The topic of lattice quantum spin systems (or 'spin systems' for short) is a f- cinating branch of theoretical physics and one of great pedigree, although many importantquestionsstillremaintobeanswered. The'spins'areatomic-sizedm- netsthatarelocalisedtopointsonalatticeandtheyinteractviathelawsofquantum mechanics. Thisintrinsicquantummechanicalnatureandthelarge(usuallyeff- tivelyin nite)numberofspinsleadstostrikingresultswhichcanbequitedifferent fromclassicalresultsandareoftenunexpectedandindeedcounter-intuitive. Spinsystemsconstitutethebasicmodelsofquantummagneticinsulatorsandso arerelevanttoawholehostofmagneticmaterials. Furthermore,theyareimportant asprototypicalmodelsofquantumsystemsbecausetheyareconceptuallysimple and yet stilldemonstrate surprisingly rich physics. Low dimensional systems, in 2Dandespecially1D,havebeenparticularlyfruitfulbecausetheirsimplicityhas enabledexactsolutionstobefoundwhichstillcontainmanyhighlynon-trivialf- tures. Spinsystemsoftendemonstratephasetransitionsandsowecanusethemto studytheinterplayofthermalandquantum uctuationsindrivingsuchtransitions. Ofcoursetherearemanycasesinwhichwecan ndnoexactsolutionandinthese casestheycanbeusedasatestinggroundforapproximatemethodsofmodern-day quantummechanics. Thesequantumsystemsthusprovideagreatvarietyofint- estinganddif cultchallengestothemathematicianorphysicalscientist. Thisbookwaspromptedbyaseriesoftalksgivenbyoneoftheauthors(JBP)at asummerschoolinJyvaskyla,Finland. Thesetalksprovidedadetailedviewofhow onegoesaboutsolvingthebasicproblemsinvolvedintreatingandunderstanding spinssystemsatzerotemperature. Itwasthislevelofdetail,missingfromothertexts inthearea,thatpromptedtheotherauthor(DJJF)tosuggestthattheselecturesbe broughttogetherwithsupplementarymaterialinordertoprovideadetailedguide 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- 'areatomic-sizedm- netsthatarelocalisedtopointsonalatticeandtheyinteractviathelawsofquantum mechanics. Thisintrinsicquantummechanicalnatureandthelarge(usuallyeff- tivelyin nite)numberofspinsleadstostrikingresultswhichcanbequitedifferent fromclassicalresultsandareoftenunexpectedandindeedcounter-intuitive. Spinsystemsconstitutethebasicmodelsofquantummagneticinsulatorsandso arerelevanttoawholehostofmagneticmaterials. Furthermore,theyareimportant asprototypicalmodelsofquantumsystemsbecausetheyareconceptuallysimple and yet stilldemonstrate surprisingly rich physics. Low dimensional systems, in 2Dandespecially1D,havebeenparticularlyfruitfulbecausetheirsimplicityhas enabledexactsolutionstobefoundwhichstillcontainmanyhighlynon-trivialf- tures. Spinsystemsoftendemonstratephasetransitionsandsowecanusethemto studytheinterplayofthermalandquantum uctuationsindrivingsuchtransitions. Ofcoursetherearemanycasesinwhichwecan ndnoexactsolutionandinthese casestheycanbeusedasatestinggroundforapproximatemethodsofmodern-day quantummechanics. Thesequantumsystemsthusprovideagreatvarietyofint- estinganddif cultchallengestothemathematicianorphysicalscientist. Thisbookwaspromptedbyaseriesoftalksgivenbyoneoftheauthors(JBP)at asummerschoolinJyvaskyla,Finland. Thesetalksprovidedadetailedviewofhow onegoesaboutsolvingthebasicproblemsinvolvedintreatingandunderstanding spinssystemsatzerotemperature. Itwasthislevelofdetail,missingfromothertexts inthearea,thatpromptedtheotherauthor(DJJF)tosuggestthattheselecturesbe broughttogetherwithsupplementarymaterialinordertoprovideadetailedguide 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 ...