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Unformatted text preview: 581 Final: Due May 15, 2009 Instructions: Work independently and turn in exam by 5:00pm May 15. Turn in directly to me in my office or place in envelope outside my office. Problem 1: Quantum phase transitions occur at zero temperature and arise funda- mentally from the uncertainty principle. Hence, present in any Hamil- tonian that admits a quantum phase transition must be two non- commuting operators. You are to consider the following Hamiltonian H = J summationdisplay i parenleftBig g x i + z i z i +1 parenrightBig (1) where i is a Pauli matrix, i refers to site i and J and g are coupling constants. (a) Identify the two ordering tendencies of this Hamiltonian. That is, identify the ground state for g = 0 and g = . (b) For g 1, a fairly accurate solution for the wavefunction is | i ) = |) i productdisplay j negationslash = i | + ) j |) i = ( | ) i | ) i ) / 2 . (2) Show that the energy of the state | k ) = 1 N summationdisplay i e ikx i | i ) , (3) is given by k = Jg parenleftBigg 2 2 g cos k + O (1 /g 2 ) + ... parenrightBigg . (4) (c) This problem can be solved exactly using the Jordan-Wigner trans- formation: x i = 1 2 n i , z i = productdisplay j<i (1 2 n j )( c i + c i ) , y i = i productdisplay l<i (1 2 n l )( c i c i ) (5) 1 where the...
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