1、T. Senthil (MIT) Subir Sachdev Matthias Vojta (Karlsruhe),Quantum phases and critical points of correlated metals,Transparencies online at http:/pantheon.yale.edu/subir,cond-mat/0209144,Outline Kondo lattice models Doniachs phase diagram and its quantum critical point A new phase: FL* Paramagnetic s
2、tates of quantum antiferromagnets: (A) Bond order, (B) Topological order. Lieb-Schultz-Mattis-Laughlin-Bonesteel-Affleck-Yamanaka- Oshikawa flux-piercing arguments Extended phase diagram and its critical points Conclusions,I. Kondo lattice models,I. Doniachs T=0 phase diagram for the Kondo lattice,J
3、K / t,“Heavy” Fermi liquid with moments Kondo screened by conduction electrons. Fermi surface obeys Luttingers theorem.,FL,SDW,Local moments choose some static spin arrangement,Luttingers theorem on a d-dimensional lattice for the FL phase,Let v0 be the volume of the unit cell of the ground state,nT
4、 be the total number density of electrons per volume v0.(need not be an integer),A “large” Fermi surface,Arguments for the Fermi surface volume of the FL phase,Fermi liquid of S=1/2 holes with hard-core repulsion,Arguments for the Fermi surface volume of the FL phase,Alternatively:,Formulate Kondo l
5、attice as the large U limit of the Anderson model,Quantum critical point between SDW and FL phases,Spin fluctuations of renormalized S=1/2 fermionic quasiparticles, (loosely speaking, TK remains finite at the quantum critical point),Gaussian theory of paramagnon fluctuations:,J.A. Hertz, Phys. Rev.
6、B 14, 1165 (1976).,J. Mathon, Proc. R. Soc. London A, 306, 355 (1968); T.V. Ramakrishnan, Phys. Rev. B 10, 4014 (1974); 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
7、 48, 7183 (1993).,Characteristic paramagnon energy at finite temperature G(0,T) T p with p 1.Arises from non-universal corrections to scaling, generated by term.,Quantum critical point between SDW and FL phases,Critical point not described by strongly-coupled critical theory with universal dynamic r
8、esponse functions dependent on In such a theory, paramagnon scattering amplitude would be determined by kBT alone, and not by value of microscopic paramagnon interaction term.,Additional singular corrections to quasiparticle self energy in d=2,Ar. Abanov and A. V. Chubukov Phys. Rev. Lett. 84, 5608
9、(2000); A. Rosch Phys. Rev. B 64, 174407 (2001).,S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).,(Contrary opinions: P. Coleman, Q. Si),Outline Kondo lattice models Doniachs phase diagram and its quantum critical point A new phase: FL* Paramagnetic states of quantum antiferromagnets: (A) Bon
10、d order, (B) Topological order. Lieb-Schultz-Mattis-Laughlin-Bonesteel-Affleck-Yamanaka- Oshikawa flux-piercing arguments Extended phase diagram and its critical points Conclusions,II. A new phase: FL*,Reconsider Doniach phase diagram,II. A new phase: FL*,This phase preserves spin rotation invarianc
11、e, and has a Fermi surface of sharp electron-like quasiparticles. The state has “topological order” and associated neutral excitations. The topological order can be easily detected by the violation of Luttingers theorem. It can only appear in dimensions d 1,Precursors: L. Balents and M. P. A. Fisher
12、 and C. Nayak, Phys. Rev. B 60, 1654, (1999); T. Senthil and M.P.A. Fisher, Phys. Rev. B 62, 7850 (2000);S. Burdin, D. R. Grempel, and A. Georges, Phys. Rev. B 66, 045111 (2002).,It is more convenient to consider the Kondo-Heiseberg model:,Work in the regime JH JK,Determine the ground state of the q
13、uantum antiferromagnet defined by JH, and then couple to conduction electrons by JK,Ground states of quantum antiferromagnets,Begin with magnetically ordered states, and consider quantum transitions which restore spin rotation invariance,Two classes of ordered states:,(A) Collinear spins,(B) Non-col
14、linear spins,(A) Collinear spins, bond order, and confinement,N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).,(A) Collinear spins, bond order, and confinement,Bond-ordered state,N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).,State of conduction electrons,Perturbation theory in J
15、K is regular and so this state will be stable for finite JK,However, because nf=2 (per unit cell of ground state) nT= nf+ nc= nc(mod 2), and Luttingers theorem is obeyed.,At JK= 0 the conduction electrons form a Fermi surface on their own with volume determined by nc,FL state with bond order,(B) Non
16、-collinear spins, deconfined spinons, Z2 gauge theory, and topological order,Solve constraints by writing:,Other approaches to a Z2 gauge theory: R. Jalabert and S. Sachdev, Phys. Rev. B 44, 686 (1991); S. Sachdev and M. Vojta, J. Phys. Soc. Jpn 69, Suppl. B, 1 (2000). X. G. Wen, Phys. Rev. B 44, 26
17、64 (1991). T. Senthil and M.P.A. Fisher, Phys. Rev. B 62, 7850 (2000). R. Moessner, S. L. Sondhi, and E. Fradkin, Phys. Rev. B 65, 024504 (2002). L. B. Ioffe, M.V. Feigelman, A. Ioselevich, D. Ivanov, M. Troyer and G. Blatter, Nature 415, 503 (2002).,Vortices associated with p1(S3/Z2)=Z2,S3,(A) Nort
18、h pole,(B) South pole,x,y,(A),(B),Can also consider vortex excitation in phase without magnetic order, : vison,A paramagnetic phase with vison excitations suppressed has topological order. Suppression of visons also allows za quanta to propagate these are the spinons.,State with spinons must have to
19、pological order,State of conduction electrons,Perturbation theory in JK is regular, and topological order is robust, and so this state will be stable for finite JK,So volume of Fermi surface is determined by (nT -1)= nc(mod 2), and Luttingers theorem is violated.,At JK= 0 the conduction electrons fo
20、rm a Fermi surface on their own with volume determined by nc,The FL* state,Outline Kondo lattice models Doniachs phase diagram and its quantum critical point A new phase: FL* Paramagnetic states of quantum antiferromagnets: (A) Bond order, (B) Topological order. Lieb-Schultz-Mattis-Laughlin-Bonestee
21、l-Affleck-Yamanaka- Oshikawa flux-piercing arguments Extended phase diagram and its critical points Conclusions,III. Lieb-Schultz-Mattis-Laughlin-Bonesteel-Affleck-Yamanaka Oshikawa flux-piercing arguments,III. Lieb-Schultz-Mattis-Laughlin-Bonesteel-Affleck- Yamanaka-Oshikawa flux-piercing arguments
22、,Lx,F,M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000).,Unit cell ax , ay. Lx/ax , Ly/ay coprime integers,M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000).,Effect of flux-piercing on a topologically ordered quantum paramagnet,N. E. Bonesteel, Phys. Rev. B 40, 8954 (1989). G. Misguich, C. Lhuillier, M. Mam
23、brini, and P. Sindzingre, Eur. Phys. J. B 26, 167 (2002).,1,2,3,Lx-1,Lx-2,Lx,F,Effect of flux-piercing on a topologically ordered quantum paramagnet,N. E. Bonesteel, Phys. Rev. B 40, 8954 (1989). G. Misguich, C. Lhuillier, M. Mambrini, and P. Sindzingre, Eur. Phys. J. B 26, 167 (2002).,1,2,3,Lx-1,Lx
24、-2,Lx,vison,Flux piercing argument in Kondo lattice,Shift in momentum is carried by nT electrons, where,nT = nf+ nc,In topologically ordered, state, momentum associated with nf=1 electron is absorbed by creation of vison. The remaining momentum is absorbed by Fermi surface quasiparticles, which encl
25、ose a volume associated with nc electrons.,Outline Kondo lattice models Doniachs phase diagram and its quantum critical point A new phase: FL* Paramagnetic states of quantum antiferromagnets: (A) Bond order, (B) Topological order. Lieb-Schultz-Mattis-Laughlin-Bonesteel-Affleck-Yamanaka- Oshikawa flu
26、x-piercing arguments Extended phase diagram and its critical points Conclusions,IV. Extended phase diagram and its critical points,IV. Extended T=0 phase diagram for the Kondo lattice,JK / t,FL,SDW,Magnetic frustration,FL*,SDW*,Hertz Gaussian paramagnon theory,Quantum criticality associated with the
27、 onset of topological order described by interacting gauge theory. (Speaking loosely TK vanishes along this line),* phases have spinons with Z2 (d=2,3) or U(1) (d=3) gauge charges, and associated gauge fields.Fermi surface volume does not distinguish SDW and SDW* phases.,Because of strong gauge fluc
28、tuations, U(1)-FL* may be unstable to U(1)-SDW* at low temperatures.Only phases at T=0: FL, SDW, U(1)-SDW*.,IV. Extended T=0 phase diagram for the Kondo lattice,JK / t,FL,SDW,Magnetic frustration,SDW*,SDW*,Hertz Gaussian paramagnon theory,Quantum criticality associated with the onset of topological
29、order described by interacting gauge theory. (Speaking loosely TK vanishes along this line),U(1) fractionalization (d=3),Because of strong gauge fluctuations, U(1)-FL* may be unstable to U(1)-SDW* at low temperatures.Only phases at T=0: FL, SDW, U(1)-SDW*. Quantum criticality dominated by a T=0 FL-F
30、L* transition.,U(1) fractionalization (d=3),Mean-field phase diagram,C/T ln(1/T),(cf. A. Georges),Strongly coupled quantum criticality with a topological or spin-glass order parameter,Order parameter does not couple directly to simple observables,Dynamic spin susceptiblity,Non-trivial universal scal
31、ing function which is a property of a bulk d-dimensional quantum field theory describing “hidden” order parameter.,Superconductivity is generic between FL and Z2 FL* phases.,JK / t,FL,SDW,Magnetic frustration,FL*,SDW*,Hertz Gaussian paramagnon theory,Superconductivity,Z2 fractionalization,Z2 fractio
32、nalization,Pairing of spinons in small Fermi surface state induces superconductivity at the confinement transition,Small Fermi surface state can also exhibit a second-order metamagnetic transition in an applied magnetic field, associated with vanishing of a spinon gap.,FL*,FL,Mean-field phase diagram,Conclusions,New phase diagram as a paradigm for clean metals with local moments. Topologically ordered (*) phases lead to novel quantum criticality.New FL* allows easy detection of topological order by Fermi surface volume,
