magnetic phases in heavy fermion systems...striped ferromagnet ferromagnet density of states -...

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