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An Introduction to Quantum Spin Systems
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An Introduction to Quantum Spin Systems
Current price: $51.59
Original price: $64.49
An Introduction to Quantum Spin Systems
Current price: $51.59
Original price: $64.49
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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