Table Of ContentLecture Notes in Physics 884
V.K.B. Kota
Embedded
Random Matrix
Ensembles
in Quantum
Physics
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V.K.B. Kota
Embedded Random Matrix
Ensembles in Quantum
Physics
V.K.B.Kota
PhysicalResearchLaboratory
Ahmedabad,India
ISSN0075-8450 ISSN1616-6361(electronic)
LectureNotesinPhysics
ISBN978-3-319-04566-5 ISBN978-3-319-04567-2(eBook)
DOI10.1007/978-3-319-04567-2
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To myparents
Preface
Random matrix theory was introduced into physics by E.P. Wigner in 1955, and
consolidatedwithdeeperandwiderranginginvestigationsinthelastthreedecades,
ithasbecomeanintegralpartofquantumphysics.AsaptlystatedbyH.A.Weiden-
müllerinarecentcommentary:“althoughusedwithincreasingfrequencyinmany
branchesofphysics,randommatrixensemblessometimesaretoounspecifictoac-
countforimportantfeaturesofthephysicalsystemathand.Onerefinementwhich
retainsthebasicstochasticapproachbutallowsforsuchfeaturesconsistsintheuse
ofembeddedensembles.”Thisnewclassofrandommatrixensembles,theembed-
ded random matrix ensembles, were introduced in the context of the nuclear shell
model in early 1970. As stated by J.B. French: “the GOE, now almost universally
regarded as a model for a corresponding chaotic system, is an ensemble of multi-
body,nottwo-bodyinteractions.Thisdifferenceshowsupbothinone-point(density
of states) and two-point (fluctuations and smoothed transition strengths) functions
generatedbythenuclearshellmodel.Forabetterapriorimodelwecanchoosean
ensembleofk-bodyinteractions(k=2isaninterestingcase)bygeneratingaGOE
in k-particle space and using it in the space of m-particles. For most purposes the
resultingembeddedGOE(orEGOE)isverydifficulttodealwith,butbygoodluck,
we can use it to study the questions we have posed and the answers are different
from,andmuchmoreenlighteningthan,thosewhichwouldcomefromGOE.”
Research over the last two decades in particular has produced a large body of
newresultsforembeddedensemblesanditisclearthattheserandommatrixensem-
bles are indispensable in the study of finite many-particle quantum systems such
asatoms,nuclei,quantumdots,smallmetallicgrains,latticespinmodelsforquan-
tum computers,and so on. In this book, starting with an easy-to-read introduction
togeneralrandommatrixtheory,allthenecessaryconceptsforembeddedrandom
matrix ensembles are developed from scratch and the reader is then carried to the
frontiersofpresent-dayresearch.Thefirstchaptergivesageneralintroductionand
thenexttwochaptersdealwithsomegeneralaspectsofclassicalrandommatrixen-
sembles.Eightchaptersintheremainingpartofthebookgiveresultsforavariety
of embedded ensembles, mainly classified according to the Lie symmetries of the
vii
viii Preface
Hamiltonianofafinitemany-bodyquantumsystem,whilefourchaptersaredevoted
toapplications.Thelastchapterprovidesasummaryandfutureprospects.
The starting point for this book was a series of lectures given by the author at
AndhraUniversity,Visakhapatnam(India)in2002.Effortshavebeenmadetogive
enoughdetailineverychaptertoensurethatanadvancedgraduatestudentcanfol-
lowthemathematicsandunderstandtheresultsof‘computerexperiments’forem-
bedded ensembles. On the other hand, the book gives an exhaustive review of the
fieldsothataresearchstudentcanusethematerialtostartworkingonnewquestions
inthesubjectofembeddedensemblesitselfandintheirapplicationtomany-body
quantumphysics.
Over the last three decades I have had the pleasure of collaborating with many
people,anddiscussedthetopicsofthisbookwithmanyothers.Firstofall,Iwould
liketospeciallythankthelateJ.B.Frenchforalongandprofitablecollaborationon
statistical nuclear spectroscopy. Embedded random matrix ensembles have grown
out of this subject and the present work is complementary to the book Statistical
Spectroscopy and Spectral Distributions in Nuclei by R.U. Haq and myself, pub-
lished in 2010 by World Scientific. The influence of J.B. French on my way of
thinkingaboutrandommatrixtheoryinphysicsissurelyvisibleinseveralpartsof
thepresentbook.
IwasfortunateinhavingA.Pandey,J.C.Parikh,V.Potbhare,andS.Tomsovic
ascollaboratorsinmyearlyyearsofresearchonrandommatrixtheory.Regarding
thetopicsdiscussedinseveralchaptersofthisbook,Ihavecollaboratedintensively
withR.Sahu,N.D.Chavda,andmyformerPh.D.studentMananVyas.Withoutthat
collaboration, this book would not have been possible. I have also benefited from
collaboration and discussions with many colleagues, friends, and students, and in
particular with Dilip Angom, B. Chakrabarti, J.M.G. Gomez, R.U. Haq, K. Kar,
D. Majumdar, the late J. Retamosa, S. Sumedha, and Y.M. Zhao. I am especially
indebtedtoH.A.WeidenmüllerandthelateO.Bohigasfordiscussionsandencour-
agement. I am thankful to N.D. Chavda and Manan Vyas for preparing some of
the figures and thank Manan Vyas once again for typing some parts of the book.
ThanksarealsoduetothedirectorsofthePhysicalResearchLaboratory(Ahmed-
abad,India)forfacilitiesandsupport.Therearemanyotherswhohavedirectlyor
indirectly contributed to my work on embedded ensembles and I sincerely thank
them.Copyrightpermissionforusingsomeofthefigures,fromtheAmericanPhys-
ical Society, the American Institute of Physics, Elsevier Science, the Institute of
Physics, Springer-Verlag, and World Scientific is gratefully acknowledged. I am
alsothankfultoalltheauthorswhohavegivenpermissiontousefiguresfromtheir
publications.SpecialthanksareduetotheeditorsatSpringer-Verlagfortheirefforts
inbringingoutthisbook.Andlastlyandmostimportantly,Iamindebtedtomywife
Vijayaforherunfailingsupportsince1980.
PhysicalResearchLaboratory, V.K.B.Kota
Ahmedabad,India
November2013
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 ClassicalRandomMatrixEnsembles . . . . . . . . . . . . . . . . . 11
2.1 HamiltonianStructureandDyson’sClassificationofGOE,GUE
andGSERandomMatrixEnsembles . . . . . . . . . . . . . . . 11
2.1.1 2×2GOE . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.2 2×2GUE . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.3 2×2GSE . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 OneandTwoPointFunctions:N×N Matrices . . . . . . . . . 21
2.2.1 OnePointFunction:Semi-circleDensity . . . . . . . . . 21
2.2.2 TwoPointFunctionSρ(x,y) . . . . . . . . . . . . . . . 24
2.2.3 FluctuationMeasures:NumberVarianceΣ2(r)
andDyson-MehtaΔ Statistic. . . . . . . . . . . . . . . 27
3
2.3 StructureofWavefunctionsandTransitionStrengths . . . . . . . 28
2.3.1 Porter-ThomasDistribution . . . . . . . . . . . . . . . . 28
2.3.2 NPC,Sinfo andStrengthFunctions . . . . . . . . . . . . 30
2.4 DataAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 UnfoldingandSampleSizeErrors . . . . . . . . . . . . 32
2.4.2 PoissonSpectra . . . . . . . . . . . . . . . . . . . . . . 33
2.4.3 AnalysisofNuclearDataforGOEandPoisson. . . . . . 34
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 InterpolatingandOtherExtendedClassicalEnsembles. . . . . . . 39
3.1 GOE-GUETransition . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 2×2MatrixResults . . . . . . . . . . . . . . . . . . . . 41
3.1.2 N×N EnsembleResultsforΣ2(r)andΔ (r) . . . . . . 43
3
3.1.3 ApplicationtoTRNIinNucleon-NucleonInteraction . . 45
3.2 PoissontoGOEandGUETransitions . . . . . . . . . . . . . . . 46
3.2.1 2×2MatrixResultsforPoissontoGOETransition . . . 46
3.2.2 2×2ResultsforPoissontoGUETransition . . . . . . . 48
ix
x Contents
3.2.3 RelationshipBetweenΛParameterforPoissontoGOE
andtheBerry-RobnikChaosParameter . . . . . . . . . . 49
3.2.4 PoissontoGOE,GUETransitions:N×N Ensemble
ResultsforΣ2(r) . . . . . . . . . . . . . . . . . . . . . 51
3.2.5 OnsetofChaosatHighSpinsviaPoissontoGOE
Transition . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 2×2PartitionedGOE . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.1 IsospinBreakingin26Aland30PNuclearLevels . . . . . 52
3.4 Rosenzweig-PorterModel:AnalysisofAtomicLevelsand
+ +
Nuclear2 and4 Levels . . . . . . . . . . . . . . . . . . . . . 54
3.5 CovarianceRandomMatrixEnsembleXXT:EigenvalueDensity 55
3.5.1 ASimple2×2PartitionedGOE:p-GOE:2(Δ) . . . . . . 56
3.5.2 MomentsandtheEigenvalueDensity
forp-GOE:2(Δ=0) . . . . . . . . . . . . . . . . . . . . 57
3.5.3 EigenvalueDensityforGOE-CRME . . . . . . . . . . . 59
3.6 FurtherExtensionsandApplicationsofRMT . . . . . . . . . . . 61
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 EmbeddedGOEforSpinlessFermionSystems:EGOE(2)and
EGOE(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1 EGOE(2)andEGOE(k)Ensembles:DefinitionandConstruction 69
4.2 EigenvalueDensity:GaussianForm . . . . . . . . . . . . . . . . 71
4.2.1 BasicResultsfromBinaryCorrelationApproximation . . 71
4.2.2 DiluteLimitFormulasfortheFourthandSixthOrder
MomentsandCumulants . . . . . . . . . . . . . . . . . 77
4.3 Average-FluctuationSeparationandLower-OrderMoments
oftheTwo-PointFunction . . . . . . . . . . . . . . . . . . . . . 80
4.3.1 LevelMotioninEmbeddedEnsembles . . . . . . . . . . 80
4.3.2 S2 inBinaryCorrelationApproximation . . . . . . . . . 81
ζ
4.3.3 Average-FluctuationsSeparationintheSpectraofDilute
FermionSystems:ResultsforEGOE(1)andEGOE(2) . . 82
4.3.4 Lower-OrderMomentsoftheTwo-PointFunctionand
CrossCorrelationsinEGOE. . . . . . . . . . . . . . . . 84
4.4 TransitionStrengthDensity:BivariateGaussianForm . . . . . . 85
4.5 StrengthSumsandExpectationValues:RatioofGaussians . . . 95
4.6 LevelFluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5 Random Two-Body Interactions in Presence of Mean-Field:
EGOE(1+2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.1 EGOE(1+2):DefinitionandConstruction . . . . . . . . . . . . 101
5.2 UnitaryDecompositionandTracePropagation . . . . . . . . . . 102
5.2.1 UnitaryorU(N)DecompositionoftheHamiltonian
Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 102
