Exact Duality and Magic Circles of the Dissipative Hofstadter Model

Transcription

1 Exact Duality and Magic Circles of the Dissipative Hofstadter Model Taejin Lee 1,2, G. Semenoff 3, P. Stamp 3 Seungmuk Ji 1, M. Hasselfield 3 Kangwon National University 1 CQUeST 2 PiTP (UBC) 3 Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

2 References T. Lee and G. W. Semenoff Fermion representation of the rolling tachyon boundary conformal field theory, JHEP 0505, 072 (2005) [hep-th/ ]. M. Hasselfield, Taejin Lee, G.W. Semenoff, P.C.E. Stamp Critical Boundary Sine-Gordon Revisited in press, Ann. Phys. (2006) [hep-th/ ] Seungmuk Ji, Ja-Yong Koo and T. Lee Dissipative Hofstadter Model at the Magic Points and Critical Boundary Sine-Gordon Model in press, Jour. Korean Phys. Soc. (2006) Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

3 I. Introduction Classical dissipative system: EQ of motion M d 2 q dt 2 + η dq dt + dv dq = 0 η: friction coefficient Microscopic action by Calderia-Leggett [ ( ) M dq 2 S = dt V (q) 2 dt ( + 1 (dxα ) ) 2 m α ω 2 2 dt αxα 2 α q α C α x α ] x α : bath of an infinite number of degrees of freedom or an environment (Calderia-Leggett, 1981, 1983) Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

4 Classical equation of motion M d 2 q dt 2 = dv dq α C α x α, m α d 2 x α dt 2 = m α ω 2 αx α C α q Solving for x α, (Fourier transformed) Mω 2 q(ω) = dv dq (ω) + K (ω) q(ω) K (ω) = α C 2 α ω 2 m α ωα 2 ω 2 ωα 2 Density of states J(ω) = π 2 α C 2 α m α ω α δ(ω ω α ), J(ω) = Im K (ω). Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

5 Ohmic condition J(ω) = ηω Frictional term Mω 2 q(ω) dv = (ω) + iηω q(ω) dq Quantum Mechanics: Non-local interaction [ ] dω S CL = dtdt ω t q(t) J(ω)e q(t ) 2π Imposing the Ohmic condition S CL = η 4π dtdt (q(t) q(t )) 2 (t t ) 2 Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

6 II. Dissipation and Boundary State String (boundary) theory: t σ [ π, π] S η [X] = = 1 4 π π π dσ π dσ π π dσ (X(σ) X(σ )) 2 (σ σ ) 2 dσ (X(σ) X(σ )) 2 sin 2 (σ σ ) 2 = 4π 2 α n x n x n where α 1 X(σ) = x + (x n e inσ + x n e inσ), 2 n n=1 Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

7 Boundary State X = n { exp 1 } 2 x n x n a nã n + a nx n + x n ã n 0 0 X = e Sη[X] Partition Function Z = ( D[X] exp S η + ) dσv [X] = Z Disk = 0 B, ( B = D[X] exp ) dσv [X] X (Callan and Thorlacius, 1990) Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

8 Boundary State Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

9 III. Schmid Model Caldeira-Leggett model with a periodic potential S SM = η T /2 4π dtdt (X(t) X(t )) 2 (t t ) 2 T /2 T /2 V 0 T /2 dt cos 2πX a. String theory where t = T 2π σ, X a 2π X S SM = η ( a ) 2 π 4π 2π V 0 T 2π π π (X(σ) X(σ ))2 dσdσ (σ σ ) 2 π dσ 1 2 ( e ix + e ix ). Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

10 Comparison: η ( a ) 2 1 = 4π 2π 8π 2 α, V 0 T 2π = g 2, 1 α = α Renormalization Critical Point: ( µ 1/α 1 g 0 (µ) = g 0 (Λ) Λ) α = α = 1 This is also the self-dual point of the duality (Schmid 1983) α 1/α At the critical point S SM coincides with the full brane action of the rolling tachyon Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

11 IV. Schmid Model at Critical Point: Fermionization Fermion in terms of boson ψ L (0, σ) = : e i 2φ L : = e P n=1 1 n einσ α n e ix L e iσ(p L+ 1 2 ) e P n=1 1 n e inσ α n In order to rewrite the potential using fermion fields, we need to introduce an auxiliary boson φ 1 = 1 2 (X + Y ), φ 2 = 1 2 (X Y ) (Polchinski and Thorlacius (1994) for open string) Fermions: ψ 1L (z) = ζ 1L : e 2iφ 1L (z) : ψ 2L (z) = ζ 2L : e 2iφ2L (z) : ψ 1R ( z) = ζ 1R : e 2iφ1R ( z) : ψ 2R ( z) = ζ 2R : e 2iφ 2R ( z) : where ζ L/R are co-cycles (Lee and Semenoff (2005): boundary state formulation) Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

12 Boundary condition for bosons: [ 1 2π τ X(0, σ) + i g 2 eix(0,σ) i ḡ ] 2 e ix(0,σ) B, D = 0 Y (0, σ) B, D = 0 Note: B, D = Boundary state for X Boundary state for Y. Boundary condition for fermions: [ ( : ψ L σ3 ψ L : : ψ R σ3 ψ R : +πgψ L 1 + σ 3) ψ R πḡψ L (Hasselfield, Lee, Semenoff and Stamp, 2006) (1 σ 3) ψ R ] B, D = 0 Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

13 V. Exact Boundary State for Schmid Model at Critical Point Gluing condition = Boundary condition ( ) ψ R (0, σ) + iσ 1 Uψ L (0, σ) BD > = 0 ( ψ R (0, σ) + ψ L (0, σ)u 1 iσ 1) BD > = 0 If we choose ( U = e 2πiA 1 π 2 gḡ iπḡ iπg e 2πiA 1 π 2 gḡ ) where A is topological parameter. U is unitary when A is real, ḡ = g, g < 1/π Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

14 Exact Boundary State in the NS-sector BD > NS = ] exp [ψ r U 1 iσ 1 ψ r ψ r iσ1 Uψ r 0 > r= 1 2 and in the R-sector BD > R = n=1 ] exp [ψ n U 1 iσ 1 ψ n ψ n iσ1 Uψ n ] exp [ψ 0 U 1 iσ 1 ψ0 + + > Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

15 VI. Exact Calculation of Mobility Mobility: 0 X (σ)x (σ ) B, D where X = J 3 L + J3 R We choose B, D = B, D NS since 0 belongs to the NS-sector Some exact calculations: 0 JL 3 (σ)j3 R (σ ) B, D = 1 16 tr(σ3 U 1 σ 1 σ 3 σ 1 U) sin 2 (σ σ ) 2 [ ] 0 JL 3 (σ)j3 L (σ ) + JR 3 (σ)j3 R (σ ) B, D = 1 (σ 4 sin 2 σ ) 2 Exact Result 0 X (σ)x (σ ) B, D = 1 2 (1 π2 gḡ) sin 2 (σ σ ) 2 Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

16 Remarks The mobility does not depend on the topological parameter A The result does not agree with the previous works of Callan and Freed (1991) and Callan, Felce and Freed (1992) which employed the bosonic theory. The mobility has been also calculated by Callan, Klebanov, Maldacena and Yegulalp (1995). But the renomalization effect due to the ordering of operators has been ignored. If it is properly renormalized, their result reduces to the exact result obtained here e R ( g 2 ψ 1 L 2 (1+σ3 )ψ R + ḡ 2 ψ L 1 2 (1 σ3 )ψ R ) ND > = e χ(g,ḡ ) e i R ( g 2 ψ L σ+ ψ L + ḡ 2 ψ L σ ψ L ) ND > sin 2 π ḡ g = π 2 g ḡg, ḡ = ḡ g Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

17 VII. Dissipative Hofstadter Model The Dissipative Hofstadter Model (Wannier-Azbel-Hofstadter (WAH) Model): Electron moving in two dimensions subjects to a magnetic field, a square lattice potential and dissipative force Phase transitions between localized and delocalized long-time behavior of the electron: Phase diagram is fractal Applications: Quantum dynamics of SQUID, Josephson junction, Polaronic motion through conductors, Tunneling in QM Hall System, New solutions of open string theory in non-trivial background of tachyons and gauge fields String theory (Rolling Tachyon): Decay of unstable D-brane in the presence of NS B-field. Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

18 Dissipative Hofstadter Model The Action Choosing S DHM = η T /2 4π + ieb H 2 c V 0 T /2 T /2 T /2 T /2 T /2 t = T 2π σ, ( X dtdt i (t) X i (t ) ) 2 (t t ) 2 dt ( t X 1 X 2 t X 2 X 1) dt (cos 2πX 1 a X i a 2π X i, and introducing dimensionless parameters 2πα = ηa2, 2πβ = eb H c a2, + cos 2πX 2 ). a g = V 0T π Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

19 VIII. Boundary State for DHM we get where α = 1/α, S DHM = α π 8π 2 β +i 4π g 2 β = 2πB π π π π π ( X dσdσ i (σ) X i (σ ) ) 2 (σ σ ) 2 dσ ( σ X 1 X 2 σ X 2 X 1) dσ ( cos X 1 + cos X 2). It appears in string theory as an action for the open string in the background of the magnetic field and the tachyon potential Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

20 String Theory S = 1 4πα i gπ 2 d 2 ξe ij ( τ + σ )X i ( τ σ )X j dσ (e ) ix i + e ix i 2π M i where E ij = (g + 2πα B) ij. Boundary State { } B = D[x, x] exp S DHM [x, x] a nã n + a nx n + x n ã n 0 α where X(σ) = x + 2 n=1 1 ( xn n e inσ + x n e inσ). Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

21 In the absence of the periodic potentials B reduces to B E which satisfies (δ ij τ X j βα ɛ ij σ X j ) B E = 0. B E = det E exp ( n=1 1 ( ) ) n αi n g(e) 1 E T α j n 0 ij Boundary state in the presence of the tachyon potential [ dσ ( B = exp gπ e ix 1 + e ix 1 + e ix 2 + e ix 2)] B E M 2π ( g ) n 1 n = dσ 1... dσ n exp [ iq j X(σ j ) ] B E 2 n! n=0 q j =±i,±j j=1 Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

22 IX. Magic Circles O(2, 2, R) Transformation: T-dual Transformation α i n = (G(E) 1) i jβn, j α n i = (G(E T ) 1) i j β j n In terms of the new oscillator basis the boundary condition for B E is transcribed into the Neumann condition ( ) β n i + β n i B E = 0. The oscillators {β, β} respect the world-sheet metric G ( ) 2 G = E T E = 1 + β α 0 ( ) β α [ ] βn, i βm j = (G 1 ) ij nδ(n + m), [ β n, i β m] j = (G 1 ) ij nδ(n + m) Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

23 L = = 1 4πα E ij ( τ + σ ) X i ( τ σ ) X j 1 4πα G ij ( τ + σ ) Z i ( τ σ ) Z j where α Z i (0, σ) = x i + ω i 1 ( ) σ + i βn i β n i e inσ 2 n n 0 Relation between two oscillator bases is summarized as X i (σ) = (δ ij βα ) ɛij Z (δ jl (σ) + ij + βα ) ɛij Z j R (σ) Boundary state B E B E = det E n=1 exp ( 1 ) n βi ng ij β j n 0. Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

24 Note B = n 1 ( g ) n n n! 2 j=1 dσ j exp [ iq j X(σ j ) ] B E Using the Baker-Hausdorff Lemma, e A e B = e B e A e [A,B], n exp [ iq j X(σ j ) ] B E j=1 = exp [iq n X(σ n )]... exp [iq 1 X(σ 1 )] B E = exp iπq i (2πα 2 BG 1 ) q j sign(σ i σ j ) i>j n exp [ iq j Z(σ j ) ] B E j=1 Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

25 By some algebra exp i>j iπq i (2πα 2 BG 1 ) q j sign(σ i σ j ) = exp i>j = exp i β ( ) iπ α 2 + β 2 qi 1 qj 2 qi 2 qj 1 sign(σ i σ j ) β 2πi α 2 + β 2 q1 i. σ i >σ j q 2 j Magic circles: Since qi 1, qi 2 = 0, ±1 for i = 1, 2, if is an α 2 +β 2 integer, this phase due to the magnetic field reduces to 1. ( α 2 + β 1 ) 2 ( ) 1 2 =, n Z 2n 2n β Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

26 On the magic circles, the dissipative Hofstadter model can be mapped into the boundary sine-gordon model. exp ( S DHM ) = D[Z ] [ exp 1 4πα dτdσg ij ( τ + σ ) Z i ( τ σ ) Z j M + g ( dσ e iz 1 + e iz 1 + e iz 2 + e iz 2)]. 2 M Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

27 X. Critical Circle It may be convenient to scale β i, β i and G such that β i n α β i n, β i α β i, G 1 α G = ( α 2 +β 2 α 0 α 0 2 +β 2 α ). The points where the effective world-sheet metric becomes a unit metric forms a circle called the critical circle"; α 2 + β 2 α = 1. Magic Points: The points where the magic circles meet the critical circle, are magic points. At the magic points the DHM is equivalent to a set of two independent critical boundary sine-gordon models. Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

28 Figure: Critical Circles Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

29 XI. Exact Duality Two DH models are equivalent if the following conditions are satisfied: exp β 2πi α 2 + β 2 q1 i q 2 j i σ i >σ j = exp β 2πi ᾱ 2 + β 2 q1 i q 2 j i σ i >σ j or Same metric β α 2 + β 2 β = n, n Z ᾱ 2 + β 2 αg(α, β) ij = ᾱg(ᾱ, β) ij or α 2 + β 2 α = ᾱ2 + β 2 ᾱ Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

30 T-Dual Transformation Both conditions can be summarized as 1 z = 1 + ni, where z = α + βi, z = ᾱ + βi z O(2, 2, R) Transformation ( a b c d E = (ae + b)(ce + d) 1 ) T ( ) ( ) 0 I a b = I 0 c d ( 0 I I 0 ). The left and right movers transform as α n (E) (d ce T ) 1 α n (E ), α n(e) α n(e )(d T Ec T ) 1, α n (E) (d + ce) 1 α n (E ), α n(e) α n(e )(d T + E T c T ) 1 Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

31 T-dual transformation from the DH model with (α, β) and that with (ᾱ, β): ( ) ( ) T 1 I 0 I 0 (ᾱ, β)t (α, β) = ᾱ β αβ ᾱ 2 + β ɛ I ɛ I 2 α 2 +β ( 2 ) I 0 = ( ) αβ ᾱ β ɛ I α 2 +β 2 ᾱ 2 + β 2 The periodic boundary potential is invariant under this T-dual transformation. Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32

32 XII. Summary The dissipative Hofstadter model describes quantum particles moving in two dimensions subject to a uniform magnetic field, a periodic potential and a dissipative force. We discuss the dissipative Hofstadter model in the framework of the boundary state formulation in string theory and construct exact boundary states for the model at the magic points using the fermion representation. The exact duality of the dissipative Hofstadter model is shown to be equivalent to the subgroup of T-duality symmetry group in string theory unbroken by the boundary periodic potential. Taejin Lee (KNU) Exact Duality and Magic Circles Banff / 32