Quantum vortices and competing orders

Transcription

1 Talk online: Google Sachdev Quantum votices and competing odes cond-mat/ and cond-mat/ Leon Balents (UCSB) Loenz Batosch (Yale) Anton Bukov (UCSB) Subi Sachdev (Yale) Kishnendu Sengupta (Toonto)

2 Distinct expeimental chacteistics of undedoped cupates at T > T c Measuements of Nenst effect ae well explained by a model of a liquid of votices and anti-votices N. P. Ong, Y. Wang, S. Ono, Y. Ando, and S. Uchida, Annalen de Physik 13, 9 (2004). Y. Wang, S. Ono, Y. Onose, G. Gu, Y. Ando, Y. Tokua, S. Uchida, and N. P. Ong, Science 299, 86 (2003).

3 LDOS of Bi 2 S 2 CaCu 2 O 8+δ at 100 K. M. Veshinin, S. Misa, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Science, 303, 1995 (2004). Distinct expeimental chacteistics of undedoped cupates at T > T c STM measuements obseve density modulations with a peiod of 4 lattice spacings

4 Is thee a connection between voticity and density wave modulations? Density wave ode---modulations in paiing amplitude, exchange enegy, o hole density. Equivalent to valence-bond-solid (VBS) ode (except at the special peiod of 2 lattice spacings)

5 Votex-induced LDOS of Bi 2 S 2 CaCu 2 O 8+δ integated fom 1meV to 12meV at 4K 7 pa 0 pa b Votices have halos with LDOS modulations at a peiod 4 lattice spacings 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). Pediction of VBS ode nea votices: K. Pak and S. Sachdev, Phys. Rev. B 64, (2001).

6 Landau-Ginzbug-Wilson theoy of multiple ode paametes: Votex/phase fluctuations ( pefomed pais ) Complex supeconducting ode paamete: Ψ sc Ψ sc Ψ e i θ sc symmety encodes numbe consevation Chage/valence-bond/pai-density/stipe ode Ode paametes: ρ Q ρ ρ i e ϑ Q i. ( ) ρ e = Q Q Q encodes space goup symmety

7 Landau-Ginzbug-Wilson theoy of multiple ode paametes: LGW fee enegy: [ ] F = F [ ] sc Ψ sc + F chage ρ Q + F 2 4 F Ψ = Ψ + u Ψ + sc sc 1 sc 1 sc int sc 2 4 Fchage ρq = 2 ρq + u2 ρq + F 2 = v Ψ ρ + Distinct symmeties of ode paametes pemit couplings only between thei enegy densities (thee ae no symmeties which otate two ode paametes into each othe) Fo lage positive v, thee is a coelation between votices and density wave ode Q 2 int

8 Ψ sc Supeconducto Pedictions of LGW theoy Fist ode tansition Chage-odeed insulato

9 Ψ sc Supeconducto Ψ sc Supeconducto Pedictions of LGW theoy Fist ode tansition Coexistence (Supesolid) Chage-odeed insulato Chage-odeed insulato

10 Ψ sc Supeconducto Ψ sc Supeconducto Pedictions of LGW theoy Fist ode tansition Coexistence (Supesolid) Chage-odeed insulato Chage-odeed insulato Ψ sc Supeconducto " Disodeed " ( topologically odeed) Ψ = 0, ρ = 0 sc Q Chage-odeed insulato

11 Ψ sc Supeconducto Ψ sc Supeconducto Pedictions of LGW theoy Fist ode tansition Coexistence (Supesolid) Chage-odeed insulato Chage-odeed insulato Ψ sc Supeconducto " Disodeed " ( topologically odeed) Ψ = 0, ρ = 0 sc Q Chage-odeed insulato

12 Non-supeconducting quantum phase must have some othe ode : Chage ode in an insulato Femi suface in a metal Topological ode in a spin liquid This equiement is not captued by LGW theoy.

13 Needed: a theoy of pecuso fluctuations of the density wave ode of the insulato within the supeconducto. i.e. a connection between votices and density wave ode

14 Outline A. Supefluid-insulato tansitions of bosons on the squae lattice at factional filling Quantum mechanics of votices in a supefluid poximate to a commensuate Mott insulato B. Application to a shot-ange paiing model fo the cupate supeconductos Competition between VBS ode and d-wave supeconductivity

15 A. Supefluid-insulato tansitions of bosons on the squae lattice at factional filling Quantum mechanics of votices in a supefluid poximate to a commensuate Mott insulato

16 Bosons at density f = 1 Weak inteactions: supefluidity Stong inteactions: Mott insulato which peseves all lattice symmeties LGW theoy: continuous quantum tansitions between these states M. Geine, O. Mandel, T. Esslinge, T. W. Hänsch, and I. Bloch, Natue 415, 39 (2002).

17 Bosons at density f = 1/2 (equivalent to S=1/2 AFMs) Weak inteactions: supefluidity Ψ sc 0 Stong inteactions: Candidate insulating states = 1 ( + ) 2 ( ) e i. All insulating phases have density-wave ode ρ = with ρ 0 C. Lannet, M.P.A. Fishe, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Pak, Annals of Physics, 298, 58 (2002) Q Q Q

18 Ψ sc Supeconducto Ψ sc Supeconducto Pedictions of LGW theoy Fist ode tansition Coexistence (Supesolid) Chage-odeed insulato Chage-odeed insulato Ψ sc Supeconducto " Disodeed " ( topologically odeed) Ψ = 0, ρ = 0 sc Q Chage-odeed insulato

19 Ψ sc Supeconducto Ψ sc Supeconducto Pedictions of LGW theoy Fist ode tansition Coexistence (Supesolid) Chage-odeed insulato Chage-odeed insulato Ψ sc Supeconducto " Disodeed " ( topologically odeed) Ψ = 0, ρ = 0 sc Q Chage-odeed insulato

20 Boson-votex duality τ y x Quantum mechanics of twodimensional bosons: wold lines of bosons in spacetime C. Dasgupta and B.I. Halpein, Phys. Rev. Lett. 47, 1556 (1981); D.R. Nelson, Phys. Rev. Lett. 60, 1973 (1988); M.P.A. Fishe and D.-H. Lee, Phys. Rev. B 39, 2756 (1989);

21 Boson-votex duality z Classical statistical mechanics of a dual theedimensional supeconducto, with ode paamete ϕ : tajectoies of y votices in a x magnetic field Stength of magnetic field on dual supeconducto ϕ = density of bosons = f flux quanta pe plaquette C. Dasgupta and B.I. Halpein, Phys. Rev. Lett. 47, 1556 (1981); D.R. Nelson, Phys. Rev. Lett. 60, 1973 (1988); M.P.A. Fishe and D.-H. Lee, Phys. Rev. B 39, 2756 (1989);

22 Boson-votex duality Cuent of ϕ boson votex e i2 π The wavefunction of a votex acquies a phase of 2π each time the votex encicles a boson Stength of magnetic field on dual supeconducto ϕ = density of bosons = f flux quanta pe plaquette C. Dasgupta and B.I. Halpein, Phys. Rev. Lett. 47, 1556 (1981); D.R. Nelson, Phys. Rev. Lett. 60, 1973 (1988); M.P.A. Fishe and D.-H. Lee, Phys. Rev. B 39, 2756 (1989);

23 Boson-votex duality Statistical mechanics of dual supeconducto ϕ, is invaiant unde the squae lattice space goup: T, T : Tanslations by a lattice spacing in the x, y diections x R y : Rotation by 90 degees. Magnetic space goup: TT = e TT 2πif x y y x R T R= T ; R T R= T ; R = y x x y Stength of magnetic field on dual supeconducto ϕ = density of bosons = f flux quanta pe plaquette ;

24 Boson-votex duality Hofstadte spectum of dual supeconducting ode ϕ At density f = p/ q ( p, q elatively pime integes) thee ae q species of votices, ϕ (with =1 q), associated with q gauge-equivalent egions of the Billouin zone Magnetic space goup: TT = e TT 2πif x y y x R T R= T ; R T R= T ; R = y x x y ;

25 Boson-votex duality Hofstadte spectum of dual supeconducting ode ϕ At density f = p/ q ( p, q elatively pime integes) thee ae q species of votices, ϕ (with =1 q), associated with q gauge-equivalent egions of the Billouin zone The q votices fom a pojective epesentation of the space goup T : ; T : e x 2πif ϕ ϕ + 1 y ϕ ϕ 1 R: ϕ q q m= 1 ϕ e m 2πimf See also X.-G. Wen, Phys. Rev. B 65, (2002)

26 Boson-votex duality The q votices chaacteize both ϕ supeconducting and density wave odes Supeconducto insulato : ϕ = 0 ϕ 0

27 Boson-votex duality The q votices chaacteize both ϕ supeconducting and density wave odes Density wave ode: Status of space goup symmety detemined by 2π p density opeatos ρq at wavevectos Qmn = mn, q T q iπmnf * 2πi mf ρmn = e ϕ ϕ + ne = 1 iqixˆ x : ρq ρqe ; Ty : R: ρ ( Q) ρ( RQ) ρ Q ρ e Q ( ) Each pinned votex in the supefluid has a halo of density wave ode ove a length scale the zeo-point quantum motion of the votex. This scale diveges upon appoaching the Mott insulato iqi yˆ

28 Votex-induced LDOS of Bi 2 S 2 CaCu 2 O 8+δ integated fom 1meV to 12meV at 4K 7 pa 0 pa b Votices have halos with LDOS modulations at a peiod 4 lattice spacings 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). Pediction of VBS ode nea votices: K. Pak and S. Sachdev, Phys. Rev. B 64, (2001).

29 Ψ sc Supeconducto Ψ sc Supeconducto Pedictions of LGW theoy Fist ode tansition Coexistence (Supesolid) Chage-odeed insulato Chage-odeed insulato Ψ sc Supeconducto " Disodeed " ( topologically odeed) Ψ = 0, ρ = 0 sc Q Chage-odeed insulato

30 Analysis of extended LGW theoy of pojective epesentation Fluctuation-induced, Ψ sc weak, fist ode tansition Supeconducto ϕ = 0, ρ mn = 0 Chage-odeed insulato ϕ 0, ρ mn 0

31 Analysis of extended LGW theoy of pojective epesentation Fluctuation-induced, Ψ sc weak, fist ode tansition Supeconducto ϕ = 0, ρ mn = 0 Ψ sc Supeconducto ϕ = 0, ρ mn = 0 Supesolid ϕ = 0, ρ mn 0 Chage-odeed insulato ϕ 0, ρ mn 0 Chage-odeed insulato ϕ 0, ρ mn 0

32 Analysis of extended LGW theoy of pojective epesentation Fluctuation-induced, Ψ sc weak, fist ode tansition Supeconducto ϕ = 0, ρ mn = 0 Ψ sc Supeconducto ϕ = 0, ρ mn = 0 Ψ sc Supeconducto ϕ = 0, ρ mn = 0 Supesolid ϕ = 0, ρ mn 0 Chage-odeed insulato ϕ 0, ρ mn 0 Chage-odeed insulato ϕ 0, ρ mn 0 Second ode tansition Chage-odeed insulato ϕ 0, ρ mn 0

33 Analysis of extended LGW theoy of pojective epesentation Spatial stuctue of insulatos fo q=4 (f=1/4 o 3/4) a b unit cells; q, q, ab, a b q all integes

34 B. Application to a shot-ange paiing model fo the cupate supeconductos Competition between VBS ode and d-wave supeconductivity

35 Phase diagam of doped antifeomagnets g = paamete contolling stength of quantum fluctuations in a semiclassical theoy of the destuction of Neel ode La 2 CuO 4 Neel ode

36 Phase diagam of doped antifeomagnets g o VBS ode La 2 CuO 4 Neel ode N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fishe, Science 303, 1490 (2004).

37 Phase diagam of doped antifeomagnets g o Dual votex theoy fo inteplay between VBS ode and d-wave supeconductivity VBS ode La 2 CuO 4 Neel ode Hole density δ

38 A convenient deivation of the dual theoy fo votices is obtained fom the doped quantum dime model H dqd ( ) = J + ( ) t + Density of holes = δ E. Fadkin and S. A. Kivelson, Mod. Phys. Lett. B 4, 225 (1990).

39 Duality mapping of doped dime model shows: Votices in the supeconducting state obey the magnetic tanslation algeba with TT = e TT 2πif x y y x f p = = q 1δ 2 whee δ is the density of holes in the poximate MI Mott insulato (fo δ = 1/ 8, f = 7 /16 q = 16) MI Most esults of Pat A on bosons can be applied unchanged with q as detemined above MI

40 Phase diagam of doped antifeomagnets g VBS ode δ = 1 32 La 2 CuO 4 Neel ode Hole density δ

41 Phase diagam of doped antifeomagnets g VBS ode δ = 1 16 La 2 CuO 4 Neel ode Hole density δ

42 Phase diagam of doped antifeomagnets g VBS ode δ = 1 8 La 2 CuO 4 Neel ode Hole density δ

43 Phase diagam of doped antifeomagnets VBS ode g d-wave supeconductivity above a citical δ La 2 CuO 4 Neel ode Hole density δ

44 Conclusions I. Desciption of the competition between supeconductivity and density wave ode in tem of defects (votices). Theoy natually excludes disodeed phase with no ode. II. III. Votices cay the quantum numbes of both supeconductivity and the squae lattice space goup (in a pojective epesentation). Votices cay halo of chage ode, and pinning of votices/anti-votices leads to a unified theoy of STM modulations in zeo and finite magnetic fields. IV. Conventional (LGW) pictue: density wave ode causes the tanspot enegy gap, the appeaance of the Mott insulato. Pesent pictue: Mott localization of chage caies is moe fundamental, and (weak) density wave ode emeges natually in theoy of the Mott tansition.