Order and quantum phase transitions in the cuprate superconductors Eugene Demler (Harvard) Kwon Park (Maryland) Anatoli Polkovnikov Subir Sachdev T. Senthil (MIT) Matthias Vojta (Karlsruhe) Ying Zhang (Maryland) Colloquium article in Reviews of Modern Physics, July 2003, cond-mat/0211005. Talk online: Sachdev Parent compound of the high temperature superconductors: La 2 CuO 4 Band theory k k

La O Cu Half-filled band of Cu 3d orbitals ground state is predicted to be a metal. However, La2CuO4 is a very good insulator Parent compound of the high temperature superconductors: La 2 CuO 4 A Mott insulator H J ij Si S j ij Ground state has long-range magnetic Nel order,

or collinear magnetic (CM) order ix i y Nel order parameter: 1 Si 0 ; S i 0 Introduce mobile carriers of density by substitutional doping of out-ofplane ions e.g. La 2 Sr CuO 4 S 0 Exhibits superconductivity below a high critical temperature Tc Superconductivity in a doped Mott insulator

BCS superconductor obtained by the Cooper instability of a metallic Fermi liquid Pair wavefunction ky k x2 k y2 kx S 0

(Bose-Einstein) condensation of Cooper pairs Many low temperature properties of the cuprate superconductors appear to be qualitatively similar to those predicted by BCS theory. BCS theory of a vortex in the superconductor Pairs Pairs are are disrupted disrupted and and Fermi Fermi surface surface isis revealed. revealed. Vortex core Superflow of

Cooper pairs Superconductivity in a doped Mott insulator Review: S. Sachdev, Science 286, 2479 (1999). Hypothesis: Competition between orders of BCS theory (condensation of Cooper pairs) and Mott insulators Needed: Theory of zero temperature transitions between competing ground states. Minimal phase diagram Paramagnetic Mott Insulator

S 0 High temperature superconductor Paramagnetic BCS Superconductor S 0 S 0 S 0 Magnetic Mott Insulator Magnetic

BCS Superconductor La 2 CuO 4 Quantum phase transitions Magnetic-paramagnetic quantum phase transition in a Mott insulator Coupled ladder antiferromagnet N. Katoh and M. Imada, J. Phys. Soc. Jpn. 63, 4529 (1994). J. Tworzydlo, O. Y. Osman, C. N. A. van Duin, J. Zaanen, Phys. Rev. B 59, 115 (1999). M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002). S=1/2 spins on coupled 2-leg ladders H J ij Si S j ij

0 1 J J close to 1 Square lattice antiferromagnet Experimental realization: La2CuO4 Ground state has long-range collinear magnetic (Neel) order i x i y Si 1 N 0 0 Excitations: 2 spin waves p cx 2 px 2 c y 2 p y 2

close to 0 Weakly coupled ladders 12 Real space Cooper pairs with their charge localized. Upon doping, motion and condensation of Cooper pairs leads to superconductivity Paramagnetic ground state Si 0

close to 0 Excitations Excitation: S=1 exciton (vector N particle of paramagnetic state ) S=1/2 spinons are confined by a linear potential. Energy dispersion away from cx2 px2 c y2 p y2 antiferromagnetic wavevector p 2 T=0 Neel order N0

c Spin gap 1 Neel state S N0 Magnetic order as in La2CuO4 Quantum paramagnet S 0 Electrons in charge-localized

Cooper pairs in cuprates Bond and charge order Mott insulator Paramagnetic ground state of coupled ladder model Can such a state with bond order be the ground state of a system with full square lattice symmetry ? Resonating valence bonds Resonance in benzene leads to a symmetric configuration of valence bonds (F. Kekul, L. Pauling) The paramagnet on the square lattice should also allow other

valence bond pairings, and this leads to a resonating valence bond liquid (P.W. Anderson, 1987) Origin of bond order Quantum entropic effects prefer bond-ordered configurations in which the largest number of singlet pairs can resonate. The state on the upper left has more flippable pairs of singlets than the one on the lower left. These effects lead to a broken square lattice symmetry near the transition to the magnetically ordered states with collinear spins. The quantum dimer model (D. Rokhsar and S.A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988); E. Fradkin and S. A. Kivelson, Mod. Phys. Lett. B 4, 225 (1990)) and semiclassical theories provide dual

descriptions of this physics N. Read and S. Sachdev, Phys. Rev. B 42, 4568 (1990). (Slightly) Slightly Technical interlude: Quantum theory for bond order Key ingredient: Spin Berry Phases A e iSA (Slightly) Slightly Technical interlude: Quantum theory for bond order Key ingredient: Spin Berry Phases A e

iSA Aa oriented area of spherical triangle formed by N a , N a , and an arbitrary reference point N 0 N0 Aa Na N a Aa oriented area of spherical triangle formed by N a , N a , and an arbitrary reference point N 0 N0 N 0

Change in choice of n0 is like a gauge transformation a Aa Aa a a Aa (a is the oriented area of the spherical triangle formed by Na and the two choices for N0 ). a Aa Na N a The area of the triangle is uncertain modulo 4 and the action is invariant under Aa Aa 4

These principles strongly constrain the effective action for Aawhich provides description of the paramagnetic phase Simplest effective action for Aa fluctuations in the paramagnet 1 1 i Z dAa exp 2 cos Aa Aa a Aa 2 2 a a, 2e a 1 on two square sublattices. This is compact QED in d +1 dimensions with static charges 1 on two sublattices. 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 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). Bond order in a frustrated S=1/2 XY magnet A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett. 89, 247201 (2002) First large scale numerical study of the destruction of Neel order in a S=1/2 antiferromagnet with full square lattice symmetry g= H 2 J Six S jx Siy S jy K ij S ijkl

i S j S k Sl Si S j S k Sl See also C. H. Chung, Hae-Young Kee, and Yong Baek Kim, cond-mat/0211299. Experiments on the superconductor revealing order inherited from the Mott insulator Effect of static non-magnetic impurities (Zn or Li) Zn Zn Zn Spinon confinement implies that free S=1/2 moments form near each impurity

S ( S 1) impurity (T 0) 3k BT Zn Spatially resolved NMR of Zn/Li impurities in the superconducting state 7 Inverse local susceptibilty in YBCO Li NMR below Tc J. Bobroff, H. Alloul, W.A. MacFarlane, P. Mendels, N. Blanchard, G. Collin, and J.-F. Marucco, Phys. Rev. Lett. 86, 4116 (2001).

S ( S 1) Measured impurity (T 0) with S 1/ 2 in underdoped sample. 3k BT This behavior does not emerge out of BCS theory. A.M Finkelstein, V.E. Kataev, E.F. Kukovitskii, G.B. Teitelbaum, Physica C 168, 370 (1990). Tuning across the phase diagram by an applied magnetic field Neutron scattering of La 2-xSrx CuO 4 at x=0.1 B. Lake, H. M. Rnnow, N. B. Christensen,G. Aeppli, Kim Lefmann, D. F. McMorrow, P. Vorderwisch, P. Smeibidl, N. Mangkorntong, T. Sasagawa, M. Nohara, H. Takagi, T. E. Mason, Nature, 415, 299 (2002). Talk today at 11:00 AM

H Hc2 Solid line - fit to : I ( H ) a ln Hc2 H Theoretical prediction by E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, 067202 (2001). STM around vortices induced by a magnetic field in the superconducting state J. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002). D iffe re n tia l C on d u ctan ce (nS ) 3.0 Local density of states Regular QPSR

Vortex 2.5 2.0 1.5 ( 1meV to 12 meV) at B=5 Tesla. 1.0 0.5 0.0 -120 1 spatial resolution image of integrated LDOS of

Bi2Sr2CaCu2O8+ -80 -40 0 40 80 120 Sample Bias (mV) S.H. Pan et al. Phys. Rev. Lett. 85, 1536 (2000). Vortex-induced LDOS of Bi2Sr2CaCu2O8+ integrated from 1meV to 12meV Our interpretation:

LDOS modulations are signals of bond order of period 4 revealed in vortex halo 7 pA b 0 pA 100 J. Hoffman E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002). See also: S. A. Kivelson, E. Fradkin, V. Oganesyan, I. P. Bindloss, J.

M. Tranquada, A. Kapitulnik, and C. Howald, condmat/0210683. Conclusions Conclusions I.I. Cuprate Cupratesuperconductivity superconductivityisisassociated associatedwith with doping dopingMott Mottinsulators insulatorswith withcharge

chargecarriers. carriers. II. II. Order Orderparameters parameterscharacterizing characterizingthe theMott Mott insulator insulatorcompete competewith withthe theorder orderassociated associatedwith with the theBose-Einstein Bose-Einsteincondensation

condensationof ofCooper Cooperpairs. pairs. III. III. Classification Classificationof ofMott Mottinsulators insulatorsshows showsthat thatthe the appropriate appropriateorder orderparameters parametersare arecollinear collinear magnetism magnetismand

andbond bondorder. order. IV. IV. Theory Theoryof ofquantum quantumphase phasetransitions transitionsprovides provides semi-quantitative semi-quantitativepredictions predictionsfor forneutron neutron scattering scatteringmeasurements measurementsof ofspin-density-wave spin-density-wave

order orderin insuperconductors; superconductors;theory theoryalso alsoproposes proposesaa connection connectionto toSTM STMexperiments. experiments.