An Introduction to Quantum Spin Systems, Paperback Book

An Introduction to Quantum Spin Systems Paperback

Part of the Lecture Notes in Physics series

Description

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 ...

Information

  • Format: Paperback
  • Pages: 154 pages, 22 Illustrations, black and white; XI, 154 p. 22 illus.
  • Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Publication Date:
  • Category: Low temperature physics
  • ISBN: 9783642132896

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