An Introduction to Quantum Spin Systems by John B. ParkinsonAn Introduction to Quantum Spin Systems by John B. Parkinson

An Introduction to Quantum Spin Systems

byJohn B. Parkinson, Damian J. J. Farnell

Paperback | September 20, 2010

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Title:An Introduction to Quantum Spin SystemsFormat:PaperbackDimensions:154 pagesPublished:September 20, 2010Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3642132898

ISBN - 13:9783642132896

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Table of Contents

Spin Models.- Quantum Treatment of the Spin-½ Chain.- The Antiferromagnetic Ground State.- Antiferromagnetic Spin Waves.- The XY Model.- Spin-Wave Theory.- Numerical Finite-Size Calculations.- Other Approximate Methods.- The Coupled Cluster Method.- Quantum Magnetism.