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Theory of bad metals. Solvable model.

Theory of bad metals. Solvable model. Alexei Tsvelik Shimul Akhanjee ( postdoc ). Published in PRB. Signs of power laws in 3D systems. Long crossover to FL in d -wave metals. . Fractional power laws for a.) theoretical self-energy S( i w n ) ~ w a and

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Theory of bad metals. Solvable model.

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  1. Theory of bad metals. Solvable model. Alexei Tsvelik ShimulAkhanjee (postdoc) Published in PRB.

  2. Signs of power laws in 3D systems. Long crossover to FL in d-wave metals. Fractional power laws for a.) theoretical self-energy S(iwn) ~ waand b.) optical conductivity s(w)~(iw-S(w))-1in Fe- chalcogenides Fe Te 1-x Se xand ruthenatesSr2RuO4 and SrRuO3. From Yin, Haule, Kotliar (2012).

  3. Kotliaret.al.: treat bad metals as Kondo lattices. Kondo physics: coexistence of localized and itinerant electrons, the itinerant ones gradually screening the local moments. The band structure is T-dependent! Lowest energies – Fermi liquid (or some kind of order), but the crossover can be long! Fe-based, physics is like in Heavy Fermion materials! They are neither itinerant, nor local moments.

  4. DynamicalMeanFieldTheory: electron self energy depends only on frequency. The lattice problem is treated as a problem of isolated impurities with self-consistent Green’s function of the bulk electrons. If DOS is constant the self-consistency is redundant and the lattice problem ~ single impurity :

  5. Kotliar-Haule description of Fe pnictides and chalcogenides: one does not even need to modify the band DOS. The dominant interactions: on-site Hubbard U ~5.0 eV, the Hund’s ferromagnetic exchange JH ~0.8 eV. Crystal fields are small. Multi-stage Kondo effect: since J1 >J2,3, the orbital degrees of freedom are screened first. Unstable QCP (SU2 (5) Wess-Zumino model universality class). J2 –term makes the QCP unstable, the system continues towards Fermi liquid strong coupling, but the crossover is long.

  6. Motivation: Fe- and Ru-compounds In Fe-pnictides and chalcogenidesFe-ion is surrounded by tetrahedron of pnictogen or chalcogen and the crystal field is weak. 5-fold degenerate d –orbital. Inruthenates the coordination is octohedral: only t2g –orbitals remain. Kondo lattice Hamiltonian (Kotliar et. al): Localized d-electrons (6 in total) carry spin and orbital index n,m = 1,…M (M=5 for Fe, M=3 for Ru). Hund’s rule: S=2. U(1)xSU(M)xSU(2) symmetry. Still too complicated!

  7. Solvable model single impurity model: electrons carry spin and 2 orbital indices: U(1)xSU(2)xSU(2): We study the case g1 >> g2 >0, g3 <0 (as Kotliaret. al. did). g2 =0 - orbital and spin sectors decouple. The orbital sector: 2-channel over-screened Kondo model – Quantum Critical. The spin sector: unscreened spin (ferromagnetic coupling). g2 –term is relevant and drives the system to full screening.

  8. For single impurity problem the bulk may have any dimensionality D PROVIDED DOS is constant. D=1 is the most convenient. It allows touse non-Abelianbosonization: x,c – Majorana fermions, f – U(1) bosonic field. Notice that fdoes not enter in the interaction with orbital T and spin S!

  9. Non-Abelianbosonization provides the most convenient form of the theory: Notice that fdoes not enter in the interaction with T and S! Here it is clear that when g2 =0 the model splits into two independent parts. These parts are 2-channel Kondo models!

  10. Consider g2 =0. The theory splits into 2 independent 2-channel Kondo models.2-channel Kondo model with T=1/2 has nontrivial critical point with nonzero ground state entropy (Wiegmann and Tsvelik, 1984).The critical action for energies < Tkondo is (Affleck, Ludwig, Ioffe et. al. 1994) Tkondo~ W g1 exp(- pv/g1). eis a local zero energy Majorana mode responsible for the ground state entropy.

  11. Insert in the interaction, taking into account the retardation: Introduce new local Majoranas: So that the resulting Lagrangian is quadratic + irrelevant terms:

  12. The magnetic susceptibility and the specific heat at T << TK (the orbital Kondo temperature).

  13. Conclusions • The Kondo model with weak spin-orbital coupling g2 has a long crossover from the non-FL QCP at temperatures ~ TK to FL fixed point at T < g22TK. • The orbital degrees of freedom are quenched at TK, • The spin ones are quenched at E0 = g22TK. • DMFT expands these arguments from single impurity to lattice systems. Bad metal behavior is a crossover phenomenon emerging at small g2.

  14. Two-channel Kondo model.

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