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Magnetic Phases in Heavy FermionSystems
Robert PetersJuly 2014
Magnetic Phases in Heavy Fermion Systems Robert Peters
Outline
• Introduction: model and method
• the phase diagram of the Kondo lattice model• weak coupling: RKKY dominated phases• strong coupling: Kondo dominated phases
Magnetic Phases in Heavy Fermion Systems Robert Peters
Heavy fermion systemsDue to the existence of strongly localized f-electrons, one canobserve a variety of long-range ordered phases in f-electronsystems.
P. Gegenwart, Q. Si, F. Steglich: Nature Physics 2008
competing effects in f-electron materials
RKKY interaction
magnetic interaction between different
magnetic moments
T > TK T < TK
Kondo effect
Screening between magnetic
moments and conduction electrons
Doniach phase diagram
tem
per
atu
re
Pressure (changing the interaction strength
AFSC
HF
NFL
AF: Antiferromagnetism
SC: Superconductivity
HF: Heavy fermion physic
NFL: Non-Fermi-liquid
RKKY
dominated
Kondo
dominated
Magnetic Phases in Heavy Fermion Systems Robert Peters
Heavy fermion systems
Due to the existence of strongly localized f-electrons, one canobserve a variety of long-range ordered phases in f-electronsystems.
P. Gegenwart, Q. Si, F. Steglich: Nature Physics 2008
competing effects in f-electron materials
RKKY interaction
magnetic interaction between different
magnetic moments
T > TK T < TK
Kondo effect
Screening between magnetic
moments and conduction electrons
Doniach phase diagram
tem
per
atu
re
Pressure (changing the interaction strength
AFSC
HF
NFL
AF: Antiferromagnetism
SC: Superconductivity
HF: Heavy fermion physic
NFL: Non-Fermi-liquid
RKKY
dominated
Kondo
dominated
Magnetic Phases in Heavy Fermion Systems Robert Peters
Heavy fermion systems
Due to the existence of strongly localized f-electrons, one canobserve a variety of long-range ordered phases in f-electronsystems.
P. Gegenwart, Q. Si, F. Steglich: Nature Physics 2008
competing effects in f-electron materials
RKKY interaction
magnetic interaction between different
magnetic moments
T > TK T < TK
Kondo effect
Screening between magnetic
moments and conduction electrons
Doniach phase diagram
tem
per
atu
re
Pressure (changing the interaction strength
AFSC
HF
NFL
AF: Antiferromagnetism
SC: Superconductivity
HF: Heavy fermion physic
NFL: Non-Fermi-liquid
RKKY
dominated
Kondo
dominated
Magnetic Phases in Heavy Fermion Systems Robert Peters
Heavy fermion systems
Due to the existence of strongly localized f-electrons, one canobserve a variety of long-range ordered phases in f-electronsystems.
P. Gegenwart, Q. Si, F. Steglich: Nature Physics 2008
competing effects in f-electron materials
RKKY interaction
magnetic interaction between different
magnetic moments
T > TK T < TK
Kondo effect
Screening between magnetic
moments and conduction electrons
Doniach phase diagram
tem
per
atu
re
Pressure (changing the interaction strength
AFSC
HF
NFL
AF: Antiferromagnetism
SC: Superconductivity
HF: Heavy fermion physic
NFL: Non-Fermi-liquid
RKKY
dominated
Kondo
dominated
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems• lattice with conduction electrons
• lattice of magnetic moments
• local antiferromagnetically couplingbetween both
H = t∑<i ,j>,σ
c†iσcjσ + J∑i
~si~Si
Magnetic Phases in Heavy Fermion Systems Robert Peters
Dynamical mean field theorydynamical mean field theory (DMFT)
DMFT maps the lattice model onto a self-consistent
impurity calculation
DMFT: PRL, W. Metzner and D. Vollhardt (1989); RMP, A. Georges et al. (1996)
half filled Kondolattice
phase diagram calculated by susceptibilities
Otsuki et al., J. Phys. Soc. Jpn. 78 (2009) 034719
• How do the SDWs look like?
• Are there other long-range orderedphases?
Magnetic Phases in Heavy Fermion Systems Robert Peters
Dynamical mean field theory
dynamical mean field theory (DMFT)
DMFT maps the lattice model onto a self-consistent
impurity calculation
DMFT: PRL, W. Metzner and D. Vollhardt (1989); RMP, A. Georges et al. (1996)
half filled Kondolattice
phase diagram calculated by susceptibilities
Otsuki et al., J. Phys. Soc. Jpn. 78 (2009) 034719
• How do the SDWs look like?
• Are there other long-range orderedphases?
Magnetic Phases in Heavy Fermion Systems Robert Peters
Dynamical mean field theory
dynamical mean field theory (DMFT)
DMFT maps the lattice model onto a self-consistent
impurity calculation
DMFT: PRL, W. Metzner and D. Vollhardt (1989); RMP, A. Georges et al. (1996)
half filled Kondolattice
phase diagram calculated by susceptibilities
Otsuki et al., J. Phys. Soc. Jpn. 78 (2009) 034719
• How do the SDWs look like?
• Are there other long-range orderedphases?
Magnetic Phases in Heavy Fermion Systems Robert Peters
Dynamical mean field theory
dynamical mean field theory (DMFT)
DMFT maps the lattice model onto a self-consistent
impurity calculation
DMFT: PRL, W. Metzner and D. Vollhardt (1989); RMP, A. Georges et al. (1996)
half filled Kondolattice
phase diagram calculated by susceptibilities
Otsuki et al., J. Phys. Soc. Jpn. 78 (2009) 034719
• How do the SDWs look like?
• Are there other long-range orderedphases?
Magnetic Phases in Heavy Fermion Systems Robert Peters
inhomogeneous DMFT
Inhomogeneous DMFT is the local approximation of themodel, where each lattice site can have a different local
self-energy.
It has been used to describe cold atoms in a trap,interfaces, superlattices, and surfaces. Furthermore, it canbe used in situations where the lattice symmetry is broken
spontaneously.
Each lattice site of a finite cluster is mapped onto its ownimpurity model, Gij =
(ω− Hi ′j ′ − Σi ′j ′(ω)
)−1
ij
vertical SDW in the doped Hubbard model
H = t∑<i ,j>,σ c
†iσcjσ + U
∑i ni↑ni↓
U = 8t, 〈n〉 = 0.9
RP and N. Kawakami; PRB 2014
Magnetic Phases in Heavy Fermion Systems Robert Peters
inhomogeneous DMFT
Inhomogeneous DMFT is the local approximation of themodel, where each lattice site can have a different local
self-energy.
It has been used to describe cold atoms in a trap,interfaces, superlattices, and surfaces. Furthermore, it canbe used in situations where the lattice symmetry is broken
spontaneously.
Each lattice site of a finite cluster is mapped onto its ownimpurity model, Gij =
(ω− Hi ′j ′ − Σi ′j ′(ω)
)−1
ij
vertical SDW in the doped Hubbard model
H = t∑<i ,j>,σ c
†iσcjσ + U
∑i ni↑ni↓
U = 8t, 〈n〉 = 0.9
RP and N. Kawakami; PRB 2014
Magnetic Phases in Heavy Fermion Systems Robert Peters
inhomogeneous DMFT
Inhomogeneous DMFT is the local approximation of themodel, where each lattice site can have a different local
self-energy.
It has been used to describe cold atoms in a trap,interfaces, superlattices, and surfaces. Furthermore, it canbe used in situations where the lattice symmetry is broken
spontaneously.
Each lattice site of a finite cluster is mapped onto its ownimpurity model, Gij =
(ω− Hi ′j ′ − Σi ′j ′(ω)
)−1
ij
vertical SDW in the doped Hubbard model
H = t∑<i ,j>,σ c
†iσcjσ + U
∑i ni↑ni↓
U = 8t, 〈n〉 = 0.9
RP and N. Kawakami; PRB 2014
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarizationDensity of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarizationDensity of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarizationDensity of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarizationDensity of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarization
Density of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarization
Density of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarizationDensity of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarizationDensity of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarizationDensity of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarizationDensity of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarizationDensity of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarizationDensity of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarizationDensity of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
new iDMFT calculations, also stabilizing incommensurate SDW states
occupation polarization
Neel state at half filling
Density of States
occupation polarization
RKKY dominated SDW state
density and polarizationDensity of States
occupation polarization
RKKY dominated SDW state
Density of States
wavelengths
occupation polarization
Striped ferromagnet
Density of States
occupation polarization
Ferromagnet
Density of States - majority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Density of States - minority electrons
RP, N. Kawakami, T. Pruschke; PRL (2012)
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
heavy fermion paramagnet
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
Kondo dominated SDW, close to QCP
NO CONVERGENCE
polarizationpolarizationpolarizationpolarization
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
Kondo dominated SDW, close to QCP
NO CONVERGENCE
polarization
polarizationpolarizationpolarization
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
Kondo dominated SDW, close to QCP
NO CONVERGENCE
polarization
polarization
polarizationpolarization
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
Kondo dominated SDW, close to QCP
NO CONVERGENCE
polarizationpolarization
polarization
polarization
Magnetic Phases in Heavy Fermion Systems Robert Peters
Magnetic phase diagram of heavy fermion systems
Kondo dominated SDW, close to QCP
NO CONVERGENCE
polarizationpolarizationpolarization
polarization
Magnetic Phases in Heavy Fermion Systems Robert Peters
Summary
• By using the iDMFT, I have studied the magneticphases of the Kondo lattice model
• There are several different types of SDW phases
• Such SDWs are a mixture of antiferromagnetic bondsand ferromagnetic bonds
• There are accompanied by a charge density wave.
• Close to the quantum critical transition, even iDMFTseems not to converge
Magnetic Phases in Heavy Fermion Systems Robert Peters
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