Diagrammatic Analysis of T-matrix Theories of the Attractive Hubbard Model Iiwin S. D.Beach .A r tiesis submitted to the Department of Physics in conformity wit 11 the rccpirernents for the degree of Master of Science Queen's University Iiingstori. Ontario. Canada September. 1999 @ Kevin S. D.Beach. 1999 National Library Bibliotheque nationale I*I ofCanada du Canada Acquisitions and Acquisitions et Bibliographic Services services bibliographiques 395 Wellmgton Street 395, rue Wellington Ottawa ON K 1A ON4 Ottawa ON K 1A ON4 Canada Canada Your h~eVow reference Our fm Mtre reldwnce The author has granted a non- L'auteur a accorde une licence non exclusive licence allowing the exclusive permettant a la National Library of Canada to Bibliotheque nationale du Canada de reproduce, loan, distribute or sell reproduire, preter, distribuer ou copies of this thesis in microform, vendre des copies de cette these sous paper or electronic formats. la forme de microfiche/filrn. de reproduction sur papier ou sur format electronique. The author retains ownership of the L'auteur conserve la propriete du copyright in this thesis. Neither the droit d'auteur qui protege cette these. thesis nor substantial extracts from it Ni la these ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent ttre imprimes reproduced without the author's ou autrement reproduits sans son permission. autorisation. Abstract 111 r his uork, r~~~iny-bocfre(-hniqu~s are used to st.udy rho effwt of ~lectroripair c-orr~latious on th(1 nor111;d~tate prop~rti~s of th~t~vo-clirn~nsi~naI attractiv~ Hubbarci ntociel npar t tlr c.ritical tt~r~iperstrirc( T.) of the sr~p~rcon~iuctingtransition. The role clt' .;elf-t.onsistrn(*y ir~t hr st ,zndxr[i T-matrix npprosirnation is pxarninrd ivi t h particular at t~ntionpaid to its c.orltriLution to the suppression of the r-ritical temprrature and to the depletion of spertrai 1i.cxiqht. Ind~ed.ivt) show that the ~~r~forc.crnrntof .;elf-consistency ~irarnatic-allyxlttlrs t hcl valu~oS I'. 1 from its r~on-self-(-onsistont value) and that t he choice of self-c-onsisttwc.y~h~tw cmrnpl~telydetprrriines the predicted density of states at T,,. Mmovcr, at low tenlpt.rnturw Iitaitr T-, non-self-consistent cornpu tational results suggest that the depletion of ~pc!c*trid ivvight at the fern~ilevel is prunounced ard leads to the opening of a pseudoqap. a prwursor to t he cl!icr%y gap of the superconducting state. In xdditior~,we critique L Pad6 analytic continucztion met hod whereby a rational poly- nornial function is fit to a set of' input points by means of a single matrix inversion. The r~let,llo(lis applied to the problerri of determining the spectral function of a one-particle propagator known only st a finite number of llstsubara frequencies. This is dono for prop- agators constructed from two example self-energies, drawn from the T-matrix theory of the attractive Hubbard model. which are calculated to extremely high accuracy using a novel symbolic corn putation algorithm. Cl'e present a systematic analysis of the effects of error in the input points and discuss how the convergence of the polynomial coefficients to allowed values can serve as a quantitative measure of the goodness-of-fit. Statement of originality Acknowledgements Ihis tvork ivas carried out 11ncic.rthe supervision of Dr. Robert d. Gooding (Qsrtri'.q I'rlirrr- .i,t// ) to LV horn the a11t hor o~~sa great ~lebt of gratitude for his guidance and cncouragernrrit. I'hx~ksare illdo doe to Dr. Frank llnrsiglio f-nr~.rrsit!jof .-1lb~rtnlfor many helpful tiis- ~-l~ssions.rnd for qrariously pro\*i(iing nnm~ri(.illrwdts ~vhichivr.ere r~quir~dto ti~bug.in11 vrrify the computer code used in this project. Fillally, the author wishes to nckno~vlniqe the unconditional support. of his family and of his partner, Diana Drappel. Notational convent ions .\ v~ctorcluarltity is (lenoted by an arrow arrd its c:ornponents by supt\rsc.ript..;: rb.g. t.he s!*rrrbol i: = ( rl,rl.. uD) represents a vector of dimension D. :In operator in rht: ~wcupationr~unibcfr forr~idism is marked by a, caret. except for the fernlion wrni hilation ;L% possible). P.E. ti = cfv. .-Ijrrmronrr operator consists of an odd number uf c nnql r."i ivhortw .a br).wr~tcoperator consists of an even nurnher. - The cornrnutntor of two operators .iand b is given by [.-i.B] = .4B - H.4 and rhc ~lr~trt.or~lrrl~rtntorby {.-i.b} = ..id + 6.i. \Ye shall use brackets wirh a ii%a iut~scripr. to ~l~nut,cc'ittl~r the commutator or anticornnlut.ator according to [..i.B]* = .ib + fj.-i. -4 useful consequence of this nition is the identity = - .41 def [.-iB,i'] .-i[d. i'] [i'. _ 8. ivhich holds for both kinds of statistics. Generally. whenever we treat fernli and hose c;tses jimdtnneously using + and 7. the fernii case will be represented by the upper sign. The herrnitian conjugate (or adjoint) is denoted by a dagger (t) and the con~plexcon- jugate by an asterisk (*). An operator .iis called hermitinn (or self-adjoint) if .-it = .Lie Every matrix element of s herrnitian operator satisfies Sotice that if we choose lo) = 10') to be an eigenstate of .i.then the equation above tells 11s that the eigenvalues of a hermitian operator are real. rl is inlplicitlv cv;rhlat.ed in the limit rl -. 0'. l\'e occasionally employ the shorthand -r, - r + 11. In t.ern~sof the infinitesimal rl. the Heaviside and Dirac t'unctions can be represented by set union set intersection set subtraction csrtesian product direct sum absolute value norm NSC' non-self-consistent pp 1 partic~e-prticI~ I I particle-hole (JhIC' quantum 1Iorltc C'arlo TCL Thouless criterion line Table of contents Abstract Statement of originality Acknowledgements Notational conventions 1v ... Table of contents Vlll List of figures 1 Attractive Hubbard model 1.1 [ntroduction . I Uoclel harniltonian . L .?.I Real space reprcsentrtt,ion . 1 \C',zvevector representation . 1 Spin representation . , . 1. Exact solutions . 1.-I Thermody narnics of the Hubbard model . l .j S~mmetriesof the haniiltonian . - . 1. Limiting cases . - . L.6.1 Band limit . , . 1.6 Atomic limit . 1.6.3 >on-trivial values of IV/t. , , . 2 Formalism of diagrammatic field theory 2.1 Approaching the strongly correlated electron problem . 2. Functional differentiation . 2. Equation of motion method . 2.4 Fourier representation . 2.5 T-matrix approximation . 2.6 Transition to the superconducting state . 3 rllt~alysisof the T-matrix approximation 67 1 O\.~rviciv ..................................... ci: -I A new approach to analytic continuation 115 1. 1 Ovt1rvil\w ..................................... 11 .: 1.2 Green's t'unctiori forrnalisrn ........................... llti 1.3 Pati6 approxinlmts ................................ 118 !. Sunlerical rcsdts ................................. 123 t.5 Summary ..................................... 131 5 Conclusions and future work 13-1 A Basic formalism 137 .\. 1 Occupation number formalism (second quantization) ............. 1;jT A.2 Real space representation ............................ LXl .4 .3 It'avevector representation ............................ 142 :\..1 Spin operator ................................... 1-15 ..I..? Fluxoperator ................................... 143 .A .rj Local number conservation ........................... 1-52 .A .; Density of levels ................................. 1.34 B Many-body formalism 161 B.1 Green's function formalism ........................... 161 B . 1.1 Time ordering operator ......................... 161 8.1.2 Equations of motion ........................... 162 8.1.3 Green's functions ............................. 163 B . 1.4 Fourier series for temperature functions ................ 166 8.1..5 Fourier respresentations ......................... 167 B.1.6 Analytic continuation .......................... 171 B.1.7 1Iornents of the spectral function .................... 172 B.2 Occupation functions ............................... 171 B.2.1 Basic occupation functions ....................... 174 B.2.2 Extended occupation functions ..................... 179 8.2.3 Partial occupation functions ....................... 132 C Functional differentiation 193 '1 F111lct.ionaltiitfcr~ntiiitiorl . 193 ('.2 5kclcto11 clspanslorl . I07 D Equation of motion method 300 D.1 Lirlt!ar t'unc.t.ionnls . 200 2 Evaluating t tie corllrnutntor ul the ~ranti~~~nonical harniltoniarl ~vit11 all ;in- !lihilation operator . 204 D.3 Equation of motiur~. 205 L). I Terrilinating the hierarch! 01' equations . .)()*- D.5 C;cneraiized intt>ractiori . 2 I I E h1APLE code 215 E.l Pndr. package . 215 I:.? €luhbi1rd2ci package . 2 l(i Vitae 254 List of figures [ortic xr~erlirlgof r.1t.c trorls ........................... .) (I;eneric loiv.tertlperaturt.i plla.~~liagrnrn of t.he h~gh.1.. I-ripr:itcs ........1 Levrl ~rrri'acwof the dispersion relation in tile Brillouin zone ........ '1 ~lomcnturr~transfer in a singlcl dectrorl-electron x;lttcrinq ovetlt....... 10 Li~lerrysp~ctrurn for ;in ~lcctrnn pair ...................... 1ri t3h;tscl ~iiagrarn(A' tllc t'rcle c:lectron ga ..................... -'q- Density of levtals it\ une . t.tvo. and three ltirn~nsior\s .............. :10 Phase tliagranl of the attractive Hubbard nlmlel in the atomic limit .... 3-4 Doubie occupancy in the atomic limit ..................... 3.5 2.1 Sktlleturl vspa~~sionof t.he Hubbard sdfen~rg,y ................ 1-1 2.2 Skt.l~torltlupansion of t.he opposit~-.; pin t~vo-particlerorrelator ....... li 2.3 Skeleton expansion of the equal-spin two-particle corrrlator ......... lfj 2 . I Dccon~positionof