Competing orders: beyond Landau-Ginzburg-Wilson theory

Transcription

1 Talk online: Google Sachdev Competing odes: beyond Landau-Ginzbug-Wilson theoy Colloquium aticle in Reviews of Moden Physics 75, 913 (2003) Leon Balents (UCSB) Loenz Batoh (Yale) Anton Bukov (UCSB) Eugene Demle (Havad) Matthew Fishe (UCSB) Anatoli Polkovnikov (Havad) Kishnendu Sengupta (Yale) T. Senthil (MIT) Ashvin Vishwanath (MIT) Matthias Vojta (Kalsuhe)

2 Talk online: Google Sachdev Putting competing odes in thei place nea the Mott tansition Leon Balents (UCSB) Loenz Batoh (Yale) Anton Bukov (UCSB) Eugene Demle (Havad) Matthew Fishe (UCSB) Anatoli Polkovnikov (Havad) Kishnendu Sengupta (Yale) T. Senthil (MIT) Ashvin Vishwanath (MIT) Matthias Vojta (Kalsuhe)

3 Possible oigins of the pseudogap in the cupate supeconductos: Phase fluctuations, pefomed pais Complex ode paamete: Ψ i Ψ Ψ symmety encodes numbe consevation e θ Chage/valence-bond/pai-density/stipe ode i Ode paametes: ρ = ρ e (density ρ epesents any obsevable invaiant unde spin otations, time-evesal, and spatial invesion) ρ ρ i e ϑ Spin liquid ( ). encodes space goup symmety

4 Ode paametes ae not independent Ginzbug-Landau-Wilson appoach to competing ode paametes (combine ode paametes into a supespin ): [ ] F = F [ ] Ψ + F chage ρ + F 2 4 F Ψ = Ψ + u Ψ int 2 4 Fchage ρ = 2 ρ + u2 ρ + 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) 2 S. Sachdev and E. Demle, Phys. Rev. B 69, (2004). int

5 Pedictions of LGW theoy Supeconducto Ψ 0, ρ = 0 Fist ode tansition Chage-odeed insulato Ψ = 0, ρ Supeconducto Ψ 0, ρ = 0 Supeconducto Ψ 0, ρ = 0 Coexistence (Supesolid) Ψ 0, ρ 0 " Disodeed " ( topologically odeed) Ψ = 0, ρ = 0 Chage-odeed insulato Ψ = 0, ρ Chage-odeed insulato Ψ = 0, ρ 0 1 2

6 Pedictions of LGW theoy Supeconducto Ψ 0, ρ = 0 Fist ode tansition Chage-odeed insulato Ψ = 0, ρ Supeconducto Ψ 0, ρ = 0 Supeconducto Ψ 0, ρ = 0 Coexistence (Supesolid) Ψ 0, ρ 0 " Disodeed " ( topologically odeed) Ψ = 0, ρ = 0 Chage-odeed insulato Ψ = 0, ρ Chage-odeed insulato Ψ = 0, ρ 0 1 2

7 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.

8 Outline A. Supefluid-insulato tansitions of bosons on the squae lattice at factional filling Dual votex theoy and the magnetic space goup. B. Application to a shot-ange paiing model fo the cupate supeconductos Chage ode and d-wave supeconductivity in an effective theoy fo the spin S=0 secto. C. Implications fo STM

9 A. Supefluid-insulato tansitions of bosons on the squae lattice at factional filling Dual votex theoy and the magnetic space goup.

10 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. Hänh, and I. Bloch, Natue 415, 39 (2002).

11 Bosons at density f = 1/2 (equivalent to S=1/2 AFMs) Weak inteactions: supefluidity Ψ 0 Stong inteactions: Candidate insulating states = 1 ( + ) 2 ( ) e i. All insulating phases have density-wave ode ρ = 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)

12 Pedictions of LGW theoy Supeconducto Ψ 0, ρ = 0 Fist ode tansition Chage-odeed insulato Ψ = 0, ρ Supeconducto Ψ 0, ρ = 0 Supeconducto Ψ 0, ρ = 0 Coexistence (Supesolid) Ψ 0, ρ 0 " Disodeed " ( topologically odeed) Ψ = 0, ρ = 0 Chage-odeed insulato Ψ = 0, ρ Chage-odeed insulato Ψ = 0, ρ 0 1 2

13 Supefluid-insulato tansition of had coe bosons at f=1/2 (Neel-valence bond solid tansition of S=1/2 AFM) A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett. 89, (2002) Lage ale (> 8000 sites) numeical study of the destuction of supefluid (i.e. magnetic Neel) ode at half filling with full squae lattice symmety VBS insulato Supefluid (Neel) ode g= ij ( + + ) ( ) i j i j i j k l i j k l = + + H J S S S S K S S S S S S S S ijkl

14 Boson-votex duality τ y x uantum 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);

15 Boson-votex duality z Classical statistical mechanics of a dual theedimensional supeconducto: votices in a magnetic field y x Stength of magnetic field = 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);

16 Boson-votex duality Statistical mechanics of dual supeconducto 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 = density of bosons = f flux quanta pe plaquette ;

17 Boson-votex duality Hofstäde spectum of dual supeconducting ode 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 ;

18 Boson-votex duality Hofstäde spectum of dual supeconducting ode 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)

19 Boson-votex duality The ϕ fields chaacteize both supeconducting and chage ode Supeconducto insulato : ϕ = 0 ϕ 0 Chage ode: Status of space goup symmety detemined by 2π p density opeatos ρ at wavevectos m n = mn, q x ρ mn = e iπmnf q = 1 ϕ ϕ iixˆ : ; y : * 2πim f + n T ρ ρ e T ρ ρ e R: ρ e ρ R ( ) ( ) ( ) ii yˆ

20 Boson-votex duality The ϕ fields chaacteize both supeconducting and chage ode Competition between supeconducting and chage odes: " Extended LGW" theoy of the ϕ fields with the action invaiant unde the pojective tansfomations: : ϕ ϕ ; : ϕ ϕ 2 if Tx 1 Ty e π + R: ϕ 1 q q m= 1 ϕ e m 2πimf Immediate benefit: Thee is no intemediate disodeed phase with neithe ode (o without topological ode).

21 Analysis of extended LGW theoy of pojective epesentation Supeconducto Ψ 0, ρ = 0 Fist ode tansition Chage-odeed insulato Ψ = 0, ρ Supeconducto Ψ 0, ρ = 0 Supeconducto Ψ 0, ρ = 0 Coexistence (Supesolid) Ψ 0, ρ 0 " Disodeed " ( topologically odeed) Ψ = 0, ρ = 0 Chage-odeed insulato Ψ = 0, ρ Chage-odeed insulato Ψ = 0, ρ 0 1 2

22 Analysis of extended LGW theoy of pojective epesentation Supeconducto ϕ = 0, ρ mn = 0 Fist ode tansition Chage-odeed insulato ϕ 0, ρ mn Supeconducto ϕ = 0, ρ mn = 0 ϕ Coexistence (Supesolid) = 0, ρ 0 mn Chage-odeed insulato ϕ 0, ρ mn Supeconducto ϕ = 0, ρ mn = 0 Second ode tansition Chage-odeed insulato ϕ 0, ρ mn 0 1 2

23 Phase diagam of S=1/2 squae lattice antifeomagnet o Neel ode ϕ σ * ~ zα αβzβ 0 Bond ode Ψ 0, S = 1/ 2 spinons z confined, S = 1 tiplon excitations α g

24 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

25 B. Application to a shot-ange paiing model fo the cupate supeconductos Chage ode and d-wave supeconductivity in an effective theoy fo the spin S=0 secto.

26 A convenient deivation of the effective theoy of shot-ange pais is povided by 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).

27 Duality mapping of doped dime model shows: (a) Supefluid, insulato, and supesolid gound states of a theoy which obeys the magnetic algeba TT = e TT 2πif x y y x 1 δ with f = 2

28 Duality mapping of doped dime model shows: (b) At δ = 0, the gound state is a Mott insulato with valence-bond-solid (VBS) ode. This associated with f=1/2 and the algeba TT = TT x y y x o

29 Duality mapping of doped dime model shows: (c) At lage δ, the gound state is a d-wave supefluid. The stuctue of the extended LGW theoy of the competition between supefluid and solid ode is identical to that of bosons on the squae lattice with density f. These bosons can theefoe be viewed as d-wave Coope pais of electons. The phase diagams of pat (A) can theefoe be applied hee. TT = e TT 2πif x y y x with f 1 δ = 2

30 Global phase diagam g = paamete contolling stength of quantum fluctuations in a semiclassical theoy of the destuction of Neel ode La 2 CuO 4 Neel ode

31 Global phase diagam g o VBS ode La 2 CuO 4 Neel ode N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).

32 Global phase diagam g o Pesent Extended LGW theoy fo inteplay between chage ode and d-wave supeconductivity VBS ode La 2 CuO 4 Neel ode Hole density δ

33 Global phase diagam g o VBS ode La 2 CuO 4 Neel ode Hole density δ

34 Global phase diagam g o VBS ode La 2 CuO 4 Neel ode Hole density δ

35 Global phase diagam g o VBS ode La 2 CuO 4 Neel ode Hole density δ

36 Global phase diagam o g Micoopic mean-field theoy in the lage N limit of a theoy with Sp(2N) symmety VBS ode 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, Phys. Rev. B 66, (2002). La 2 CuO 4 Neel ode Hole density δ

37 Global phase diagam o VBS ode g Micoopic mean-field theoy in the lage N limit of a theoy with Sp(2N) symmety 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, Phys. Rev. B 66, (2002). La 2 CuO 4 Neel ode Hole density δ

38 C. Implications fo STM

39 Votex-induced LDOS of Bi 2 S 2 CaCu 2 O 8+δ integated fom 1meV to 12meV 7 pa b 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).

40 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).

41 Enegy integated LDOS (between 65 and 150 mev) of stongly undedoped Bi 2 S 2 CaCu 2 O 8+δ at low tempeatues, showing only egions without supeconducting coheence peaks K. McEloy, D.-H. Lee, J. E. Hoffman, K. M Lang, E. W. Hudson, H. Eisaki, S. Uchida, J. Lee, J.C. Davis, cond-mat/

42 STM of LDOS modulations (filteed) in Bi 2 S 2 CaCu 2 O 8+δ C. Howald, H. Eisaki, N. Kaneko, M. Geven,and A. Kapitulnik, Phys. Rev. B 67, (2003).

43 Pinning of chage ode in a supeconducto The ϕ votex fields ae fluctuating, and we can use the action pin ( [ ] [ ]) 2 S = d d L0 + Lpin L 0 [ ] ( 2 2 ϕ s ) µ ϕ ϕ q ϕ = pin ii iπmnf * 2πi mf ρ with ρ = e mn ϕϕ+ ne = 1 [ ] ( ) L V e τ ϕ ϕ = + + The pojective tansfomation popeties of votices imply that each votex caies the quantum numbes of density wave ode. The vacuum fluctuations of votex-anti-votex poduce density wave modulations which ae obsevable nea pinning sites at wavevectos mn 2π p = mn, = 2 π f mn, q ( ) ( )

44 Chage ode in a magnetic field The ϕ votex fields ae fluctuating, and we can use the action pin ( [ ] [ ]) 2 S = d d L0 + Lpin L 0 [ ] ( 2 2 ϕ s ) µ ϕ ϕ q ϕ = pin ii iπmnf * 2πi mf ρ with ρ = e mn ϕϕ+ ne = 1 [ ] ( ) L V e τ ϕ ϕ = + + Recompute modulation in same theoy but in secto with "chage" = numbe of votices. ϕ Additional density wave ode paamete appeas as a halo aound pinned votices.

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