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Fermi Liquid at B =0, Ν = Eff 2

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Fermi Liquid at B =0, Ν = Eff 2 Quasiparticle excitations in frustrated quantum matter: The case of metallic quantum magnets Peter Wölfle Institute for Theory of Condensed Matter (TKM) Institute for Nanotechnology (INT) Karlsruhe Institute of Technology Germany Outline • Electrons in metals: the first instance of Quantum Matter • Emergence of quasiparticles in ordered states: „The death and rebirth of quasiparticles“ • The fate of quasiparticles near Quantum Critical Points: „Critical quasiparticles“ • Frustrated quantum magnets: „Fractional quasiparticles“ • Electrons in two dimensions in a strong magnetic field: “Composite fermions” in the fractional Quantum Hall effect (FQHE) „Exotic quasiparticles“ • Conclusion and outlook Electrons in metals: first example of Quantum Matter • Arnold Sommerfeld (1927) : Electrons in metals modelled by a Fermi gas: a system of identical noninteracting quantum particles Weak excitations are possible only near the Fermi energy EF (Pauli principle) Specific heat C ~ T Spin susceptibility χ ~ const • Felix Bloch (1930) : Bandstructure of energy spectrum of electrons in a crystalline solid • Walter Kohn (1964): Density functional theory of electrons in solids: approximate mapping on to non-interacting system One-particle theory of electrons in solids Successfully describes properties of many normal solids Problem: Coulomb interaction between electrons unimportant? Landau’s Fermi liquid theory L.D. Landau (1957) free electron Interaction - - - e- qp adiabatically 1 Long lived “Landau quasi particle” ω 2 τω() Landau qp has all the quantum numbers of free particle Effect of interaction absorbed in a few parameters (effective mass m*, Landau parameters) Electron spectral function and quasiparticle weight Z What is required for Fermi liquid idea to be applicable: Z is not zero and the quasi-particles establish their energy before they decay: Γ<Ep Quasiparticle picture of normal quantum matter Model description of weakly excited states of systems of interacting quantum particles by mapping on to system of nearly free quasiparticles Fermions (Spin ½) : Landau qp and Bosons (Spin 0,1) : Zero sound, Phonons, Plasmons, Excitons, …. Quasiparticle picture of normal quantum matter Model description of weakly excited states of systems of interacting quantum particles by mapping on to system of nearly free quasiparticles Fermions (Spin ½) : Landau qp and Bosons (Spin 0,1) : Zero sound, Phonons, Plasmons, Excitons, …. Landau Fermi liquid interaction may lead to instability: transition to ordered state Quasiparticle picture of normal quantum matter Model description of weakly excited states of systems of interacting quantum particles by mapping on to system of nearly free quasiparticles Fermions (Spin ½) : Landau qp and Bosons (Spin 0,1) : Zero sound, Phonons, Plasmons, Excitons, …. Landau Fermi liquid interaction may lead to instability: transition to ordered state What happens to Landau quasiparticles when the system undergoes a phase transition into an ordered state? The superfluid phases of Helium 3 3He-atoms (quasiparticles) form Cooper pairs BCS-theory of electrons in superconductors Bardeen, Cooper, Schrieffer (1957) Orbital ang. Mom. L=1 3x3 sub states Spin S=1 Order parameter matrix Î-Vektor Nobel prizes: Exp.: Lee, Osheroff, Richardson (1996) Theory: Leggett (2003) Vollhardt, Wölfle, “The superfluid phases of Helium 3” (1990, 2013) Quasiparticles in superfluid Helium3 Emergence of order parameter cc kk σσ − ' has the consequences: • Landau qp die and are reborn as Bogoliubov qp Cooper pair - number not conserved 22 - energy gap Ekkσσ=(εµ − ) +∆ | k |] Quasiparticles in superfluid Helium3 Existence of order parameter cc kk σσ − ' has the consequences: • Landau qp die and are reborn as Bogoliubov qp Cooper pair - number not conserved 22 - energy gap Ekkσσ=(εµ − ) +∆ | k |] • “Elasticity” of order parameter field allows for wave-like excitations: - “acoustical” : Goldstone modes with dispersion ω=ck or similar: Anderson-Bogoliubov sound (transformed zero sound), spin waves, orbital waves - “optical” : massive modes ω=const. , k → 0 (pair vibration modes) → new bosonic quasiparticles Quasiparticles in superfluid Helium3 Existence of order parameter cc kk σσ − ' has the consequences: • Landau qp die and are reborn as Bogoliubov qp Cooper pair - number not conserved 22 - energy gap Ekkσσ=(εµ − ) +∆ | k |] • “Elasticity” of order parameter field allows for wave-like excitations: - “acoustical” : Goldstone modes with dispersion ω=ck or similar: Anderson-Bogoliubov sound (transformed zero sound), spin waves, orbital waves - “optical” : massive modes ω=const. , k → 0 (pair vibration modes) → new bosonic quasiparticles • Defects in the order parameter field represent localized qp usually protected by topological constraints (domain walls, vortices) → new topological quasiparticles Quasiparticles in heavy fermion metals Metallic compounds containing localized spins (Ce, Yb, Fe, Co, U, ..) exchange coupled (J) antiferromagnetically to the conduction electron spins Kondo effect: localized spin gets screened by surrounding conduction electron spins J below a temperature TK - (many-body resonance state). e At each Kondo ion the electron effective mass is enhanced through resonance scattering by a factor m*/m=TF/TK Magnetization of CeCu6 Kondo lattice: the coherently scattered electrons form a heavy Fermi liquid below a temperature T coh m*/m ~ 500 Schlager, v. Loehneysen et al. (1993) Quasiparticles in heavy fermion metals Metallic compounds containing localized spins (Ce, Yb, Fe, Co, U, ..) exchange coupled (J) antiferromagnetically to the conduction electron spins Kondo effect: localized spin gets screened by surrounding conduction electron spins J below a temperature TK - (many-body resonance state). e At each Kondo ion the electron effective mass is enhanced through resonance scattering by a factor m*/m=TF/TK Magnetization of CeCu6 Kondo lattice: the coherently scattered electrons form a heavy Fermi liquid below a temperature T coh m*/m ~ 500 Instabilities of Kondo lattice? Schlager, v. Loehneysen et al. (1993) Traditional view: breakdown of Kondo effect at QCP Local quantum criticality: Si, Ingersent, Smith, Rabello, 2001 Deconfined criticality: Hermele et al., 2004; Senthil et al. 2004 Quantum phase transition in heavy fermion compounds H. v. Löhneysen et al. (1996) F. Steglich et al. (2002) Quantum phase transition in heavy fermion compounds H. v. Löhneysen et al. (1996) F. Steglich et al. (2002) Quasiparticles disappear near the quantum critical point (correlation length ξ -> ∞) Fermi liquid regime: 1 zz2 −−η z ξω, Z(0) ~ ξ→< 0 , ω ω ξ critical exponents z,η τω() c Quantum critical regime: 1 η |ω |, Z( ω ) ~| ω |→ 0, ωω > c Landau quasiparticles die: τω() Non-Fermi liquid behavior Moriya, 1970, J. Hertz (1976), A. Millis (1993), S. Sachdev (2002) H. v. Löhneysen, M. Vojta, A. Rosch, P. Wölfle, RMP (2007) Quasiparticles beyond Landau Fermi liquid: examples Quasiparticle weight factor Z: Z-1 =1 - dΣ(ω)/dω Power law non-Fermi liquid (NFL) 1−η ΣNFL ()ωω ∝−ii ( − ) Z(ωω )∝ (| |)η Γ(ω ) =ZE Im Σ∝ | ωω |, = p Γ/Ep = tan(ηπ / 2) < 1, 0 <<η 1/ 2 Our approach: Critical heavy quasiparticles on the basis of robust Kondo screening Observed properties contradicting the Kondo breakdown scenario: • effective mass is not decreasing on moving to the AFM ordered state • the observed magnetic moments in the ordered state are small • no convincing sign of abrupt change of Fermi volume is observed Instead, our theory focuses on • the effects of the scattering of heavy quasiparticles by critical spin- and energy fluctuations • the changes in the spin/energy fluctuation spectrum induced by critical quasiparticles. Self-consistent strong coupling theory of qp effective mass • Hyperscaling • Fractional power laws • Excellent agreement with experiment. E. Abrahams, PW, PRB (2011), PNAS (2012); EA, J. Schmalian, PW (2014) Critical heavy quasiparticles modify spin fluctuations. Scattering by spin fluctuations drives qps critical N0 χω(,q )= vertex λQ ~ m*/m → ∞ 22iω 2 r +ξ0 ()qQ −− λQ vQF Landau damping Σ= m*/m E. Abrahams, PW, PRB (2011), PNAS (2012); EA, J. Schmalian, PW (2014) Critical heavy quasiparticles modify spin fluctuations. Scattering by spin fluctuations drives qps critical N0 χω(,q )= vertex λQ ~ m*/m → ∞ 22iω 2 r +ξ0 ()qQ −− λQ vQF Landau damping Σ= m*/m spin fluctuations q=0 Q -Q energy fluctuation E. Abrahams, PW, PRB (2011), PNAS (2012); EA, J. Schmalian, PW (2014) Specific heat : YbRh2Si2 T 3d AFM fluctuations ) 2 Inside the critical cone: * (J/mol K (J/mol C mT() −1/4 T ∝∝T / Tm C B Theory: E. Abrahams, PW, PNAS (2012) T (K) Exp.: N. Oeschler et al., Physica B 403, 1254 (2008) Outside the critical cone: −0.33 γ 0 ∝−||BBc −1/3 C|∝−HHc | T B (T) Exp.: Custers et al., Nature (2003) Scaling behavior of S(q,E): CeCu5.9Au0.1 Dynamical structure factor measured by inelastic neutron scattering 1 (/)ET3/4 SQET( , ; )∝+ (1 nE ( / T )) T3/4 1(/)+ ET3/2 quasi-2d AFM fluctuations SQETT(,;)3/4 Th: Abrahams, Schmalian, PW, (2013) ET/ Exp.: Schröder et al. (2000) Quasiparticles in metallic antiferromagnets • In the ordered AF phase new bosonic qp arise: Spin waves (Goldstone modes) ω = cq • Landau qp are transformed: rebirth of fermionic qp - spin of qp is no longer conserved spin-flip by scattering off transverse spin waves - energy spectrum acquires a gap 1 Eh=[ε ++ ε σε ( − ε )]22 + kσ 2 k kQ++ k kQ s “Standard model” of interacting Fermi systems Fermi liquid theory + spontaneous symmetry breaking = one of the most successful concepts in condensed
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