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 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 3d AFM fluctuations T ) 2 Inside the critical cone: * C mT() −1/4 K (J/mol ∝∝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 matter physics Beyond the “standard model”: exotic quasiparticles Quantum fluctuations in reduced dimensions may destroy • Landau quasiparticles • long range order not only at a critical point, but in a whole region of parameter space Examples: • Frustrated magnetic system: spin 1 excitations break up in spin ½ particles (spinons) Phase diagram of J1-J2-Heisenberg model on the square lattice Pseudofermion Functional Renormalization Group Spin operators obey non-canonical commutation relations; express in terms of pseudo-fermions µµ1 † † Si= 2 ∑ ff ii,,ασ αβ β subject to constraint Qi=∑ ff ii,,αα =1 αβ, α Fulfill constraint on average Qi =1 error exponentially small at T< Spin operators obey non-canonical commutation relations; express in terms of pseudo-fermions µµ1 † † Si= 2 ∑ ff ii,,ασ αβ β subject to constraint Qi=∑ ff ii,,αα =1 αβ, α Fulfill constraint on average Qi =1 error exponentially small at T< Take into account three-particle vertex in form of self-energy corrections J. Reuther and P. Wolfle, Phys. Rev. B 81, 144410 (2010) RG-flow of static susceptibility of the J1-J2 model J. Reuther and P. Wolfle, Phys. Rev. B 81, 144410 (2010) RG-flow of pseudo-fermion line-width Do pseudo-fermions become critical quasi-particles? J1-J2-J3-model: J. Reuther, PW, et al, PR B 81, 046416 (2011) honeycomb latt.: J. Reuther, D.A. Abanin, R. Thomale, PR B 84, 014417 (2011) Heisenberg-Kitaev: J. Reuther, R. Thomale, S. Trebst, PR B 84, 100406(R) (2011) Interact. QSH: J. Reuther, R. Thomale, S. Rachel, PR B 86, 155127 (2012) J1-J2 kagome: R. Suttner, C. Platt, J. Reuther, R. T., PR B 89, 020408(R) (2014) cluster FRG BHM: J. Reuther, R. Thomale, PR B 89, 024412 (2014) Beyond the “standard model”: exotic quasiparticles Quantum fluctuations in reduced dimensions may destroy • Landau quasiparticles • long range order not only at a critical point, but in a whole region of parameter space Examples: • Frustrated magnetic system: spin 1 excitations break up in spin ½ particles (spinons) • Electrons in 1d: separation of spin and charge Landau quasiparticle breaks up in two: spinon and holon 1d Quantum wires: Luttinger liquid model Single particle Quantum fluctuations in 1-d Fermi systems lead Spectral function to break-up of Landau quasi-particles into spinon holon • charge-carrying quasi-particles : “holon” • spin-carrying quasi-particles: “spinon” Fractional power laws Tomonaga, Luttinger, 1950s; Kane, Fisher, 1993 … 1d Quantum wires: Luttinger liquid model Single particle Quantum fluctuations in 1-d Fermi systems lead Spectral function to break-up of Landau quasi-particles into spinon holon • charge-carrying quasi-particles : “holon” • spin-carrying quasi-particles: “spinon” Fractional power laws Tomonaga, Luttinger, 1950s; Kane, Fisher, 1993 … Critical quasi-particles lead to fractional power law behavior of conductance G Carbon nanotube junction K ≈ 0.55 Exp: Mehbratu et al., 2013 Theory: D. N. Aristov, P. Wölfle, 2008, 2009, … Fermionic RG: Yue, Glazman, Matveev, 1995 Beyond the “standard model”: exotic quasiparticles Quantum fluctuations in reduced dimensions may destroy • Landau quasiparticles • long range order not only at a critical point, but in a whole region of parameter space Examples: • Frustrated magnetic system: spin 1 excitations break up in spin ½ particles (spinons) • Electrons in 1d: separation of spin and charge Landau quasiparticle breaks up in two: spinon and holon • Quantum Hall effect in 2d: Landau qp transforms into “composite fermion” and breaks up in “fractional” quasiparticles , depending on density • Quantum Hall effect Plateaus of Hall resistance at Multiples of “Quantum resistance” h R = Q e2 Integer QHE 1 RRint = , ν =1,2,.. xyν Q Fractional QHE fract 1 RRxy = q Q , ν p = q p q 2, 4,.. ν p = , pq +1 p =1, ± 2, .. V. Umansky und J. Smet (2000) “Composite Fermion” = Electron + q Flux quanta q=2 : magnetic flux quantum h/e Composite Fermion “absorbs” part of the magnetic flux: Effective magnetic field: BBeff =(1 − 2ν ) 1 • Fermi liquid at B =0, ν = eff 2 J. Jain (1989), Halperin, Lee and Read (1995) QHE of “Composite Fermions” ()ν = 1 2 Coulomb p • integer QHE of CFs corresponds to νν*=pp , = , integer fractional QHE of electrons pq +1 J. Jain (1989) Transport properties of Composite Fermions Composite fermions effectively behave like weakly interacting particles moving in random magnetic field/potential created by remote donors Demonstration of quasiparticle properties of Composite Fermions QH-bar with stripe pattern focussing experiment jx Smet et al. (1996) Weiss, von Klitzing, et al. (1989) Mirlin, Wölfle. (1998) Fractional quasi particles and shot noise Laughlin quasi particles: In the ν=1/3 FQHE-state → 3 flux quanta per electron Fractional quasi particle: Vortex excitation carrying 1 flux quantum → 1/3 electron Detectable through shot noise of current IR S=2e* IR effective fractional charge e*=e/3 Exp.: Glattli et al. (1997) fractional statistics ? de Picciotto et al. (1997) Summary and outlook • The transmutation of quasi-particles in the “standard model” of quantum matter is well understood. • Critical quasi-particles, characterized by diverging effective mass, may emerge in metals near a quantum critical point. • Critical fermionic quasi-particles (spinons) possibly exist in higher dimensional frustrated quantum spin systems. • Exotic quasi-particles associated with dimensional or topological constraints (spinons and holons in 1d systems, vortices in superconductors, composite fermions in the QHE, …) are known to exist, but are in most cases less well understood. • The role of fractional quasi-particles - if any - in higher dimensional (d > 1) metallic systems remains largely unknown (although many proposals exist !). It can not be ruled out at present that such excitations play a role in the physics of High Temperature Superconductors, or other strongly correlated systems.