An Introduction to Quantum Spin Systems, Paperback / softback Book

An Introduction to Quantum Spin Systems Paperback / softback

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 / softback
  • 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

Other Formats

£31.99

£27.69

 
Free Home Delivery

on all orders

 
Pick up orders

from local bookshops

Also by John B. Parkinson