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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). Spinspin 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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This note was uploaded on 07/17/2008 for the course PHYS 848 taught by Professor Stroud during the Fall '05 term at Ohio State.
 Fall '05
 STROUD
 Physics

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