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Unformatted text preview: Physics 848: Problem Set 2 Due Tuesday, October 18 at 11:59 PM Note: each problem is worth 10 points unless otherwise stated. 1. (10 points). Spin-spin correlation function for the Ising model in 1d. Consider the ferromagnetic Ising model in 1D with zero applied magnetic field. The Hamiltonian is H =- J N- 1 X n =1 S n S n +1 , (1) where S n takes on the values 1, J > 0, and we assume FREE bound- ary conditions, so that S 1 and S N are not attached to any neighbors. (a) The partition function for this special case of B = 0 can be easily calculated by defining new variables V n = S n S n +1 , with n running from 1 to N- 1 These V n s also take on the allowed values 1 and are independent of one another. Hence, the partition function can be written Q N ( T ) = 2 V 1 = 1 .... V N- 1 = 1 exp(- H/k B T ), where H =- J N- 1 n =1 V n . The factor of 2 in front of the partition function allows for two possible orientations of the spin S 1 . Since H is now the sum of independent variables, the partition function can be easily calculated. Use this simple transformation to calculate the partition function for a chain of N spins. (b). Calculate the correlation function h S p S q i , where q > p but | q- p | N . ( p and q are site indices.) Hint: write S p S q = ( S p S p +1 )( S p +1 S p +2 ) .... ( S q- 1 S q ) and use the change of variables described in (a). Why is it possible to write S p S q in this way?...
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- Fall '05