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hw07soln - ψ gives ψ = r a b T c − T K = gψ 2 = ga b T...

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1 Homework 7 (due December 6) Problem 7-1 (5 pts.) Find the correlator of a scalar order parameter, below the critical temperature ( T < T c ): G = h η ( r ) η ( r 0 ) i Assume the Ginzburg—Landau functional to have the following form (with all positive coe cients): F = Z d 3 r a ( T T c ) η 2 + b 2 η 4 + g ( η ) 2 ¸ Solution: η 2 0 = a b ( T c T ) δF = Z d 3 r μ 2 a ( T T c ) + 12 2 0 2 δη 2 2 + g ( δη ) 2 ¸ = X q ¡ 2 a ( T T c ) + gq 2 ¢ η q η q G = h ( η 0 + δη ( r )) ( η 0 + δη ( r 0 )) i = η 2 0 + T 2 V X q exp ( i q ( r r 0 )) 2 a ( T c T ) + gq 2 = a b ( T c T ) + T 8 π 2 g exp ( | r r 0 | ) | r r 0 | ξ = r g 2 a ( T c T ) . Problem 7-2 (6 pts.) Find the average value of a complex order parameter, Ψ = ψ exp ( ) , at ( T < T c . Take into account only the fl uctuations of phase φ , i. e. assume magnitude ψ to be constant. Note that φ has a Gaussian distribution function. Assume its average to be h φ i = φ 0 , and determine the width of the distribution, ­ φ 2 ® , from the Landau-Ginzburg free energy: F = Z d r a ( T T c ) | Ψ | 2 + b 2 | Ψ | 4 + g | Ψ | 2 ¸ Take the microscopic cut-o ff length to be λ . Consider two cases: a) 3D system; b) 2D system of a typical size L . Solution: F = Z a ( T T c ) ψ 2 + b 2 ψ 4 + g ³ ( ψ ) 2 + ψ 2 ( φ ) 2 ´ ¸ d r = F ( ψ ) + Z K ( φ ) 2 d r (1) Minimization in
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Unformatted text preview: ψ gives: ψ = r a b ( T c − T ); K = gψ 2 = ga b ( T c − T ) Assuming q max = 2 π/λ and q max = 2 π/L , we obtain: ­ δφ 2 ( r ) ® ' Tb 2 πga ( T c − T ) ½ 1 /λ , i n 3 D log ( L/λ ) , in 2D 2 h Ψ i = ψ h exp ( iφ ) i = ψ R dφ exp ³ iφ − ( φ − φ ) 2 2 h δφ 2 i ´ R dφ exp ³ − ( φ − φ ) 2 2 h δφ 2 i ´ = ψ exp à iφ − ­ δφ 2 ® 2 ! h Ψ i 3 D ' r a b ( T c − T ) exp μ − Tb 4 πλga ( T c − T ) + iφ ¶ h Ψ i 2 D ' r a b ( T c − T ) exp ( iφ ) μ λ L ¶ Tb/ (4 πga ( T c − T )) Problem 7-3 (4 pts.) Determine the temperature of Kosterlitz-Thouless transition for the system from Problem 7-2(b). Solution: T BKT = πK = πgψ 2 = πga b ( T c − T BKT ) T BKT = T c 1 + b/πga...
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