hw07soln - gives: = r a b ( T c T ); K = g 2 = ga b ( T c 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 ( ) ,a t( T<T c . Take into account only the f 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 f 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)
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Unformatted text preview: gives: = r a b ( T c T ); K = g 2 = ga b ( T c T ) Assuming q max = 2 / and q max = 2 /L , we obtain: 2 ( r ) ' Tb 2 ga ( T c T ) 1 / , i n 3 D log ( L/ ) , in 2D 2 h i = h exp ( i ) i = R d exp i ( ) 2 2 h 2 i R d exp ( ) 2 2 h 2 i = exp i 2 2 ! h i 3 D ' r a b ( T c T ) exp Tb 4 ga ( T c T ) + i h i 2 D ' r a b ( T c T ) exp ( i ) L Tb/ (4 ga ( 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 = K = g 2 = ga b ( T c T BKT ) T BKT = T c 1 + b/ga...
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This note was uploaded on 09/28/2008 for the course PHYS 510 taught by Professor Anon during the Winter '04 term at Cornell University (Engineering School).

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hw07soln - gives: = r a b ( T c T ); K = g 2 = ga b ( T c T...

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