Magnetic phases and critical points of insulators and superconductors

Transcription

1 Magnetic phases and critical points of insulators and superconductors Colloquium article in Reviews of Modern Physics, July 2003, cond-mat/ cond-mat/ Quantum Phase Transitions Cambridge University Press Talks online: Sachdev

2 What is a quantum phase transition? Non-analyticity in ground state properties as a function of some control parameter g E E True level crossing: Usually a first-order transition g Avoided level crossing which becomes sharp in the infinite volume limit: g second-order transition

3 Why study quantum phase transitions? T Quantum-critical g c Theory for a quantum system with strong correlations: describe phases on either side of g c by expanding in deviation from the quantum critical point. Critical point is a novel state of matter without quasiparticle excitations g Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures. Important property of ground state at g=g c : temporal and spatial scale invariance; characteristic energy scale at other values of g: ~ z g g ν c

4 I. Quantum Ising Ising Chain chain II. Outline Coupled Dimer Antiferromagnet A. Coherent state path integral B. Quantum field theory near critical point Single order parameter. III. Coupled dimer antiferromagnet in a magnetic field Bose condensation of triplons IV. Magnetic transitions in superconductors Quantum phase transition in a background Abrikosov flux lattice V. Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins, bond order and confined spinons in d=2 Class B: Z 2 gauge theory: non-collinear spins, RVB, visons, topological order, and deconfined spinons Multiple order parameters. VI. Conclusions

5 I. Quantum Ising Chain Degrees of freedom: j = 1 N qubits, N "large", 1 1 or =, j 2 2 j j ( + ) = ( ) j j j j j Hamiltonian of decoupled qubits: H = Jg σ 2Jg 0 j x j j j

6 leads to entangled states at g of order unity Coupling between qubits: H = J σ σ + z z 1 j j 1 j ( + )( + ) Prefers neighboring qubits are Full Hamiltonian j j j+ 1 j+ 1 either or j j+ 1 j j+ 1 (not entangled) ( x z z g ) j j j H = H + H = J σ + σ σ j

7 Ground state: G = Weakly-coupled qubits Lowest excited states: 1 2g j = j + Coupling between qubits creates flipped-spin quasiparticle states at momentum p p j ipx j = e pa Excitation energy ε 2 Excitation gap = 2gJ 2J + O g j 2 1 ( p) = + 4Jsin + O( g ) 1 ( ) ( g 1) ε ( p) π a p π a Entire spectrum can be constructed out of multi-quasiparticle states

8 S( p, ω) Dynamic Structure Factor : Cross-section to flip a to a (or vice versa) while transferring energy ω and momentum p S( p, ω ) Zδ ω ( p) ( ε ) Quasiparticle pole Weakly-coupled qubits ( g 1) Three quasiparticle continuum Structure holds to all orders in 1/g ~3 ω At T > 0, collisions between quasiparticles broaden pole to a Lorentzian of width 1 τ where the phase coherence time is given by ϕ τ 1 ϕ = 2kT B e π kbt τ ϕ S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)

9 Ground states: G = g 2 Second state G obtained by G and G mix only at order g Lowest excited states: domain walls Strongly-coupled qubits( g 1) N Ferromagnetic moment z N0 = G σ G 0 d j = + Coupling between qubits creates new domainwall quasiparticle states at momentum p ipx = j p e d j j j 2 2 ( p) = + Jg + O( g ) Excitation energy ε 4 sin 2 ( ) Excitation gap = 2J 2gJ + O g pa 2 ε ( p) π a p π a

10 S( p, ω) Dynamic Structure Factor : Strongly-coupled qubits ( g 1) Cross-section to flip a to a (or vice versa) while transferring energy ω and momentum p S( p, ω ) N 2 ( π) δ ( ω) δ ( p) Two domain-wall continuum Structure holds to all orders in g ~2 ω At T > 0, motion of domain walls leads to a finite phase coherence time τ, 1 2kT B B and broadens coherent peak to a width 1 τϕ where e = τ π S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997) ϕ k T ϕ

11 Entangled states at g of order unity Flipped-spin Quasiparticle weight Z g c g ( ) 1/4 Z g g ~ c A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, (1994) N ~ ( ) 1/8 0 gc g moment N 0 Ferromagnetic P. Pfeuty Annals of Physics, 57, 79 (1970) g c g Excitation energy gap ~ g gc g c g

12 S( p, ω) Dynamic Structure Factor : Critical coupling ( g = g c ) Cross-section to flip a to a (or vice versa) while transferring energy ω and momentum p S( p, ω ) ( ) ω 7/8 ~ c p cp ω No quasiparticles --- dissipative critical continuum

13 ( x z z gσ + σ σ ) H = I J i i i+ 1 i Quasiclassical dynamics Quasiclassical dynamics i χω ( ) = dt σ j () t, σk ( 0) e = T k 7/4 0 A ( 1 iω / Γ +...) π kt B Γ R = 2 tan 16 z z iωt S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997). R σσ z j z k ~ j 1 k P. Pfeuty Annals of Physics, 57, 79 (1970) 1/4

14 Outline I. Quantum Ising Chain II. Coupled Dimer Dimer Antiferromagnet A. Coherent state path integral B. Quantum field theory near critical point III. IV. Coupled dimer antiferromagnet in a magnetic field Bose condensation of triplons Magnetic transitions in superconductors Quantum phase transition in a background Abrikosov flux lattice V. Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins, bond order and confined spinons in d=2 Class B: Z 2 gauge theory: non-collinear spins, RVB, visons, topological order, and deconfined spinons VI. Conclusions Single order parameter. Multiple order parameters.

15 II. Coupled Dimer Antiferromagnet M. P. Gelfand, R. R. P. Singh, and D. A. Huse, Phys. Rev. B 40, (1989). 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, (2002). S=1/2 spins on coupled dimers H = < ij> J ij S i S j 0 λ 1 J λ J

16 λ close to 1 Square lattice antiferromagnet Experimental realization: La CuO 2 4 Ground state has long-range magnetic (Neel) order S i = ( ) i 1 x + iy N 0 0 Excitations: 2 spin waves (magnons) ε = c p + c p p x x y y

17 λ close to 0 Weakly coupled dimers 1 = 2 ( ) Paramagnetic ground state S = 0 i

18 λ close to 0 Weakly coupled dimers 1 = 2 ( ) Excitation: S=1 triplon (exciton, spin collective mode) Energy dispersion away from antiferromagnetic wavevector ε spin gap p = + c p x x y y 2 c p

19 λ close to 0 Weakly coupled dimers 1 = 2 ( ) S=1/2 spinons are confined by a linear potential into a S=1 triplon

20 T=0 Neel order N 0 λ c Spin gap λ 1 Neel state S = N 0 Quantum paramagnet S = 0 δ in cuprates?

21 II.A Coherent state path integral Path integral for quantum spin fluctuations Key ingredient: Spin Berry Phases A isa e

22 II.A Coherent state path integral Path integral for quantum spin fluctuations Key ingredient: Spin Berry Phases A isa e

23 II.A Coherent state path integral See Chapter 13 of Quantum Phase Transitions, S. Sachdev, Cambridge University Press (1999). Z = Path integral for a single spin ( H[ S] / T Tr e ) ( is A d d H S ) D N τ δ N 1 exp ( τ) τ τ N τ ( ) ( 2 = ) ( ) τ ( ) Aτ τ dτ = Oriented area of triangle on surface of unit sphere bounded Action for lattice antiferromagnet ( τ) ( τ + dτ) 0 by N, N,and a fixed reference N N n L ( τ ) = η ( x, τ) + ( x, τ) j j j j η = ± 1 identifies sublattices j n and L vary slowly in space and time

24 Integrate out L and take the continuum limit ( ) ( 2 Z = D n x, τ δ n 1) exp is ηjaτ ( xj, τ) dτ j η =± 1 identifies sublattices j 1 ( 2 2 ( ) ( ) ) τ n c xn 2 2 d xdτ + 2g g g < g c λ > λ > g c c λ < λ c Discretize spacetime into a cubic lattice ( 2 ) 1 i Z = d n δ n 1exp n n η A a A oriented area of spherical triangle aµ a a a a+ µ a aτ g a, µ 2 a formed by n, n, and an arbitrary reference point n a a+ µ 0 a cubic lattice sites; µ xy,, τ;

25 Integrate out L and take the continuum limit ( ) ( 2 Z = D n x, τ δ n 1) exp is ηjaτ ( xj, τ) dτ j η =± 1 identifies sublattices j 1 ( 2 2 ( ) ( ) ) τ n c xn 2 2 d xdτ + 2g g g < g c λ > λ > g c c λ < λ c Z Discretize spacetime into a cubic lattice ( 2 ) 1 = d n n 1exp n n a aδ a a a+ µ g a, µ Berry phases can be neglected for coupled dimer antiferromagent a cubic lattice sites; µ xy,, τ; (justified later) Quantum path integral for two-dimensional quantum antiferromagnet Partition function of a classical three-dimensional ferromagnet at a temperature g Quantum transition at λ=λ c is related to classical Curie transition at g=g c

26 II.B Quantum field theory for critical point S φα λ close to λ c : use soft spin field 1 u = d xdτ φ + φ + λ λ φ + φ 2 4! (( ) 2 c ( ) 2 ( ) ) ( ) 2 α τ α α α b x c 3-component antiferromagnetic order parameter Oscillations of φ α about zero (for λ < λ c ) spin-1 collective mode Im χ ( p, ω ) ε p = + c p T=0 spectrum = λ λ c c ω

27 Dynamic spectrum at the critical point Critical coupling ( λ = λ c ) ( p ω ) Im χ, ( ω ) ~ c p (2 η )/2 cp ω No quasiparticles --- dissipative critical continuum

28 I. Quantum Ising Chain Outline II. Coupled Dimer Antiferromagnet A. Coherent state path integral B. Quantum field theory near critical point III. Coupled dimer Dimer antiferromagnet Antiferromagnet in a in magnetic a magnetic field field Bose condensation of triplons IV. Magnetic transitions in superconductors Quantum phase transition in a background Abrikosov flux lattice V. Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins, bond order and confined spinons in d=2 Class B: Z 2 gauge theory: non-collinear spins, RVB, visons, topological order, and deconfined spinons VI. Conclusions Single order parameter. Multiple order parameters.

29 Both states are insulators T=0 H S j SDW = N1 cos. + N sin. 2 ( K r ) j Collinear spins: N N = ( K r ) j Non-collinear spins: N N S j = 0 Pressure, exchange constant,. Quantum critical point Evolution of phase diagram in a magnetic field

30 Effect of a field on paramagnet Energy of zero momentum triplon states 0 Bose-Einstein condensation of S z =1 triplon H

31 III. Phase diagram in a magnetic field. H SDW gµ B H = Spin singlet state with a spin gap 1 Tesla = mev 1/λ Related theory applies to double layer quantum Hall systems at ν=2

32 III. Phase diagram in a magnetic field. Zeeman term leads to a uniform precession of spins ( i H )( i H ) 2 * τφα τφα εασρ σφρ τφα εαβγ βφγ Take H oriented along the z direction. Then ( )( 2 2) ( 2)( 2 2 λ ) c λ φx + φy λc λ H φx + φy 2 For λ > λc, φx ~ λ λc + H, while for λ < λc, Hc = ~ λc λ. H SDW gµ B H = Spin singlet state with a spin gap 1 Tesla = mev 1/λ Related theory applies to double layer quantum Hall systems at ν=2

33 III. Phase diagram in a magnetic field. Zeeman term leads to a uniform precession of spins ( i H )( i H ) 2 * τφα τφα εασρ σφρ τφα εαβγ βφγ Take H oriented along the z direction. Then ( )( 2 2) ( 2)( 2 2 λ ) c λ φx + φy λc λ H φx + φy 2 For λ > λc, φx ~ λ λc + H, while for λ < λc, Hc = ~ λc λ. Elastic scattering intensity [ ] I H = H I[ 0] + a J 2 H SDW gµ B H = H ~ λ λ c Spin singlet state with a spin gap c 1 Tesla = mev 1/λ Related theory applies to double layer quantum Hall systems at ν=2

34 I. Quantum Ising Chain Outline II. III. Coupled Dimer Antiferromagnet A. Coherent state path integral B. Quantum field theory near critical point Coupled dimer antiferromagnet in a magnetic field Bose condensation of triplons IV. Magnetic transitions in superconductors IV. Magnetic transitions in superconductors in superconductors Quantum phase transition in a background Abrikosov flux lattice V. Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins, bond order and confined spinons in d=2 Class B: Z 2 gauge theory: non-collinear spins, RVB, visons, topological order, and deconfined spinons VI. Conclusions Single order parameter. Multiple order parameters.

35 S j SDW = N1 cos. + N sin. 2 ( K r ) j ( K r ) j T=0 S j = 0 Collinear spins: N N = Non-collinear spins: N N Quantum critical point Pressure, carrier concentration,. We have so far considered the case where both states are insulators

36 S j SC+SDW = N1 cos. + N sin. 2 ( K r ) j ( K r ) j T=0 SC S j = 0 Collinear spins: N N = Non-collinear spins: N N Quantum critical point Pressure, carrier concentration,. Now both sides have a background superconducting (SC) order

37 Magnetic transition in a d-wave superconductor If K does not exactly connect two nodal points, critical theory is as in an insulator Otherwise, new theory of coupled excitons and nodal quasiparticles L. Balents, M.P.A. Fisher, C. Nayak, Int. J. Mod. Phys. B 12, 1033 (1998).

38 Interplay of SDW and SC order in the cuprates T=0 phases of LSCO k y π/a Insulator 0 π/a k x 0 Néel 0.02 SDW SC+SDW ~ SC δ (additional commensurability effects near δ=0.125) J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 278, 1432 (1997). S. Wakimoto, G. Shirane et al., Phys. Rev. B 60, R769 (1999). 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. Rev. B 63, (2001).

39 Interplay of SDW and SC order in the cuprates T=0 phases of LSCO k y π/a Insulator 0 π/a k x 0 Néel 0.02 SDW SC+SDW ~ SC δ (additional commensurability effects near δ=0.125) J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 278, 1432 (1997). S. Wakimoto, G. Shirane et al., Phys. Rev. B 60, R769 (1999). 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. Rev. B 63, (2001).

40 Interplay of SDW and SC order in the cuprates T=0 phases of LSCO k y π/a Superconductor with T c,min =10 K 0 π/a k x 0 Néel 0.02 SDW SC+SDW ~ SC δ (additional commensurability effects near δ=0.125) J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 278, 1432 (1997). S. Wakimoto, G. Shirane et al., Phys. Rev. B 60, R769 (1999). 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. Rev. B 63, (2001).

41 Collinear magnetic (spin density wave) order Sj = N cos K. r + N sin K. r ( ) ( ) j j 1 2 Collinear spins K = ( ππ, ) ; N 2 = 0 K = ( 3π 4, π) ; N 2 = 0 K = 3π 4, π N ( ) ( 2 1) = ; N 2 1

42 Interplay of SDW and SC order in the cuprates T=0 phases of LSCO H k y π/a Superconductor with T c,min =10 K 0 π/a k x 0 Néel 0.02 SDW SC+SDW ~ SC δ Use simplest assumption of a direct second-order quantum phase transition between SC and SC+SDW phases Follow intensity of elastic Bragg spots in a magnetic field

43 Dominant effect of magnetic field: Abrikosov flux lattice r v s 1 r Spatially averaged superflow kinetic energy v H 3H ln 2 c2 s Hc2 H

44 Effect of magnetic field on SDW+SC to SC transition (extreme Type II superconductivity) Φ = N + in α 1α 2α Quantum theory for dynamic and critical spin fluctuations S b 1/ T g g2 τ r α τ α α α α ( ) = dr d Φ + c Φ + sφ + Φ + Φ v Sc = drdτ Φα ψ ( ) = Φ(, τ) Z ψ r D r e ψ ( r) ( r) δ ln Z = δψ 0 F GL S b S c ψ FGL = d r ψ + + r ia 2 ( ) ψ 2 Static Ginzburg-Landau theory for non-critical superconductivity

45 Triplon wavefunction in bare potential V 0 (x) Bare triplon potential ( r) = + v ( r) 2 V s ψ 0 Energy Spin gap 0 Vortex cores x D. P. Arovas, A. J. Berlinsky, C. Kallin, and S.-C. Zhang, Phys. Rev. Lett. 79, 2871 (1997) proposed static magnetism (with =0) localized within vortex cores

46 Wavefunction of lowest energy triplon Φ ( r) = ( r) + Φ ( r) 2 after including triplon interactions: V V g α 0 α Bare triplon potential ( r) = + v ( r) 2 V s ψ 0 Energy Spin gap 0 Vortex cores x Strongly relevant repulsive interactions between excitons imply that triplons must be extended as 0. E. Demler, S. Sachdev, and Y. Zhang, Phys. Rev. Lett. 87, (2001). A.J. Bray and M.A. Moore, J. Phys. C 15, L7 65 (1982). J.A. Hertz, A. Fleishman, and P.W. Anderson, Phys. Rev. Lett. 43, 942 (1979).

47 Phase diagram of SC and SDW order in a magnetic field Spatially averaged superflow kinetic energy r v s 1 r v H 3H ln 2 c2 s Hc2 H The suppression of SC order appears to the SDW order as a uniform effective "doping" δ : δ eff H 3H H = C H H c2 ( ) δ ln c2 E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, (2001).

48 Phase diagram of SC and SDW order in a magnetic field Elastic scattering intensity [, δ] I[ 0, δ ] I H [ δ ] eff H 3Hc 0, ln 2 I + a H c2 H δ H eff ( H ) = δ ( δ δc) ~ ln 1/ c ( ( δ δc )) E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, (2001).

49 Magnetic order parameter Structure of long-range SDW order in SC+SDW phase E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, (2001). δ 2 f0 Hln(1/ H) s s c = -0.3 Dynamic structure factor S 3 2 ( k, ω) ( 2π) δ ( ω) f δ ( k G) = + G G G reciprocal lattice vectors of vortex lattice. k measures deviation from SDW ordering wavevector K

50 Neutron scattering of La 2-xSrxCuO 4 at x=0.1 B. Lake, H. M. Rønnow, N. B. Christensen, G. Aeppli, K. Lefmann, D. F. McMorrow, P. Vorderwisch, P. Smeibidl, N. Mangkorntong, T. Sasagawa, M. Nohara, H. Takagi, T. E. Mason, Nature, 415, 299 (2002). H Hc Solid line - fit to : I( H) a ln H H 2 = c2 See also S. Katano, M. Sato, K. Yamada, T. Suzuki, and T. Fukase, Phys. Rev. B 62, R14677 (2000).

51 Phase diagram of a superconductor in a magnetic field Neutron scattering observation of SDW order enhanced by superflow. δ H eff ( H ) = δ ( δ δc ) ~ ln 1/ c ( ( δ δ c )) Prediction: SDW S = fluctuations 1 triplon energy enhanced by superflow and bond order pinned by vortex Hcores 3 ( ) ( ) (no Hc2 ε H = ε 0 b ln spins in vortices). H cshould 2 H be observable in STM K. Park and S. Sachdev Physical Review B 64, (2001); E. Demler, Y. Zhang, S. Sachdev, E. Demler and and Ying S. Zhang, Sachdev, Phys. Physical Rev. Lett. Review 87, B , (2001). (2002).

52 Vortex-induced LDOS of Bi 2 Sr 2 CaCu 2 O 8+δ integrated from 1meV to 12meV 7 pa b Our interpretation: LDOS modulations are signals of bond order of period 4 revealed in vortex halo 0 pa 100Å 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, cond-mat/

53 Fourier Transform of Vortex-Induced LDOS map K-space locations of vortex induced LDOS K-space locations of Bi and Cu atoms J. Hoffman et al. Science, 295, 466 (2002). Distances in k space have units of 2π/a 0 a 0 =3.83 Å is Cu-Cu distance

54 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 M. Vojta, Phys. Rev. B 66, (2002); C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik, Phys. Rev. B 67, (2003). Theoretical modeling shows that this spectrum is best obtained by a modulation of bond variables, such as the exchange, kinetic or pairing energies. D. Podolsky, E. Demler, K. Damle, and B.I. Halperin, Phys. Rev. B in press, condmat/

55 I. Quantum Ising Chain Outline II. III. IV. Coupled Dimer Antiferromagnet A. Coherent state path integral B. Quantum field theory near critical point Coupled dimer antiferromagnet in a magnetic field Bose condensation of triplons Magnetic transitions in superconductors Quantum phase transition in a background Abrikosov flux lattice V. Antiferromagnets with an odd number V. Antiferromagnets with an odd number of S=1/2 spins per per unit cell. unit cell Class A A: Compact U(1) gauge theory: collinear spins, bond order and confined spinons in d=2 Class B: Z 2 gauge theory: non-collinear spins, RVB, visons, topological order, and deconfined spinons VI. Conclusions Single order parameter. Multiple order parameters.

56 V. Order in Mott insulators Magnetic order S cos (. ) sin (. ) j = N1 K r j + N2 K r j Class A. Collinear spins K = ( ππ, ) ; N 2 = 0 K = ( 3π 4, π) ; N 2 = 0 K = 3π 4, π N ( ) ( 2 1) = ; N 2 1

57 V. Order in Mott insulators Magnetic order Sj = N cos K. r + N sin K. r ( ) ( ) j j 1 2 Class A. Collinear spins Key property 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.

58 Class A: Collinear spins and compact U(1) gauge theory Write down path integral for quantum spin fluctuations Key ingredient: Spin Berry Phases A isa e

59 Class A: Collinear spins and compact U(1) gauge theory Write down path integral for quantum spin fluctuations Key ingredient: Spin Berry Phases A isa e

60 Class A: Collinear spins and compact U(1) gauge theory S=1/2 square lattice antiferromagnet with non-nearest neighbor exchange H = J S S i< j ij i j Include Berry phases after discretizing coherent state path integral on a cubic lattice in spacetime ( 2 ) 1 i Z = dn δ n 1 exp n n η A a ηa ± 1 on two square sublattices ; na ~ ηasa Neel order parameter; A oriented area of spherical triangle aµ a a a a+ µ a aτ g a, µ 2 a formed by na, n, a + µ and an arbitrary reference point n 0

61 ( 2 ) 1 i Z = d n δ n 1exp n n η A a a a a a+ µ a aτ g a, µ 2 a Small g Spin-wave theory about Neel state receives minor modifications from Berry phases. Large g Berry phases are crucial in determining structure of "quantum-disordered" phase with n = 0 Integrate out n a to obtain effective action for a A a µ

62 n 0 A a µ n a na + µ

63 n 0 n 0 Change in choice of n 0 is like a gauge transformation A A γ + γ aµ aµ a+ µ a γ γ a a + µ (γ a is the oriented area of the spherical triangle formed by n a and the two choices for n 0 ). n a A a µ A a µ The area of the triangle is uncertain modulo 4π, and the action is invariant under Aaµ Aaµ + 4π na + µ These principles strongly constrain the effective action for A aµ which provides description of the large g phase

64 Simplest large g effective action for the A aµ 1 1 ( ) i Z = da exp cos A A a, µ with e ~ g This is compact QED in d +1 dimensions with static charges ± 1 on two sublattices. η A aµ 2 2 µ aν ν aµ a aτ e a 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, (2002).

65

66 For large e 2, low energy height configurations are in exact one-toone correspondence with dimer coverings of the square lattice 2+1 dimensional height model is the path integral of the Quantum Dimer Model There is no roughening transition for three dimensional interfaces, which are smooth for all couplings There is a definite average height of the interface Ground state has bond order.

67 V. Order in Mott insulators Paramagnetic states S j = 0 Class A. Bond order and spin excitons in d=2 1 = 2 ( ) S=1/2 spinons are confined by a linear potential into a S=1 spin triplon Spontaneous bond-order leads to vector S=1 spin excitations N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).

68 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, (2002) First large scale numerical study of the destruction of Neel order in a S=1/2 antiferromagnet with full square lattice symmetry ( x x y y) ( ) i j i j i j k l i j k l = + + H 2J S S S S K S S S S S S S S ij ijkl g=

69 I. Quantum Ising Chain Outline II. III. IV. Coupled Dimer Antiferromagnet A. Coherent state path integral B. Quantum field theory near critical point Coupled dimer antiferromagnet in a magnetic field Bose condensation of triplons Magnetic transitions in superconductors Quantum phase transition in a background Abrikosov flux lattice V. Antiferromagnets with an odd number V. Antiferromagnets with an odd number of S=1/2 spins per per unit unit cell. cell Class A: Compact U(1) gauge theory: collinear spins, bond order and confined spinons in d=2 Class B: B Z 2 gauge theory: non-collinear spins, RVB, visons, topological order, and deconfined spinons VI. Conclusions Single order parameter. Multiple order parameters.

70 V.B Order in Mott insulators Magnetic order Sj = N cos K. r + N sin K. r Class B. Noncollinear spins ( ) ( ) j j 1 2 K = π π ( 3 4, ) (B.I. Shraiman and E.D. Siggia, Phys. Rev. Lett. 61, 467 (1988)) K = π π ( 4 /3,4 3) N = N, N.N =

71 V.B Order in Mott insulators Magnetic order Sj = N cos K. r + N sin K. r Class B. Noncollinear spins Solve constraints by expressing N in terms of two complex numbers z, z 2 2 z z 2 2 N1 + in2 = i( z + z ) 2zz Order in ground state specified by a spinor, 1,2 ( ) ( ) j j 1 2 N = N, N.N = ( z z ) This spinor can become a S=1/2 spinon in paramagnetic state. (modulo an overall sign). Order parameter space: 3 2 Physical observables are invariant under the 2 gauge transformation a a S Z Z z ± z A. V. Chubukov, S. Sachdev, and T. Senthil Phys. Rev. Lett. 72, 2089 (1994)

72 V.B Order in Mott insulators Magnetic order Sj = N cos K. r + N sin K. r Class B. Noncollinear spins Vortices associated with π 1 (S 3 /Z 2 )=Z 2 (visons) ( ) ( ) j j 1 2 (A) North pole y (A) (B) (B) South pole S 3 x Such vortices (visons) can also be defined in the phase in which spins are quantum disordered. A Z 2 spin liquid with deconfined spinons must have visons supressed N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)

73 S Model effective action and phase diagram = J σ z z + h.c. K ij σ ij ij αi α j σ ij First order transition Z 2 gauge field Magnetically ordered (Derivation using Schwinger bosons on a quantum antiferromagnet: S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)). Confined spinons Free spinons and topological order P. E. Lammert, D. S. Rokhsar, and J. Toner, Phys. Rev. Lett. 70, 1650 (1993) ; Phys. Rev. E 52, 1778 (1995). (For nematic liquid crystals)

74 V.B Order in Mott insulators Paramagnetic states S j = 0 Class B. Topological order and deconfined spinons A topologically ordered state in which vortices associated with π 1 (S 3 /Z 2 )=Z 2 [ visons ] are gapped out. This is an RVB state with deconfined S=1/2 spinons z a N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991). X. G. Wen, Phys. Rev. B 44, 2664 (1991). A.V. Chubukov, T. Senthil and S. S., Phys. Rev. Lett.72, 2089 (1994). T. Senthil and M.P.A. Fisher, Phys. Rev. B 62, 7850 (2000). P. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974). G. Misguich and C. Lhuillier, Eur. Phys. J. B 26, 167 (2002). R. Moessner and S.L. Sondhi, Phys. Rev. Lett. 86, 1881 (2001). Recent experimental realization: Cs 2 CuCl 4 R. Coldea, D.A. Tennant, A.M. Tsvelik, and Z. Tylczynski, Phys. Rev. Lett. 86, 1335 (2001).

75 V.B Order in Mott insulators Paramagnetic states S j = 0 Class B. Topological order and deconfined spinons Direct description of topological order with valence bonds Number of of valence bonds cutting line line is is conserved modulo Changing sign sign of of each each such such bond bond does does not not modify state. state. This This is is equivalent to to a Z 2 gauge 2 transformation with with za za on on sites sites to to the the right right of of dashed line. line. D. Rokhsar and S.A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988); N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989).

76 V.B Order in Mott insulators Paramagnetic states S j = 0 Class B. Topological order and deconfined spinons Direct description of topological order with valence bonds Number of of valence bonds cutting line line is is conserved modulo Changing sign sign of of each each such such bond bond does does not not modify state. state. This This is is equivalent to to a Z 2 gauge 2 transformation with with za za on on sites sites to to the the right right of of dashed line. line. D. Rokhsar and S.A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988); N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989).

77 V.B Order in Mott insulators Paramagnetic states S j = 0 Class B. Topological order and deconfined spinons Direct description of topological order with valence bonds Terminating the the line line creates a plaquette with with Z 2 flux 2 flux at at the the X a vison. X D. Rokhsar and S.A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988); N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989).

78 Effect of flux-piercing on a topologically ordered quantum paramagnet N. E. Bonesteel, Phys. Rev. B 40, 8954 (1989). Φ G. Misguich, C. Lhuillier, M. Mambrini, and P. Sindzingre, Eur. Phys. J. B 26, 167 (2002). D = L y Ψ = a D D D L x -2 L x -1 L x 1 2 3

79 Effect of flux-piercing on a topologically ordered quantum paramagnet N. E. Bonesteel, Phys. Rev. B 40, 8954 (1989). vison G. Misguich, C. Lhuillier, M. Mambrini, and P. Sindzingre, Eur. Phys. J. B 26, 167 (2002). D = Ψ = a D Number of bonds ( ) cutting dashed line L x -2 L x -1 L x L y D D After flux insertion D 1 D ; Equivalent to inserting a vison inside hole of the torus. This leads to a ground state degeneracy.

80 I. Quantum Ising Chain II. III. IV. VI. Conclusions Coupled Dimer Antiferromagnet A. Coherent state path integral B. Quantum field theory near critical point Coupled dimer antiferromagnet in a magnetic field Bose condensation of triplons Magnetic transitions in superconductors Quantum phase transition in a background Abrikosov flux lattice V. Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins, bond order and confined spinons in d=2 Class B: Z 2 gauge theory: non-collinear spins, RVB, visons, topological order, and deconfined spinons VI. Cuprates are best understood as doped class A Mott insulators. Single order parameter. Multiple order parameters.

81 Competing order parameters in the cuprate superconductors 1. Pairing order of BCS theory (SC) (Bose-Einstein) condensation of d-wave Cooper pairs Orders (possibly fluctuating) associated with proximate Mott insulator in class A 2. Collinear magnetic order (CM) 3. Bond/charge/stripe order (B) (couples strongly to half-breathing phonons) S. Sachdev and N. Read, Int. J. Mod. Phys. 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, (2002).

82 Evidence cuprates are in class A

83 Evidence cuprates 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. Rev. B 63, (2001).

84 Evidence cuprates are in class A Neutron scattering shows collinear magnetic order co-existing with superconductivity Proximity of Z 2 Mott insulators requires stable hc/e vortices, vison gap, and Senthil flux memory effect S. Sachdev, Physical Review B 45, 389 (1992) N. Nagaosa 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. Liang, W. N. Hardy, J. R. Kirtley, and K. A. Moler, Phys. Rev. Lett. 87, (2001).

85 Evidence cuprates are in class A Neutron scattering shows collinear magnetic order co-existing with superconductivity Proximity of Z 2 Mott insulators requires stable hc/e vortices, vison gap, and Senthil flux memory effect Non-magnetic impurities in underdoped cuprates acquire a S=1/2 moment

86 Effect of static non-magnetic impurities (Zn or Li) Zn Zn Zn Zn Spinon confinement implies that free S=1/2 moments form near each impurity χ impurity ( T 0) = SS ( + 1) kt 3 B

87 Spatially resolved NMR of Zn/Li impurities in the superconducting state 7 Li NMR below T c Inverse local susceptibilty in YBCO J. Bobroff, H. Alloul, W.A. MacFarlane, P. Mendels, N. Blanchard, G. Collin, and J.-F. Marucco, Phys. Rev. Lett. 86, 4116 (2001). SS ( + 1) Measured χ impurity( T 0) = with S = 1/ 2 in underdoped sample. 3kT This behavior does not emerge out of BCS theory. B A.M Finkelstein, V.E. Kataev, E.F. Kukovitskii, G.B. Teitel baum, Physica C 168, 370 (1990).

88 Evidence cuprates are in class A Neutron scattering shows collinear magnetic order co-existing with superconductivity Proximity of Z 2 Mott insulators requires stable hc/e vortices, vison gap, and Senthil flux memory effect Non-magnetic impurities in underdoped cuprates acquire a S=1/2 moment

89 Evidence cuprates are in class A Neutron scattering shows collinear magnetic order co-existing with superconductivity Proximity of Z 2 Mott insulators requires stable hc/e vortices, vison gap, and Senthil flux memory effect Non-magnetic impurities in underdoped cuprates acquire a S=1/2 moment Tests of phase diagram in a magnetic field

90 Phase diagram of a superconductor in a magnetic field Neutron scattering observation of SDW order enhanced by superflow. δ H eff ( H ) = δ ( δ δc) ~ ln 1/ c ( ( δ δc )) E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, (2001).

91 Phase diagram of a superconductor in a magnetic field Neutron scattering observation of SDW order enhanced by superflow. δ H eff ( H ) = δ ( δ δc) ~ ln 1/ c ( ( δ δc )) Possible STM observation of predicted bond order in halo around vortices K. Park and S. Sachdev Physical Review B 64, (2001); E. Demler, Y. Zhang, S. Sachdev, E. Demler and and Ying S. Zhang, Sachdev, Phys. Physical Rev. Lett. Review 87, B , (2001). (2002).

92 VI. Doping Class A Doping a paramagnetic bond-ordered Mott insulator systematic Sp(N) theory of translational symmetry breaking, while preserving spin rotation invariance. T=0 d-wave superconductor Superconductor with co-existing bond-order S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). Mott insulator with bond-order

93 Vertical axis is any microscopic parameter which suppresses CM order A phase diagram Microscopic theory for the interplay of bond (B) and d-wave superconducting (SC) order Pairing order of BCS theory (SC) Collinear magnetic order (CM) Bond order (B) S. Sachdev and N. Read, Int. J. Mod. Phys. 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, (2002).