved spectral functions
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A(k,w=0)
Momentum distribution function
Zquasiparticle
renormalizationfactor
2D cubic lattice DOS:
Caveat: obviously, DMFT is not a good approximation for 2D problems. But 2D cubic lattice is nice for plotting...
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Ordered phases
• Ferromagnetism: break spin symmetry (i.e., add spin index s, use QSZ symmetry type)
• Superconductivity: introduce Nambu structure, compute both standard G and anomalous G (use SPSU2 symmetry type)
• Antiferromagnetism: introduce AB sublattice structure, do double impurity calculation (one for A type, one for B type)
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Hubbard model: phase diagram
Zitzler et al.
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Hubbard model on the Bethe lattice,
PM phase
inner band-edge features
See also DMRG study,Karski, Raas, Uhrig,PRB 72, 113110 (2005).
High-resolution spectral functions?
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Z. Osolin, RZ, 2013
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Hubbard model on the Bethe lattice,
AFM phase
spin-polaron structure
(“string-states”)
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Thermodynamics
Bethe lattice:Note:
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Transport in DMFTVertex corrections drop out, because S is local, and vk and ek have different parity.
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X. Deng, J. Mravlje, R. Zitko, M. Ferrero, G. Kotliar,A. Georges, PRL 2013.
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qmc
nrg
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Other applications
• Hubbard (SC, CO)• two-orbital Hubbard model• Hubbard-Holstein model (near half-filling)• Kondo lattice model (PM, FM, AFM, SC phases)• Periodic Anderson model (PAM), correlated
PAM