Notes on Renormalization Group: Berezinskii-Kosterlitz-Thouless (BKT) transition and Sine-Gordon model

Transcription

1 Notes on Renormalization Group: Berezinskii-Kosterlitz-Thouless (BKT) transition and Sine-Gordon model Yi Zhou (Dated: December 4, 05) We shall discuss BKT transition based on +D sine-gordon model. I. BKT TRANSITION IN A NUTSHELL Berezinskii-Kosterlitz-Thouless transition describe a number of phenomena in D or +D, including O() spin model (XY model) in D, D superfluid film, + D Coloumb plasma, and sine-gordon model in +D. The nature of such transitions is binding-anti-binding transition of topological defects, say, votrices pairs. XY model. The action is given by S[θ = J ij cos(θ i θ j ), where J > 0. High temperature expansion, Z = D[θe S[θ = D[θ ij [ + J cos(θi θ j ) + O(J ). The spin correlation function, with ξ = ln(/j). S( r) S(0) = cos (θ r θ 0 ) ( J ) r = exp [ r ξ Low temperature expansion, neglecting topological constraint, continuum theory, S = J d r( ϕ). The spin correlation function is dominated by the Gaussian fluctuations, S( r) S(0) = cos (ϕ r ϕ 0 ) = e (ϕ r ϕ 0 ) / ( ) a πj, r which is algebraically decayed and describe a phase with quasi-long-ranged order (or algebraic-long-range order). Vortices and topological phase transition The energy for a system with a single vortex will diverge as E πj ln L a, where L is the system linear size and a is the size of vertex core.,

2 The energy for a system with a pair of vortex-anti-vortex, E E core (a) + πj ln R a, where R is the distance between the two vortex cores. The entropy for a single vortex, The free energy per vortex is estimated as S ln ( ) L = ln L a a. F = E T S (πj k B T ) ln L a. When T > T c = πj k B, the entropy will always win. The system undergo a topological phase transition associated with vortex binding. II. SINE-GORDON MODEL IN +D: RG ANALYSIS The action for +D Sine-Gordon model reads: [ S = d r ( ϕ) g cos(ζϕ). () Firstly rewrite the scalar field ϕ as ϕ Λ ( r) = ϕ Λ( r) + h( r), ϕ Λ( r) = (π) e i p r ϕ( p), 0 p < Λ h( r) = (π) e i p r ϕ( p). Λ p <Λ Substitute the above into the action, we obtain that [ S = d r ( ϕ Λ) + ( h) g cos(ζϕ Λ + ζh). The next step is to integrate out the rapid varying section h( r). To do it, we expand the partition function to the second order of the coefficient g, say, S Z = [dϕ Λ[dhe [ = [dϕ Λ[dhe S0 e S h + g d r cos(ζϕ Λ( r) + ζh( r)) + g d r d r cos(ζϕ Λ( r ) + ζh( r )) cos(ζϕ Λ( r ) + ζh( r )) The first order term reads cos(ζϕ Λ( r) + ζh( r)) h = eiζϕ Λ ( r) e iζh( r) h + c.c.. The expectation value e iζh( r) can be evaluated as the following, h e iζh( r) h = n ( ζ ) n (n)! h( r) n h = n ( ζ ) n (n)! (n)! n n! n h(0) = e ζ h(0),

3 3 where h ( r) is the Green function of h, h ( r) = = (π) e i p r Λ p <Λ p (π) = π Λ Λ Λ Λ h (0) = Λ ln π Λ. cos θ eipr dpdθ p dp J 0(pr), () p Here J 0 is the zero order Bessel function. For Λ Λ, Assume that Λ = Λ + dλ = Λ dλ, we have Introducing a function B(r), h ( r) = π J 0(Λr) dλ Λ, h (0) = π dλ Λ. B(r) = e ζ h(r), we have a simple formula for the first term, The second order term reads cos(ζϕ Λ( r) + ζh( r)) h = cos (ζϕ Λ( r)) B(0). cos(ζϕ Λ( r ) + ζh( r )) cos(ζϕ Λ( r ) + ζh( r )) h = 4 eiζϕ Λ ( r)+iζϕ Λ ( r) e iζh( r)+iζh( r) + h 4 eiζϕ Λ ( r) iζϕ Λ ( r) e iζh( r) iζh( r) + c.c. h = 4 eiζϕ Λ ( r)+iζϕ Λ ( r) e ζ ( h (0)+ h ( r r )) + 4 eiζϕ Λ ( r) iζϕ Λ ( r) e ζ ( h (0) h ( r r )) + c.c. = e ζ h (0) [ cos(ζϕ Λ( r ) + ζϕ Λ( r ))e ζ h ( r r ) + cos(ζϕ Λ( r ) ζϕ Λ( r ))e ζ h ( r r ). To compute the effective action to the order of g, what we need is the following integral, d r d [ r cos(ζϕ Λ( r ) + ζh( r )) cos(ζϕ Λ( r ) + ζh( r )) h cos(ζϕ Λ( r ) + ζh( r )) cos(ζϕ Λ( r h ) + ζh( r )) h = [ h (0) d r e ζ d r cos ζ(ϕ Λ( r ) + ϕ Λ( r ))e ζ h ( r r ) + cos ζ(ϕ Λ( r ) ϕ Λ( r ))e ζ h ( r r ) cos ζϕ Λ( r ) cos ζϕ Λ( r ) = [ h (0) d r e ζ d r cos ζ(ϕ Λ( r ) + ϕ Λ( r ))e ζ h ( r r ) + cos ζ(ϕ Λ( r ) ϕ Λ( r ))e ζ h ( r r ) cos ζ(ϕ Λ( r ) + ϕ Λ( r )) cos ζ(ϕ Λ( r ) ϕ Λ( r )). Becauese h ( r) only contain the moment component Λ p < Λ, we can think h ( r) 0 in the range r > Λ. For r Λ, we can expand an relative somooth function ϕ Λ( r ) ϕ Λ( r ) ( r r ) ϕ Λ [( r + r )/. By variables

4 4 transformation and up to the second order of r r, we have d r d [ r cos(ζϕ Λ( r ) + ζh( r )) cos(ζϕ Λ( r ) + ζh( r )) h cos(ζϕ Λ( r ) + ζh( r )) cos(ζϕ Λ( r h ) + ζh( r )) h = [ h (0) d R d r cos ζ(ϕ Λ( R + r/) + ϕ Λ( R r/))(e e ζ ζ h ( r) ) + cos ζ(ϕ Λ( R + r/) ϕ Λ( R r/))(e ζ h ( r) ) = [ B(0) d R d r cos(ζϕ Λ( R))(B(r) ) + ( ζ ( r ϕ Λ( R)) )(B(r) ) = [ B(0) d R d r cos(ζϕ Λ( R))(B(r) ) + ( 4 r ( ϕ Λ( R)) )(B(r) ) [ = d R B(0) a 3 cos (ζϕ Λ( R)) + 8 ζ B(0) a ( ϕ Λ( R)) + const., with the coefficients a and a 3 are given by a = a 3 = d r(b(r) )r, d r(b(r) ). Up to the seconder order of g, we obtain that S eff = d r {[ + 8 } g ζ B(0) a ( ϕ Λ( r)) gb(0) cos ζϕ Λ( r) + g B(0) a 3 cos ζϕ Λ( r). Ingoring the cos ζϕ Λ( r) term, and rescaling the field ϕ Λ( r) as ϕ Λ( r) ϕ Λ( r) = + 8 g ζ B(0) a ϕ Λ( r), we obtain that with S eff = [ d r ( ϕ Λ) g cos ζ ϕ Λ, (3) Now let us examine the behavior of function B(r), for infinitesmall dλ, g = B(0)g, ζ = ζ/ + 8 g ζ B(0) a. (4) B(r) = e ζ h(r) = ζ 4π J 0(Λr) dλ Λ, B(0) = + ζ 4π dλ Λ, B(r) = ζ π J 0(Λr) dλ Λ. The above results in divergence in a, which is caused by the slowly-decayed h (r), which can be cured by using smoothing momentum space slicing instead of the abrupt cut-off in principle. Assuming this can be done, we may

5 replace the Bessel function J 0 by a new function J 0 that decrease strongly at r Λ. With the help of dimensionless parameter α, α = dρ J 0 (ρ)ρ 3 > 0, we can replace the original a by Then we have the following differential equaitons, a = ζ α dλ Λ 5. dg = dζ = Defining dimensionless parameters x and y satisfying that Eqs. (5) becomes ζ ζ π g dλ Λ, 6 g ζ 5 dλ α Λ 5. (5) = 4π( + x), y = α gλ, 5 These flow equations can be solved through When x = y, we have dy d ln Λ = xy, dx d ln Λ = y. (6) d d ln Λ (x y ) = 0. x(λ) x(µ) = y(µ) = + x(λ) ln Λ. µ It is easy to see that IR fixed points are given by y = 0, x > 0, while UV fixed line yields y = 0, x < 0. [ M. E. Peskin and D. V. Schroeder, An introduction to quantum field theory, Westview Press (995). [ A. M. Polyakov, Phys. Lett. B59, 79 (975). [3 A. M. Polyakov, Gauge fields and strings, Harwood Academic Publishers GmbH, Switzerland (987). [4 John B. Kogut, Rev. Mod. Phys. 5, 659 (979).

6 6 y x FIG. : RG flow for sine-gordon model.