behaves above and below T c \u03c7 A T T c \u03b3 T T c \u03c7 A T c T \u03b3 T T c Determine the

# Behaves above and below t c χ a t t c γ t t c χ a

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behaves above and below T c : χ A ( T - T c ) - γ , T > T c χ A 0 ( T c - T ) - γ 0 , T < T c Determine the values of γ , γ 0 , and A/A 0 . 1
(d) Similarly, the behavior of the specific heat, C V = - T ( 2 F/∂T 2 ) at h = 0 is specified by C V B ( T - T c ) - α , T > T c C V B 0 ( T c - T ) - α 0 , T < T c Determine the values of α , α 0 , and B and B 0 . 2. Landau theory for the XY model: This is expressed in terms of a vector field ~m ( r ) = ( m 1 ( r ) , m 2 ( r )). In zero applied field, the Landau free energy has the form F = Z d 3 r K 2 X i = x,y,z X a =1 , 2 ( i m a ) 2 + α 2 X a =1 , 2 m 2 a + β 4 X a =1 , 2 m 2 a 2 (1) and the critical point is at α = 0 (Assume, K, β > 0).Determine the correlation (0) i - h m 1 ( r ) ih m 1 (0) i and G 22 ( r ) = h m 2 ( r ) m 2 (0) i - h m 2 ( r ) ih m 2 (0) i for both signs of α . You can compute these correlation functions by applying an external field ~ h ( r ) to the system, under which F → F - Z d 3 r X a =1 , 2 h a ( r ) m a ( r ) , (2) and then computing the change in h m a i due to the presence of the field. As shown in class for the Ising model, we have to linear order in ~ h h m a ( r ) i| ~ h = h m a ( r ) i| ~ h =0 + 1 k B T Z d 3 r 0 G aa ( r - r 0 ) h a ( r 0 ) + . . . (3) By writing ~m in the above form, you can read off the values of G aa . Above the critical

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