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1、Talk online at http:/pantheon.yale.edu/subir,Eugene Demler (Harvard) Kwon Park (Maryland) Anatoli Polkovnikov Subir Sachdev T. Senthil (MIT) Matthias Vojta (Karlsruhe) Ying Zhang (Maryland),Understanding correlated electron systems by a classification of Mott insulators,Colloquium article in Reviews

2、 of Modern Physics, July 2003, cond-mat/0211005.Annals of Physics 303, 226 (2003),Strategy for analyzing correlated electron systems (cuprate superconductors, heavy fermion compounds ),Start from the point where the break down of the Bloch theory of metals is complete-the Mott insulator.Classify gro

3、und states of Mott insulators using conventional and topological order parameters.Correlated electron systems are described by phases and quantum phase transitions associated with order parameters of Mott insulator and the “orders” of Landau/BCS theory. Expansion away from quantum critical points al

4、lows description of states in which the order of Mott insulator is “fluctuating”.,Outline Order in Mott insulators Class A: Compact U(1) gauge theory: collinear spins, bond order and confined spinons in d=2 Class B: Z2 gauge theory: non-collinear spins, visons, topological order, and deconfined spin

5、ons Class A in d=2 The cuprates Class A in d=3 Deconfined spinons and quantum criticality in heavy fermion compounds Conclusions,Class A: Compact U(1) gauge theory: collinear spins, bond order and confined spinons in d=2,I. Order in Mott insulators,Magnetic order,Class A. Collinear spins,I. Order in

6、 Mott insulators,Magnetic order,Class A. Collinear spins,Order specified by a single vector N. Quantum fluctuations leading to loss of magnetic order should produce a paramagnetic state with a vector (S=1) quasiparticle excitation.,Key property,Class A: Collinear spins and compact U(1) gauge theory,

7、Key ingredient: Spin Berry Phases,Write down path integral for quantum spin fluctuations,Class A: Collinear spins and compact U(1) gauge theory,Key ingredient: Spin Berry Phases,Write down path integral for quantum spin fluctuations,Class A: Collinear spins and compact U(1) gauge theory,S=1/2 square

8、 lattice antiferromagnet with non-nearest neighbor exchange,Include Berry phases after discretizing coherent state path integral on a cubic lattice in spacetime,The area of the triangle is uncertain modulo 4p, and the action is invariant under,These principles strongly constrain the effective action

9、 for Aam which provides description of the large g phase,Simplest large g effective action for the Aam,This theory can be reliably analyzed by a duality mapping. d=2: The gauge theory is always in a confining phase and there is bond order in the ground state. d=3: A deconfined phase with a gapless “

10、photon” is possible.,N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990). K. Park and S. Sachdev, Phys. Rev. B 65, 220405 (2002).,I. Order in Mott insulators,Paramagnetic states,Class A. Bond order and spin excitons in d=2,N. Read a

11、nd S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).,S=1/2 spinons are confined by a linear potential into a S=1 spin exciton,Spontaneous bond-order leads to vector S=1 spin excitations,A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett. 89, 247201 (2002),Bond order in a frustr

12、ated S=1/2 XY magnet,g=,First large scale numerical study of the destruction of Neel order in a S=1/2 antiferromagnet with full square lattice symmetry,Class B: Z2 gauge theory: non-collinear spins, visons, topological order, and deconfined spinons,I. Order in Mott insulators,Magnetic order,Class B.

13、 Noncollinear spins,(B.I. Shraiman and E.D. Siggia, Phys. Rev. Lett. 61, 467 (1988),A. V. Chubukov, S. Sachdev, and T. Senthil Phys. Rev. Lett. 72, 2089 (1994),I. Order in Mott insulators,Paramagnetic states,Class B. Topological order and deconfined spinons,D.S. Rokhsar and S. Kivelson, Phys. Rev. L

14、ett. 61, 2376 (1988) N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991); R. Jalabert and S. Sachdev, Phys. Rev. B 44, 686 (1991); X. G. Wen, Phys. Rev. B 44, 2664 (1991). T. Senthil and M.P.A. Fisher, Phys. Rev. B 62, 7850 (2000).,Number of valence bonds cutting line is conserved modulo 2 this

15、 is described by the same Z2 gauge theory as non-collinear spins,RVB state with free spinons,P. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974).,I. Order in Mott insulators,Paramagnetic states,Class B. Topological order and deconfined spinons,D.S. Rokhsar and S. Kivelson, Phys. Rev. Lett. 61, 2376

16、 (1988) N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991); R. Jalabert and S. Sachdev, Phys. Rev. B 44, 686 (1991); X. G. Wen, Phys. Rev. B 44, 2664 (1991). T. Senthil and M.P.A. Fisher, Phys. Rev. B 62, 7850 (2000).,Number of valence bonds cutting line is conserved modulo 2 this is described

17、 by the same Z2 gauge theory as non-collinear spins,RVB state with free spinons,P. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974).,I. Order in Mott insulators,Class B. Topological order and deconfined spinons,S3,(A) North pole,(B) South pole,x,y,(A),(B),Vortices associated with p1(S3/Z2)=Z2 (viso

18、ns) have gap in the paramagnet. This gap survives doping and leads to stable hc/e vortices at low doping.,N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) T. Senthil and M.P.A. Fisher, Phys. Rev. B 62, 7850 (2000). S. Sachdev, Physical Review B 45, 389 (1992) N. Nagaosa and P.A. Lee, Physica

19、l Review B 45, 966 (1992),Paramagnetic states,II. Evidence cuprates are in class A,Competing order parameters,1. Pairing order of BCS theory (SC),Bose-Einstein condensation of d-wave Cooper pairs,Doping a paramagnetic bond-ordered Mott insulator,systematic Sp(N) theory of translational symmetry brea

20、king, while preserving spin rotation invariance.,S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991).,Mott insulator with bond-order,T=0,II. Doping Class A,A phase diagram,Pairing order of BCS theory (SC) Collinear magnetic order (CM) Bond order (B),S. Sachdev and N. Read, Int. J. Mod. Phys.

21、B 5, 219 (1991). M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999); M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. B 62, 6721 (2000); M. Vojta, Phys. Rev. B 66, 104505 (2002).,Vertical axis is any microscopic parameter which suppresses CM order,Evidence cuprates are in class A,Evidence cupra

22、tes are in class A,Neutron scattering shows collinear magnetic order co-existing with superconductivity,J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). Y.S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999). S. Wakimoto, R.J. Birgeneau, Y.S. Lee, and G. Shirane, Phys. Re

23、v. B 63, 172501 (2001).,Evidence cuprates are in class A,Neutron scattering shows collinear magnetic order co-existing with superconductivityProximity of Z2 Mott insulators requires stable hc/e vortices, vison gap, and Senthil flux memory effect,S. Sachdev, Physical Review B 45, 389 (1992) N. Nagaos

24、a and P.A. Lee, Physical Review B 45, 966 (1992) T. Senthil and M. P. A. Fisher, Phys. Rev. Lett. 86, 292 (2001). D. A. Bonn, J. C. Wynn, B. W. Gardner, Y.-J. Lin, R. Liang, W. N. Hardy, J. R. Kirtley, and K. A. Moler, Nature 414, 887 (2001). J. C. Wynn, D. A. Bonn, B. W. Gardner, Y.-J. Lin, R. Lian

25、g, W. N. Hardy, J. R. Kirtley, and K. A. Moler, Phys. Rev. Lett. 87, 197002 (2001).,Evidence cuprates are in class A,Neutron scattering shows collinear magnetic order co-existing with superconductivityProximity of Z2 Mott insulators requires stable hc/e vortices, vison gap, and Senthil flux memory e

26、ffectNon-magnetic impurities in underdoped cuprates acquire a S=1/2 moment,Effect of static non-magnetic impurities (Zn or Li),J. Bobroff, H. Alloul, W.A. MacFarlane, P. Mendels, N. Blanchard, G. Collin, and J.-F. Marucco, Phys. Rev. Lett. 86, 4116 (2001).,Inverse local susceptibilty in YBCO,7Li NMR

27、 below Tc,A.M Finkelstein, V.E. Kataev, E.F. Kukovitskii, G.B. Teitelbaum, Physica C 168, 370 (1990).,Spatially resolved NMR of Zn/Li impurities in the superconducting state,Evidence cuprates are in class A,Neutron scattering shows collinear magnetic order co-existing with superconductivityProximity

28、 of Z2 Mott insulators requires stable hc/e vortices, vison gap, and Senthil flux memory effectNon-magnetic impurities in underdoped cuprates acquire a S=1/2 moment,Evidence cuprates are in class A,Neutron scattering shows collinear magnetic order co-existing with superconductivityProximity of Z2 Mo

29、tt insulators requires stable hc/e vortices, vison gap, and Senthil flux memory effectNon-magnetic impurities in underdoped cuprates acquire a S=1/2 momentTests of phase diagram in a magnetic field (talk by E. Demler, Microsymposium MS IV, May 28, 11:40),E. Demler, S. Sachdev, and Ying Zhang, Phys.

30、Rev. Lett. 87, 067202 (2001).,E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, 067202 (2001).,100,b,Vortex-induced LDOS of Bi2Sr2CaCu2O8+d integrated from 1meV to 12meV,J. Hoffman E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (20

31、02).,Our interpretation: LDOS modulations are signals of bond order of period 4 revealed in vortex haloSee also: S. A. Kivelson, E. Fradkin, V. Oganesyan, I. P. Bindloss, J. M. Tranquada, A. Kapitulnik, and C. Howald, cond-mat/0210683.,C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik, Phys. Rev. B

32、 67, 014533 (2003).,Spectral properties of the STM signal are sensitive to the microstructure of the charge order,Measured energy dependence of the Fourier component of the density of states which modulates with a period of 4 lattice spacings,Conclusions Two classes of Mott insulators: (A) Collinear

33、 spins, compact U(1) gauge theory; bond order and confinements of spinons in d=2 (B) Non-collinear spins, Z2 gauge theory Doping Class A in d=2 Magnetic/bond order co-exist with superconductivity at low doping Cuprates most likely in this class. Theory of quantum phase transitions provides a description of “fluctuating order” in the superconductor. Class A in d=3 Deconfined spinons and quantum criticality in heavy fermion compounds (cond-mat/0209144 and cond-mat/0305193),