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# Final - 581 Final Due Instructions Work independently and...

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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 x i + σ z i σ z i +1 parenrightBig (1) where σ μ 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 ǫ 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: σ x i = 1 2 n i , σ z i = productdisplay j<i (1 2 n j )( c i + c i ) , σ y i = i productdisplay l<i (1 2 n l )( c i c i ) (5) 1

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where the c i ( c i ) are fermionic annihilation (creation) operators and n i = c i c i . You are to apply this transformation to the Hamiltonian for
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Final - 581 Final Due Instructions Work independently and...

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