whichmightbeofuse,perhapstoagraduatestudentstartingworkinthisarea. Thebookisorganisedintochaptersthatdeal rstlywiththenatureofquantum mechanicalspinsandtheirinteractions. Thefollowingchaptersthengiveadetailed guidetothesolutionoftheHeisenbergandXYmodelsatzerotemperatureusing theBetheAnsatzandtheJordan-Wignertransformation,respectively. Approximate methodsarethenconsideredfromChap. 7onwards,dealingwithspin-wavet- oryandnumericalmethods(suchasexactdiagonalisationsandMonteCarlo). The coupledclustermethod(CCM),apowerfultechniquethathasonlyrecentlybeen vii viii Preface appliedtospinsystemsisdescribedinsomedetail. The nalchapterdescribesother work,someofitveryrecent,toshowsomeofthedirectionsinwhichstudyofthese systemshasdeveloped. Theaimofthetextistoprovideastraightforwardandpracticalaccountofall of the steps involved in applying many of the methods used for spins systems, especiallywherethisrelatestoexactsolutionsforin nitenumbersofspinsatzero temperature. Inthisway,wehopetoprovidethereaderwithinsightintothesubtle natureofquantumspinproblems. Manchester,UK JohnB. Parkinson January2010 DamianJ. J. Farnell Contents 1 Introduction.................................................. 1 References..................................................... 5 2 Spin Models................................................... 7 2. 1 SpinAngularMomentum................................... 7 2. 2 CoupledSpins............................................ 10 1 2. 3 TwoInteractingSpin- 's................................... 11 2 2. 4 CommutatorsandQuantumNumbers......................... 14 2. 5 PhysicalPicture........................................... 16 2. 6 In niteArraysofSpins..................................... 16 1 2. 7 1DHeisenbergChainwith S = andNearest-Neighbour 2 Interaction............................................... 18 References..................................................... 19 1 3 Quantum Treatment of the Spin- Chain......................... 21 2 3. 1 GeneralRemarks.......................................... 21 3. 2 AlignedState............................................. 22 3. 3 SingleDeviationStates..................................... 23 3. 4 TwoDeviationStates....................................... 27 3. 4. 1 FormoftheStates................................. 33 3. 5 ThreeDeviationStates..................................... 36 Z N 3. 5. 1 BetheAnsatzforS = ?3....................... 36 T 2 3. 6 StateswithanArbitraryNumberofDeviations................. 37 Reference...................................................... 38 4 The Antiferromagnetic Ground State............................ 39 4. 1 TheFundamentalIntegralEquation........................... 39 4. 2 SolutionoftheFundamentalIntegralEquation................. 43 4. 3 TheGroundStateEnergy................................... 45 References..................................................... 47 ix x Contents 5 Antiferromagnetic Spin Waves.................................. 49 5. 1 TheBasicFormalism...................................... 49 5. 2 MagneticFieldBehaviour..................................
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