Breakdown of the Landau-Ginzburg-Wilson paradigm at quantum phase transitions

Transcription

1 Brekdown of the Lndu-Ginzburg-Wilson prdigm t quntum phse trnsitions Science 303, 1490 (2004); cond-mt/ cond-mt/ Leon Blents (UCSB) Mtthew Fisher (UCSB) S. Schdev (Yle) T. Senthil (MIT) Ashvin Vishwnth (MIT)

2 Outline A. Mgnetic quntum phse trnsitions in dimerized Mott insultors Lndu-Ginzburg-Wilson (LGW) theory B. Mott insultors with spin S=1/2 per unit cell Berry phses, bond order, nd the brekdown of the LGW prdigm

3 A. Mgnetic quntum phse trnsitions in dimerized Mott insultors: Lndu-Ginzburg-Wilson (LGW) theory: Second-order phse trnsitions described by fluctutions of n order prmeter ssocited with broken symmetry

4 TlCuCl 3 M. Mtsumoto, B. Normnd, T.M. Rice, nd M. Sigrist, cond-mt/

5 Coupled Dimer Antiferromgnet M. P. Gelfnd, R. R. P. Singh, nd D. A. Huse, Phys. Rev. B 40, (1989). N. Ktoh nd M. Imd, J. Phys. Soc. Jpn. 63, 4529 (1994). J. Tworzydlo, O. Y. Osmn, C. N. A. vn Duin, J. Znen, Phys. Rev. B 59, 115 (1999). M. Mtsumoto, C. Ysud, S. Todo, nd H. Tkym, 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

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7 λ close to 0 Wekly coupled dimers

8 λ close to 0 Wekly coupled dimers 1 = 2 ( ) Prmgnetic ground stte S = i 0

9 λ close to 0 Wekly coupled dimers 1 = 2 ( ) Excittion: S=1 triplon

10 λ close to 0 Wekly coupled dimers 1 = 2 ( ) Excittion: S=1 triplon

11 λ close to 0 Wekly coupled dimers 1 = 2 ( ) Excittion: S=1 triplon

12 λ close to 0 Wekly coupled dimers 1 = 2 ( ) Excittion: S=1 triplon

13 λ close to 0 Wekly coupled dimers 1 = 2 ( ) Excittion: S=1 triplon

14 λ close to 0 Wekly coupled dimers 1 = 2 ( ) Excittion: S=1 triplon (exciton, spin collective mode) Energy dispersion wy from ntiferromgnetic wvevector ε spin gp p = + c p x x y y 2 c p

15 TlCuCl 3 triplon N. Cvdini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer nd H. Mutk, Phys. Rev. B (2001).

16 Coupled Dimer Antiferromgnet

17 λ close to 1 Wekly dimerized squre lttice

18 λ close to 1 Wekly dimerized squre lttice Excittions: 2 spin wves (mgnons) ε = c p + c p p x x y y Ground stte hs long-rnge spin density wve (Néel) order t wvevector K= (π,π) ϕ 0 Si spin density wve order prmeter: ϕ = ηi ; ηi = ± 1 on two sublttices S

19 TlCuCl 3 J. Phys. Soc. Jpn 72, 1026 (2003)

20 T=0 λ c = (3) M. Mtsumoto, C. Ysud, S. Todo, nd H. Tkym, Phys. Rev. B 65, (2002) Néel stte ϕ 0 Quntum prmgnet ϕ = 0 λ 1 λc Pressure in TlCuCl 3 The method of bond opertors (S. Schdev nd R.N. Bhtt, Phys. Rev. B 41, 9323 (1990)) provides quntittive description of spin excittions in TlCuCl 3 cross the quntum phse trnsition (M. Mtsumoto, B. Normnd, T.M. Rice, nd M. Sigrist, Phys. Rev. Lett. 89, (2002))

21 LGW theory for quntum criticlity S ϕ Lndu-Ginzburg-Wilson theory: write down n effective ction for the ntiferromgnetic order prmeter ϕ by expnding in powers of ϕ nd its sptil nd temporl derivtives, while preserving ll symmetries of the microscopic Hmiltonin u 2 = d xdτ ϕ + ϕ + λ λ ϕ + ϕ 2 4! (( ) 2 ( ) 2 ( ) ) ( ) 2 x c τ c S. Chkrvrty, B.I. Hlperin, nd D.R. Nelson, Phys. Rev. B 39, 2344 (1989)

22 LGW theory for quntum criticlity S ϕ Lndu-Ginzburg-Wilson theory: write down n effective ction for the ntiferromgnetic order prmeter ϕ by expnding in powers of ϕ nd its sptil nd temporl derivtives, while preserving ll symmetries of the microscopic Hmiltonin u 2 = d xdτ ϕ + ϕ + λ λ ϕ + ϕ 2 4! (( ) 2 ( ) 2 ( ) ) ( ) 2 x c τ c S. Chkrvrty, B.I. Hlperin, nd D.R. Nelson, Phys. Rev. B 39, 2344 (1989) For λ < λ, oscilltions of ϕ bout ϕ = 0 c constitute the triplon excittion A.V. Chubukov, S. Schdev, nd J.Ye, Phys. Rev. B 49, (1994)

23 Key reson for vlidity of LGW theory Clssicl sttisticl mechnics: There is simple high temperture disordered stte with ϕ =0 nd exponentilly decying correltions Quntum mechnics: There is " quntum " non-degenerte ground stte with disordered ϕ =0 nd n energy gp to ll excittions

24 B. Mott insultors with spin S=1/2 per unit cell: Berry phses, bond order, nd the brekdown of the LGW prdigm

25 Mott insultor with two S=1/2 spins per unit cell

26 Mott insultor with one S=1/2 spin per unit cell

27 Mott insultor with one S=1/2 spin per unit cell Ground stte hs Neel order with ϕ 0

28 Mott insultor with one S=1/2 spin per unit cell Destroy Neel order by perturbtions which preserve full squre lttice symmetry e.g. second-neighbor or ring exchnge. The strength of this perturbtion is mesured by coupling g. Smll g ground stte hs Neel order with ϕ 0 Lrge g prmgnetic ground stte with ϕ = 0

29 Mott insultor with one S=1/2 spin per unit cell Destroy Neel order by perturbtions which preserve full squre lttice symmetry e.g. second-neighbor or ring exchnge. The strength of this perturbtion is mesured by coupling g. Smll g ground stte hs Neel order with ϕ 0 Lrge g prmgnetic ground stte with ϕ = 0

30 Mott insultor with one S=1/2 spin per unit cell Possible lrge g prmgnetic ground stte with ϕ = 0

31 Mott insultor with one S=1/2 spin per unit cell Ψ bond Possible lrge g prmgnetic ground stte with ϕ = 0 Such stte breks the symmetry of rottions by nπ / 2 bout lttice sites, nd hs Ψbond 0, where Ψbond is the bond order prmeter irctn( r ) bond () i S j r Ψ = i iisje ij

32 Mott insultor with one S=1/2 spin per unit cell Ψ bond Possible lrge g prmgnetic ground stte with ϕ = 0 Such stte breks the symmetry of rottions by nπ / 2 bout lttice sites, nd hs Ψbond 0, where Ψbond is the bond order prmeter irctn( r ) bond () i S j r Ψ = i iisje ij

33 Mott insultor with one S=1/2 spin per unit cell Ψ bond Possible lrge g prmgnetic ground stte with ϕ = 0 Such stte breks the symmetry of rottions by nπ / 2 bout lttice sites, nd hs Ψbond 0, where Ψbond is the bond order prmeter irctn( r ) bond () i S j r Ψ = i iisje ij

34 Mott insultor with one S=1/2 spin per unit cell Ψ bond Possible lrge g prmgnetic ground stte with ϕ = 0 Such stte breks the symmetry of rottions by nπ / 2 bout lttice sites, nd hs Ψbond 0, where Ψbond is the bond order prmeter irctn( r ) bond () i S j r Ψ = i iisje ij

35 Mott insultor with one S=1/2 spin per unit cell Ψ bond Possible lrge g prmgnetic ground stte with ϕ = 0 Such stte breks the symmetry of rottions by nπ / 2 bout lttice sites, nd hs Ψbond 0, where Ψbond is the bond order prmeter irctn( r ) bond () i S j r Ψ = i iisje ij

36 Mott insultor with one S=1/2 spin per unit cell Ψ bond Possible lrge g prmgnetic ground stte with ϕ = 0 Another stte breking the symmetry of rottions by nπ / 2 bout lttice sites, which lso hs Ψbond 0, where Ψbond is the bond order prmeter irctn( r ) bond () i S j r Ψ = i iisje ij

37 Mott insultor with one S=1/2 spin per unit cell Ψ bond Possible lrge g prmgnetic ground stte with ϕ = 0 Another stte breking the symmetry of rottions by nπ / 2 bout lttice sites, which lso hs Ψbond 0, where Ψbond is the bond order prmeter irctn( r ) bond () i S j r Ψ = i iisje ij

38 Mott insultor with one S=1/2 spin per unit cell Ψ bond Possible lrge g prmgnetic ground stte with ϕ = 0 Another stte breking the symmetry of rottions by nπ / 2 bout lttice sites, which lso hs Ψbond 0, where Ψbond is the bond order prmeter irctn( r ) bond () i S j r Ψ = i iisje ij

39 Mott insultor with one S=1/2 spin per unit cell Ψ bond Possible lrge g prmgnetic ground stte with ϕ = 0 Another stte breking the symmetry of rottions by nπ / 2 bout lttice sites, which lso hs Ψbond 0, where Ψbond is the bond order prmeter irctn( r ) bond () i S j r Ψ = i iisje ij

40 Mott insultor with one S=1/2 spin per unit cell Ψ bond Possible lrge g prmgnetic ground stte with ϕ = 0 Another stte breking the symmetry of rottions by nπ / 2 bout lttice sites, which lso hs Ψbond 0, where Ψbond is the bond order prmeter irctn( r ) bond () i S j r Ψ = i iisje ij

41 Quntum theory for destruction of Neel order Ingredient missing from LGW theory: Spin Berry Phses A isa e

42 Quntum theory for destruction of Neel order Ingredient missing from LGW theory: Spin Berry Phses A isa e

43 Quntum theory for destruction of Neel order

44 Quntum theory for destruction of Neel order Discretize imginry time: pth integrl is over fields on the sites of cubic lttice of points

45 Quntum theory for destruction of Neel order Discretize imginry time: pth integrl is over fields on the sites of cubic lttice of points Recll ϕ = 2η S ϕ = (0,0,1) in clssicl Neel stte; η ± 1 on two squre sublttices ; ϕ 0 ϕ ϕ+µ ( µ = x, y, τ )

46 Quntum theory for destruction of Neel order Discretize imginry time: pth integrl is over fields on the sites of cubic lttice of points Recll ϕ = 2η S ϕ = (0,0,1) in clssicl Neel stte; η ± 1 on two squre sublttices ; A µ hlf oriented re of sphericl tringle formed by ϕ, ϕ, nd n rbitrry reference point ϕ + µ 0 ϕ 0 2A µ ϕ ϕ +µ

47 Quntum theory for destruction of Neel order Discretize imginry time: pth integrl is over fields on the sites of cubic lttice of points Recll ϕ = 2η S ϕ = (0,0,1) in clssicl Neel stte; η ± 1 on two squre sublttices ; A µ hlf oriented re of sphericl tringle formed by ϕ, ϕ, nd n rbitrry reference point ϕ + µ 0 ϕ 0 ϕ ϕ +µ

48 Quntum theory for destruction of Neel order Discretize imginry time: pth integrl is over fields on the sites of cubic lttice of points Recll ϕ = 2η S ϕ = (0,0,1) in clssicl Neel stte; η ± 1 on two squre sublttices ; A µ hlf oriented re of sphericl tringle formed by ϕ, ϕ, nd n rbitrry reference point ϕ + µ 0 γ ϕ 0 ϕ ϕ +µ

49 Quntum theory for destruction of Neel order Discretize imginry time: pth integrl is over fields on the sites of cubic lttice of points Recll ϕ = 2η S ϕ = (0,0,1) in clssicl Neel stte; η ± 1 on two squre sublttices ; A µ hlf oriented re of sphericl tringle formed by ϕ, ϕ, nd n rbitrry reference point ϕ + µ 0 ϕ 0 γ γ + µ ϕ ϕ +µ

50 Quntum theory for destruction of Neel order Discretize imginry time: pth integrl is over fields on the sites of cubic lttice of points Recll ϕ = 2η S ϕ = (0,0,1) in clssicl Neel stte; η ± 1 on two squre sublttices ; A µ hlf oriented re of sphericl tringle formed by ϕ, ϕ, nd n rbitrry reference point ϕ + µ 0 2A 2A γ + γ µ µ + µ Chnge in choice of ϕ0 is like guge trnsformtion γ ϕ 0 ϕ γ + µ ϕ +µ

51 Quntum theory for destruction of Neel order Discretize imginry time: pth integrl is over fields on the sites of cubic lttice of points Recll ϕ = 2η S ϕ = (0,0,1) in clssicl Neel stte; η ± 1 on two squre sublttices ; A µ hlf oriented re of sphericl tringle formed by ϕ, ϕ, nd n rbitrry reference point ϕ + µ 0 2A 2A γ + γ µ µ + µ Chnge in choice of ϕ0 is like guge trnsformtion γ ϕ 0 ϕ γ + µ ϕ +µ The re of the tringle is uncertin modulo 4π, nd the ction hs to be invrint under A A + 2 µ µ π

52 Quntum theory for destruction of Neel order Ingredient missing from LGW theory: Spin Berry Phses exp i η A τ Sum of Berry phses of ll spins on the squre lttice.

53 Quntum theory for destruction of Neel order Prtition function on cubic lttice 1 ( ) 1 1exp Z = dϕδ ϕ ϕ ϕ + + i η A 2 + µ µ τ gg, µ, µ LGW theory: weights in prtition function re those of clssicl ferromgnet t temperture g Smll g ground stte hs Neel order with ϕ 0 Lrge g prmgnetic ground stte with ϕ = 0

54 Quntum theory for destruction of Neel order Prtition function on cubic lttice ( 2 ) 1 Z = dϕδ ϕ 1exp ϕ ϕ + i η A + µ τ g, µ Modulus of weights in prtition function: those of clssicl ferromgnet t temperture g Smll g ground stte hs Neel order with ϕ 0 Lrge g prmgnetic ground stte with ϕ = 0 Berry phses led to lrge cncelltions between different time histories need n effective ction for A t lrge g µ S. Schdev nd K. Prk, Annls of Physics, 298, 58 (2002)

55 Simplest lrge g effective ction for the A µ 1 Z = da exp cos( A A ) + i η A, µ with e ~ g 2 2 µ 2 µ ν ν µ τ 2e This is compct QED in 3 spcetime dimensions with sttic chrges ± 1 on two sublttices. N. Red nd S. Schdev, Phys. Rev. Lett. 62, 1694 (1989). S. Schdev nd R. Jlbert, Mod. Phys. Lett. B 4, 1043 (1990). S. Schdev nd K. Prk, Annls of Physics, 298, 58 (2002)

56 N. Red nd S. Schdev, Phys. Rev. Lett. 62, 1694 (1989).

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58 For lrge e 2, low energy height configurtions re in exct one-toone correspondence with nerest-neighbor vlence bond pirings of the sites squre lttice There is no roughening trnsition for three dimensionl interfces, which re smooth for ll couplings D.S. Fisher nd J.D. Weeks, Phys. Rev. Lett. 50, 1077 (1983). There is definite verge height of the interfce Ground stte hs bond order. N. Red nd S. Schdev, Phys. Rev. Lett. 62, 1694 (1989).

59 ( 2 ) 1 Z = dϕδ ϕ 1exp ϕ ϕ + i η A + µ τ g, µ or 0 Neel order ϕ 0? Bond order Ψ bond 0 Not present in LGW theory of ϕ order g