UNIVERSITY OF CAMBRIDGE MATHEMATICAL TRIPOS Part III Tuesday, 9 June, 2015 1:30 pm to 4:30 pm PAPER 81 QUANTUM CONDENSED MATTER FIELD THEORY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS Cover sheet Treasury Tag Script paper SPECIAL REQUIREMENTS None You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
UNIVERSITY OF CAMBRIDGE 2 1 The one-dimensional spin S quantum Heisenberg antiferromagnet is specified by the Hamiltonian, n=1 N ? = J YSn · ?n+1 , where J > 0, the number of lattice sites N is even, and periodic boundary conditions are imposed such that SN+1 = S1. (a) Define the classical ground state degeneracy of the system and specify one of the possible classical antiferromagnetic ground states. Explain why the classical ground state is not an eigenstate of the quantum Hamiltonian. (b) Applying a suitable canonical transformation, show that the Hamiltonian can be brought to the form 1 ? = - J N n=1 - + 1 ? ? 2 , where St = SIT + i Sy denote spin raising and lowering operators. (c) Show that the Holstein-Primakoff transformation, §+ = (2S)1/2 (1- a?a 2S 1/2 a. ?- = (ÎS+)+, ? = S - ata, is consistent with the quantum spin algebra, [S+, S-] = 2 Sz (where we have set ~= 1). (d) Taking the spin to be large S I 1, show that to first order in S, the antiferromagnetic Hamiltonian can be written as a bilinear in lattice boson operators an and an. (e) Bringing the Hamiltonian to diagonal form, show that the excitation spectrum of the resulting Hamiltonian is given by wk = 2JS| sin k|. Define the quantum numbers k and comment on the form of the low-energy spectrum. (f) Show that the ground state sublattice magnetisation is given by N n=1 >(-1)" Salg.s.i I S- | dk 1 27 2 1 - 1 hg.s.IT 1 . | sin k| Comment on the physical implications of this result. Part III, Paper 81
UNIVERSITY OF CAMBRIDGE 3 2 In the BCS approximation, the pairing Hamiltonian of a superconducting electron system is given by 1 EkCka Cko + > k? k,k Vk,kb?kbk, where by = C'kiC-ki, and the fermion operators c'ko and Cko respectively create and annihilate electrons with wavevector k and spin o. (a) Neglecting terms of order (bk - hbki)2, where hbki = hg.s.|bk|g.s.i denotes the ground state expectation value, show that ? - uÑ = 1 SkCko Cko - k? k E (Akb + h.c.) - > k,k' where Sk = Ek - u and Ak = - Ek Vk,k'hbx'i. Vk,k'hbihbKi, (b) Under what conditions on the real coefficients uk and vk is the following transforma- tion canonical, Ckt + C-k+ = ! Uk -UK Uk Uk Ykt + Y-k! ! . (c) Taking Ak to be real, use this transformation to show that the BCS pairing Hamiltonian can