Post on 08-Jul-2020
Ryotaro ARITA (RIKEN)
LDA+DMFT study for LiV2O4 at T→0
Phys. Rev. Lett. 98 166402 (2007)
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K. Held(Max Planck Institute, Stuttgart)
A.V. Lukoyanov(Ural State Technical Univ.)
V.I. Anisimov(Institute of Metal Physics)
Collaborators
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Outline
Introduction LiV2O4: 3d heavy Fermion system
heavy (sharp) quasi-peak in A(w) for low T
Development of PQMC QMC for T→0 Application to DMFT calculation
Results LDA+DMFT(PQMC) for LiV2O4
Discussion Origin of HF behaviors
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Crossover at T*~20K
・ resistivity: r =r0+AT2 with an enhanced A
・ specific heat coefficient: anomalously largeg(T→0)~190mJ/V mol・K2
(Kadowaki-Woods relation satisfied)
・ c: broad maximum (Wilson ratio～1.8)
cf) CeRu2Si2 ~350mJ/Ce mol・K2
UPt3 ~420mJ/U mol・K2
T*・・ onset of the formation of the heavy mass quasiparticles(m*～25mLDA)
T*
(Urano et al. PRL85, 1052(2000))
LiV2O4: 3d heavy Fermion system
Incoherenet metal
FL(T2 law)
g(T→0)~190mJ/Vmol・K2
CW law at HT S=1/2 per V ion
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(Shimoyamada et al. PRL 96 026403(2006))
PhotoEmission Spectroscopy Magnetization curves(Niitaka et al. ‘06)
A sharp peak appearsfor T<26K
w=4meV, D~10meV
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
M (m
B/f.u
.)
6050403020100H (T)
H//110 1.3K 4.2K 10K 20K 30K
LiV2O4: 3d heavy Fermion system
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Kondo scenario
• Anisimov et al, 99• Eyert et al, 99• Matsuno et al, 99• Kusunose et al, 00• Lacroix, 01
• Shannon, 01 • Fulde et al, 01• Burdin et al, 02• Hopkinson et al, 02• Fujimoto, 02
Geometrical Frustration
hybridization→Kondo?
Hund
• Tsunetsugu, 02• Yamashita et al, 03• Laad et al, 03• Nekrasov et al, 03• Yushankhai et al, 07
Anisimov et al, PRL 83 364(1999)
a1g(localized)
eg(itinerant)
Theoretical studies so far
Short range (local) correlationsbecome dominant⇒ DMFT expected to be a good approx.
T=750K
LDA+DMFT by Nekrasov et al, PRB 67 085111 (2003)
Calculation forT→0
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LDA+DMFT Successful beyond-LDA method
Applications to various strongly correlated materials
Solver for the effective impurity model QMC (Hirsch-Fye)
Numerically exact Restricted to high T
(Numerically expensive for Low Teffort ~ 1/T3)
Development of Projective QMC for T→0 and its application to LDA+DMFT calculations for LiV2O4
S(w)
Impurity model
Lattice model
LDA+DMFT(QMC)
DMFT
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Purpose
Reproduce the sharp (heavy) quasi-particle peak in A(w) at T→0 by LDA+DMFT(PQMC)
Clarify the origin of HF behaviors
(Shimoyamada et al. PRL 96 026403(2006))
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1 2
1 2
1 2
..., ...
... : arbitrary operator L
L
L
s s ss s s
s s sA
ZA A
Zá ñ = á ñå
Suzuki-Trotter decomposition
Hubbard-Stratonovich transformation for Hint
Many-particle system = (free one-particle system + auxiliary field)
1 2
1 2
......
L
L
ss sss s
Z Z= å 0
1 2 ...1
12
Tr [ ]l
L
LH s n
s s sl
LZ e es
st ls
s
- D
=
º Õ Õ
0 intL
l 1= ( 1/ )Tr Tr HHH T LZ e e e ttb b t-D-D-
== = D= Õ
12( )
1
( )[ ] 12s
U s n nn n n ne et l ¯- +
=±
¯ ¯-D -= å (cosh( ) exp[ /2])Ul t= D
Monte Carlo sampling
å
= å
Auxiliary-field QMC
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0<t1,t2<b=1/T, b=LDt
Projective QMC
Finite-T QMC for the Anderson impurity model (Hirsch-Fye 86)
Projective QMC for T=0
Size of G =L2
Effort ~L3
(calculation for low T is numerically expensive)
†1 2 1 2
{ } { } 1 2{ }
( , ) ( ) ( )
( , )p p
s ss
G T c c
w Gt s st t t t
t t
= -
= å
Integrate out the conduction bandsand calculate G of the impurity
Calculate G{s}(t1,t2) from G0(t1,t2)
Feldbacher, KH, Assaad, PRL 93 136405(2004)
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0<t1,t2<b=1/T, b=LDt
Projective QMC
Finite-T QMC for the Anderson impurity model (Hirsch-Fye 86)
Projective QMC for T=0
Size of G =L2
Effort ~L3
(calculation for low T is numerically expensive)
†1 2 1 2
{ } { } 1 2{ }
( , ) ( ) ( )
( , )p p
s ss
G T c c
w Gt s st t t t
t t
= -
= å
Integrate out the conduction bandsand calculate G of the impurity
Calculate G{s}(t1,t2) from G0(t1,t2)
Feldbacher, KH, Assaad, PRL 93 136405(2004)
R.AritaR.Arita
0<t1,t2<b=1/T, b=LDt
Projective QMC
Finite-T QMC for the Anderson impurity model (Hirsch-Fye 86)
Projective QMC for T=0
Size of G =L2
Effort ~L3
(calculation for low T is numerically expensive)
†1 2 1 2
{ } { } 1 2{ }
( , ) ( ) ( )
( , )p p
s ss
G T c c
w Gt s st t t t
t t
= -
= å
Integrate out the conduction bandsand calculate G of the impurity
Calculate G{s}(t1,t2) from G0(t1,t2)
Feldbacher, KH, Assaad, PRL 93 136405(2004)
q b+ 0
InteractionIsing fields no interaction
tq
→∞ →∞
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Projective QMC
Interaction U only in red part
for sufficiently large P:Accurate information onG for light red part
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DMFT(PQMC)
PQMC
1 2( )G t t,1
01() ( ) )(i G i G iw w w- -S = -
( )( )( )n n
DG i di i
ew e
w m w e=
+ - S -ò1 1
0 ( ) ( ) ( )G i G i iw w w- -= + S
0 1 2( )G t t,
1 ex( p( ))) (A dG t w wt wp
= -ò(( ) 1 )
nn d
iAG i w w
p ww
w=
-ò
Maximum Entropy Method
ProblemG(t)→FT→G(iw)? No
only G(t),t<q obtained by PQMC
(T=0)(T=0)
DMFT self-consistent loop
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DMFT(PQMC)
PQMC
1 2( )G t t,1
01() ( ) )(i G i G iw w w- -S = -
( )( )( )n n
DG i di i
ew e
w m w e=
+ - S -ò1 1
0 ( ) ( ) ( )G i G i iw w w- -= + S
0 1 2( )G t t,
1 ex( p( ))) (A dG t w wt wp
= -ò(( ) 1 )
nn d
iAG i w w
p ww
w=
-ò
Maximum Entropy Method
ProblemG(t)→FT→G(iw)? No
only G(t),t<q obtained by PQMC
(T=0)(T=0)
DMFT self-consistent loop
1/T
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DMFT(PQMC)
PQMC
1 2( )G t t,1
01() ( ) )(i G i G iw w w- -S = -
( )( )( )n n
DG i di i
ew e
w m w e=
+ - S -ò1 1
0 ( ) ( ) ( )G i G i iw w w- -= + S
0 1 2( )G t t,
1 ex( p( ))) (A dG t w wt wp
= -ò(( ) 1 )
nn d
iAG i w w
p ww
w=
-ò
Maximum Entropy Method
ProblemG(t)→FT→G(iw)? No
only G(t),t<qP obtained by PQMC
(T=0)(T=0)
DMFT self-consistent loop
1/T
R.AritaR.Arita
DMFT(PQMC)
PQMC
1 2( )G t t,1
01() ( ) )(i G i G iw w w- -S = -
( )( )( )n n
DG i di i
ew e
w m w e=
+ - S -ò1 1
0 ( ) ( ) ( )G i G i iw w w- -= + S
0 1 2( )G t t,
1 ex( p( ))) (A dG t w wt wp
= -ò(( ) 1 )
nn d
iAG i w w
p ww
w=
-ò
Maximum Entropy Method
ProblemG(t)→FT→G(iw)? No
only G(t),t<q obtained by PQMC
(T=0)(T=0)
DMFT self-consistent loop
1/TqP 1/T-qP
Calculate G only for t<qPLarge t: Extrapolation byMaximum Entropy Method
R.AritaR.Arita
DMFT(PQMC)
PQMC
1 2( )G t t,1
01() ( ) )(i G i G iw w w- -S = -
( )( )( )n n
DG i di i
ew e
w m w e=
+ - S -ò1 1
0 ( ) ( ) ( )G i G i iw w w- -= + S
0 1 2( )G t t,
1 ex( p( ))) (A dG t w wt wp
= -ò(( ) 1 )
nn d
iAG i w w
p ww
w=
-ò
Maximum Entropy Method
ProblemG(t)→FT→G(iw)? No
only G(t),t<q obtained by PQMC
(T=0)(T=0)
DMFT self-consistent loop
1/TqP 1/T-qP
Calculate G only for t<qPLarge t: Extrapolation byMaximum Entropy Method
FAQ:Why can we discuss A(w→0)even if we do not calculateG(t→∞) explicitly?
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DMFT(PQMC)
PQMC
1 2( )G t t,1
01() ( ) )(i G i G iw w w- -S = -
( )( )( )n n
DG i di i
ew e
w m w e=
+ - S -ò1 1
0 ( ) ( ) ( )G i G i iw w w- -= + S
0 1 2( )G t t,
1 ex( p( ))) (A dG t w wt wp
= -ò(( ) 1 )
nn d
iAG i w w
p ww
w=
-ò
Maximum Entropy Method
ProblemG(t)→FT→G(iw)? No
only G(t),t<q obtained by PQMC
(T=0)(T=0)
DMFT self-consistent loop
1/TqP 1/T-qP
Calculate G only for t<qPLarge t: Extrapolation byMaximum Entropy Method
Sufficiently large qP needed
FAQ:Why can we discuss A(w→0)even if we do not calculateG(t→∞) explicitly?
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0 0
DMFT(PQMC)A
(w)
A(w
)
T→0
w0 w0
T>0
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0 0
DMFT(PQMC)A
(w)
A(w
)
T→0
w0
G(t
)
0 t
～exp(-wt0) G(t
)
0 t
Slow decay component
w0
T>0
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Single-band Hubbard modelHF-QMC vs. PQMC
DMFT(PQMC)
M I
HF-QMC
insulating metallic
b=16 b=40
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Single-band Hubbard modelHF-QMC vs. PQMC
Convergence w.r.t q is much better than b
M I
PQMC
q=16 q=40
DMFT(PQMC)
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Single-band Hubbard modelHF-QMC vs. PQMC
Convergence w.r.t q is much better than b
M I
PQMC
q=16 q=40
DMFT(PQMC)Orbital selective Mott transition in the two-orbital Hubbard model
DCA(PQMC) study for anisotropicpairing in the t-t’ Hubbard model
RA and KH, PRB 73 064515(2006)
RA and KH, PRB 72 201102(2005)
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12 (=4x3) band Hamiltonian
egp
a1g
egp
a1g
Hij(k)=
8 (=4x2) band model
Hij(k)=
DMFT self-consistent eq.
hybridization (Hij, i≠j) taken into account explicitly
ne/V=1.5
Effective low energy Hamiltonian
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LDA+DMFT Result (T=1200K, HF-QMC)
U=3.6, U’=2.4, J=0.6
U U’ U’-J Hund coupling = Ising
a1g
egp
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U=3.6, U’=2.4, J=0.6
U U’ U’-J Hund coupling = Ising
LDA+DMFT Result (T=300K, HF-QMC)
a1g
egp
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U=3.6, U’=2.4, J=0.6
U U’ U’-J Hund coupling = Ising
LDA+DMFT Result (T=0, PQMC)
a1g
egp
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LDA+DMFT Result (T=0, PQMC)
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Kondo scenario
• Anisimov et al, 99• Eyert et al, 99• Matsuno et al, 99• Kusunose et al, 00• Lacroix, 01
• Shannon, 01 • Fulde et al, 01• Burdin et al, 02• Hopkinson et al, 02• Fujimoto, 02
Geometrical Frustration
hybridization→Kondo?
Hund
• Tsunetsugu, 02• Yamashita et al, 03• Laad et al, 03• Nekrasov et al, 03• Yushankhai et al, 07
Anisimov et al, PRL 83 364(1999)
a1g(localized)
eg(itinerant)
Theoretical studies so far
Short range (local) correlationsbecome dominant⇒ DMFT expected to be a good approx.
T=750K
LDA+DMFT by Nekrasov et al, PRB 67 085111 (2003)
Calculation forT→0
R.AritaR.Arita
hybridization→Kondo?
DMFT self-consistent eq.
a1g-egp hybridization not considered explicitly
Kondo coupling between a1g and egp
not reason for peak above EF
Effect of a1g-egp hybridization
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Origin of the peak: a1g =slightly doped Mott insulator ?
Th. Pruschke et al, PRB 47 3553 (1993)
DMFT for the single band Hubbardn=0.97 (U=4)
na1g ~ 0.95na1g~ neg
LDA LDA+DMFT
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Origin of the peak: a1g =slightly doped Mott insulator ?
Th. Pruschke et al, PRB 47 3553 (1993)
DMFT for the single band Hubbardn=0.97 (U=4)
na1g ~ 0.95na1g~ neg
LDA LDA+DMFT
Question:Strong renormalization can survivethe presence of short-range correlationbeyond DMFT?
cf) A.Toschi, A. Katanin, K. Held (PRB 75 045118 (2007))DGA study for cubic lattice
Damping of the peak:Irrelevant for 3D frustrated lattice(?)
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Development of PQMC and its application to LDA+DMFT calculations for LiV2O4
Origin of the sharp peak just above EF
a1g = slightly doped Mott Insulator
Future problems beyond-DMFT
Conclusions