Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Leon Balents (UCSB)Matthew Fisher (UCSB)Subir Sachdev (Yale)T. Senthil (MIT)Ashvin Vishwanath (Berkeley)Matthias Vojta (Karlsruhe)Quantum phase transitions out of the heavy Fermi liquidPhys. Rev. Lett. 90, 216403 (2003).Science 303, 1490 (2004).cond-mat/0409033.Talk online: SachdevThe Kondo lattice+Local moments fsConduction electrons cs† †'K ij i j K i i fi fi fji j i ijH t c c J c c S J S Ss s s ss st<= + � + �� � �r r rrNumber of f electrons per unit cell = nf = 1Number of c electrons per unit cell = ncOutlineOutlineA. The heavy Fermi liquid (FL) B. The metallic antiferromagnet Local moment magnetic metal (LMM),Spin density wave metal (SDW).C. FL to SDW transitionD. The route from FL to LMM The fractionalized Fermi liquid (FL*)E. Detour: Deconfined criticality in insulatorsLandau forbidden quantum transitionsF. Deconfined criticality in the Kondo lattice ?A. The heavy Fermi liquid (FL)Obtained in the limit of large JKThe Fermi surface of heavy quasiparticles encloses a volume which counts all electrons. Fermi volume = 1 + ncArgument for the Fermi surface volume of the FL phase Single ion Kondo effect implies at low energiesKJ � �( )( ) ( )Fermi surface volume density of holes mod 2 1 1 mod 2c cn n=-=- - = +Fermi liquid of S=1/2 holes with hard-core repulsion( )† † † †0i i i ic f c f� � � �-†0 , =1/2 holeif S�Operator approach †Define a bosonic field which measures thehybridization between the two bands: The absence of charge fluctuations on the sitesimplies an emergent compact U(1) gauge thei i ib c ffs ss�:( ) ( )ory, associated with the gauge transformations ; i ii ii i i if f e b b ef t f ts s� �Operator approach DecoupledFLThe FL state is the “Higgs” phase of the U(1) gauge theory.Because of the dispersionless f band in the decoupled case, the ground state is always in the Higgs phase.1Fk cV n= +0b = 0b �B. The metallic antiferromagnetTwo possible states:(A)The local moment magnetic metal (LMM)(B) Spin density wave metal (SDW)(A) The local moment magnetic metal (LMM)The local f moments order antiferromagnetically, and this halves the volume of the Brillouin zone.LMMFk0b =(A) The local moment magnetic metal (LMM)The local f moments order antiferromagnetically, and this halves the volume of the Brillouin zone.LMMFkMagnetic Brillouin zone boundaryThere is an electron-like Fermi surface at with .The Luttinger Theorem is obeyed becausethere are now two electrons per unit cell.The electrons are not part of the FermispLMMFLMMF ckk V nff=here.0b =(B) The spin density wave metal (SDW)There is incomplete Kondo screening of the local f moments in the FL state, and the static moments order antiferromagnetically. This halves the volume of the Brillouin zone.0b �(B) The spin density wave metal (SDW)SDWFkMagnetic Brillouin zone boundaryThere is an hole-like Fermi surface at with 1 .The Luttinger Theorem is again obeyed but the topology of the Fermi surface is different from the LMM metal. The electrons are SDWFSDWF ckk V nf= -part of the Fermi sphere.There is incomplete Kondo screening of the local f moments in the FL state, and the static moments order antiferromagnetically. This halves the volume of the Brillouin zone.0b �C. The FL to SDW quantum phase transition( )( ) ( )( ) ( )222 21,42ddK Kcdd qd uS q q J J d rdjwj w w t jp+= + + - +� �urrKrJ. Mathon, Proc. R. Soc. London A, 306, 355 (1968); T.V. Ramakrishnan, Phys. Rev. B 10, 4014 (1974); M. T. Beal-Monod and K. Maki, Phys. Rev. Lett. 34, 1461 (1975); J.A. Hertz, Phys. Rev. B 14, 1165 (1976).T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism, Springer-Verlag, Berlin (1985); G. G. Lonzarich and L. Taillefer, J. Phys. C 18, 4339 (1985); A.J. Millis, Phys. Rev. B 48, 7183 (1993).LGW theory for quantum critical pointWrite down effective action for SDW order parameter jr fluctuations are damped by mixing with fermionic quasiparticles near the Fermi surfacejrFluctuations of abo put 0 aramagnonsj j = �r rD. The route from FL to LMM:the fractionalized Fermi liquid (FL*)FLFL*Because of direct exchange J between local moments, allow f band to disperse in the decoupled limit.For large J/JK, the gauge theory can enters its deconfined phase, and the fractionalized Fermi liquid (FL*) is obtained.0b =0b �HiggsDeconfinedWork in the regime with small JK, and consider destruction of magnetic order by frustrating (RKKY) exchange interactions between f momentsGround state has Neel order with 0j �rWork in the regime with small JK, and consider destruction of magnetic order by frustrating (RKKY) exchange interactions between f momentsDestroy SDW order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.Destroy SDW order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.Work in the regime with small JK, and consider destruction of magnetic order by frustrating (RKKY) exchange interactions between f momentsWork in the regime with small JK, and consider destruction of magnetic order by frustrating (RKKY) exchange interactions between f moments2A ground state with 0 and no broken lattice symmetries. Such a state has emergent excitations described by a or U(1) gauge theoryspin liquidZj =rP. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974). N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991); X. G. Wen, Phys. Rev. B 44, 2664 (1991).+Local moments fsConduction electrons cs  † †',ij i j K i i fi H fi fji j i i jH t c c J c c S J i j S S           r r rrInfluence of conduction electronsDetermine the ground state of the quantum antiferromagnet defined by JH, and then couple to conduction electrons by JKChoose JH so that ground state of antiferromagnet is a Z2 or U(1) spin liquid+Local moments fsConduction electrons csInfluence of conduction electronsPerturbation theory in JK is regular, and so this state will be stable for finite