2016 Kosterlitz-Thouless Haldane Dept. of Phys., Kyushu Univ. 2016 11 29
2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER
( ) ( ) (Dirac, t Hooft-Polyakov)
BKT J. M. Kosterlitz, and D.J. Thouless: Journal of Physics C 6 p. 1181-1203 (1973). 2 XY 2 U(1) XY O(2) ) 2 (Mermin-Wagner (1966)) But, 2 XY 2 ( ( ) Berezinskii: Sov. Phys. JETP, 32, p.493 (1971); Sov. Phys. JETP, 34, p.610 (1972)
BKT Kosterlitz (1973) Kosterlitz-Thouless (1973) Kosterlitz (1974) J. M. Kosterlitz: J. Phys. C, 7, pp. 1046-1060 (1974). Anderson-Yuval :Phys. Rev. Lett, 23, (1969) 89 ; Anderson-Yuval-Hamann :Phys. Rev. B, 1, (1970) 4464 2
Haldane F.D.M. Haldane: Phys. Letters A, 93, p.464 (1983); Physical Review Letters, 50, p.1153 (1983) 1950-1970 1 S=1/2 1 ) ( )
TKNN D. J. Thouless, Mahito Kohmoto, M.P. Nightingale, and M Den Nijs: Physical Review Letters, 49(6):405, (1982) ( )
1. 2. 1. 3. 2. (winding number) Figure: : -2, : -1, : 0 Figure: :1, : 2, : 3
2 3. 2. ( ) 3.1 ( 1 ( x 2π C r 2 dy y ) r 2 dx 3.2 ( ) 1 2πi dz z a C (1) (2) 3.3 ( ) ( )
2 XY XY : H XY = J <i,j> S i S j = J <i,j> cos(θ i θ j ) (3) ( θ j (0 θ j < 2π) S j (cos θ j, sin θ j ). < i, j > )
2 XY 2 XY
2 XY (vortex) (anti-vortex) 1. 1 XY 2. 1 XY 3. ( 4.
2 XY BKT 1. 2. 3.
2 XY BKT 1. 2. 3. 2 ( ) ( )
BKT 2 1. c 1 (T < T c ) S(r) S(r ) c 2 r r η (T = T c ) c 3 exp( r r /ξ(t )) (T > T c ) (4) 1.1 1.2 1.3 T c 2. 2 XY S(r) S(r ) { c 1 r r η(t ) (T T c ) c 2 exp( r r /ξ(t )) (T > T c ) (5)
(3) cos(θ i θ j ) 1 (θ i θ j ) 2 /2 1 ( θ) 2 /2. H = E 0 + J d 2 r( θ(r)) 2 (6) 2 (E 0 = 2JN ) (β = 1/(k B T )) ( Z = exp( βe 0 ) D[θ] exp β J ) d 2 r( θ(r)) 2 (7) 2 Green 2 ln(r) = 2πδ(r) Γ(r r) θ(r )θ(r) = 1 2π ln r r (8)
2 ( ) 1. 1 W (h) = dθ exp( 1 2 Aθ2 + ihθ) = (2π/A) 1/2 exp( 1 2 A 1 h 2 ) (9) exp(ihθ) W (h) W (0) = exp( 1 2 A 1 h 2 ) (10) 2. N exp( 1 2 N i=1 N j=1 θ ia i,j θ j ) exp(i i h i θ i ) = exp 1 2 N i=1 j=1 N h i (A 1 ) i,j h j (11)
3 ( ) 3. exp( 1 2 θ(r)a(r, r )θ(r )d d rd d r ) W (h) exp(i h(r)θ(r)d d r) ( = exp 1 ) h(r)a 1 (r, r )h(r )d d rd d r 2 (12) A 1 A 1 (r, r )A(r, r )d d r = δ d (r r ) (13) 4. θ(r )θ(r) = δ2 W (h) δh(r 1 )δh(r 2 ) = A 1 (r 1, r 2 ) (14)
4 XY S(r) S(r ) = exp(i(θ(r) θ(r ))) = exp( k BT 2πJ Γ(r r )) ( ) a kb T/2πJ = r r (15)
< θ < θ 1
< θ < θ 1 But, θ = θ + 2π ( ) v v 1 2π C dl θ(r) (16) S(r)(= exp(iθ(r))) v = n (n: )
2πn = dl θ(r) = 2πr θ (17) θ = n/r θ(r = n/r E vor E 0 = J d 2 r( θ(r)) 2 2 C = Jn2 2 2π 0 dθ L a rdr ( ) 1 2 = Jπn 2 ln L r a (18) (L a )
( +1) ( -1) ( 0 ( r 2πJ ln a) (19) (r )
( ) ( ) L L 2 F = E T S Jπ ln k B T ln a a 2 (20) T KT Jπ/(2k B )
S = 1 ( µ θ) 2 d 2 x, (θ θ + 2π) (21) 2g θ θ sw θ vortex θ(x) = θ sw (x) + θ vortex (x) (22) dθ sw (x) = 0, (23) dθ vortex (x) = v (v : ) (24) ψ ( ) ϵ µν ν ψ = µ θ vortex (25)
v = dθ vortex (x) = 1 2π 2 ψd 2 x (26) 2 ψ = 2π j v j δ(x x j ) (v j : ) (27) 2 Green 1 2π ln x ψ(x) = 2π j v j 1 2π ln x x j = j v j ln x x j (28) θ vortex (x) = Im j v j ln(x x j ) (29) ( )
2D Coulomb gas Action S vortex = 2π 2g v i v j ln z i z j (30) i,j (2 ( )) Action sine-gordon ( < χ < ) L = 1 2g ( χ)2 2 cos( 2πχ g ) (31)
2 sine-gordon ( ) α α = α exp(dl) α(1 + dl) L l 0 = ln L (1/ ln L) dy 1 (l) = y 2 dl 2(l) dy 2 (l) = y 1 (l)y 2 (l) (32) dl
: K. Nomura: J. Phys. A, Vol. 28, pp.5451-5468 (1995); Nomura and A. Kitazawa: J. Phys. A: Vol. 31 (1998) pp.7341
Haldane 1 H = J j S j S j+1 (33) Néel (S=1/2,3/2,...) (S=1,2,...) 0 π
Haldane : 1 2g dtdx ( 1 v ( ) φ 2 v t ( ) ) φ 2 x (φ (φ 1, φ 2, φ 3 ), φ 2 = 1) (34) Wick 1 2g dx 2 ( φ) 2 (35) 2 )
Haldane : 1 2g dtdx ( 1 v ( ) φ 2 v t ( ) ) φ 2 x (φ (φ 1, φ 2, φ 3 ), φ 2 = 1) (34) Wick 1 2g dx 2 ( φ) 2 (35) 2 ) massless(gapless)
Haldane : 1 2g dtdx ( 1 v ( ) φ 2 v t ( ) ) φ 2 x (φ (φ 1, φ 2, φ 3 ), φ 2 = 1) (34) Wick 1 2g dx 2 ( φ) 2 (35) 2 ) massless(gapless) But φ 2 = 1
Haldane : 2 ( ) φ
Haldane : 2 ( ) φ Figure: ( ) gap
Haldane : 2 ( ) φ Figure: ( ) gap 4 Yang-Mills ( ) 2
Haldane :, 0 π ˆφ 2i 1 2s (Ŝ2i+1 Ŝ2i) a 0, s ˆl 2i 1 2a (Ŝ2i+1 + Ŝ2i) (36) [ˆl a (x), ˆl b (y)] = iϵ abcˆlc δ(x y) [ˆl a (x), ˆφ b (y)] = iϵ abc ˆφ c δ(x y) [ ˆφ a (x), ˆφ b (y)] = iϵ abcˆlc a2 δ(x y) 0 (37) s2 δ(x y) = lim a 0 δ x,y a
Haldane : 2 ˆφ 2i ˆl 2i = 1 2s (Ŝ2i+1 Ŝ2i) 1 2a (Ŝ2i+1 + Ŝ2i) = 1 2sa (Ŝ2 2i+1 + Ŝ2i+1 Ŝ2i Ŝ2i+1 Ŝ2i Ŝ2 2i) = 0 ˆφ (38) ( ˆφ 2i ) 2 = 1 4s 2 [Ŝ2 2i + Ŝ2 2i+1 2Ŝ2i Ŝ2i+1] = 1 4s 2 [2Ŝ2 2i + 2Ŝ2 2i+1 4a 2 (ˆl 2i )) 2 ] = 1 + 1/s a 2ˆl/s 2 (39)
Haldane : 3 Ŝ 2i Ŝ2i+1 = 2a 2ˆl2 2i + s(s + 1) (40) Ŝ 2i 1 Ŝ2i = s 2 ˆφ 2i 2 ˆφ 2i as [ˆl2i 2 ˆφ 2i ˆφ 2i 2 ˆl ] 2i + a 2ˆl2i 2 ˆl 2i a 2 ( 2s 2 ( ˆφ ) 2 2s(ˆl ˆφ + ˆφ ˆl) + 2ˆl 2) s(s + 1) 2a 2 s 2 ( ˆφ ˆφ ) (41) ( ˆφ 2 = 1 + 1/s aˆl 2 /s 2 ˆφ ˆφ = ( ˆφ ˆφ) ( ˆφ ) 2 )
Haldane : 4 Ĥ = aj 2 dx[4ˆl 2 + 2s 2 ( ˆφ ) 2 2s(ˆl ˆφ + ˆφ ˆl)] (42) Ĥ = dxĥ, (43) [ Ĥ = v ( g ˆl θ ) ] 2 2 4π ˆφ + ( ˆφ ) 2 (44) g v = 2Jas, g = 2/s, θ = 2πs
Haldane : 5 Hamiltonian (44) Lagrangian L = 1 2g µ ˆφ µ ˆφ + θ 8π ϵµν ˆφ ( µ ˆφ ν ˆφ) = 1 2g ( 0φ 0 φ 1 φ 1 φ) + θ 4π φ ( 0φ 1 φ) (45) ( v = 1 [ ] Π ( ) L ( 0 φ) = 1 g 0φ + θ 4π ( 1φ φ) (46)
Haldane : 6 H 0 φ Π L = g ( φ Π θ ) 2 2 4π ( 1φ) + 1 2g ( 1φ) 2 (47) l φ Π Ĥ = v 2 [ ( dx g ˆl θ ) ] 2 4π ˆφ + ( ˆφ ) 2 g Q.E.D.
Haldane : 7 Lagrangin Wick Euclid L = 1 2g µ ˆφ µ ˆφ + iθ 8π ϵµν ˆφ ( µ ˆφ ν ˆφ) = 1 2g ( 0φ 0 φ 1 φ 1 φ) + θ 4π φ ( 0φ 1 φ) (48) ( ) ( Z = Dφ exp ) d 2 xl (49)
Haldane : (S 0 ) Q = 1 d 2 xϵ µν ˆφ ( µ ˆφ ν ˆφ) (50) 8π ( exp(s 0 + iθq) θ = 2πs gapped gapless
Haldane : 2. φ = (sin α cos β, sin α sin β, cos α) Q = 1 d 2 xϵ µν ˆφ ( µ ˆφ ν ˆφ) 8π = 1 d 2 x sin αϵ µν µ α ν β 4π = 1 D(α, β) sin α 4π D(x 0, x 1 ) dx 0dx 1 = 1 ds int (51) 4π ( D(α,β) D(x 0,x 1 ) ) : Q
Q µ φ + ϵ µν (φ ν φ) = 0 (52) φ 1 = 1 w ( w = φ 1 + iφ 2 1 + φ 3 (53) Figure:
2 (52) z w = 0, (z = x 0 + ix 1 ) (54) w(z) = Q j=1 z a j z b j (55)
Haldane : 1. S=1/2 Bethe 2. (S=3/2,5/2, ) Lieb-Schultz-Mattis I.Affleck, E.H. Lieb: Lett. Math. Phys, p. 57 (1986)
Haldane : Haldane 1. 2. 3. AKLT(Affleck-Kennedy-Lieb-Tasaki) I. Affleck, T. Kennedy, E.H.Lieb, H.Tasaki: Physical Review Letters. 59, p.799(1987).
1. 1.1 BKT 1.2 Haldane ( ) 1.3 TKNN(Thouless-Kohmoto-Nightingale-denNijs) 2.?(Umklapp )? 3. 4. 4.1 4.2 SPTP 4.3 4.4 4.5
2 1., 2. ( ) 3.