T-Matrix Theories of the Attractive Hubbard Model

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

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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 propagators 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 def nition is the identity [.-iB,i'] = .-i[d. i'] - [i'..41 . _ 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 th~rc-particle Green's function into independrnt and cor- rr.1at.d propagating c0rnponent.s ...... IS 2 .. Dictionary of propagatinr, lines ...... I9 2 . t Dysorl's equation ...... il 7 Equation Tor the two-part.icle correlatiorls ...... i? 2 .$ Generalized two-particle interaction ...... 32 2.O Equation for the generalized two-particle interaction ...... 53 2.10 Self-energy expressed in terms of the generalized two-particle interaction . . 54 - - 2 . I I Translating between space-time and wavevector-frequency coordinates ....1 I 2 . I2 Space- time and wavevector-frequew interpretations of the self-energy diagrsrn 59 2.13 Particlehole and particleparticle diagrammatic element.^ ...... 59 2.14 Ladder diagrams ...... 60 2.15 Integration contours in the upper and lower half planes ...... 65

Bound state energies as a function of interaction strength ...... 72 'Thouless criterion line ...... 7-4 Depletion of spectral weight at the fermi level ...... 79 Opening of a pseudogap in the density of states ...... $0 Density of states along a constant density contour ...... 1 Six T-matrix theories ...... 33 Density of states at the superconducting transition for odd-numbered theories 58 Spectral function at the superconducting transition for even-numbered theories 92

xii

Chapter 1

Attractive Hubbard model

.\t s~~ffitrientlvlow te~nperatures,many tnaterials can malie the transition to ;F sup~~on-

1111ctinqstate in tvllich ct~rrenttlows thro~lqhthe nlatcrial without ~lissipation.This state il;ts Ion.; been u~lderst.udiin trrms 01 t. he pairing of electrons ro itrm (-ortlposite ob jec-ra ;1! ivhic h travel wit bout being scattered. wen in the presence of crystal (.f efccts. In i:onventior~aliuperconduct.ors, this pairing is the result of an ~ffectivet~lectrorl-rlectro[i ;~ttrilc:tic,r\.rr~cdiated by the ions in t.he material. which can tie described quaiitnt.ivc~l;is f'ulluws. The ions move (in a harmonic- potential ~vcllabout their quilibriurn position in the lattice) in an attempt to scrcen out the excess negative charge of nearby plectrons. Ho~vr.cv~r. since the propagation of the fattice distortions occurs on time scales much larger than t.he t3lwtronic relaxation time. the ionic motion tends to generate regions of net positive rharqe ivhich persist even after the instigating electrons have left the vicinity. These regions act as attractive centres for subsequent electrons (see Fig I. I). Electron-electron attraction by this mechanism is tveak enough that thermal Ruct uations obliterate its effect at all but the lotvest temperatures. Only when the system is cooled sufficiently can the ion-mediated attraction appreciably alter the physical properties of the material by favouring the pairing of electrons that would otherwise repel one another. Below some critical temperature. typically T, 5 20 li. the attractive interaction causes the formation of bound electron pairs that condense into a superconducting state whose many- body wavefunction exhibits macroscopic phase coherence. The appearance of this condensate coincides with the opening of a gap at the fernli level in the one-electron density Figure 1.1: Panels A-D illustrate how the retarded ionic response gives rise to an effective attraction between electrons 1 and 2. Due to ionic screening. electron I leaves a .wake' of positive charge which attracts electron 2. anomalous nodstate

ot' brcttt3s. r he uitlt h of ivt~i(-tris qua1 to t hc pair binding energy. The ,itwnt.t~of -t,lttls

,trouncl the G'rr~~iIPVPL IPAVPS the cxlcc-trons nr the cttge of th~fcrmi wa no fItlsrirl;i.tir~!lin scattering events - whence, via fernli's golden rule, the zero resistivity to direct c-urrent th'it I-haractcriz~sa superconductor. \loreover. the phase coherence of the condensate iniparts n riqirlity t. h;~trr\ncicrs t hc ol(~trunic t ranspurt properties insensitive r o the prrvncc of i~npuritiesand ~lethcts.

T!~ispicture of superconductivity wru considered reasonably complct~until t hi ( I'lscovery ot' h~avyfernlion systems ['?I, and later novel copper-oxide supercond~lctingconipo~mds [3]. \v hi(-h t1shihit rharacteristics strikingly [iifferent from conventional suporcond tlctors. For these latter compounds, there is still no definite theory to account for their unusually high transition temperatures (currently 2.5 I< 5 Tc 5 160 li). .\loreover. they comprise a layerpd crystal structure in which a stacked sequence of Cu02 planes is separated by largely inert layers. Experiments have shoivn that this structural anisotropy leads to superconductivity which exists essentially in the CuO? planes [-I,51. That is to say. the relevant physics takes place in two dimensions. The parent compounds of these so-called high-Tc sr~perconductorsare. in fact. insulating aritiferrornagnets. It is the introduction of mobile charge carriers, via cation doping. that causes the magnetic order to give way to superconductivity. .A sketch of a generic cuprate phase diagram is given in Fig. I.'?. .An interesting - but poorly understood - feature of the underdoped cuprates is the presence of a pseudogap in the normal state. Thct pse~idoo,apis not. :t true! qnp it1 the wnse of;ln cinprgy int~rval~vith zero .pet-trnl

*,vfli$ht. h11t rathtlr ;L st11aIl rtyion of ~iin\inistltxI(but no~\-z~ro)iveiqht t-~ntr[ltiilt 1.h~ferrni

!(.\.el. S~ICIIa qap had lonq bwn t.outc.d u the explanation for the 11t111~11iil !oitlper;tt~~r~

~ieptlrr~l~lrrc-clrile tdectricitl c:ori(l~ictivit~,specific heat. and spin susceptibility irl~n~~diat.~ly ;~bo~.clr he ~ript.rconti~~c.tinqt ransition. It h~snow t~tvno hserve(l (iirwrly in phot.oor~lis-

+ion stutti~s[ti, 7j ~ndits (lxi~t~~~cc?(-onfirr~~tl(i b). 111icIclartnaqnetic rt-wnaIrw. ~~~tlrl~llirr<

r~~i,-ros~-opy.~11ti optical mr~tit~rti~~ityl~l~~~illror1~6~r1t~ -.la41. 111 I-OIII~LS~. ivith I~IIVPII~~O~IAI511-

~)c'rcor~~iilc:tors.in ~vhictr the sup~~rronciuc*tingsap tirst. appears at the t.r;111sit.io11.inti crroLv3

(from ,!pro ivicltlr) rvit h 1fct-r~~5irrgr.enlpttrature. t.hc s~perconiiii~tingqap in t. htl hi.;h- 1:.

(-upr;~trs~~vulvra ~rtlout.hly o~~t of ~.IIc 1\0r1na1titt~ PSIIU~O:~~, the ruagt~it~icl~(jf xhich is Inrqely t.pnlperatlircl indeper~da~tis]. ,\ithough tire pseudosap phenornenoo is ~xperinirntallywil-chamt:tcrizFd. a rllmrf?tiral

~~ncierstnndirr

11;s proti~icc?(fa host of cori~petirlgtheories badon several different rncchanisnls - - pairing

ti ~ictuations.prcformetl pairs. spin-charge separat.ion, phase separation into striprs -- but no c!splmat.ion proposed to date is in corrlplete agreement with the e.uperimental rpality. \I'hat these approachcs have in common is a recognitior~that the unique proptlrtiw of tlwtic. r11;tterials iike the high-T, superconducturs can only be understood in tcrrrls of !.he ~:ollcctiw behaviour of the total interacting assembly of electrons. The t.rariit ional Frtrrni liquid (FL) picture of weakly interacting but largely independent Landau qiiasi-particles [Dl

is not sufficient and must be sriperscded by rt new ~lescriptionof thp correlations between t'hc itrongly interacting electrons using the full machinery of statistical mechanics anti rnany- body theory. In our ~vork,a system of ferrnions on a two-dimensional square lattice is studied rising the attractive Hubbard model (.AHhI), the simplest rnodel with the necessary ingredients to give a superconducting transition. The appeal of this approach is that it bypasses the controversial issue of pairing mechanism and allows us to concentrate on the nature of the superconductivity and its onset in a lox-dimensional system. It is hoped that if the phase (liagram of the Hubbard model can be better understood, it might provide insight into the correlated electrons in the cuprate compounds. Yow while it is true that the Hubbard model is little more than a 'caricature' of the un- derlying physical hamiltonian. it has proved to be a useful paradigm for strong correlations: the apparent simplicity of the model belies the complexity of the electronic behaviour that 1.3 Model hamiltonian

1.2.1 Real space representation

..\[I assetrtbly of S = I/:! t'crmions moving on a lattice and interacting via a spin-ir\deper\clc.nt. t ~vo-body potential can be ~fcscribetiby the Ilarniltonian

where Latin letters label sites and Greek letters label spin. t,, is the hopping integral

I~tattv~ensites i and J and c,,,,,,~ is the matrix elen~tlntfor the interaction potential with rclsprct. to \Vannier states lij) and jr'j'). The operator c,, and its adjoint cj, [see Ap- pendix .\.I) respectively destroy and create a particle with spin a at the site i and satisfy the anticornmutation relations

However, the matrix elements t,, and ~,,,I,I are not entirely arbitrary. The hermitivity of H and the requirements of particle indistinguishability and local particle conservation impose certain restrictions on their values (see Appendices -1.2 and .A.6). The simplest possible (non-trivial) model consistent with these restrictions is the one- rncti~l-irlsulatortransitions) in narrow a' and f band materials [L'?]. In that cont~xt.I' > 0 rrprcserlts a locai. screened C'oulomb repulsion between eiect,rons in a t.ight-binding banci.

In l.(Jritrta.st,LVC! are interested in ..;tudying the superconducting instability of an (us~r~ihly ut' rnrttually attrrzcltng electrons. In the real physical system -- the high-'r,- (-uprates - t hr. oriqin of t.he attractive interaction remains controversial. The c-011 piing of t. he electrons r.0 lattice vibrations phor~ons),plasrnons. and spin Auct uations have all be~nsuggested. In qen~ral,hoivewr. we take the point of view that the details of rhe attractive rriechanism

;I~Poutside of the interests addressed in this thesis. .\ccordingly. we take the I' < 0 regime of the Hubbard model as our starting point and endeavour to understand the resultinq btlliaviour of the interacting electrons. By way of justification. we remark that an effective >hurt range electron-electron attraction can result from the elimination of local phonon (131 or internal electronic [Ill degrees of freedom. Let us make our choice of model explicit. Applying Eqs. (1.2.3) to the hamiltonian, the kinetic energy term becomes

where (ij) indicates that each n.n. pair is summed over once: written in terms of the number 1 .?.2 Wavevector representation

Hubbard haniiltonian to its wavevector fort11 by way of a simple Fourier transform of the Rravais lattice. .-I real space. D-(lirnensional lattice is spanned by a complete set of linearly ind~prn~lllr~t I~asisvC1t.t.ors {ii, : r = 1. 2. . . . . D}2nd h.as associated with it a reciprocal lattice g~n~r;ited - - hy r he complete set of basis vectors (b, : -5 = 1.2. . . . . D) satisfying iiT- b, = 2~4,~.The Brillouin zone Li is defined to be the first \C'igner-Seitz cell of the reciprocal latt,ice. i.e.

the volume aljout the origin enclosed by the set of planes that perpendicularly bisect t he reciproc.nl lattice vectors. Bloch's theorern states that there c~istsa subset h' C G ot' Brillouin zone vectors which forms a basis for all functions defined on the real space lattice that satisfy Born-C'on Karmen boundary conditions. \Ve now specialize to the geometrically most simple case. .-\ D-dimensional hypercubic lattice and its reciprocal are characterized by the basis vectors - a', = ae, ,

where (& = (ef , ef. . . . . e,D) : e: = s,,) is the set of standard orthogonal unit vectors. In ;inti t. he corn pleteness relations

-. ivhere .Fl is the position vector of site j and the sums xi are over all k E h'. In the wavevector representation, the kinetic energy part of the total hamiltonian is tiiagonal (see Appendix -4.3):i.e.

where we have introduced the dispersion relation I. Fir1 'The shaded area rrlarks the Brillouin zone G = (-T/~L.-,,'a] i-r;rr. 7.a~ot * - ;I t.\vo-dirnensional square lattice defined by the basis vectors = arl. a? = ~rt?~.Fill~tl circles represent points in the reciprocal lattice. The closed curves represent lrvel JIIIF~LCFS of ic. the {iispersion function.

;ind ilspr~ssedit in t.ernls of the band-width IV = -It D. [Sotice that r: 5 -2t D - 21 k2a2 ivtl~nk = 1 li(l is small. Thus, the positive sign of the hopping integral ensures rhar r hr. &spersion is free-electron-like about its minimum. = 6.1 iirill. the interactiou

part of t. he hsmiltonian. can he re-expressed as the surll of all possitlt: scatterings hcta.wn Bloch states of opposite spin.

[Here. the Kroniker delta imposes momentum conservation.] Thus, the full hamiltonian, Eq. ( 1.2.6), is equivalent to ~icpr~n1lin~on how \vr rhoose to parameterize the final sum. Figure 1.4 provides a si~npic pirtoriai reprw!n~ationof th~interaction. - a ~vritt~nin Eq. ( 1.2.lia). Sot.ire t. hat by replacing t. he (2 surnrnation - which e~tendsover irll possi bl~t-~~lt rr - * of rnas rnornenta of the scattered electron pairs - with Q = 0. we recover the s~-t*niltvi

pairing hnnlrltoniorr used in the original paper of Bardeen. Cooper. and Schrieffer ( BC'S) [I].

r - r his ivas the starting point for their variational treatment of the superconducting ground stare.] What Cooper had shown earlier [16] is that. for a net attractive interaction potential. the fermi sea is unstable to the creation of bound electron pairs. Since a non-zero centre of mass momentum serves to increase the energy of such pairs. he proposed that the superconducting instability is driven by the formation of bound pairs of particles with equal and opposite momenta. The restriction of the pairing hsrniltonian to 0= 6 is an approximation whereby only the terms which are dominant below the phase transition (T 5 TC)are included. The approximation scheme is justified in the sense that the neglected terms do not appreciably ('nlct~lationsalong these lincs, using a separrrbie d-wave interaction potrntial, havcl hocn perforrllcd [I#. 101. Nonet helrss. we believe that such investigations are prematurr since r i) rriany fundamental questions remain unanswered for the far impler +wave case and iii) no reliable QhIC data esists for the non-s-wave symmetries. leaving one no benchmark for judging the validity of theoretical results. Thus, we leave the treatment of anisotropic pairing t.o others.

1.3.3 Spin representation

We shall find it instructive to consider one additional representation for the Hubbard model hamiltonian, one in which its interaction part is expressed in terms of quantum spin operators. Recall that the spin operator at site i is given by ~icpendsonly on the total number of electrons ( .i' = x,xcj it,,,) and the number of doubly ~m.upicdsit~s. C'omparirlg Eqs. i 1.2.21) and ( 1.2..5), wc sec that the interaction t.crrr1 of t.tic harniItonian can be ivrir.ten ;ts

[Sotice that a global rotation of spins leaves the right hand side of Eq. ( 1..'.22) unchanged.] Taking ensemble averages of Eqs. (1.2.21) and (1.2.22). we get ark expression for t.he average double occupancy

and For the average interaction energy 1.3 Exact solutions

\l'itho~lttoo much difficulty, the .-\H\I can be solved exactly in r he iirnit IJ~irnall lattices

,~n,11;lw ~lwtIUILS. ;r\ fOrniaI wluti~nto the problr~mof r.wo ttiectrons of oppositc spin 011 a hyperr~lbic lat tire ( any dimension) has recently been found j lD] .] Such sirnplc ij.st.errls .lrc not physically relevant, but they do reveal many of the important aspects of the physic-s at, work. For roncrcteness. tve examine a svsten~of .V tllectrons on n ring of .\I iit~s.[.\ rin5 is A linear chain in wtlict~ive allow hopping from one t.nd to the other.] To begin. we define a set

rvhere each C E $1is a 'L.\I-tuple of binary coordinates. The set $3 is a generator for the basis B (complete in the canonical ensemble) made up of kets

Since the elements of '31 and B are in 1-1 correspondence. both sets have cardinality (2:!). the number of ways to populate 2JI states (JI sites x 2 spin projections) with .V electrons. With respect. to this tmis, thr harniltonian and the total spin operators have rt1atri.u rrpre-

r - 0 -2 2t 0 0' 000000 0 0 0 0 0 0 020000

-2t o 0 0 O 2t L, 001100 tl = and .SW= . ( 1.3. 1) -2 0 0 0 0 "t 0 0 110 0 O O D 000 00~02~ O O -2t 2t 0 I' 000000 - - 919 h - '3

These can be shown to commute under matrix multiplication, which verifies that fi and .!? ran be simultaneously diagonalized. By conatnlctiol~,the elerrlents of 'B' are stares of definite energy and spin: i.r.. t,h~ysi~tisf? r he t!igcnvalue ~quationsHlor) = Eli(3/)and .!?(ol) = .5'i(SI+ 1 j(oi).TIIP corresponding

I~~~~IIV~JIICS;lrc

where \I- = -It is the band-width in one dimension, and

To make clear the dependence of the solution on the Hubbard model parameters. the spectrum of energy eigenvalues. given by Eq. (1.3.6). is plotted in the top-left graph of Fig. 1..j as a function of the dimensionless quantity

(satisfying 0 5 u 5 I). The value u = 0 (C = 0) corresponds to free electrons and u = 1 (LV = 0) to localized electrons. Yotice that the two states with significant double occupancy Figure 1.5: Plotted above is the energy spectrum (as a function of u = 1l71/(1.1;+ I('\)) for a pair of electrons confined to rings of two (topleft), four (topright),six (middle-left). and eight (middle-right) sites. The bottom graph is the energy spectrum for the infinite linear chain. In each case, the unit energy on the vertical axis is kC' + (L'1. On this sliding energy scale, the unit energy is W on the left-hand side of each graph (u = 0) and IUI on the right-hand side (u = 1).

1Icre .1 = L/T is the inverse temperature. K = H - p.F7 is the grand r-anontrd hnrnlltori~rln. .ind t hc tiegeneracy functiori g( E. .\') is the number of cig~nststeshaving pnergy E iinll total particit. number .\. In going from the second to the third equality in Eq. ( i. 4.2). t lit. t rare ovclr the kcts ((o,,)} been converted into a sum over E and .I', the ~igenvalucsof rhr kcts ~vithrespect to fi and .$. 'The rl~casnredvalue of an observable u is postulated to b~ a \vr.right~davprngp of all possible t3xpectation vslucs, viz.

fhis tlefinition is equivalent to the assnrt~ptionthat the probability of finding the system in

.I qiven state with energy E and particle number .V is proportional to the Boltzrnann factor e-'(E-p.y'. but that otherwise all states are equally likely to be sampled by the system (ergodic hypothesis). Equation ( 1 .-4.3)is usually justified by noting that fluctuations about the ensemble average become small in the t hermodynarnic limit.

It is in this sense that the electron density n = (.?)/.\I and the internal energy Id = (h) have meaning as thermodynamic quantities. Thus. although we have chosen to describe the system in terms of a particular set of thermodynamic variables, others are still meanirlgful as thermodynamic averages. Con- sequently, one can determine the value of any explicit argument of R by specifying its ;dlotvs 11s to write compactly the ensemble average of an operator 0:

l~~~rtt~ermorc.partial derivatives of the statistical weight can be used to construct I he tlcriva- t ivtls 11t' c~rlsc~rriblcaverages. For instance.

provided that the operator d has no explicit p dependence. In particular, the change of \\-e rrtnarli that.. in ~,enerai.all the important physical quarltiti~sticpenti on r hi) PIPI:-

t~llr~~~'l;hto c-apture the (lori~irii~[~tc-orrrlation c~tf~cts,but sinlpl~~1io11qh to hc ~-ori~pl~t;itio~i- ally tractable.

1.5 Symmetries of the hamiltonian

In attmipting to describe the dynamics of the interactins rl~ctrons,it is also irrlport;\l~t to find i\n approximation scheme that preserves the important ..;yrt!rrietrics of the undc>riyir\g, han~iltr)riian.One class of syrnrnetries is characterized by the invariance of the harniltonian mler various spin operations. Ke have already seen in 61.2.3 that ttie hamitonian is invariant under a global rotation of spins. It is also the case that the hamiltonian rcniains

~~nchangedwhen all spins are flipped (f w i).This follows from Eq. ( 1.2.6) by remarking

that [iltt, ictL] = 0 or trivially from Eq. ( 1.2.22). :\ccordi[~gly. away from the symmetry breaking of a magnetic transition, we may take for granted that the system exhibits no not magnetization: i.e.

(1.5. la)

{ 1.5.1b)

Another symmetry is the equivalence of particle and hole excitations about half filling. In order to demonstrate this property, we introduce a modified hamiltonian that is manifestly ;ISits name suggests. The kinetic part of the hamiltonian can be re~vrittenin terms of hole operators

provided that we make a change in sign to the hopping integral t(h) = -1. Likewise, the Lt is clear. though. that Eqs. ( 1.5.") and ( l..'>.

\\'o [.a11 t. his invnrianc~portit-It -hole symnietry.

'I'lic t ronslormation .;ivcu by Eq. ( 1.3.0) is not unique. hoiv~~ver.For ~'sarnpie.a.rl r~liiv ~visht.o esprcss the particle-hoIc symmetry of the harniitonian as a transformation (if t he

(-reation and annihilation operators only. Recall that the sign of t was altered in GI. I l..5.(ji to proservc t he overall sign of the kinetic energy term. Since the summations in Ho are over n.n. sites, a staggered sign change of the operators could equally be used to preserve the sign of Ho. Thus, the hamiltonian is also invariant under the transforniation

tvith the understanding that (- 1)' assigns an opposite sign to adjacent sites. This can be

made esplicit by defining (- 1)' = e-'"'1 where i?= (ala. nla.. . .. ria). In the ivavevector representation, the transformation given by Eq. (l.5.lOI takes the ) A ot' t ictior + = -1< Equation i 1 .i.1 l anlount.s to nothing more than shift hr that. at half filling. there is perf~ctliesting of th~ferrni surface and all anti-equivalent 1)oint.s in the cxtendcd Hriorlllin zone scheme are connected via translation by ? (we Fig. 1,:3\.

cnsenlble nnqf t.hlls ivc rrlust consider the grand canonical harniitonian. \\'P first. remark that

This motivates our introducing a new grand canonical hamiltonian, i 1 ..i.ltj)

(Or 'lrbitrary choice of the operator d. This is the statement that oprntor avcraqm are i t~s~nsitiveto an arbitrary additive constant in the grand canonical haniiltoninn. F'l~rt.hvr. it implies that I;' and fc +C' give rise to identical physics. Second. r1ot.e thirr th~interaction

tc~rrnran also be written 'as Thus. for calculations in the canonical ensemble, where the expectation value of .'+: is a constant value .V representing the total number of electrons in the system. the harniltonians Ho + HI and fro + firh are equal to within a constant and hence are equivalent. However. 1.6 Limiting cases

.As we illustrated in $1.3, the complicated physics of the .-\I111is determined by the c-ornpe- tition between the kinetic energy and Huhbard interaction operators. The relative weight given to each of these terms in the hamiltonian is governed by the ratio ['It. Ideally, one would like to be able to describe the physics of the model over the full range of I-/tvalues. However. before attempting to solve the general problem, it is helpful to gain a better understanding of the two limiting cases of the model. These cases, corresponding to I' = 0 and t = 0. can be solved exactly and give the properties of the system when governed by KO and K~ alone. In what follows. we present these solutions in full. To begin. it is necessary to find an appropriate basis to describe the simultaneous eigenstates of the hamiltonian and the total number operator. Wannier states provide such a basis for the localized eigenstates 1.61 Band limit

In the band limit (1- = Oj, the grand canonical hamiltonian. Eq. (1.6.2b). reduces to

rv hich is diagonal in the wavevector representation. Consequently. the {nFcl) are good quantum numbers with which to describe its eigenstates. The set trace Tr is a surn over ail states generated the elements of 'To w here,zr; t lit. is a sum over the four configurations of the Hc subspacc:

Hence, the partition function is ;~nrlthe corresponding grand potential is

Sou. t,l~i~tLve have an analytic espression for R, all the physical properties of the non- irittlracti~iglattice fernlions can be easily calculated. First. following Eq. ( 1.-l..il, the partial

I ierivat ivtl

allo~vs11s to relate the chemical potential to the electron density. That is.

where we have defined the ferrni function /[I]= (eJx + I) -'. The constant-n contours of Eq. ( 1.6.1 1) are plotted in Fig. 1.6. I ~vith -* rucssur~din units of the band-width). In higher dinlensions it is (lciirimi rrcursivtlly

Isee Appt~ndisA.7). Specifically, g~+,is defined as a particular convolution of gl ii11{1 y1~:

fhe rw~ltingdensity of levels in one. trvo. and t.hree dimensions is plotted in Fiq. 1.7.

Sot*it.(. that the low \-iitrierisionaitieusity of lewls posseisel; ( in trgrnble 1 \iln Huv~*i[rqll- laritivs - at the band edges for D = 1 2nd at the band centre for D = 2. The coni.ointion structure of Eq. (1.6.13) ensures that at each level of recursion. these singularities are pro-

;rest-iwiy integrated out. By D = 3. t.he singularities are reduced to t~vocusps in the tia~lai: in higher dimensions, the density of levels is compietely smooth. .\not her consequence of Eq. ( 1.6.15) is that the order of the powerlaw behaviour at the band edges is a monotonic function of D:

For D = 1. the density of levels diverges at the band edges like I/&: for D = 2. it is flat: for ail D 2 3, it vanishes at least as fast as Ji. This should serve as a demonstration that the reduction of phase space available to the system can radically alter its behaviour. Finally. we note that. in the band limit, the hamiltonian can be completely decoupled Figure 1.: The density of levels is plotted for one (top-left). two (topright), and three dimensions (bottom). In one and two dimensions. the density of levels has infinite Van Hove singularities at the band edges and band centre. respectively. In three dimensions. the density of levels exhibits two small cusps. 1.6 Atomic limit

tfcre. the quantum basis is zenerated by a set '?Ti, identical to Eq. ( 1.3.1 ) rxcept for t ht. rtlst,rict,ion to a fixed nurrlbtlr of electrons (a restriction that we relax in the grand cnnor~ic-al t~~~srrnblr).The ."." elements of this set are in 1-1 correspondence with the sinlultaneol~s t.igenstates of and S.Each element of TI1 generates a kt,given by Eq. (1.3.2).havinq

The grand canonical hamiltonian can be written as

tvhere the operator kIct1c.c. t llr =rand partition function ran be ivritteri in t tie form

lvh~reth~ trace Tr is a surn over all states generated by '?I1and the trace Tr, is s .;urn over t hr folr r ~.onfiguratior~sthat span HI :

Thtls, the grand partition function is \Ye now in a position to t3trnc.t irorn I! the t>lt.ctror~fier1sit.y itrid rhc tiollhlt3 OI,I.II-

[)ilIlqF. firs^.. the ~lectronilensity is qivcn by

S~concl.I. he derivative of the grand partition funct.ion.

This allows 11s to derive an rxpression for the average number of double occupancies.

.A little algebra shows that

Let us now shift the chemical potential by (*/2. as prescribed in $1.5. so that half filling coincides with p = 0. Equation ( 1.6.26) becomes This can bc inverted to ~ivea quadratic cquation in c*jU

the (positive) root of which fixes p as a function of 11:

It is easy to verify that substituting n = I into this equation gives ,u = 0. Equation ( i.R.:l:l] is plotted in Fig. 1.8. Finally, a shift of the chemicai potential by 1-12 turns Eq. (1.6.30) into

Eliminating eJp between Eqs. (1.6.3L)and (1.6.31)gives the double occupancy jntriL) as A function of n. From the plot of this result in Fig. 1.9, we see that when T >> I[-1. thermal effects destroy the onsite correlations between electrons. In this regime. (ntnl) - n2/-l. which is the uncorrelated result of Eq. (1.6.li). As IUI is increased relative to T. the double occupancy approaches (iiTirl) = n/2, which corresponds to the highly correlated state in ivhich all electrons arc paired.

1.6.3 Non-trivial values of Wt

The barid limit describes a system of free electrons in a symmetric band of stat~s. Oc- cupation of these states is limited only by the Pauli exclusion principle and, in 211 l~t~her respects. ihc t3lectrons are independent of one another. In particular. the ~lectronsare frce to transport charge by rnoring frorn site to site. The atomic limit describes a systrrn of electrons that cannot hop between sites. Here. the only redistribution of electrons is hr- tween cash site and the particle reservoir. At high temperatures. the electrons on a given site are uncorrelated: at low temperatures. the electrons form local pairs. In each limit. the problem is solvable because the hamiltonian can be decoupleci into .I1 independent single-wavevector or single-site harniltonians. For non-trivial ratios of ('It (0 < LYjt < P), that is no longer the case and thus an analytic solution is not generally available. To handle these intermediate values of I'/t requires more than interpolating between the two limiting cases since the band and atomic limits each lack the essential competition between fio and fil that gives rise to the interesting physics of the AHSI. For intermediate values of C:/t. we know that the conflict between the itinerant and localized character of the eigenstates of fia and HI can give rise. at low temperatures, to phases exhibiting superconducting or charge density wave (CDW) long range order. neither of

Chapter 2

Formalism of diagrammatic field theory

2.1 Approaching the strongly correlated electron problem

\\'hen ulcctronic corrtllations are sutficiently strong. interacting particles in the normal state (-an become uristable (indicated by a divergence in the associated response function) a~ti ~lnrierqoa transition to a condensed phase exhibiting long range order. Such a transition is c.haracttlrized by a c-hange in the syrtimetry of the %round state of the interacting system.

The transition to a superconducting state is marked by the symmetry breaking ot' t.he particle-conservation invariance of the original hamiltonian ['? I]. Below some critical temper- i~ture.7::. the assembly of mutually attracting ferrnions becomes unstable to the formation of bour~dstate pairs and phase coherence between these pairs sets in. .-Inew

In contrast. all off-diagonal matrix elements vanish in the normal state (i.e. (ctct) = 0) since the ensemble average consists of a weighted trace of operators over states of definite particle number. From the point of view of the Green's function formalism (reviewed in Appendix 8).one expects a superconducting transition to reveal itself in the two-particle properties by the appearance of an instabilit>. in the partirl+partic.lc c-hann~land in the on+particle propcrtip.5 - t,y r hc operlirrq i hrlow T..) of a ;uper(-onducrinq qap of ~vidth 21; 1 k indepprr4lent for .+R aifl iritrrac-tions) in the ~poctralfunction. Son~thcl~ss.to capturp thew f~aturesin '1 thcor~ric-ill

1t1odr.lis a cfrlicatc t 'ask. \l'c. know chat in ortirr to liest-rib properly t hc ph~.si(-alproprrti~s of t he ~ystcrnaround the phase transition. a complete theory of the c-ollec-tive tehavio~~rof t Ire qu;ultuni illrctrons is ahaolut*ly ws~ntialj??!.

Ft>r this r~x50n.mohr tra~litionait~t-hniquw ,in3 of lit1 IP valu~.rh~ in~t~rxrriot~ h~tivrtlr\ the ~I~ctronsc-annot be trusted in a mean-field approuirnation. since rnexn-tield approac-hias

r !pically ignore or inteqratt>OII t precisely those rorrdation etfw ts we ivish to prPservcx. f h is

is nor r he (-at3for wc.,zn-firl~ft hwrics which ir~r.luliewrr~lation effects by first rtlnorrnali~ir~q

he illtvr~entaryc1xc.i tations of t hc system: tl.g,.. R('S is a rriean-field trrat mcnt of the inter- .irtion het\v~enfully forrnrd C'oop~rpairs.] Zor is standard perturbation theory a IISPCIII xpproxinlation strategy. i1.e are intercsteri in the interrricdiate coupling re5irne wher~there is no small parameter in which to expand the p~rturbationseries. Furthermore. BCS sho~ieci

I hat the superconllr~ctinqorder parameter is non-analytic in the interaction itrclnqth. r hus irr~plyingthat perturbation theory with r~spectto the normal ground state of t.he lattic-t. t'clrmions cannot possibly converge to the superconducting 5tate. Instead. what is rrq~~ir~d is an approsinlation that incorporates those WO-particle rorrelatior~swhich are ~iominnnt aro~inrithe phnse transition. In introducing the tIubbard model, we argued that the assumption of an onsite interaction is a useful physical simplifiration. Ipet.despite its highly idealized interaction potential. the Hubbard harniltonian remains non-trivial to solve. Apart from the two limitin2 cases examined in 3 1.6. there are few exact results availabie to us. Of these, the most ~videlycde- brated is the solution in one dimension, provided by Lieb and Wu in 1968 [?3]. [In practice. renormalization group approaches provide a more workable framework for calculations in one dimension [21].] Slore recently, the infinite dimensional case has been treated using a liy namical mean-field theory ('251. In intermediate dimensions (1 < D < m), however. we must resort to non-exact techniques. 'These have included slave bosons, solutions in infinite dimensions (it has been found that D + x calculations do a reasonably good job of describing the physical properties of the model in spatial dimensions as low as three), 1/N expansions, and various diagrammatic schemes. What is problematic is that none of these approaches has been made to work reliably in two dimensions. To be fair. there has been some highly successful work in ~LVO~iirnensions (esp. for pseudogap iriquiries [26. 27]).t)ut it ha generally r~((uir~11iL r10n- ,ierivxble ansatz for a renormalization of the interact.ion strength. inf~rrcdfrom QlIC ~lata. or t hr introduction of arbitrary parameters ivhich are then fisect by enforcin5 wrt ain sun1 r~llrj.D~ie to tlicir ;id hoc flavour. t.hcsr approaches are not cntird~.satisfac*tort. Iltrla!ly. w \r.ould like to haw an approsimat.ior~sc:hcrne t.hat can be derived rigorously a1111 irnpro\*rll i>*st~nmticall_v.

Onp wtit~nit~t. hat. nlccts these (-rit~riais t he .;n-~-nllt.~iclc!~~a,tinn of mot ill11 i b:O\I I rlr~~110Ii. It is Ilcrivc~iby ~.c~nsirl~ringt.he cliffrrer~tialequations that govorn t.he rriany-holly (;rrvr~'.i functions. The n~cthociis ~iysternaticin the sense [.hat it can be made to inclridc wrr~~lation r~ffect.~up t.o any liesired order.

In t.his ct1aptc.r. 1c.c outline how functional Jifferentiation u'i t h respect to a tic-titioils c.stt.rnai field ran hc uwd to gcnprate the rxact diagrammatic ~xpansionof t hc on+ ;11111 t.1r.o-particle propagators for the attractive Hu bbarci model. Then. following tiadanoff and llnrtin [2Y]. ivc employ the EOhI method to select an infinite subset of these Iliagrams for rcsum~t~ation.The dilute limit of the EOlI res:!lt. in which we neqlect all partic-le-hok ( ph) interactions, yields t.he so-called T-matrix t hcory in which only direct particle-particle 1 pp) icattering events (all orders) are retained. [.-Ifull derivation of the functional diff~rent.iation quations presented in $2.2 can be found in .-ippendix C. The det,ails of the EOhI niothod prrs~ntedin $2.3 can be found in Appendix D.]

2.2 Fu~lctionaldifferentiation

Throuqhout the next several sections. we shall employ a shorthand in which a single numerical index (m) is used to denote the pair of space-time coordinates (j,, r,,). In this notation. c,( rn) depicts a Heisen berg annihilation operator parameterized by an imaginary time coordinate

which we interpret as an operator that destroys a particle of spin a at site j, at (imaginary) time rm. Response functions of the system can be constructed from ensemble averages of products of such operators [29]. The time ordering operator T rearranges its (imaginary-time-parameterized) arguments [Zotice that we have (letined Eqs. (2.2.1)and (2.2.3) with half 'as n~anyspin intiices .is spare-rime coordinates. The extra indices are supertlrio~rsbecause the Hubbard mock1

Iiarniltonian has no terms capable of Hipping spins (like. fa.<.. fljci). Thus. t hr individual t ipin states of particles arp preserved: i.e. (c,ci,) = \c,c\)h ,,ldJjl and r j =

r Jc,jc,,,JJ.]t t\ The utility of these functions is t bat any response funrtiorl car1 ho c-onstruc'tcd from ,z Green's function taken in the appropriate limit. For instance, t ii-o IW~III st;ttir' prop~rtiesare the ~lectrondensity

and the double occupancy

[In arriving at Eqs. (2.2.4) and (2'2.5). we have made use of the spatial and temporal homogeneity of the system and exploited the fact that the time ordering operator will order operators that differ even by infinitesimal times: i.e. (4,)= (&). (.-i(r))= (.-i(rl)).and

T, [.i(r)B(T+ )] = - ~(r).i(r)= - (B.4)(7). We have also simplified the Creen's function notation by taking advantage of the spin symmetry described by Eqs. (1.3.1).] Lnfortunately, only for very simple hamiltonians can we salve for the Green's functions exactly and so. in most cases, we must resort to approximate solution methods. One of the more common techniques is to perform a diagrammatic expansion. The most familiar of :-'J:;~~I!c~Pc! a L:;~i~ii!~~i~Li~iCt.F~ y,i-i!lt:~?~~it! dii :.bit ~!~i,~~~~i?df"iiij cjf :./it? !i,iiliiii,c,iii,Lii iiibti rhc choico of i{'. For the Hubbard modrl. the two obvious 4-lioiccs arp an ~sp;insioriahur h~ banci limit in tern~sof c;lr.,cl or an expansion about the atonlic limit in tprrns of C;it=o- Rcsp~ctively.these are perturbation series ivith (-it and tjl- as small parameters and io rcquire either weak (I'/t << I) or strong ilv/t >> 1) coupling. [n the regime whprp the int.t>ract.ionstrength is comparable to ihe band-width. we do [lor cxpect 11ch nn approach to bcl of rtiuch use. Ho\vever. even if the dii~grarnrnaticjeries is not abruptly terminated and ;rt least some lliagrartls are kept to all orders. the Feynnian series is still perturbativr in thc sense that

it rnt.ails ~spnndinsthe Gr~en'afunction G in terms of t.he Grwn's f~in(-t.iorl of om^ si~nplersystern. It is not clear that such an expansion ivill always converqe to the mrrwt f11nc:tion. since the basis function C;" may not have the same symmetry ;as the true fllnct.ion. i; . .I second. more robust diagrammatic scheme is the skeleton expansion i nt ro(iuccd by Luttinger [N).LVnder this scheme. following Dyson. the difference between the full C; and ilw free G"' z is encapsulated in a function S. called the proper self-energ. The self-energy is then expanded order by order in the jull Green's function and the interaction strength. The appeal of this approach is that the expansion makes no reference to the Green's function of any other hamiltonian. Its main drawback is that expanding the Green's function in terms of itself introduces a complicated feed-back loop and thus the resulting integral equations must be solved for self-consistently. In what follows, we execute the skeleton expansion for the Hubbard model using a functional differentiation approach inspired by Baym and Kadanoff [31].

By introducing a spin-dependent. non-local external field. P:( I. 1'). we can perform a perturbation analysis of the full oneparticle propagator (with respect to vex). To second order in ;he rsternal field strcnqth. tve hair,

n hich ive ]lave chosen to ~vritein t.ernls of the two-particle correlation function (or t~vo-

know the one-particle propagator in the pres- tl!lr:t? uf a non-local external held. tve can extract the tw-particle properties by functional differentiation. [The t hree-particle properties follow From differentiation of the t~vo-particle propagator. and so on.] To make use of Dyson's equation, which in the presence of an external field takes the for rn

ive first need the generalization of the identity (d/dr)(l/ f (x))= - l//(x)'d/(r)jdx to the functional differentiation setting. That result. ivhich can be ~irnplified 115ingEq. 2.2.) to zive nn quation rr!ating the t~vo-partirlc.

Eq~i:~tinni 2.2.12) is a %enera1 rcr;ult. independent of the harniitoriinn 11ncler c-ur~sidclr-

.ition. I'o prc,clccti \vit tl the .;kcleton espansion, we now need n relationship botivwrl r hc r-orr~latorand the self-enerqy that. is specific r.0 the IIubbard model. Such a. rdarionship is ~ierivdin 32.3 ~vherewe show that

Toget her. Eqs. (2.2.12) and (2.2.13) completely determine the self-enerqy: stsrtinq wit il t hc. first order (-ontrihution, X( 1; It) = I-G(1: 1+)b( 1. 1'). each functional derivativf. of X in Eq. (2.2.12) geuerates new, higher-order correlations which feed back into the self-energy

through Eq. (2.2.13). The resulting diagrammatic expansions for S. C'+-. and (-Ic+ are siven in Figs. 2.1 - 2.3. .-\ consequence of the onsitc nature of the Hubbard interaction and the Pauli csclusiun principle is that bare interactions may only occur between particles of opposite spin. Soticc r hat in Fig. 2.2 the two main legs of C+-. having opposite spin. are free to interact directly. .-kcordingly, diagrams which exhibit a ladder-like structure tend to predominate. On the other hand, the two main legs of C++ (see Fig. 2.3) interact only indirectly via ph fluctuations. Since X is determined by C'+-. we find that many of the self-energy contributions have the form of a single particle interacting with the fermi sea via repeated scattering events. The expansion given in Fig. 2.1 contains a countably infinite number of diagrams. Since each self-energy diagram contributes a term to the integral equation that defines the one- particle propagator, it is impossible to proceed with a calculation unless the number of terms Figure 2.1: The skeleton expansion of the Hubbard model self-energy. Z(1: It), is depicted diagrammatically to fourth order in I.. Here, each solid line represents a full propagator. G(m.m') say, with its arrow indicating the direction of motion from the rn' to the m coordinate. The dashed line represents the Hubbard interaction. The upper and lower external legs on each diagram correspond to 1 and It. respectively. The coordinates of all internal vertices are contracted. Figure 2.2: The skeleton expansion of the opposite-spin two-particle correlator. C+-(12: 1'2'). to third order in Li. --a::

Figure 2.3: This series of diagrams gives the skeleton expansion of the equal-spin tivo- particle correlator. C++(l.L:1'2'1, to third order in L'. To avoid unnecessary duplication. we have employed the convention that each diagram is understood to represent both itself, as drawn. and the diagram with its two bottom legs crossed. Put another way. for each diagram above. there is another which is not shown that is identical but for the transposition of its 1' and 2' coordinates. This diagram doubling effect is a consequence of the indistinguishabiii ty of particles and thus it does not arise for C+- ( 12: 1'2') whose two particles are distinguished by their opposite spin projections. 2.3 Equation of motion method

I'he time rate of change of any Heisrnherg operator is proportional to its commlltator wit.h rhr %rand canonical hamiltonian. The EO\I rnetho(i is ii 5eneralization of this principle to r he rnany-body C; rcen's function. The E011 of the n-particle C; reenasfunction. G',. tiefined

-9 is a diffrrential equation relating G, to (.,,+I and C,-1. For the Hubbard model, it rt)arls

= ('G' ,,,,,n. ;,(I..'. . . .. n. 1; I'..?'. . . .. n'. 1') +d(l. 1')~~,,.. ;,,(2.:1.. ... 1~:2'.3',. . ,. 11')

At the one- and two-particle level. Eq. (2.3.2)specializes to Thc set of all such cquations of motion (rl = 1. 2. ;3. . . . I describes an iniiniicl hit~rarchy~jt' tw~~plmiequations. The intractable problem of solving t.h~scquarions sirr~~~ltaneousl>.

()niy i,tl ~l~nlill co~np~t tatiomiiy kaibie h~ trrrnlnntlns t tw hierarchy ot clc\llar\ons ;LI cur~w l~v~l.-Phis ran be acmn~plishedin n sysiematic way [32] by ~vritingf:ach G,, ir~rrlrrris of

r htl one-particle propaqitor G and the r-particle trorrplators C', (r = 1.2. . . . . rr ) an(! then I-hoosinq t,o n~.;lect correlations beyond some given order: i.e. by setting C', = O Yr > r.,,,:,,.

For t~saniple.accorriinq to Eq. (2.2.3).the trvo-particle Green's function is the -1irr1 of ;dl possible processes (numborin5 tivo) in which t~voparticles t.ravel independently frorn rtw space-time points 1' and 2' to the points 1 and 2, plus t.hose procesws in which the pair's r~lotionis t-orr~lxtcd.Likewise. the t hrecparticle Green's function is t tie 5ufn of ;ill poasiblc prowsses i numbering six) in which thre~partic1t.s travel independently from points 1'. 2'. 2nd :1' to points 1. 2. and 3. plus all those (numbering nine) in ~vhichone particle travels independently ivhile the motion of the remaining pair is correlated. plus those in

LV hich thr! motion of all three particles is correlated. Figure 2.4 depicts these situat.iorls (iinqrarnmatirally. The decomposition of the Green's function into independent and correlated propagating components can be carried out to any order. must keep in mind, however. that the number of terms in the decomposition of G', grows very quickly with n. Therefore. in order to rninirnize the computational burden, we want to keep the maximum order of the included correlations as low as possible while still retaining sufficient correlation effects to capture the physics of the phase transition under consideration. The lesson from BCS is that the superconducting instability is the result of strong correlations between pairs of electrons. Accordingly. we choose to include pairwise correlations and to neglect correlations between three or more particles (r,,, = 2). The first step is to rewrite the equations of motion in terms of the two-particle correlation Functions. Putting Eq. (2.2.8) into Eq. (2.3.3) yields Figure 2.4: The two- and three-particle Green's functions (hollow ovals) can he writtcn ~rniqutllyin terms of independent propagating lines and two- and threcparticlc correlation funct.ions (shaded).

Figure 2.5: Throughout the text, three kinds of propagating lines are used: non-interacting (Go),Hartree-Fock (G~~),and fully interacting (G). They are distinguished in diagrams by their arrow. The dashed line represents the bare Hubbard interaction (V). :fl\is irnplirs that the IIF approximation is a mean-field theory. iinre C; = c;'"- I-orr~spor~ds to C'+- = 0.1 The new EOll for the single particle Green's function. Eq. ('?.:I.($). ran 1 hen be integrated t.o sive

[Here, integration of the llifferential equation is equivalent to the application of the lincar functional c;''~to bot,h ides of Eq. (2.3.R): see Appendix D.I.] [n thr second linp ivp hav~ rrlacfe the identification

in order to cast Eq. ('2.3.9) a Dyson's equation. Notice that the definition of Y wed here differs slightly from that used in 32.2. Since the Hartree bubble diagram has already been absorbed into G"~,the function 5 contains only those proper self-energy corrections beyond first order in C:. At this point we have not yet made any approximations. However, when we go to rewrite Eq. (2.3.4), the two-particle EOhI. in terms of correlation functions, we shall terminate the hierarchy of equations, as discussed above, by neglecting three-particle correlations. The resulting differential equations (now approximate) describe the correlations between

[.'ie;ilrc 2.7: *rtie equation of niotion result for C'+- (shaded),calculated hy neglcct.ing t hrw- parbicle correlations.

Figure 2.3: 't'hc correlator C'+- (shaded) can be representeli in trrnls of a gencralizd two-particle interaction (lvavy line1 that acts between particles of opposite spin. ducirr~a gerieralized interaction r that acts between particles of opposite spin. \Ye take

to be its defining equation (see Fig. 2.8). Substituting this into Eq. (2.3.13) and factorirq out the external propagating lines leaves an integral equation for T:

The homogeneous term on the right-hand side of Eq. (2.3.15) is simply the bare Hubbard interaction. The four remaining terms represent various higher order correlation effects. 2.9: The generalized ttvo-partic.fr interaction i wavy line) is defined bv an irit.tlgrnl quation wtlose ~liagran~maticrepresentation is given here. The first tern1 on the right-hand .side is thc Hubbard interaction. The remaining four terms are a particlehole lacider rlllrg. ;I particle-particle laiitier rung, a vprtex correction, and a particle-hole bubble pair.

Since these wrms are expressed in terms of r itself. Eq. ('2.3.1.4) can be viewed as a recurrcnc-c rdation for r. Repeated ;tpplication of the recursion to orle of these terms done would qcn~ratea series of ph ladders, pp ladders. rer~ornlalizedvert.ices. and ph bubble catlains. rrspectively (see Fig. 2.9). This makes it cIear that the r,,, = 2 EOlI result accounts for mrrclations arising from ph and pp scattering events and density fluctuations. Finally, rather than determining the two-particle correlations by solving the intyral t~quatioafor C'+-, viz. Eq. (2.3.13). we could ~quallychoose to solve Eq. (2.3.1-5). I'his. of cwrlrsc, requires an expression for I: in terms of r in order to execute the self-consistency loop. Such an expression is casily obtained. Putting Eq. (2.3.14) into Eq. (2.3.10) arid factoring out the external lines gives the self-energy expressed in terms of T:

The diagrammatic structure of this equation, shown in Fig. 2.10. is easily understood by comparing Figs. 2.6 and 2.8.

2.4 Fourier representation

htil now, we have worked exclusively in real space and imaginary time, in large part because the formulation of the equations of motion is most convenient in such coordinates. From this point on. however. we shall work in Fourier transformed wavevector and frequency coordinates. The advantage to such a switch is twofold. (i) A wavevector representation Fiqure 2.10: The self-energy ( \') can be exprwed in terms of the %eneralizfxi iot~rrtction ( ivavv line l . is more natural for the stu(ly of superconductivity since Cooper pairins o(*rursin rwipro- t-a1 space. In particular. it allows us to access the zero-centreof-mass rornponrnt of t hr pair correlations ,uthe system approaches the superconducting transition. (ii) .A Fot1rir.r rr;lnslorni .iirnplifirs caic~~lar.iorlstmxd on Eq. (2.3.15) by i!liminating all intryrations over irtlaginary time (ivhich arc irrlplicit in the contracted variables). il'e have already described in $1.2 how physical quantities that are periodic lunct,ioris of spatial position can be transformed to their wavevector represeritation bv introducing Rloch annihilation and creation operators (given in Eqs. (1.2.10)). In essence, this prow- {lurecor~stitutes a D-dimensional Fourier series expansion in a plane-ivav~basis set. I'der this transformation, sums over the sites of the lattice are converted into sums over ;i. set K of basis wavevectors in the Brillouin zone. Similarly, functions of an imaginary timp r:oor- flinate admit s one-dimensional Fourier series expansion with frequency-dependent Fourier components. Here, integrations over time are transformed into sums over a discrcte iet of hlatsubara frequencies. These frequencies come in fermionic and bosonic varieties. In the Green's function formalism, a one-particle object which is fully Fourier trans- fornlod will depend on a single (D+ 1)-dimensional coordinate

k = (kl,k'. . . ., kD. kD+') = (L,) where the first D components are those of a wavevector and the final component is a llatsubara frequency. For example. the one-particle propagator and its self-energy can be represented by - where each 1,is ~lnderstoodto irlciud~a sun\ over all k E K and a sum ovrr .dl fc3rrlii LIatsubara frequencies. {--, = ('ln - 117; .j : n i Z}.

j~ini-tion~vith Eq. (2.3.tj). implies that the LIF propagator has n Fourier reprc~sentation

[Followins $1.3. we have shifted p by (+/2in the last line of Eq. (2.4.4)in order to ohtain thr

ph symmetric result .] :Is rr convenience. the frequency-independent Hartree shift. I - ( n - 1 )/2. is often absorbed into the chemical potential. That is. the oneparticle propagator can be ivritten

ivherr = /L - I-(n- 1 )/:! and ,C; is the single-particle energy measured ivith respect to i. [.-Icliscussion of the Hartree term and its effect will be put off until $3.6. L'ntil then. we shall simply ignore the issue by allowing to serve as the chemical potential.] In turn, the Fourier representation of the full Green's function is given in terms of ~~~(k)and X(k)by the transformed Dyson's equation

The two-particle Green's function is represented in terms of the three-index object iv here 2k is the rplative energy-mon~eoturof the injected partir:lc pair. Q is t.he t>nt3rq>,- r~iumc~ltlirn of t hr i-r3ntre of mass. all11 q is the ~r~~rqy-ni~nl~nt~~r~~t ransf~rb~r~v~tw IIIV

!)artirl~s.T~P fl - i I'~cornp~~n~~ri tq of Q ;in11 q ,I~Phw~ \la t~itham frv7~~~u,-i~q{ = 2Kf'A: IZ e 2). Tlie t~vo-particleGreen's hlnrrion can be evaluated in various limits to giv~rcsponsr f'unctions of the system. Two particular limits j*ield functions that rncl'asurp the propn- $ation of particle pairs. viz. GI 1 I: 1'~').and the propasation of density iluctuations. viz.

( 1:1 Thpw haw Fouri~rrepresentations in terms of t h~ funrtions C;Pk'iQ) .in11

Gphlq) ir. bich are ~lcriwdfrom C;(kQq) by eliminating t hr k. y and k. Q deqrees of freedom. rt~spectivdy.That is.

where

One helpful feature of the Fourier transform is that it leaves diagrams unchanged. Only our convention for interpreting them need be altered. Rather than envision a line as propagating from a primed to an unprimed space-time coordinate, we picture it as carrying a certain energy-momentum (see Fig. 2.11). Each propagating line is assigned a fermionic i111lesand each intcract.ior1 line is 'tssigned a bosorlic index in such a bvay that c:onserixt.ion of rrlon~entunlarid r,ner

I-inally. any physical quantity of interest that can be tlstracted from a (:;reen's funcrior~ ran also be ext ract.ed from its Fourier rcpres~ntation. Equation (2.2.4) sives t. he eitxt ron ~lensity

ant1 Eq. (2.2.5)gives the double occupancy

Alternatively. we can use Eqs. (2.2.$)and (2.3.10)to express the double occupancy entircly in terms of one-particle quantities:

Here, the contracted variable has been eliminated by using the identity e("k')3 = 3.\.fb(k - A!). 2.5 T-matrix approximat ion

Equation (2.5.1) is derived simply by reinterpreting the diagrams of Fig. 2.9 in t.errtls of Fourier transformed variables according to the rules given in $2.4. In the same wq.thc Fourier rcprcsentations of Eqs. ('2.3.1-1) and (2.3.16) are obtained by appropriate relabelling of Figs. 2.S and 2.10. .As an example of this process. we illustrate in Fig. 2.12 how the iliagrammatic representation of the self-energy can be labelled with space-time coordinates to give Eq. (2.3.16) or with wavevector-frequency coordinates to give

At this point, what remains is to solve Eq. (2.5.1),a very difficult, self-consistent equation for the three-index object r(kQq). By neglecting three-particle correlations in the EOXI calculation, we have derived a generalized two-particle interaction which contains only a small subset of the complete set of two-particle interaction diagrams. Yet. even after this considerable simplification, the resulting equation, viz. Eq. (2.5.1),still cannot Figure 2.13: Every diagram is constructed from smaller diagrammatic elements that are flither p;trt,iclehole or particle-particle in character. The t.hree fundamental rlernrrits ;Lrfl t he particle-hole hu bble. the particle-hole \adder rung. and the particle-particle laddrr runq.

he solved in full (ils a matter of practicality). However. while a general solution is out of rmch, ccrtairi special czes are accessible. For example, we can evaluate r(kQq) in tlic dilute iirriit. n(2 - n) + 0+. where it loses its k and q dependence. Every diagram can be decomposed into fundamental units consisting of a pair of bare interactions and a pair of propagating lines. These four constituent parts can be combined in three distinct ways to form the diagrammatic elements shown in Fig. 2.13. These elements are classified as ph or pp according to ~vhethertheir propagating lines are oppositely or similarly directed. When the density of either particles or holes goes to zero. the relative weight of ph scattering events to pp events becomes vanishingly small (see the discussion in Ref. ['?9]). Thus, on the right-hand side of the equality in Fig. 2.9, only the homogeneous and the pp rung terms survive (corresponding to the first and third terms on the right-hand side of Eq. (2.5.1)).The resulting recurrence relation specifies an infinite series of ladder diagrams, as shown in Fig. 2.14. A by-product of this simplification is that the generalized two-particle interaction be- Figure 2.1.1: Ladder diagrams represent multiple particle- particle iratt~rinsFvrnts.

The rrsult. c.omnionly referred to as the T-matrix, is a renormalized interaction of rar~rlorri phase approximation ( RP:\)-like form written in terms of the ( mired) pair susceptibility

\Ve can now determine the form of the self-energy in the dilute limit by substituting T(Q)for r(kQq) in Eq. ('2.5.2). LVe find that Thtl T-matrix iipprosimation is a system of coupled equations in which the one illld twm-particle properties must be solved self-consistently. .is a first step, howei~er.let 11s isnore the self-consistency process entirely. \Ye can accomplish this by replacing all the full propasating lines in Eqs. (2.5.4)and (2.5.3) with HF lines. This gives a self-energy

where the pair propagator Gy' (Q)= (Q)/( 1 + (Q))is now written in terms of the the non-self-consistent (NSC) pair susceptibility

For every G in Eqs. (2.5.4) and (2.3.5) that has been approximated by a G~~,there is that. hu been neglcctrd. Equatiorl (2.;. 11) tells us that the tlifkrence G - G'"' is propor-

tional to the self-energy and that, t heretore. rhe NSC' approsirilation is justifiec] ~vtlrnt~v+~r

t hc wlf-cnttrsy is not too Inrse.

The nppenl of I IIP NSC' approsir~iationis that by writing bls. (?.>.!I) and (?..i.10) in

t crms ot' hart1 propagators done. [vr can r~lakec-onsidernble progrfass nnaI>.tic;dly. Sinw

c;"~ 11% ;I lil~o~vnanalytical form. r he frequency summations in thcsc cquations ran be perfornlcd using the extended occupation number formalism of Appendix B.2 (or. wi til

%reatt3rI tifFicrllty. using the more traditional contour integral approach ).

2.6 Transition to the superconducting state

There are ruo distinct approaches that one can take in order to study the superconducting [.ransition. The tirst is to start in the superconducting state and to heat the system unt.il thermal effects break up the phase coherent Cooper pairs. The second is to start in the metallic state and to cool the system until it becomes unstable to Cooper pair formation. That is to say. one can start from T < T, and pass through the t,ransition from h~lowor start from T > T, and pass through the transition from above. Since we are primarily interested in the properties of the anomalous normal state from ~vtlich t.he superconducting state emerges. for our purposes the latter approach is clearly superior. It allows us to observe possible precursor effects (such as pseudogap formation) en route to the superconducting instability. Note that although we have only now stated our preferred direction of approach, this choice of direction has been implicit throughout the formal developments of this chapter. The EO%I method is inherently a normal state calculation since, unlike BCS, the behaviour it describes is smoothly connected to that of a non-interacting assembly of lattice fermions by the parameter l' (i.e. as Lr + 0. G + @ smoothly). Its dilute limit. the T-mat rix approximation. describes the electrons in the normal state as quasi-particlelike entities dressed with pairing fluctuations. In this thesis. we employ the T-matrix approximation to describe the normal state prop erties of the system near the superconducting transition where such pairing fluctuations are potentially Inrqe. Below r he t.rnnsitior~tprnperaturr, the T-mat rix a.pproximation rrlust I'iiii. since it cannot, possibly capturf! the svninietry i-hange of tllc ground state rts the system p.uses into the superconducting p lase ( unless ire introduce anornslous propaqators [?$I 1.

Ilit.,ht at t hr transition. NT expect t.he break-ciotvn of t hc T-matrix ta bc i5nailccl by thc Ifiverqcncc of the pair correlations The sourre of this divcryncc is the Cooper instability

11f t hc> rons to pair t'orrtlat ion. .\ccorciingly, for the purpose of rnnppinq out. t.he super-

is the ivave\.ector representation of the pair annihilation operator A, = I .C,~C,~.Cfowev~r. t.o better illustrate the nature of the pairing instability, it is customary to introduce the corresponding real-time response function. Hence. we define the retarded pair propagator

i(,-zG~~,~+- (1,. t:, 1 , t') = ([&(it), .+tl)])qt - t') .

This function gives the real-time response to a pair of particles injected into the system at site j' and time ti and removed from site j at a later time t. In the normal state. a pair of particles localized on a single site is not an eigenstate of the system. Thus, as the injected pair interacts with other particles in its environment. it loses its localized character and its internal coherence as a composite particle pair. The response typically falls off exponentially in both space and time. C'H.4 PTER 2. FOR.iI.4 LISAI OF DI.4GR.4~1~I.4TIC'FIELD THEOR\'

This observation can be ~spressedas

ivhich wc have ti ritten ir~terrtis of the pair correlation lenqt h ,t,-,,,, 'ind the c-linractt~ristil- relaxation time t,,1,,. These t~voquantities give the average len%thover which particl~pairs

.trc corrrlatrd an(l the aLvPrnge t ime-scale for the (1~ci-i~of a single pair. 011 apprmch to ,t

phase transition marke(i by the forrrlation of phae croher~ntho~irlci pairs. ivp rspcc-t rliiir t lit>

pair c-orrrlations tvill h~c-omelong ranged and the pairs thenlselv~slong lived: i.tl. <,.,,,, -- s -.pp.R wid tr.r.,x x. leaving G,- (1,t: j'. t'i a (-onstant. .Is ive ~liscusseclin $2.1, an analysis of the pair correlations usually pro(-ceds in tt>rrtls of Fo~lriertransformed variabl~s.The retarded pair propagator of Eq. (2.6.3)can t~ repre wnted in terms of its Fourier transform:

[in an effort to curb the proliferation of superscripts and subscripts, we shall drop r hr pp and +- lnbcls on the pair propagator for the remainder of this chapter.] Noticr that. ;~ccordingto Eq. (2.6.5).c;~( j. t:1'. I') is a constant, independent of its position and rime wordinates. only if its transform behaves like ~~(0.J)= n'q,ab(;). This suggests that we

can track the onset of s~~perconductivityby watching ~~(0= 6.d = 0) i r as T -+ X- from above. If we let C;(Q.I) be the analytic continuation of ~~(0.d)to the coniplex plane then tve can convert the frequency integral in Eq. ('2.6.5) into a complex integral over one of the contours given in Fig. 2.15 (the upper contour when t < t' and the lower contour when 1 > ti):

.\ccording to the Residue Theorem [R3], Eq. (2.6.6) is given by the sum of the residues of the poles of e-"(t-i')~(o, z) in U when t < t' and by the sum of the residues of its poles in RUL when t > t'. ?low consider the contribution to the retarded pair propagator made by a single. simple Figure 2.15: Two integration contours are shown above. The contour on the left is c-ountctr- t-lock~visein direction whereas the contour on the right is taken to be rlocliivise. The straight line portion of each contour lies infinitesimally above the real axis. The radius of each semi-circular arc is understood to be larg~enough to encompass all the poles of the integrand.

pole at, :o. Provided that 20 is not in the upper haif plane (i.~.z,, E W u L),r he Kt~siifut~ T heorern yields

This result is entirely consistent with the Heaviside factor in the definition of the retarded pair propagator. hloreover, since Irn zo 5 0, the contribution it, makes to the pair propagator either remains constant in magnitude or decays exponentially in time (for t > t'). which is consistent with the behaviour described by Eq. (2.6.4).

However, if the pole moves into the upper half plane (i.e. 20 E W), then

This result violates causality since it gives a response before the initial injection of the particle pair at time t'. Furthermore, it represents a contribution to the retarded pair

propagator that grows without bound as t goes from -a to t' (since Im 20 > 0). The appearanct. of a rnocie I hat increases mponent idly ivit h t.irne indicates that the sy.;t.t?n~tias 1)t~conieunstable.

For the 5SC T-matrix approximation (in which feed-hack effects arc nttgiccted). I,np (-nu ricnionstrate that the s~~p~rconciucrtiriginstability appears according to th~foilo~vin%

t h~ origin btlgiri to rnove t.ou.ard one another. Thev t>vr!ntually coalesce. at a tcrnpcrature ive identifv as E.. illto ;I single pole at the orisin. t.hus givirlq (>I():.0) = (;'10.01 = x. ,\s the t.eniptlrnturp is lowrcci f~irrhcrstill. !.he poles split apart gain. hut now inove otf rt\e rral axis to conjugate points in r he lipper and loiver half c-ornplex planes. Ttl~polc

in the upper plane represents an unstable mode ~vhichexhibits the exponential form ot' Eq. (2.6.3). [:I full discussion OF this process is given by .-\mbegaokar in Ref. i:N].Sote that the above ciisct~ssion is relevant, to a finite lattice: for t.he complications t.hat. arise in the r tierrnodynamic limit. see Ref. [:15].] In 1960. Thouless rigorously demonstrated that this pairing instability actually L-orresponds to a phase transition by shoiving that the specific heat begins to diverge as T approaches T,- [XI. It is convenient that the instability manifests itself at zero frequency. This is the one

point where t htt three Green's functions

I re. [The origin is the one r:omnlon point between real- and imaginary-axis val- culations.] LVhat this tells 11s is- that we can look for the instability by using either cR(Q= G.J = 0) i s or G(Q = 8. v, = 0) -, x: as our criterion. This allows us to avoid real frequency calculations and to work exclusively in the temperature formalism. Chapter 3

Analysis of the T-matrix approximation

3.1 Overview

Our analysis in this chapter is motivated by the failure of FL theorv to account for thr unusual norrrlal state properties of the high-T.. nlaterials. There are three main ~,oals. [i) By studying the T-matrix approximation of the .AHlI. we hope to better understand t he connection between reduced dimensionality anti the breakdown of the FL description. In particular, we discuss how the formation of bound states below the band in the YSC' T-matrix approximation changes the physics of the problem at low temperatures. [ii) \VP want to know whether the T-matrix approximation can account for the observed pseudogap physics. To this end, we perform severaI NSC caiculations of the DOS for temperatures above Tc and comment on the depletion of spectral weight at the iermi level. (iii) L\*e would like to determine what effect the enforcement of self-consistency in the T-matrix equations has on the results of (i) and (ii). In Ref. [37]. Schmitt-Rink. Varma. and Ruckenstein (SVR) proposed that any effective attraction between particles in two dimensions destroys the fermi surface at low temperatures and thus gives rises to non-FL physics. The mechanism they described involves the appearance of an excluded region in the low-temperature phase diagram (due to the divergence of the pp scattering matrix) which directs constant density contours of the system toward bound states at the bottom of the two-particle continuum. As it turns out, the NSC T-matrix approximation of the AHh1 constitutes the discrete 67 lattice analoque of the 51-R phenorncnoloqy. However. we tind that the SSC T-rr~iirri~ 1c.ads to ccrtain patt~oioi$es ~vhcnthe available pt~~asespacc is miall. mnlething 5I.R ,lit1 not address in their paper. \Ve argue that. for loiv-dirncnsional systems, the ~nforrrenlent of self-consist~ncyis requircrl to correct the ~lnphysicalfeatures of the XSC' approairnation.

.\[I important qurstion that arks is whether the S1.R result is robust or rnercly nn 11rt.ifac.r id neglcctinl: wlf-c-onsist~ncy,

C'nfort unatelv. t-here is i:onsid~rablt!tlcbate in t.he literature as to which of I hrl rrlany possible vorsions of self-consistent>, ought to he cnt'orced. The source of the ciisaqrcernent is that varioiis theoretical approaches lead to 'r-mat riu-like self-energies with differ~ritconfiq- urations of bare and dressed lines. The reiative merit of each configuration is still debatd mil there is no agreement yet on which is best. In an attempt to settle the q~restion.we investigate dl ph symmetric i~ersionsof the self-consistent T-matrix approximatinn, in(-lud- ing the so-called 'non-(l~rivnble'theories. and t.est them for their ability to corr~rtiypredict t.he physical properties of t.he system. in the course of our critique. it becorncs clear t.hat. two theories stand out as potential candidates.

3.2 Non-self-consistent Thouless criterion

St'e argued in $2.6 that the superconducting instability is marked by a divergence in the- pair propagator. In the T-matrix approximation, the pair propagator has the form c;:~-(Q, un)=

i(Q. v,)/ ( 1 + l-i(o.Y,)) and becomes singular only when its denominator vanishes. Con-

4 iequently. the onset of superconductivity mrtst coincide with L + l.-t(Q.v,) = 0, although it remains to be determined which (0,v,) component first becomes unstable as the system is cooled from the rrormal state. In the YSC T-matrix approxin~ation,the pair susceptibility is

Since \(Q,Y,,) must be real and positive in order to satisfy v(~.vn) = -I/['. we can rule out those components with a. non-zero Matsu bara frequency. Moreover, writing Eq. (3.2.1) - - makes it clear that the pair suscrptibility is even in Q. i.e. \(Q.u,) = 11-0, r/,i. rtris implies [.hat 0 = 6 is an pstrernurn point and that. in the neighbollrhooti of 0 = 6.rhe

The above resdt confirms Cooper's insight that the superconducting instability is tiriven by pairs of particles ~vitbzero cent re of mass momentum. We conclude t,hat the I 0,v.,) = (0'.0) wrnponent of t.he pair propagator is the first to diverge. Tln~s.the s~~perconductin~instability may he found using the so-called Tllouless rrite- rion.

The equation xsC(F,T) = 0 describes a line in the ji-T plane. the Thouless r-rlterion lint (TCL).that demarcates the boundary of the superconduct ing phase. I-sing Eq. 3.2.1I. the Thouless criterion can be written explicitly:

tanh JCI/l - =I. 2 *\I c E,-

Since tanh(Jz/2)/x is a strictly positive function of x, the Thouless criterion can only be satisfied when I' < 0. This is consistent with our contention that an attractive interaction between electrons is a necessary ingredient for superconductivity. It also suggests that we The shape cd the TC'L is of great interest. First. ~vsnotice that it has reflect.ior1 synlrnetr!. about t.hc = 0 line . That is.

Kc;(.

I['I tanh 3kC++- c),/"? - I&-C '2.u ,--t- -- :cci - G

Here ivtl have used t. he antiperiodicity of r he dispersion relation. <<+? - -:(. and the oddness of tan h .Jx;2 and r. ['This result is rclated to the fact that our forrnulatior~of t.he T-matrix is ph symmetric.] Second. the (6.0)component of the pair susceptibility may also be written as an integral weighted by the density of levels:

.-It low t.ernperatures, where the integrand is strongly peaked about li. this says that

(,0) ( .in interesting consequence is that the TCL in the p-T plane has tivo peaks near the band edges in one dimension and a single central peak in two and three (iirnensions (cf. Fig. 1.7). Third. the TCL intersects the axis at only two points. In one and two dimensions, it intersects once just below the band and once just above the band. For a three dimensional system, the situation is more complicated. In the thermodynamic limit. the TCL starts from the lower band edge fi = -W/Z and finishes at the upper band edge 9 = +W/2.

except when (7 surpasses some critical value. in which case the TCL end-points are pushed outside the band. For a non-infinite lattice in three dimensions. finite size effects also push the TCL end-points outside the band (for all values of the interaction strength). These features can be demonstrated rigorously by solving the Thouless criterion at where g~igr(c ) - -,lot Di2-1 \b-D/' is the density of levels measured with respect to the hand

To be provocative. let 11s write this as

Equation (3.2.1I) has the form of a bound state energy eigenvalue equation ivhrre EB = "(PI - I\. is the energy difference between the chemical potential of a particle pair and the bottom of the two-particle continuurn. The TCL intersects the T = 0 axis at = -ll*/.? - EBi2 and ,ii = CV/2 + Eel'?. In one dimension (D = I). Es/CV z ~'1.'/2()(~1+irW)' and in two dimensions ID= 2). EB/I.V zz 2,/(erw/lul - 1). In three dimensions (D= 3). there is a critical value of the interaction strength, call it l;,such that a two-particle bound state appears only when (L-I > . The bound state energy has the functional form Ee/W .r e(lL-1 - 1LJ)&(1L'1 - ((;I) where C(x) is a monotonically increasing function of t. satisfying C(0) = 0. These results are plotted in Fig. 3.1. The exponential form of EB in two dimensions ensures that a bound state, albeit a shallow one, exists even for infinitesimally small values of IT. Particles which are channelled Fiqur~3.1: Tho D = 1 and D = 2 c-urws pass through the origin ivhcreas thc D = .i (-uri.e interse(-ts the horizontal axis at I1;.//IIV= 0.95. About the origin. the D = 1 line twhav~s parabolically and the D = 2 line behaves exponentially. to such states cannot be described ever1 approximately by FL r heory since hound state pairs bchavc nothing like the weakly interacting quasi-particles of FL theory. The clues- tion Lve must answer is whether these potential bound states are accessible to physicnll!; r~alisticdensities of particles. -4s it turns orit, this depends entirely on whether lin~sc~f (.onstant clensity penetrate the 'TC'L or are deflected around it. This. in turn. tieperids on the behaviour of the self-energy in the vicinity of the TCL. As the sJrstem approaches the superconducting transition. the T-matrix self-energy

- is dominated by its Q = 0. vn1 = 0 contribution. For sniall 0, we may write

where y > 0 is the constant of proportionality in Eq. (3.2.3). Thus, expanding the v,l = Q p,., ~vittl n, D-tiependent .lacobian factor:

Tuqct her rvir1h Eq. (:3.'.14). t.his rcsult irnplies that, at the superconductinc: transition. r he

sirqies out D = 3 a critical dimension for the NSC T-matrix: in three or more dinlensions the .;elf-rncrgy remains finite on the TCL. but in one and two dimensions it. ilivrrqcs. An 11nus11alconsequence of r his divergence is that, since the number liensit? is r~latr,i

the TCL is an n = L constant density contour whenever D < 3 (since n + 1 as S 4 x). IVhen D 2 3. the self-energy remains finite at the superconducting transition and the number density takes on all possible values (0 5 n 5 2) along the TCL. Put another way. all lines of constant density in the 1-T plane intersect the boundary of the superconducting region in three dimensions whereas in one and two dimensions. the lines are expelled from the superconducting region and collect at accumulation points above and below the band. This effect is illustrated in Fig. 3.2 for a large two-dimensional lattice. [Recall that the existence of an excluded region at low temperatures is an important feature of the mechanism proposed by SVR.] Fi5urc 3.2: Lines of constant density are plotted in t.he b-1' plane for a 14 I k lattice of sites. The contours shown here correspond (from [eft to right) to tic~nsitits 11 = 0.1. 0.2, 0.3. . . . . 1.S. 1.9 and nre c-alculated for I- = -ZI'/2. The Thorilcss ~'riteriorl line fornls t.hc boundary to the shadtd superconducting region. Xotice that all the dcnsit!, rontours. except the rl = L line. arc ciettected as they approach the superconductirrg region.

\!'hat this inipiies is that. in one and two dimensions, the large excluded region Forrm t Ilc ciorlsity contours aide ,as the temperature is lowered. diverting then1 into two-part,iclc bound states along the edge of the superconducting phase. If ive follow a single constant (lensity contour from T >> itr- down to T = 0, we never cross the TCL, which is to say that. tve newr encounter the superconducting transition. This indicates that the particle pairs ivhich form in this ivay (lo not achieve phase coherence amongst themselves. Strictly speaking. though, ns the temperature is lowered in the grand canonical ensemble. the system does not trace a constant density contour but rather a line of fixed chemical potential. .A vertical trajectory in Fig. 3.2 positioned at any fi in the conduction band dearly intersects the TCL. Following this trajectory from high temperature down to T,: (the temperature at which it crosses the TCL), we find that the number density approach% half filling and the self-energy diverges. We know, however. that a divergence of the self-energy is unphysical because it indicates that the electrons are able to lower their energy without bound. Furthermore, since the number of double occupancies is proportional to the average interaction energy, it too increases without bound as the TCL is approached. Indeed. This is ph~,sicallytlnrealistit- since the I touble occupancy ,.an not ~sceedni2.

The salient point here is t-hat. follo~vinsEq. (2.5.1 1 ), the SSC' approximation is only just.ified ~vhenthe self-onerg iu small. \!'hen it leads to S -- x. the NSC' approximation rli115t hcl r rcl;tt.ccl \vit ti skt>pt.ic:isrr~.for 1-[early it is ivholly rlnj~~stifi~ciiri such ~*irr:~ir~~st;~r~c~~:s. .-\ci-ordinqly, r he div~r~enceof t. he self-tlnurgy for D < 3) ticnlanrls that. self-c.01~sist.tl11~~y

IIP t~nfon-wiin iow (iimcnsions irl order t.o obtain realistic predictions in the vicinit~.of t.hr. s~iprrc:c~~~~luc.t.in.;irlstabilitv. !On an infinite lattice in three 11irn~nsii)ns.the NSC' ;i~prosi-

rl\at.ion -clt'rns to s~lfficil.For such n system. t he Ttlo~ilcssc:rit.c,rion gives prcc.ist>ly the HC'S rmu I t [;I$] .j Thcre is one final deficiency of the theory that must be addressed. The llerrnin-\\'agner rheorern [:N]states that no one- or tivo-dim~nsionslsystem with a continuous symmetry (.an r~iakea rrausition at non-zero temperatures to a state exhibitinq lonq ranrge ordcr. Therefore.

r h~ ~;uperconductinqtransition temperature 13f the AH11 in one and two (linlensions is identically zero (T,, = 0). The NSC T-matrix approximation. though. gives a non-zcro critical tcrnperat~lrein all climensions (T,: > 0). It r~mainsto be seen whether \It.rrnin- IVargner is satisfied in a self-consistent calculation.

3.3 Non-self-consistent calculations and the pseudogap

In 1.1. ue attcrnpted to motivate our study ot' the AH11 by suggesting that it might provide

n rudimentary description of the correlated electrons in the high-Z: niaterials. One feature that we were keen to see reproduced in the model was the appearance of a normal state pseudogap above T,: that evolves smoothly into the energy gap of the superconducting state. As we pointed out. this phenomenon habeen well established experimentally in the cuprate superconductors. Recently. Q51C studies [.lo] have produced evidence of pseudogap physics in the two- dimensional AHhl. This has renewed interest in finding a theoretical framework that is capable of describing the phenomenon. In particular, if the T-matrix can be shown to suppress the DOS at the fermi level for temperatures above T,. this would lend credence to the notion that the pseudogap is the result of strong pairing fluctuations. In this section. we demonstrate that the NSC T-matrix approximation, despite its abovementioned failures in one and two dimensions, does indeed give rise to a precursor energy gap that is reminiscent of the pseudogap behaviour seen in QMC simulations. To be$). lye most calculate the fully interactins DOS rvithin the NSC T-rnatrix approu- inlation. Such a calculation is rndc possible by rhe fact that ive have access to the tlu;l~-t

-elution of the US(' pair propqator for arbitrary valu~sof and T. viz. (,'Tp-((j. rd.,) =

. L ( 1 + I -\((j.~1,:)) ~vher~ vq) is 5iver-1 by Eq. (.{.2.1). Notice that thc analytir r-ontintlation of the SSC pair 91isc.eptibilitl; to thc ror~lpl~u plane.

is n rrwromorphic function with a finite nurnher of simple poles on the real axis. Likcuiw

t,ho ~nnlyticcontinuation of the pair propagator.

is also a nlerornorphic function with a finite number of simple poles. \Ve can see thar this is rruc by multiplying each (5 component of Eq. (3.3.2) by n,-(z - (; - td-;) on th~top and bottorrl of its main fraction and then expanding and simplifying the numerator and iienorninator. This casts the pair propagator as a rational polynomial with real coefficients. a consequence of which is that there are only two possibilities for the placement of its

sin2ularities: they lie either on the real axis (corresponding to the normal state) or in conjugate pairs. equally spaced above and below the real axis (corresponding to the unstable state below Tc).ILIoreover, since the degree of the polynomial in the denominator is larger by one than that of polynomial in the numerator, the high-frequency asymptotic behaviour of the pair propagator is governed by C?~P_(G,:) - 1/- as 1;) + 1~. Thus, in the normal state. the pair propagator admits a partial fraction decomposition into a series of simple poles. We write this as which arise in the frequency convolution of GtE and GHF.[Here, b[~]= (cJr hose occupation function: see Appendix B.'?.] The final result is

Once the self-energy is calculated numerically by this procedure. the Green's lunrtiori follows from Dyson's equation. 6 (l.2) = 1/ (z - <; - :((. :)). The DOS can then be cunstructed according to

Since the negative imaginary part of the terms in ~(i.J + iq) all behave like ive know that the DOS is a seiies of &function peaks. In plots. ive usually drax these pcaks ivith a srtinll art.ificia1 broadening to aid the eye. Figures :l.3 and 3.4 present the resulting DOS for a small two-dimensional lattice. The twelve plots sho~vnare calculated at fixed crhernical potential over a \vide range of temper- i~turesaborer I.,.. Notice that as the temperature is Jecrmsed. the DOS at the ferrni Ievei is pro~ressivel\..reduced (Fiq. :I.:{). cvcnt,ually leading to the op~ning(>f a pscudo.sterti approaches the s~ipercond~ictingt.ransition. \Vhat is observed in Q\IC1 sin~r~lat.ions is t,liat the pseud02ap remains roughly constant in width hut progressively (leepens r he temperature is lmvered. [{ere. as T + T,: and r;sc i 0. we observe instead that the incipient sJap $rows ever wider. \Vhen I' - T,. is taken small ~nouqh.the DOS comes to resemble two &function peaks at opposite ends of the real line. [\Ye shall demonstrate later how this is relatccl to the (iivergence of tho self-energy on the TCL.]

This particular cotnplication (lees nor arise if we restrict ourselves to points along a line of constant density. In Fig. 33. we give the DOS for several temperatures on an $ 9 lattice ~t one cighth fiiling. \Ye have deliberately chosen the lattice size and temperature valws to coincide with the example densities of states given in Fig. 2 of Ref. [-101. The I-ornparison is not, cntirdy appropriate since the QXIC calculation was performed for constant density rr = 0.9.5 whereas tve have chosen n = 0.25. (Recall that the T-matrix is a dilute limit theory and thus a realistic rendering of the DOS at half filling is likely beyond its capabilities.] Sonetheless. the results are strikingly similar. There is qood qualitative agreement both for the general shape of the DOS and for the temperature scale at which the pseudogap first appears. .-I more quantitative comparison is frustrated by the fact that the Maximum Entropy method. which was used in Ref. [-lo] to extract the spectral properties from the QhIC data, produces artificially moot h output.

3.4 Self-consist ent T-matrix theories

In the early 1970s, following the work of Martelja [-!I] and Schmid 1-12], it became apparent that the choice of self-consistency scheme can dramatically alter the outcome of a self- consistent calculation. It was found that the shape of the predicted DOS depends on which propagators are dressed and which are left bare in the self-energy. This discovery led Pat ton Figure 3.3: The density of states is plotted for a decreasing sequence of temperatures above Tc. Calculations are performed using the non-self-consistent T-matrix approximation on a 6 x 6 lattice of sites for the parameters P/W = -0.3 and L:/W = - 1/2. The depletion of spectral weight at the fermi level is evident at low temperatures. Nu) /w-1

Figure 3.4: The density of states is plotted for the same material parameters as in Figure 3.3. At temperatures very close to TJW = 0.0590318, a pseudogap opens in the density of states. As the Thouless criterion line is approached (nsc + 0 and T + T,),the density of states evolves into two sharp peaks which move apart to +oo and -m. Figure 3.5: The density of states of an 8 x 8 system with interaction strength l' = -\I"/?is plotted at four points along the n=O.25 contour. The temperatures are chosen to correspond to T/t = 1. L/3. 1/4, and 1/5. to suqgest. t.trat the self-consistency mndition be altered by hand in orller to produce a thcory n. how predictions best aqree with the known physical properties of the rnodel !-131. \V hile this may seem an ineiegant kludge, we are inclined t-oward the pragmiitir vicw that im r~lenierlt.ot' our theorj, that. varies ividely liepending on the irpprosimation xhcrne

11scd cannot he espected to have any real predictive power. Thus. which vr?rsion ot' self- t-onsistoncy t.o enforce is arbitrary in the ~nsethat several t hror~ticalapproaches rriay bt> tlw,] tn tjerivp .r-njq t yi y rhenrjclf ~yhibjri,~~g,ji!f~y~nt cp!!lbjna?i,)!l.; qf'h;?rrr ?!!r! l!r-st.~! !L!I~+.

1;or r~sarriple.t hc rnodificd T-mat rix advocated by Pat ton later riqor~tlsl~~lerivwi i). P~dersenusing a Ginzburg-Landau (GL) functional met hod [.I-!]. \VP trust the (dilute limit) EOlI prediction of ladder diagrams insofar as it is corrob- orated by other theoretical approaches that also arrive at the T-matrix. but we rccoqniz~

that thc particular combination of bare and (lressed lines in Eqs. (2.5.\j and (2.5.5)is an xrtifart of the i~yrt~metricalexpansion of the two-part.icle correlation function usmi in the EO\I method. .As it turns out. the self-consistency scheme predicted by the EOll method

- asymrnrtric internal lines (one full C and one and a full rlosing line - turns out

to htl incapable of producing a superconducting energy gap!

In previous sections, we argued that self-consistency is necessary in one and two 1 1'mm- sions and we speculated that its enforcement might correct some of the obvious dcficienrics of the NSC T-matrix approximation. In this section, we study how the re-introduction of fiwl-back into the T-mat rix equations modifies those NSC results. Hotvever, given the con- siderations discussed above, we do not restrict ourselves to the version of self-consistency ticrived in Chapt. 2. Instead, we endeavour to select from all possible versions of the T- matrix theory the one which is most successful. The question is, successful by what criteria.' \Ve might select the theory that best satisfies certain sum rules, symmetries. conservation laws. or other exact results. We choose to concentrate on those aspects that are most important for modelling the superconducting transition: the nature of the superconducting gap and the value of the critical temperature. Figure 3.6: These are the six possible T-matrix theories numbered according their srlf- I-onsistcncy condition 2( u + l:) w.

\t'e generalize the T-matrix theory as follows:

(3.I. la)

ivhere (*) is a (symmetric) convolution operator and u. L'. U! are binary parameters with O denoting the HF propagator and 1 the fully self-consistent propagator (i.e. GU= uG + ( 1 - U)G~~).These parameters span the six different theories shown in Fig. 3.6.

Version (~ 1) is simply the NSC T-matrix that we studied in the previous two sections. (-4) is the original EOM method result. (3) is the end result of Patton's fix. and also the theory predicted by Pederson's GL calculation. (6) is the so-called (@-derivable) fully self- consistent T-matrix. Versions (2) and (5) are 'non-derivable' in the sense that no one has (yet) produced a theory that calls for either of them. These theories fall into two important classes: even numbered theories have the self-energy closed with a fully dressed propagating lint.: mid nurnbered theories are closed ivith a bare line.

3.4.1 Closing the self-energy with GHF

For norv. w* lpavp t. hr self-co!~sistencycondition or the pair suscept.ibili ty 11 nspecifird ,11111

(-onsidcr ail the (o(ld-numbered)r.wcs in Fig. i1.6 ivhere t he self-energy has the lorn1

\\.P ~vishr.0 better understand the hehaviour of this self-energy near the supercontl~~ct.iri< instability.

tkall !,hat the Thouless criterion Frirlction r;s~= I i (-l(O) vanishes at the rritic-al

tc,r!lperature. I:.. of the superconducting transition. For temperatures just above 'T,:. 6s~is snlall and strictly positive. Thus. \((I)= ( I - r;sc )/\['I > 0. \Vith this rt?sult in mind. ivr (.an part,it.ion the self-energy into its dominant Q = 0 component plus a non-singular part.

5"" that rcrrlains bounded as r;Sc -+ 0:

The singular component which dominates near the transition can be written as

.-it non-zero temperatures on a finite lattice (3,.If < x),the ratio Sns/Ybecomes vanishingly small as the system approaches the superconducting transition. Therefore, on the TCL. the complete T-matrix self-energy is S = Xs. According to Eqs. (3.4.lb) and (:3.4.ld),such a self-energy is the result of a pair propa- C'F-I.-\ PTER :3. .-\S.4LI-SlS OF THE T-lI.4TRI.Y .-\PPROSIAf.\TIOS qnt.or 1vit.h a A function p~akat C) = O:

iThe form uT Eq. [:1.1.5) cems to suggest that the suptlrconductir~qinstability 1 rnarketi by

1.11~(iiv~rqenct. of the pair propaqntor) i-a11only occur ivhen one of .j. .\I. or 1' jiiv~rgr>s.:

Ttlc sigrrificanct. of r he pair propagat.or'i &function form is elucidat-rd by t-on.iidtlrirlu, irs

Fourit~rtmnsforrn. Followinq Eq. ('?..I.':). we can apply the .ilirnrnation ( I,'.jII!xQ t'Qtl-"' ro both sides of Eq. (3.4.5)in order t.o rrcovpr its space-time repr~s~ntatior~.\fwhat 1b.e finti is that the pair propagator. strongly peaked in Fourier transformed variables. btmrri~s.r wnstant. Gr- ( 1 1: 1'1') = A'!(-'. independent of the space-time coordinates 1 and 1'. [This is indicative of the Ions range correlations and critical slowing dorvn that one ~sp~ctst.0 find ;it a phase transition: recall our (liscl~ssionin $2.6.1 it'ritten mot-her way.

Equation 1 :l..i.6) tells us that 1' is proportional to the magnitude of the pairing Hort,~~ations. Returning to the one-particle properties. we find that the self-enerqy Y(C)= -1'GHF (4 gives rise to a one-particle propagator

Since the HF propagators used in Eq. (3.4.7) have the form

(we have made use of the fact that the non-interacting fermions move in a symmetric band OF states. i.e. :-< = <; ) . t. he self-en~rqyand the one-particle propagator can be ~vrit.t.t3!1 - +splicitly as funct.ions of k ant-! --,:

The analytic continuation of the propagator from discrete llatsubara frequenci~son r he imaginary axis t,o the t-on!pl~xplane can be executed trivially via the substitution id,, -+ z. The rcsult is

Sot.ic~[hat this furlction trarlsfornls smoothly into the bare propagator w A' vanishrs.

which is to be expected since 1'. as defined in Eq. (3.4.4). is proportional to ('. Notice also that, at the fernli level (ECp = 0). Eq. (3.4.11) can be written as a sum of two 5irnple poles.

olle at +A and one at -1 on the real axis. The corresponding spectral function is

Therefore. a~ the parameter A grows from zero, the state at the fermi level is split in tivo. leaving a gap of width 'LA. \low qcn~rally.Eq. 19. I. 1 1 ) ,111ruitsa partial fraction ~ierornposition

consists of two unequally w~ightrid4-

ii'ritten more compactly, this is

.4 summation over the angular degrees of freedom. .V(d) = (2/11) 1;.-l(c. d). yields the fully interacting DOS Fir3.7: On the left is the non-interactins densit). of states measured with respect to t ht. 1-hc.111icalpotential, fi/ll- = -0.25. On the right is the fully interacting DOS at T h~ juperconduct,ing transition. It. displays a gap of ~vidth21/I€' = 0.2 at t.h~fprnri Irv~l.

Then, nlakirlg use of the identity

the r iritegration in Eq. (:I.-I.L$) can be e.uecuted to give

The resuiting DOS, plotted in Fig. 3.7. displays a BCS-style gap of width 21. Spectral weight is expelled from the region immediately around the fermi level and collects at the sap edges where the DOS goes as The iniplication of his rcsult is that. at the superconducting transition, the parameter 1 p1q.s the role of the s~~perconductingorder parameter. Accordins to Eq. (3.-!.ti),howct.rlr, it..;, sq~~ar~ITW~SU~PS t.he magnitude of the pairing iluctuations in t-he normal 5tat.c. That is to 53.. 1 c-llnrxctorizes the rna~nitudeof these fluctuations above T,. and characterizes the ivicith of t.hc sllperconducting energ>-gap at T,-. Therefore. it' 1 remains bounricci near x.

,1r1~1is r.~r~tiriuou~across the phase boundary. the superconducting gap must appear !'~~ily t;)rrn~ciat 11.. tri~~ciilik~ the high-T,. cuprates .;nil r!uitc unlike BC'S ~u~erconciuc.torsi ~vhic-n c>shibit a s~iperconciuctinggap of zero width clxactly at T,.). It is corlc~ivabltl that suuh 3 t'lllly forrrlecl zap might evolve out of a pseudogap in the normal state. [n tieriving Eq. (3.4.20).we have taken advantase of the fact that on the TCL many of t.he propertics of tht. system are inserisitive to t.he form of the pair susceptibility and ~lep~ncl

only on t hc choice of t.he external line in the T-matrix self-energy. As it turns out. the value of 1 cannot be determined until the pair susceptibility is specified. The simplest case where this can be done is the 3SC T-matrix approximation hr which 1 = cHF*G'IF. Here. must diverge at the TCL in spatial dimensions levs than three. The ~lcplanationgoes as follows. In qi3.2 we showed that the 3SC' Thouless criterion predicts a non-zero critical temperature (see Fig. 3.2) at which the self-energy diverges in one and two dimensions. Since the self-energy approaches the form of Eq. (3.4.9) as i7 + 'T,. A' must. be unboundeci in the same limit. i\.'hen 1 is much larger than the band-width. J<$+ A? zz A and the spectral function in Eq. (3.4.171 becomes largely independent. The corresponding DOS comes to resrmble .L'(;) =z i)'(d- -1) + d(d+ A). Since the electron density can be obtained from an integration of the DOS weighted by the fermi occupation function,

Lve get n = /[A] + /[-A]. In the limit A -+ n.the DOS becomes two &-function peaks moving tc ico and -m and n = f [A] + f[-A] + I. This explains why the NSC TCL is characterized by n = I: half of the total number of available states (Jdz .Vp(x) = 2) is fully occupied ( f [-m] = 1) while the other half is unoccupied ( f [m] = 0 j . It also explains the shape of the DOS in the bottom-right plot of Fig. 3.4! In the general case, when y is renormalized and A is not necessarily large, it is much easier to calculate the electron density directly from Eq. (2.4.12). Since the one-particle propagator takes on such a simple form on the TCL. viz. that of Eq. (3.-1.10). the r~quired \latsl~barafrequency iunl can be wccuted analytically to give

3.4.2 Closing the self-energy with C;

The remaining (even numbered) self-energies in Fig. 3.6 are of the form

The effect of closing Eq. (3.4.241rvit h a full propagating line is that. nt the superconducting t.ransition. the self-energ? now beconles

Putting Eq. (3.4.25) into Dyson's equation. we find that the C;reen's function is given by

ivhich. in contrast with Eq. (3.4.7).must be solved self-consistently for G. From one point of view. Eq. (3.1.16) is a simple recurrence relation which yields an infinite continued fraction Iiunwer. Eq. (3.-1.26)is also s quadratic equation in G ivhic-h can be solvcci easily. lt'~find that

.dthough sou~t.cart. must be exercised in choosing which root to take. [One root %i\.~sstrictly positive spcctrai tveiqht while r he other qives st.rict.!y negative tveiqht .j In fact. Eq. 3.1.2;) c.orlvero,w to

Once aqain. Eqs. (3.-I.$)ran be used to write our results as explicit furlctions of k and

-I,, . Equation (3.4.27) becomes

inlplyirig that

and Eq. i 3.4 .?g) becornes

The latter expression can now be analytically continued to the complex plane to give a function - re3.: The spectral function .-l(k.~)(plotted here with tF = It./' arrd 1 = :)\1', 4) vshibits a strong peak at ;= tC..\ subsidiary hump of spectral weight sits at the sarrie position on the opposite side of the J = O axis.

;~r~alyticrvery~vhere in C escept for two .;mall branch cuts on the open interval

Evaluated just above the real axis, t' looks like

the imaginary part of which yields the spectral function

This spectral function is plotted in Fig. 3.8. We observe that interactions have smeared out 1l1e uon-interacting 4-function peak ar ;= 1iV,/2,an effect ~vhichbecon~es greater

~vhicilsays that. stares near t.hc ferrni level have non-zero xeiqht. In partir~llar.rh~y corl- - trihw .likr-.OI = 1/72, to tho ~pcctralweight. This gives 11s a hint that. Lve hoi~ldriot t1spccr to find a. BCS-stbple gap at the fernli Ievel. From Eq. (3.4.36).the DOS is given bv

Immediately around the fermi level (;' g 11'1, we have

~vhichshows that the DOS goes linearly with distance from the fermi surface (see Fig. 3.9). \\'hen dL > 41'. t.he DOS is rssentiaily a distorted version of the non-interacting DOS:

Finally. ~t high enough enerqies (2; >> -112),the DOS is indistinguishable from that of Eq. (3.4..'0). the result when the self-energy is closed with GHF:

.As we have defined it, is a c-independent quantity, given by 1' = !iml,l+m c(6.z):. which describes the high-frequency asy mptotics of the self-energy. By studying the moments of the spectral function (see .Appendix B.1.7), we can also show that

This says that 1' is a measure of the amount of spectral weight (as compared to the non- interacting case) that has been shifted away from the fermi level. -4s we have seen, though. how that weight is shifted depends on the version of self-consistency that is enforced. Figure 3.9: The fully int,eracting DOS ( right) retains the general shape of the non-interacting DOS (Idt) with the psccption of a deep trough that goes to zero (linearly in 1;1) at t.hc f'errni Icvel.

\!'hen the self-energy is closed with a bare propagating line. the spectral weight is shifted in such a way that the DOS develops a clean. BCS-style gap. This is guaranteed by the Iieaviside dependency of Eq. (:l.-I.H): .t'(d)- t)(d2- A') . On the other hand. when the self-energy is closed with a full propagating line, the spectral weight is shifted to produce a linear gap: .l'(d)- (&(/A. In this case. the DOS does not exhibit a proper superconducting 5ap at T,,. On this basis, we choose to disregard the even numbered self-consistency schemes of Fig. ;].ti.

3.4.3 Analytic form of the propagator

The analytic form of the one-particle propagator is determined by its spectral function through the relation

The spectral function itself. defined by

is proportional to the number of eigenstates of the system that (i) are connected by the

addition of a single particle of wavevector and (ii) differ in energy by J. [See Appen- I iicca B. 1.3 irnil B.:j. 1 for detail~il(lerivations of Eqs. (3.4.-!3) and r :l.4.-l4j.] In a -?it em like the (\HlI that is purely illectrorlic (i.e. the electrons interact amongst t.hernselves nnri

;UP not coupled to any additional degrees of freedom). ferrni statistics restricts the nurnber ot' cisenst.ates xssociat.eti ivit.11 each site. Thus. on rt finite lattice. the exact spectral func- r ion .A( G. ;) consists of a finite numbt?r. 1.: say. of Munction peaks and G(C. r 1. r hro~lgi

Eq. (:j.-1. I:{),consists of An equal number of simple poles. Let us call this t.he Finite Fernli S~.stc~r~lsrrls~ilt. 1l.e ~01hilike to choose our approximation scheme such that this propprr.;b. is preserved. In :$:I. 1.1 and b:{.-k.?. we invcstiqateci t.he one-partirle propqitt.or at the superc-(:,r~cir~c,t.ir~q r rimsition (T = I::) and (derived its corresponding DOS. Here, we show that. it is also

possible 1.0 learn about the analytic form of the propagator in the normal state IT> 'I:.) by develupin~a few simple inequalities based on pole counting ar2uments.

4 \\P have already supposed that G(k.:) has rg poles. Let 11s further slippose that the rwo intcrnal lines and the one closing line of the T-matrix self-energy have r':. r:l. itnd r" kk poies, respectively. By this notation. we mean that r;j = r; when o = 1 and r". 1 ivhen k a = 0. [The latter case rorresponds to the single pole of CHF(i. :).I Then. in the T-rnatrix ;ippronimation. GrP- (0,z) has SQ poles where

'The justification for this inequality is that the pair propagator can have at most the number of poles in the pair susceptibility or. due to cancellation of poies in the numerator and denominator, as few as one. Each term in the summation of the self-energy is a frequency convolution between the

pair propagator with sd poles and a closing line with moles.y-E Consequently. the self-

lower bound. the outer maximum accounts for the possible degeneracy of poles associated with different Q components and the inner maximum accounts for any overlap between the poles of the pair propagator and those of the closing line. Since Dyson's equation implies that the number of poles in z(i, ;) is r; - I, we are left with

W I + max { max (sp,ri-;)) 5 5 1 + 1~~r~-~. 4 Q It is clei~rt.hat whenever one of rL and L' is non-zero. the right-hand inequality placcs rro restriction on the value of rg and we are left with 2 5 r; 5 x. Thus. in the normal state, t. he one-particle propagator has rc 2 2 poles. iVe saw in $3.-1.1that the lower bound. r.; = 2, corresponds to the number of poles of d{c.2) at the supercondocr.ing transition (see Eq. (3.4.15)).

.\;ow consider the case where the self-energy is closed with a fully dressed line I tll = I I. In this case. Eq. (3.4.A';) reads

By definition. r; 5 maxp rg,. Together with the left-most inequality in Eq. (3.4.49). this implies that 1 + rnaxg, rb, < rc j maxg, r;, which is satisfied if and only if r; + x. In this context, an infinite number of poles implies that G'(<. I) has a branch cut somewhere along the real axis. This is precisely what we found in Eq. (3.433). The proof here. though. is more general since it applies throughout the normal state and not just on the TCL. The arguments we have outlined here provide additional incentive to discard the even numbered self-consistency schemes: they all give rise to non- rneromorp hic one-particle prop agators and thus violate the Finite Fermi Systems result. Furthermore, as we shall discover in Chapt. 4, functions with branch cuts pose a serious problem for numerical analytic continuation methods. 3.4.4 Calculating the critical temperature

In 5:3.-!. 1 and $3.-1.'2.[ye st.udied t.he one-particle properties of the system at the supercon- ~l~lctiny:transition. Except in the SSC' case. we left rlnanswer~dthe question of ivhert. sue-h it transition [night occur. To remedy this. we must turn our ;ittention to the r.tvo-particle properties which determine the location of the superconductinq instability.

To calculate the one-particle properties. -+ye relied on the &-function form of t.hc pair propagator alonz t.he 'TC'L. The filterin? property of the (j-iunct.ion peak erisurps t-hat the form of the self-energy at t.he superconducting transirion is entirely deter~ninedby the chuiw of thc rlosinq line (i.~.by the value of u1 in Eq. (3.4.ld)). The detailed form oE the pair propagator. hoivcvcr. is rdated to the self-consistency condition of the internal lines of t.he

T-matrix self-energy (i.e. to the values of u and ti in Eq. (3.4.1~)).To deternlinc? whew the pair propagator becomes singular requires that we specifv the pair susceptibility and solvc t-he Thouless criterion function self-consistently. \\'e know that the Thouless criterion is satisfied when the (6.0) component of the pair s~~sceptibilitybecomes sufficiently large. For = C?' G", this component can he xrittcn as

is a smooth Function with its maximum at the origin (see Appendix B.2). Equation (3.4.50) indicates that the value of r(B,O) comes from the spectral functions .qU and ..I",weighted by ).V and integrated over all energies.

In the NSC case (ti = v = 0). -4" and .Au are non-interacting spectral functions with their peaks at the free single-particle energy EL. Under these conditions, Eq. (3.4.50)simply reproduces the known result (8.0)= (1/.\1) Cg(l- 2 f [[G])/2~s. In contrast, when at least one of the lines in the pair susceptibility is dressed ( u + v > 0). the value of k(6.0) depends

on the fully interacting spectral function A(&~l). At high temperatures, the function ).V is slowly varying and thus the integrals in Eq. (:).-I..% i broadly sample spectral wight from every energy range. .-it low t.tlm pera- Iurcs. t,tlough. )I' becomes wry stronqly peaked. 'The height of this peak is ).V[O. O] = li.1 ;~rlilits ividth oes as i/J so t.hat, in the larqe 3 limit. )L'[x. IJ] - 6(~)d(~].C'ons~~uently. the vanishing of ~;c= L + 1'\(6.0) at low t.enlpcrature.: is governed by the b~hsviourof 4 - .-\IL( k. -*I and .-lY(k, in th~!vicinity of = 0. \Ve have already est nblished. horvever. that the Fully interacting jpectral funcation %ow

Lt.ro as 11'~ to ~t the fermi l~vel the sv=tcm approaches the superronductinsc- transition. ~ho~vedthat the DOS behaves like .l'(d) = I 2l.11) '7-.-L(k. d) d(-rL - A') or ;I, 1 Lk - - on the TCL. This can only mean that the self-consistent critical temperature is luw~rr ha11 r he SSC value: zi KS~approaches zero. we exp~cta gap to open at the t'ermi level: in r II rn. this reduction in spectral iseight at - = 0 curbs the growth of \(6.0) and restrains xSc'j descent to zero. This is an example of feed-hack effects working to suppress the ('ooper i nstahility. This roush picture can be made more precise. T~Pdefinition of the paranleter A' in Eq. (3.4.4). viz.

can be inverted to give an expression for the Thouless criterion function, ~56in the vicinity

of KSC = 0:

Consequently. on any finite lattice (?rl< X) the following must be true.

(ii) If T, = 0 then as T -t Tc from above. rcsc + 0 provided A' is non-zero.

Since these conditions are exhaustive, and we know that T. - A: at the superconducting instability, each version of the T-matrix must have either a, non-zero transition temperature and a divergent self-energy or a zero transition temperature and a non-zero self-energy [Zero self-energy corresponds to no transition at all.] Having surmised that the T-matrix self-energy diverges whenever the transition temperature is non-zero. we anticipate that any effort to moderate that divergence will necessarily lead to ;l suppr~ssionof the critical temperature. In particuiar. we specu!atc t.hat the rn- lorremerit of self-roniist~ncyivill prevent the self-energ from becoming singular and thus push TI-to zero.

..Isfor the SSC' T-matrix i 1 = GktFs GHF1, we have already esrablishd in $:I..? r hat its critical ternperaturc is non-tcro and that its self-enere;>,diverges at the TC'L. However. ~VP now have the tools t.o explain this behaviorir in terms of the lack of self-consistency in the pair

slisceptibility. To begin, E~J.r 3.-1.5:)) and i:I.'.f l ran hc iis~dto :vritl. = 1 + I '\I (7. 0)

Since there is nu fwd-back in the pair susceptibility, the right-hand side of this quation is independent of A. Let 11s evaluate both sides of Eq. (3.4.54) at the critical temperature. The expression to the right of the equality vanishes by definition and we are left ivith il*l/(J,-.\[A' + I[-I) = 0 where 3,. = i/T. < x. Clearly, this can only be satisfied if 1 + x at the NSC' TC'L. In the remaining versions of the T-matrix, the pair susceptibility includes at least one 11111 G. Since A' feeds into i; (though E) and then into t, it must be that in these cases 1' appears explicitly on the right-hand side of ~sc= 1 + ('[(C 0). Indeed. when \ = G' * [iHF. ivt. have

=--C1 tanh :,/

'To start. we know that A' = x is not a solution sirlc-e Eq. 13.4.56)evaluatefi irl [ha. lif~~it of 1' -+ x produces 0 = I. Hotvever. if 1' remains bounded on the TCL thrn. ;icrorclinq to condition (i) follo~vingEq. (3.-I.5:1).T, = 0. Evaluating Eq. (J.-I.T,Gi in t.he limit .1 4 s. ivc. find that both sides vanish provided that

Equivalently. this can be written as

~vhichwe recognize as the well-known BC'S gap equation [38]. It determines the value of 1 imci hence the size of the gap at Tc. Since i/q'r2+ L2is strongly peaked around x = 0 ~ve can approximate the gap equation by

This equation can be inverted to give the width of the gap as a function of the interaction strength:

This demonstrates one of the most interesting features of superconductivity: there is no critical threshold of interaction strength that must be passed in order for the electrons to undergo a transition to the superconducting state. Let us now consider the one remaining permutation of dressed lines. When y = G * G. i he (6.0)component of t. he pair suscept.ibility looks like

This result implies [,hat r;sc = 1 + I',(G.O) can be written as

.As before. ive find that this expression cannot be satisfied by A' = r.which implies that Tc = 0. Taking the 3 + x limit, we are left with a new gap equation

which bve can cast as the BCS gap equation plus a correction that is small when < IF. 3.5 Zero temperature calculations based on two candidate theories

In t hr (.ollrw of ollr critiqrlc of t he ;is possible T-mat ris r heories, two havt. t1rllerscd po- ttlntial candidates. Follo~i,inqthe numberin(: in Fig. ii.6. they corrrspond to self-(:or~sistrncy schemes (:I I anif (-5). \.VP have ruled out, ail even-numbered versions on the a,rol~il&t h;ft thf? (lo not qi\,r. rise lo ;L sllpc~rcr31l(i11c-tinqFap in thc DOS an~ithat thrv Iraci to bran(-h cSut,sill thta 0n+partic11~propagator, twn for finite systems. Li'e have ruleci o~itv~rsiorl I 11. t.he NSC' rase. because it contairls no feed-back effects at all. By process of elimination. oniv versions i:)) nrrd (5)are wirclble. They iead t,o the following set of equations:

For each of the t~vocandidate theories given above. hell-cronsistenry iupprcsses the C'ilopcr instability at ail non-zero temperatures and thus the self-consistent TC'L lies along the T = O axis in the @-T plane. .it zero temperature. however. the equation

is no longer exact. because the result derived in Eq. (3.4.4)relies on the assumption of a non- zero Tc. In the zero-temperature limit. the Matsubara points merge into a continuous line of frequencies and the Matsubara sum Cvngoes over to an integral S dv. Consequently, the &-function peak of Eq. (3.4.5) is broadened into a Lorentzian with respect to its frequency variable. Once the Q = 0 filtering property of the pair propagator is weakened in this manner. Eq. (3.5.2) is no longer the only component of the self-energy that survives at the juperconducting transition. Sonetheless, to the degree that the pair propagator is still strongly peaked about Q = 0. Eq. (3.3.2) provides a good approximation to the true T-matrix self-energy. Thus, taking Eq. (:1.?.21 as a self-energy ansatz. let 11s t-alculate i.he properties of the zero-t.~n~perat~lrP

To begin. iv\.e use the expression for t,he number density provided by Eq. (:J..I.2:]).In the rcro tcrllprraturr limit. r.hc rrumber density and its [lerivate xith rwpect to fi bPcorrlr2

tanh i 1' i +,/:;- k n = 1 - --C:; .\ 1 - ."c; + 1' 1; 1;

Sotc t.hat the value of A' in these equat.ions must be chosen to satisfy the appropriatr qnp quation: i.e. either Eq. (3.4.57)or Eq. (3.4.64). depending on the self-consistency achen~r

Xest. we derive an expression for the double occupancy by putting Eq. (3.5.2)into Eq. ('>.-I.14). At zero temperature. it becomes

+ I.1 tanh 4 $i + l2 = -+ -- 4 C"2.W k' pp

Again. A' must be chosen to satisfy the gap equation. However. when = G * cHF.the rvhich LVP recognize .as a factor in !.he ktterm of Eq. (3.5.5i. .\laking thc appropriatr stlhst.itution. ive nrr left ivit h [.he simple expression

H~re.t he do11ble occupancy is 5ivt.n by the non-interacting res~~lt.n2, I. plus ir (:orrtv:t.iu~l

{jue to cl~ctronic (.orrelations. On the other hand. when = C * C;. the zap equation has: the form

which ~liffersfrom Eq. (:l..i.fi) by a Term that happens to be proportional t.o i)nii)P (vC. Eq. (3.5.4)).That is.

;tnd thus the double occupancy can be written as

For small interaction strengths, the physical properties of the zero-temperature superconducting state are largely independent of the renormalization of the internal lines in the T-mat rin self-energy. Most importantly, since the superconducting order parameter vanishes as (-+ 0. the two gap equations. Eqs. (3.5.6) and (3.5.J), become indistinguishable in the limit of small C. For intermediate values of the interaction strength, the differ- ences between the two self-consistency schemes become apparent. Calculations based on schemes (3) and (5) (corresponding to y = G * eHFand y = G r G, respectively) are Figure 3.10: The square of the superconducting gap width, the derivative an/ajl. and the double occupancy are plotted above as a function of the electron density. These quantities are calculated in the T-matrix approximation at zero temperature for two different self- consistency schemes. The lattice size is 14 x 14 and the interaction strength is I; = -\1'/2. The shaded region in the bottom graph indicates the allowed values of the double occupancy: n2j4 5 (iitiil.) < n/2. prrr-er\t~(ii[! Fiq. 3.10. Thclre. the square of the qtp tvidt h. the ~icrivntiv~On; db, an11 t hr

1 lo~ible occupancy arc plot t ~d as a filnction of the rlectrun density for a t wo-dinlensionai .iystc.rn tvhose interaction strength is half the band-xidt h.

Sol ice t hat in these self-c-onsistcnt calculations 1' is a well- be hav~ci.boun~dct_l func-t ion uf the tllectron ~lerwity. Lik~tvise,thc double occupancy is a wil-b~havdfunction tvhic-h rtlsprcts its llpper and lo~vert)ounds: 11' 1-1 < ( ictnL) 5 n/2. This desirable tatr OC ~tfairs t-ot~~walm~it h~ca11w th~ wIV-O~~P~~ tjmx qnt tji\*~rynn the r(-'[-.LG !t ,jqo< i" Yq(' t-nlculations. \i'ith the introductiun of df-consistency, feed-back ~fTects ~vorkto keep r hc

-r.lf-i>nt.rqy bot~nlidin the vic-irrity of the sl~pcrronciuctingtransition. ( 'onseqiipnti~.r hrl I'CL thc (iocs not hehavc like an rz = 1 csc~ntourand does not divert lines of constant 1frn.sity nuay from the superconiiuctinq phase. Having (ien~onstratedthat sclf-consistency ran inJeed c-orrect the unphysicsi feat urrs 01' the ?;SC T-matrix iipproxirnation at r,. tve ivoulci like to .;eo these self-ronsistent T- matrix calculations extended to the normal state (T > O). C'~~fortunately,time limitations have prevented 11s from carrying out such a. calculation. 1'0 the best of our krrowl~ciqe. normal statp calculations based on self-consistency scheme (.5) have never been pcrformcd. Sdwrne (3).however, has been well studied. Proposed by Patton more than twenty-five ?clan 'igo. this version of the setf-consistent T-matrix has been t,he subject of renewed interest in the wake of the discovery of the pseudogap phenomenon in the underdoped cuprates. In rwent times. Patton's formulation has been rriost vigorously championed by Levin .ind co~vorkers[l A. 4.5. 461. Their ent husiasrn for this particular version of self-consistency is based on the claim that it satisfies three important criteria [-Is!. two of which we believe deserve comment: ti) the Patton scheme preserves the law of particle conservation and (ii) it is consistent with the precise BCS result in weak coupling. First. we disagree that scheme (3) satisfies criterion (i). Baym and Kadanoff derive a detailed set of conditions [3 11 that must be met in order to ensure that a given diagrammatic approximation conserves number, energy, and angular momentum. In particular, they demonst rate that an approximation is number conserving only if it satisfies GaJ(12:1+2+) = GJ,(21: PI+).However. the asymmetric manner in which bare and dressed functions enter the pair propagator in scheme (3) ensures that the two-particle Green's function does not exhibit this symmetry. In comment 21 of Ref. [IS], Levin and coworkers acknowledge this criticism, but dismiss it (on the basis of private communication with V. Ambegaokar and B. Patton). Note that. due to the symmetric form of its pair propagator. scheme (5) does conserve particle number. 3.6 Accounting for the Hartree term

3.6.1 Hartree-Fock approximation

S tic simplest approxir~~atiunurie car1 niake is to ne~lcctdecr ronic (-orrelations ~ntirely..\s

IVP sho~vdirl $2.3. this is the so-called Hartree-Fock ( HF) approximation. Hartree-Fock is s mean-field t.reatment in which each single-particle energy i; is shifted by the averaqe iri~~raction euerqy of one particle with the background of particles having opposite spin.

.\t*cordin~to Eq. (2,!,.j). t lie one-particle propagator in the H F approximation looks like

where fi = p - (.(n - 1 )/2 and (; is the single-particle energy measured with respect to h. \Ye have already argued that no mean-field approximation can capture the C'oop~rin- 5tability that signals the transition to a superconducting state. Indeed. in the HF approsi- rustion t h~recan be no divergence in the pair propagator since the two-particle correlator is taken to be zero. Thus. insofar as we confine our investigation to the superconducting iran- sition, the HF state is of little value except as a basis for diagrammatic expansion. There is. however, the possibility that superconductivity may be in conllict with some other phase transition. In this section, we show that the HF approximation provides evidence for a competing instability at low temperatures. Since the HF propagator has the same analytic form as the free propagator.

the electron density has the form of Eq. (1.6.1 1): iVhen the Hubbard interaction is attractive (I' < 0). the compressibility has an instability corresponding to i)n/i)ji = 2/ll*j. For sufficiently large T. i)n/i& < ?/)I*]and i)n/ilp is finite. but as the wrtiperature is lowered. ilrridp -r x cu i)nii)P -+ 2/II'/ from hc4ow. \\'c interpret this as a phase scparathn in analogy with the infinite fluid I-omprflssitjilit>.

(-3\-/dP 4 x)at the classical gas-liquid transition (via Maxwell's construction). The phase separation boundary can be mapped out in the p-T plane in the fo!loh.ing way. Since the difference p - is a constant along lines of fixed density, a plot of the (.ontours of Eq. (3.6.3) in the fi-T plane (identical to Fig. 1.6) can be transformed into the

~:quivalent 11-T plot by shifting each line horizontallv toward the p = 6 = 0 asis by the amount \iv(n- l)i'?\. Iloreover, since each constant density line is shifted by a different amount (the n = 1 line is stationary, the n = 0 and n = 2 lines are shifted most). there is the possibility that lines may overtake one another. Where two lines cross marks a point of infinite compressibility (see Fig. 3.1 1). [This technique for mapping out the boundaries of the phase separation region can be applied even when the approximation scheme is carried beyond HF.] According to HF, the onset of phase separation does not require that the interaction

5t rengt h surpass some critical value. With an infinitesimal attractive L;, the phase separation region appears as a point at the origin (p = T = 0) and grows with increasing \I' . For small interaction strengths, only density contours near half filling encounter the phase separation; those away from half filling arrive in the conduction band at T = 0. By the time the interaction strength is as large as C' = -CV, the phase separation instability has Figure 3.11: The topleft and topright graphs depict lines of constant density in the p-T and p-T planes, respectively. The dotted lines mark the boundaries of the regions that are magnified in the lower set of graphs. The shaded region in the bottom-right graph marks the phase separation instability (for 17/W = -3/4). Figure :3.12: The c.ompressibility calculated in t.he random phase appro:iirnation.

or. quivalently. i)rr/"l)fi = ZIl(0) where

is the factor associated with each polarization insertion ( ph bu bbie). The resulting c-orn- pressibility (see Fig. 3.12) is given by

\I'e recognize this as RP.-\. It predicts an instability when density fluctuations beconic resonant, i.e when KPS 1 + C-n(O)= 0. Thus, this instability may represent a transition to a CDW state (signalled by GP~(~)-+ x in much the same way that the transition to a

superconducting state is signalled by GPP(Q = 0) 4 m). Nonetheless, it remains to be determined whether the predicted phae separation is real or merely an artifact of the (admittedly severe) HF approximation. In support of the latter. Emery and Kivelson argue that a system of lattice ferrnions feeling only an onsite attraction cannot separate into low- and high-density phases since the interaction is saturated when a local pair is formed [-IT]. In addition, hlonte Carlo studies report no evidence for phase separation at low temperatures [.IS. 491. The consensus view seems to be that the RP.4 result is valid only when jl:(ih/ap < 1. In this regime it correctly describes t hr t endenc!. of att ractivr interactions to incr~aethe compressibility. It fails. ho~vev~r,at low terllporaturt?s ir-hrr~it predicts a spr~rio~~sinstability.

There is one further difficulty with the HF approximation. Equation ( 1 .-I. 1 1 \ r.clls 11s that thr rnrtlpr~ssibilityis dcternlined by correlations in the ph charlrlel of the rwo-partit-le propaptor. Therefore. Eq. (3.6,$) must be t. he result of an indirect interaction hettvwg ~>lm:tronpair.;. rncciiated by density fluc.tuations. C'l~arly.this cannot be rt~canc:ilcd tvith

t lip ti F a.5~~nl pt ion of %erntwrr~la t in~i. TIIP in11 rvp of this tiisrr~panr.yis t h~ (art t 11 il t t 11'3

pi~rt.ialderivative i)/ilp acts like a functional Jerivative d/&ce"(i, 2) tvith respect to ;L locrnl. sirrluitaneoua external field re"!1.2) = pi( 1. 2). Thus. whenever i)n/i)lr is calculated ~iirm-t.ly from n via partial differentiation. it is ~ffectivdydetermined by correlation functions at one

higher level of approximation ( in the sense of Baym and Kadanoff ) than the Green 's function

which first. determind n. [In (heir paper on conserving approximations [:Ill. Baym and liadnnoif denlonst,rate t.hat functionai differentiation of the HF self-energy leads to liPh for

t.he two-particle correlations.] This discrepancy becomes less acute as the npproxinlation for

1.h~self-energy is improved (and disappears entirely when the self-energy is known ~xactIy ).

3.6.2 Hartree shift in the T-matrix approximation

In our discussion of the NSC T-rnatrix approxiniation. we neglected all freque~lcy-intleprndcrlt energy contributions by plotting the low-temperature phase diagram with respect to p: see. e.g. Fig. :l.2. The HF mean-field energy (which we absorbed into p) is accounted for by rrnnsforrning such a. plot from b-T to g-7' coordinates. L1.e accompiish this by shifting t-hr constant density lines by I'(n - I)/'. as we did in $3.6.1. In three or more dimensions. this has the effect of narrowing the superconducting region, since all points on the TCL are translated toward the p = 0 axis. In one and two dimensions. the TCL remains unchanged. since it is characterized by n = 1, and thus the shifted constant density lines now penetrate the superconducting region (see Fig. 3.13, topright). Thus, the SVR result is destroyed when we account for the first order HF energy shift in this way. In performing such a shift. we must contemplate the possibility of an infinite compressibility. :Is it turns out. the divergence of the self-energy encourages the phase separation instability near half filling. First, we observe that Figure 3.13: In top row of plots. lines of constant density are plotted in the ji-T and p-T planes. The shaded area marks the superconducting region whose boundary is the Thouless criterion line. The bottom row of plots illustrates a typical line crossing that results when lines of constant density are shifted by p - p. Sincr \'(k\ is unbounded in a compact reeion of the 1-I' plane. viz, the TCL. it muat b~

I liar ilS ( k),i)p( -- x in a neighbourhoocl of t hat rp~ion.Consequently. dn !i);~ incr~asm ii i: tiout hound .LS t hc tPmperaturP is lo\ver~(iand. at a line of points above r h~ T( 'L. dn ilji

In p-7' c*oortiinntes. t. hese points mark tile crossing of I-onstant (lensit! lines, an riecnr ~vltichorx-urs whcrever t he value of

is inti nite. Interestingly. almost all of the phase sparation points. which lie just orlt.

the top-right plot of Fig. 3.13. This tells 11s t,hat the phase separation instability wins out ovcr s11p~rcont1uctivit.yonly near half fi llirrg.

In ct self-consistent T-matrix calculation. the Hartree term is generally less trou bl~sonle to (leal with. Since the TCL is a horizontal line segment in the p-T plane. horizontal shifts of the density contours do not aKwt whether they penetrate the auperronducting phase. Indeed. we haw already established that the density varies continuously along the TCI. so that ther~is no excluded region like that observed in the SSC case. Sonetheless. there is still the possibility that a line crossing might occur if i)n/i)fi becomes sufficiently large. The middle plot of Fig. 3.10 shows that da/i)fi has a maximum at

TI = 1, but in that example, i)n/8b < '1/(1'(= 4/ECW(I,* = -l1/'/2). One can show. however. that the height of the an/ap maximum grows with the interaction strength and thus a phase separation instability is possible provided that the interaction strength is made to be large enough. That is to say, there is a critical value of the interaction strength. call it l'ps,such that an instability occurs at p = T = 0 when I(-/ first surpasses I(Psl We find that (is= -2lV for = G * G~~ and lbs = -LY for = G * G. [It seems that a greater

degree of renormalization (G * G vs. G iG~~) tends to suppress superconducting order (cf. the t~voline in the top plot of Fig. 3.10) and to enhance the ph fluctuations that drive the phase separation instability (cf. the two lines in the middle plot of Fig. 3.10).] Chapter 4

A new approach to analytic continuation

4.1 Overview

In the NSC' T-matrix approximation. the one-particle DOS can be ralculateci to arbi-

trary accuracy for any value of the temperature and the chemical potential. Aftrr thp reintroduction of feed-back into the T-matrix equations. it is no longer possible to so easily determine the spectral properties of the system. The exception. of course. is for points on the TC'L where the simplified form of the T-matrix self-energy makes analytic;?/ progress possible. For temperatures above TC.no such simplification presents itself and thus the T- matrix calculations must be executed numerically. This involves solving for the one-particle propagator self-consistently at a finite number of Matsubara frequencies on the imaginary axis and then analytically continuing from these points to the real axis in order to extract the spectral functions and the DOS. The purpose of this chapter is to present a new method for analytic continuation which we believe represents a considerable advance over existing methods. Analytic continuation arises in the many-body problem whenever real time dynamics are to be recovered from a response function calculated at non-zero temperatures in the Matsubara formalism. In that case, the function whose value is known only at a discrete set of points on the imaginary axis must be continued to the real axis. .An analytic continuation of a function f defined on a subset A c C is a function that coincides with J on A and is analytic on a domain containing A. Usually, we are interested 1l5 in thr analytic continuation f with thr largeat such domain. For then / is the greatest ;~nalyticextension of j to the conlplcs plane. Since there exists no general prescript.ion for finding f from /, there is no choice but ~.o resurr ro approxinlat~techniques. Currently, the state of the art is to interpolate het~vcen kno~vnpoints using fitting functions capable of reproducing the analytic structure of / in the complex plane. :I serious difficulty is that the analytic structure of j is not ilsually

l.!!n~t.n ;\ prinri.

.-\ ~videly11scd technique is t. he Pad4 approxinian t rnet hod in u. hich rational polynomials (or terminating continued fractions) are used as fitting fn nctions. Several Pad4 scherncs exist. The most common scheme, a recursive algorit h nl called Thiele's Reciprocal Difference llet hod [.jO]. was first introduced by L'idberg and Serene 1.5 11 in the context of the Eliashberq

quations. lht, despite twenty .cars of widespread use. the Pad6 approxirnant met hod remains somewhat of a black art. There is still no reliable. quantitative mesure of the quality of a Pade result. The prevailing wisdom is that a Pade fit can be considered 'good' when the o11t.put function is stable with respect to the addition of more input points. The results of this chapter rnakt! it clear that such a criterion is insuificient. The various Pade schemes can be divided into two broad classes: (I) those which return

the value of the continued function point by point in the complex plane (/(A).:) r, /(:I and (11) those which yield the function itself f(A) ct f by returning the polynomial (or c-ontintled fraction) coefficients. Thiele's method is class I, as are most numerical rnetho(ds. In this chapter we present a robust Pad6 scheme that is class 11 and propose a goodness-of-fit criterion based on the convergence of the polynomial coefficients to allowed values. One advantage of our approach is that we formulate the problem as a matrix equation, allowing us to make use of existing, highly efficient routines for matrix inversion. In contrast. a na'ively implemented recursion algorithm can lead to a severe propagation of error since repeated operations are performed on terms of very different orders of magnitude.

4.2 Green's function formalism

The oneparticle propagator or Green's function can be formulated using real or imaginary time operators. In real time, the retarded Green's function (lest-rib~s ho~[.he system responds whcn a particle is added at rime zero and renloved at timr t. Its imaginary time counterpart. the temperature Green's function

is not so clearly physically motivated. [is main advantages are its mathenlatical rl~gar1c.e and sorrlputational ease. Since it is Mined in terms of the time ordering operator T.L't dnlit~.a (liagrarnn~aticexpansion via \Vickls theorem. .Iloroover. tvhercas the rctar(it.cl

Green's Eunction (iR(t)is i~pcriodicirl t [it has a lone discontinuity st I = 0).the t.~~nlpernt.ur~

( ;reen's function is periodic in r \vith period 23. The two Cheen's functions have Fourier representations: the first a Fourier transform

and the second, as a consequence of its periodicity, a Fourier series

l.\,hich. in Eq. (-l.?.-lb), we have recast as a sun1 over the hlatsubara frequencies {;,, = ('ln - I)T/'J: n E Z} of some new Fourier component G(;,). The formal connection between the real and imaginary time formalisms is the following. There exists a unique function 6 : c, @ with asymptotic form

which takes on the values of the Fourier components of the temperature Green's function at Jlatsubara points on the imaginary axis ~(iw,)= G(;,) and gives the Fourier transform of the retarded Green's function just above the real axis G(J+iq) = G~(;). That is, Eqs. (42.3) and (4.2.lb) can be written as Clearly. all the infornlation one can potentially extract from these fmct ions is contained in

The ~'IInc-tion Cr' has several interestins properties. First. it is analytic cver.\vht:rc in t.ht: c.ornple-ls plant. ivit h the exception of the real axis; this is a c-ausality r~quirer~ierit. S~(-orl,j,

the value of in t.he upper and lower half planes is related by G'(zm)= (>(:)-. ivhich is

ii statement of the time reversal symmetry between the retarded and advanced Grwn's

frl nctions. Its inirned iatc consequence is that the imaginary part of C; [nay be discontinuous :tcross the real asis. It also implies that we need only know the function in either thr llpprr or the lower half plane since the other is a conjugated reflection of the tirst,. Third. C; ran be written as a- Stieltjes transform

where the spectral function. given by the magnitude of the discontinuity in across the real axis. viz.

is non-negative and normalized to unity

Typically, we are working in the LIatsubara formalisnl and we calcdate G(;,) from its self-energy (via G(lt,)-' = iw, -( - X(d,) j which is in turn calculated from an approximate theory based on, e.g.. a diagrammatic expansion of the propagator. From here the route to real time dynamics is somew hat circuitous: The Pad6 met hod is baed on the assumption that L' can be written as a rational polynomial or tcrrnirlating continued fraction. Since theories are most, commonly specified by a choice of self-energy. the continued fraction form turns out to be the more useful, at least for inv~stigatingquestions of a mat hematical nature (e.g.analytic structure). In particular. ~vc .;hail find it helpful to consider 'T (in the upper half plane) a continued fraction of .lac,ohi !i)rni [53] (.I-frac). That is.

where the A, and en are complex constants. By comparison with Dyson's equation. given in Eq. (4.3.lb), we make the identification A = 1 and eo = <, where < is just the free particle energy measured with respect to the chemical potential. [.As usual, we have chosen to absorb all frequency independent terms of the self-energy into E.] Then. we find that '(,,(z) is itself a continued fraction

The justification for this continued fraction form is a theorem due to Wall and Wetzel [S] which assures us that a positive definite J-frac has a spectral representation with non- lhrc are tivo special cues ~vorthmentioning. If thc A,, and e, are ;rll real rIit?r~ thc. .I-frac is positive licfinitr and (an be cast as a surn of simple poles [J:)]

ivith real. distinct ener5ies E, and positive residues R, > 0. The .l-frac is also positive definite if the A, are real and none of the t.,, sits in the upper half complex plane ( Eq. (4.3.3) is satisfied by setting all g, = O or 1). in tvhich case the Function is characterized by iirnple poles resting on or below the rcal axis. In the general case. all the continued i'ractiori coefficients have the potential to be complex. with the exception of A; = I. to = <. and A:. 'rlnct.7. r" = < has no imaginary part. Eq. (-1.3.3) implies that the coefficient A: must alivn!-s hc rcal and positive. It is clear that by observing the values of the A,, en coefficients, one can learn a great deal xbout the analytic properties of C(,+,).For example. if some en has a positive imaginary part (and no A, = 0 for rn < n] then cc,+,,may have a pole in the upper half plane. Soch a function would be noncausal and have negative spectral weight. Nonet,heless. despite the usefulness of the continued fraction form, for computational purposes it is actually much easier to work with rational polynomials. Conveniently. every terminating continued fraction is equivalent to a rational polynomial. For instance. a .J-frac with r stories, Eq. (4.3.2) say. can be written as the ratio of two polynonlials P, Q defined rccursiv~11;by the formulas

(for rl = 1, 2.3, . . . ) Lvith base

\-irrit,ingout the leading order terms of P and Q

tliakes it clear that the polynor~lial P is of order r - 1 in : tvhiie the polynonlial Q is ot'

rer. [.-\ccordingiy. we r~ferto c(,,in Eq. 1-1.3.5)as ii [[r - ~jr]rational polynomial.) lloreover, it suggests that we write the self energy explicitly as a rational polynomial 01' t.h~ form

It is straightforward to relate the old and new coefficients to one another via Eqs. (4.3.61

id(-1.:1.7): e.g. hi = p,. r 1 = p,- 1 jp, - q,, etc. The coefficients p,, q, can be determined hy specifying the value of Xi,,at 2r points. i.e. by solving the set of 2r linear equations

[The system of equations is linear in the (p,, q,) basis but not it the (A,, en) basis.] If we ~vheren,; = X(;,) are the known values of the self-energy at 'lr hlatsubara frcqucncies. imri

~quivalentto the system of equations given by Eq. (-t.:).10),then the entire prot:Pss r~f' analytic continuation is reduced to a single matrix inversion

which provides the polynomial coefficients necessary to construct

What 1vr.e propose is f hat. having determined the p,, q, coefficients. we recover the A,. e., coefficients and then use the criteria provided by \Val1 and Wetzel's theorem to determine whether the matrix inversion produced a G(,+,, with an acceptable analytic form. As a first step. we investigate what can be learned from A:, the first non-trivial J-lrac coefficient. Xi is equal to the sum of the residues of the poles in the self-energy and as such it gives the high frequency asymptotic behaviour of the self-energy via ?&) -- A:/--. A necessary condition for positive definiteness is that A: be real and positive. We shall see that the t-onv~rgeoceof lm A: to rrro as a function of the rulniber r of poles in the Pad4 firtinq t'11ncrion (-all providc information on the quality of the fit and on the anaktir struct.ure of

t h~ true c-orltinua~ionG'.

1.4 Numerical results

flie proccdurc~u.e havv out.linotl in :j.l.:j is .\ ip~cidizat.io[iof the followi~lqyneral Psrl'

pmc-ecfurr. Given a fririction / ant1 a set .-I of '21. input points. apesuppose t.hat 1r.p (.an ;rp- proximate the analytic continuation j by a [r. - 1 jr] rational polynomial f(,).the cocHicier~ts

~rhichare l.let.~rrninetlby solving the linear system of equations {/(fI(~)= fin) : a i .-I).

This problem (:a11 be rut as a matrix inversion in rvhich the kernel .y has elements ~vith ratios as large as

-1 . Thus to reliably perform the inversion ive need a numerical range - \-. 1.e. 2 loqIui;(Iecirnal (iigits of numerical precision. This analysis is general in that no other Pad6 algorit hrrl have less stringent precision requirements. For the case of a self-energy 1.knoivn at the first 2r Llatsubara frequencies above the real line on the irnaqinary axis. Lve have shown that the matrix .Y is given by Eq. (4.3.12).Sincc %,) - &,,. the ratio of the largest to smallest terms in .Y is < = Id2,)' = ((4r- l)dlr. the square of which gives an estimate of the amount of precision needed to invert .Y. Here. that corresponds to

(iecimd digits. To achieve a sufficient level of precision for our numerical work. we implement the Pad4 algorithm using the symbolic computation package MAPLE. Cnder hIAPLE, expression evaluation takes place in software and thereby transcends the limits imposed by hardware floating-point. All computations are performed in base ten to any desired level of precision (we specified Digits := 250 ; ). Sloreover. hIr\PLE is an ideal environment for rapid prototyping since high level matrix data types and routines are available as primitives. and [''is the XSC' pair susceptibilitv

- II[E,-,] + f[Eq-,rtl - 1)f[:,--;l - f K;tif!:,--~l S(k.;,) = - c c [.I. 1.7 lW2E .\I2 \+;I (j;' is, + b-;- \p -

Since the tCare real, the analytic continuation of the self-energy is a meromorphic function with a finite number of simple poles. ail situated- along- the real axis. Calculated in two dimensions on an 3 w Y (If = 64) lattice, its P = 0 component possesses ro = 26 poles. [The number of poles is determined by counting the number of distinct elements in the set C : {b- <,-) - kf p. 01.1 For a particular set of parameters - interaction strength C/W = -0.5, chemical p*

tential p/\V = -0.25. and temperature T/IV = 0.0875 - the test function. Eq. (4.4.3). is calculated in two different ways for the Slatsubara frequencies .~2,.. . . d2r}. First. it

is calculated exactly, as prescribed by Eq. (4.4.7). but with a small, random error (1 + C) applied to each point. Second, it is calculated by truncating the Matsubara sum at an arbitrary cutoff frequency vp >> 1 (much larger than the relevant energy scale of the problem) and then systematically adding back the high frequency contributions up to a given order. That is.

\v here

?;est.. rye let the self-energy, evaluated nt the first 2r 1,latsuabara frequencies according to the trvo scllemes described above, .;erve as the input to the Pade procedure. The resulting

Pade approxiniant is compared to that of the exact function using the logarithmic measure

In practice we choose r) to be a small, but noninfinitesimal quantity (we use q/CC' = 0.008). which has the effect of introducing a slight artificial broadening to the &function peaks of the spectral function. The results of this comparison (for the L = d component) are presented in Fig. 4.1 where F is plotted as a function of r for different values of the random error E = - log,,c (Fig. 4.1, left) and the systematic error E = - loglo l/ju,)* = rn logl, v, (Fig. 4.1, right]. In each graph. a vertical dashed line marks the exact number of poles (ro= 26) in the true self-energy. The most distinctive feature of both graphs is that, at high accuracy (large E), the F curves exhibit a large step at the point r = ro. In the random error case. the E = 120 curve jumps by four decades. This represents an improvement in the Pad6 fit of nearly 40 Figure -1.1: For various levels uf random (left) and systematic error (right1, the quality ot' r he Pad6 fit. cu measured by F, is plotted with respect to the number of poles in the Pai6 npprosirnarlt. The vertical ~dwlledline indicates the number of poles (rll= 26) in th~trw Green's function. In the left-hand plot. error bars (representing the standard deviation of t he 11;ita points over a set of initial random seeds) are smaller than the symbols marking the data points and are not shown. In the right-hand ptot. the dotted line is the best lincar fir through the maximum values of F. orders of' magnitude. In the systematic error case. the result is even more dramatic: thc L: = LOO and E = 120 curves jump by roughIy four and seven decades, respectively. .it these large accuracies, the only inhibiting factor for the Pade inversion is the lack of a sufficient number of poles to reproduce the analytic structure of the true funct,ion. The ,harp step observed in the large E curves marks the point, r = ro, at which the number of poles in the Pade approximant exactly matches the required number. In contrast. when the input points are known to relatively low accuracy, no such feature is observed. instead. the F curves pass smoothly through ro. This would seem to suggest that for self-energies calculated to 'LO. -10, or even 60 decirnal digits of accuracy, the level of error in the input points is still the main obstacle to a successful Pade fit. The usual response to this situation is to increase the number of Pad6 points in an attempt to overcome the intrinsic error limitations (by making the system of equations more and more overcomplete). However, whatever advantage this additional information brings to the Pade inversion is soon outweighed by the accompanying complicatiuns. When a rational polynomial of degree [r - l/r] is used to fit a function with ro < r poles, r-ra zeros of the numerator must coincide with an equal number of zeros in the denominator in order to cancel the extraneous poles. As r - ro grows, it is less and less likely that this cancellation ivill be (.on~pletr.:\ slight nlisplacc~~~entof zeros leads to 'defects' in rvhirh the function r11ot.t~~b~tiv~t~rl 0 ;lti(l x in ;L .small nciqhbourhod. hlorcovcr. it. cannot bc. prdictwl ivhcri.

r - t hpse zero-zero pairs ivill appear. Theorems 110 t?uist conc~rninqtheir overall ~listribur. ion in ilio t.or~lplesplane: see C'h:tpts. 1 I anti 13 of Ref. [5O].j For t.he purposes of ralrulatinq it ip(-t.ral function, they are or little ronscqupnce proviclecl that they lie deep in the c-orr1plt.u plant.. I-Ioivewr. ~vtienthey arc nor. 50 far removed frorrr the rml axis. they can dist.ort r he s$)ectral function aivav from ils ~rorwr. . shape. \\'hen they lip on or 11clar the r~alaxis. rhf?. can give rise to deep troughs of riegati\.e spectral weight and otht2r ?;purious. non-physical features.

I'hp riotcrioration of the Pad6 fit. AS ~liescribecl above. is c.vidcnt in Fi5. -1.1 in ivhic-h

marry of' the F rllrves reach rrlasirna at points rbrst > r13 and t.hen quicklv besin to fall otf for larger r. 1nt.er~stingjy.this b~haviouris much more pronounced in the systematic +Inor rvnse ivhere such maxima occur for each curve. In the random error case. the curves below some error threshold are essentially Aat for ail r. The primary lesson that one should draw from these results is that the addition of PadP points nvI1 beyond the required number is not a useful strategy for improving the Pati4 fit. Unless the exact analytic continuation is already known, there is no way to predict t.hr

value of rbest. LC'(? believe that better results arr achieved by fixing the number of Pad6 points at 2ro and working towards increasing the accuracy with which those input points are csicnlated. Even a small effort there can result in an improvement of several orders of rnagniturie in the fit. Consider Fig. 4.2 in which the spectral function of a Pad6 approxirnant with 26 poles (calculated by specifying the value of the seif-energy at 5'2 hlatsurbara frequencies) is compared to the exact spectral function. In the topleft plot, the accuracy of the input points is given by E = 16 (random error). roughly the number nf digits in a double precision Fortran variable. The fit is particularly poor. Here, the effect of insufficient accuracy is to produce a washed out version of the spectral function which completely lacks fine structure. Even at E = 30 (Fig. -1.2, topright), corresponding to the number of digits available in the largest Fortran data type, the Pad6 inversion is only just beginning to distinguish the main peaks of the spectral function. The bottom-left plot shows the result for E = YO and the bottom-right, the result for E = 120. Yotice that in the bottom-right plot. the fit is essentially exact: even the smallest peaks have been reproduced faithfully. In this example, with r = ro, the Pad6 approximant provides an excellent fit to the 5 I 5 -l : E=BO --- Appmximant : E=120 --- Approxixnant 4 - -Eract 4 - -Exact

3 - 3 - ~(fl,w) : 4K4 : 2 - 2 -

0- 0- A AAA -1 0 1 -1 0

Figure 4.2: The spectral function of the Pade approximant is compared to the exact spectral Function for different levels of error on the initial input points. true lunctiori whenever rhe accuracy of I he input points is better than E - 110. 1.11~ liifficult!- in translating our success in this specific cae to the general problem is that. i!~ real applications. one has no way to judge when sufficient accuracy hubeen achieved. .\lso. in [nost instances. the number ot' poles in the self-ei~ergyis 111tknown.

In rvhat follows, WP ~OPPto ~L~~~TPSSthese ileficierlcies. ll'e b~qinby (i~fininsa loqnrithrrlir rwnsurP of the imaginary part of T hr .J-frac coefficient A;:

\\'e arsned in $4.3that Xi ought, to be real and positive. In a Padi ralculntiorl. ho~vev~r.it is rral-valued only to ~vithinsome small fraction which characterizes the numerical scnsit.ivity of the matrix inversion. .Is ~vcshall soon discover, the convergence of the imaginary part

01' A; to zero (.\ -t x)can ht. used (i) to (letermine when the threshold of accuracy for ilrl (Isact fit has been reached and (ii) to infer the value of r,-, if it is unknown. In Fig. 4.3, we plot .\ ai a function of r for the random and systematic error rases. Over twh plot is superimposed a reference line given by Eq. (-4.4.2).\Vhat we observe is a set. of .\ curvcs that initially follow the reference line but later fan out. spaced according to r heir E values. Our claim is that these curves provide the quantitative measure of Pad6 succts~ rhat has heretofore been lacking. the essential point being that the shape of the curve rcveals the performance characteristics of the Pade inversion in the various r regimes. \Vhen 0 < r < ro, the Pade inversion is matrix dominated and the behaviour of .\ is governed by .\ - 2r log,,(-lr - 1)nT. In this regime. the Pad4 approximant has too few poles to fit the true function and thus the matrix inversion must judiciously arrange the available poles (sometimes apportioning one pole to a region where there should be two or three) to give the best possible fit. In the opposite limit, r > ro, the Pade inversion is error dominated. In this regime, there are more than enough poles to perform an exact fit. but the proper placement of those poles and the determination of their residues is hampered by the finite accuracy to which the input points are known. Me find this reflected in the .\ curves which, for large r, saturate at a value :\ - E (roughly). XIost interesting, though, is the behaviour of :\ in the vicinity of r = ro where the .\ curves in Fig. 4.3 first cross the reference line. In those plots, we see that the ,I curves corresponding to small values of E closely follow the reference line until finite accuracy becomes a limiting factor. The curves then fail below the reference line and become more Fiqurc 4.3: For various levels of rantlorn (left) and systematic prror ( riqht 1. the pararnrttlr .\ is plotted with respect to the number of poles in the Pade approxinlant. The vertical ~i~ashcdline indicates the number of poles (ro= 26) in the true Green's function. The did line ori

or less flat. .-\is E is increased, the r coordinate at which a given .\ furve first deviates from t.he reference line moves to the right until (for some accuracy. Eo say) it coincides 1vit.h

r.0. Hcre, there is a stidden change in behaviour: all .\ curves corresponding t+o accuracies E > Etl cross the reference line at r = ro. Such a crossing signals that there are now both suficient poles in the approximant and sufficient accuracy on the input points to fit I: more or less exactly. lye can verify this interpretation by appealing to Fig. -I. 1. It clearly shoivs

,i large jump st r,, for precisely the same curves that demonstrate a crossing in Fig. 1.3. The results we have described are extremely general and do not dependent on the choicr! of test furlction. For example. we may replace Eq. (4.4.3)with the full NSC T-matrix self- energy

Here. the frequency sums cannot be performed analytically (except as an infinite series of iacreasingly more difficult sums) and thus we do not have a closed form analytical expression for the self-energy. [This is more representative of the usual situation in which the Pad6 method might be applied.] in this case. we know only that its analytic continuation has a finite number of poles along the real axis. \,i]

\'>) -*..

- (1LJ,,+ -r (tun): + - \,:j, - +u,2,\,,) w;l, - + +... 1 + q % + lt21 \I J\ 1 V, i i~/,) tun [lLJn)2 + 3+ . - .I

This ctxpansion is trivial to irrlplernerlt in 11.-IPLE usin5 the computational resources required to execute it are s csplosion of terms at high order. The Pade approximant method can then he appliec to Eq. (4.4.12) calculated in this way. \Ye find that the resulting plot of .I vs. r is identical to that of Fig. 4.3 (rio;ht)vxcept that th~(-rossin~ of the reference line at high accuracy now occurs at r = 1.36. This allows us to deduce that the function has ro = 156 poles. significantly more -.than the 26 poles of Eq. (-1.4.7). [This is a consequence of the lifting of degeneracy in each Q component bruught. lrbout by the renormalization I/( i + c*(Q,vn)) .] \Ye also find that the approxiniant spectral function converges with increasing accuracy to the numerically exact spectral function as calculated by a non-Pad8 method due to Slarsiglio et al. [55]. [This non-PadP method is of limited application since it requires the self-energy to have a very specific form, but for those cases where it is applicable, it can outperform the Pad6 method.] Finally. one interesting feature that could potentially be exploited is that for self-energy values calculated using the 8 function expansion, the value of r which gives the maximum

value of .I roughly tracks rbest (cf. Figs. 4.1 (right) and 4.3 (right)).

4.5 Summary

The Pade procedure is very sensitive to the numerical precision with which the matrix inversion is performed and to the intrinsic error on the input points. Sufficient precision is difficult to achieve in traditional computer languages (e.g. C. Fortran) and so. in many instances, it may be necessary to make use of a symbolic computation package capable of supporting very large precision data types. Likewise, sufficient accuracy is difficult to achieve without a sophisticated computational scheme (e.g. the 8 function expansion) that not's h~!.o11(.1 il simple t runcarion of the llatsu bara Crequency sunls in the self-ener%y. The rrquired lewl of precisior~and accuracy depends on the temperature 'T, which controls the spacing of the llatsubara points. and on the pole count ru. ..in insritfici~rltlevel of accuracy leads to an approxirnant spectral function that lacks tine

-3t r11(.t11rnl (letail or, xorse. onp that exhibits spurious spikes or 1-rollqhs of spectral iveight.

This poses a problem lvhenever ive are interested in the presence of a specific featrlrcl in t ht! . . ~:?w.tr?! f~~~~:ti~>~jn.~.r !!a r>!!.;~t ,>f ;? !ln~nl;tl~tz!~ n

(r. 2 rhcsr). [nstead. wc recommend the use of a Pad6 approximant function having the m-ne number of poles u, the function to be fit. The exact number of poles, when it. is not known, can be lieternlined from the crossing point in a .\ vs. r plot. The crossing also indicates that a sufficient level of numerical accuracy in the input points has been achieved. There are several caveats to the procedure we have outlined. (i) If the true Green's function has a branch cut along the real axis arising from transcendental functions then no .I crossing will ever be observed. since a branch cut of that kind can only be represented by an infinity of poles (ro = x). (ii) The self-energy of the Green's function we are trying to reproduce must have the correct asymptotic form and must be analytic on C\ R:otherwise. the rational polynomial (or continued fraction) form of the approximant cannot reproduce its analytic structure. (iii) The Pad6 method is often used to model a function that is smooth in some region of interest (well away from its poles) and such calculations are rarely performed with more than machine accuracy. Our numerical analysis of the Pad6 inversion. with its prediction of extremely high accuracy requirements, is not meant to invalidate these results. We have applied the Pad6 method to the particularly difficult problem of reproducing the sharp peak structures characteristic of a spectral function whose Green's function has its poles along the real axis. In that case, the poles lie in the region of interest. The precision and accuracy requirements of the Pad6 inversion are greatly reduced if the poles of the Green's function lie deep in the complex plane. Finally, let 11s r~n~eniberthat the startin5 point for our new Pade approach ~vasthe rt1;tlizatiorl that thc c.otlvpr~cnceof the continued fraction coefficients to .allowed' values ran provide a criterion for judging the quality of a Pade approuimant, even if the analytic

3tr1icturc of the function IV~are trying to fit is unknown. In 14.4. ive (iernonstratcd the

~ltilityof this idea usins the A, coefficient. Hotvever. IV~knoiv that there is much additional

~~~tbrrnationthat. car1 htb ~strnctedfrom the renlaininq continued fraction r-oeffir-icnts. In f'i~t~ir~.p~rhaps oilr analvsis ran he tlxt~r~(i~dto inr1111j~ t ! . A!. el. ~tr. Chapter 5

Conclusions and future work

The rornphted ~kctrondynamics of the Hubbard rnodel emerges from tho corrlperitio~l

betiveen the kinetic energy and interaction terms of its hamiltonian. The model describes ari asembly ofelectrons. nioving in a symmetric band ofstates. which interact with one mother ty means of an onsite potential of strength L*. In the AHXI. the tendency of electrons to extend ti~roughoutthe solid is tempered by a negative-[--induced local act ract ion t. hat fitvuors the forrrlation of bound electron pairs. \\.'hen cooled sufficiently. the systenl rriay undergo a transition to a new state exhibiting long-range superconducting (or perhaps CDLV) order.

.-\ t. heoretical liescription of the normal state behaviour of the .AH11 at low ternperntlires is complicated by the strong correlations between the electrons. In Chapt. 2. Lye attempted to address this issue by performing a non-perturbative diagrammatic expansion in which contributions were kept to ail orders in I:. This expansion was based on the E05I method which n-e hoped would provide a controlled, systematic approximation to the hH.11. In the first level of approximation beyond H F (in which tweparticle correlations are included. but three-particle correlations are neglected), the dilute limit of the E041 result yields the so-called T-mat rix approximation in which the two-particle correlations are modelled by repeated pp scattering events. We discovered, however, that the EOhl method is not really a controlled approach to the problem. LVhile it predicts what we believe is the correct 'coarse' diagrammatic structure of the model self-energy. the more subtle issue of which propagating lines in the self-energy ought to be dressed seems to be beyond its reach. This is an important consideration because the renormalization condition of the Green's functions determines the self-consistencv scheme (i.e. it determines the form of the fed-back loop betwwn thp one- and two-particle properties j. \t'e established in Chapt. 3 t. hat the v~rsionof wlf- c-otlsist~ncy~~nforccd cornpleteiy lete ermines the shape of the DOS at the 511pert-ontirlc-tir~g t rrrnsitiorl. T tlc partic~rlar combination of bare and dressed Green's functions predicted by t. he EOlI rnethol-1 c-orrcsponcis to one of six possible T-matrix theories. Of those six. the t.hrw that have a full (.losing line in t.h~self-ener~y are incxpable of producinc a true ~~~per[-o~\(ir~r-rirllr qap at 1::. In dcfition. they rive rise to an unusual (non-merornorphic) analytic ,tr~ict~lr~ for the one-particle propagator which is not amenable to Pade analytic continuation. The EO\I result falls into this group of poorly performing theories. On the other hand. the theorips that have a bare closing line t.xhibit A BC'S-style gap at T,-and are iv~ll-hehavt?ci

,tnalvtically. :\ccordingly. we selected two of those theories - based on self-(:onsistrnry

5chenles (3)and (5) (in the terminology of Fig. 3.6) - as the best candidates. [\'I?rtlrnark that scheme (5)has the additional merit of being number ronserving. The one theory we have yet to rliscuss is the deeply flawed 5SC' T-matrix. so-called because it has bare lines everywhere anti thus no self-consistency effects at all. In C'hapt. :I, ive shoived that the NSC result is equivalent to the phenomenology advanced by SC-R to

ticscribe t, he failure of FL t heocy in attractive two-dimensional systems. The basic physical idea was that, at low temperatures. the electrons are forced into tweparticle bound states below the band by an ercluded region in the phase diagram. even for arbitrarily wcak interactions. \Vhat SVR failed to point out. though, is that the cause of the excluded region is the divergence of the self-energy at T,:. This divergence transforms the TC'L into

it n = L contour through which no other densities can pass. We emphasized that this divergence is unphysical and that it must be corrected. .As it turns out, the SVR mechanism is not at all robust. Even the inclusion of the Hartree shift in r he T-matrix self-energy eviscerates the SVR physics. We suggest. however. that the NSC T-matrix approximation be 'rehabilitated' by introducing feed- back effects. We have found that enforcement of self-consistency constrains the (previously unchecked) growth of the self-energy in the vicinity of the superconducting transition. It does this in such a way that the self-energy remains finite on the TCL. Consequent,ly. the constant density contours of the system move back into the conduction band and the two-particle SVR bound states are destroyed. Moreover, self-consistency pushes back the onset of the pairing instability so that T, is suppressed to zero and thus even the Merrnin-Wagner theory is satisfied. In an etFort to clarify the nature of the resulting superconducting state. wr performed a self-consistent r-alculation at mro temperature usins schemes (3) and (5). \t'e were able io (lo r his ;111altically. \Vhat LVP found is that the wlf-energy and (fouble cwci~pancy;Lrr tl~.er~..rvherebounded and that the densit!, varies continuously along, the self-consistent TC'L. \!'hat rerriains is to estend this ~vorkto t.enlperatures above T,. = 0. We would like to know ivhct hcr r hc pseildngap h~haviol~rthat rv~fo~lnd in YSC' ralc.l~larionspt.r.;i.;t-: in a wlf-

I-onsistcnt 'T-mat rix formulation. To perforni .;t~ch a calculation involves itcratinq t h~ r- rnatrix equatiorls numerically until the! have converged to a self-consistent soh1 tion. It' the one-particle propagator can be determined in this way to very high accuracy at a suficient number of llatsubara points. the Pad6 analytic continuation technique of C'hapt. 4 can bc. ils~dTO fast ract t he DOS. Appendix A

Basic formalism

A.1 Occupation number formalism (second quantization)

\f'e begin by defining an operator ci,, with site index i and spin index u rvhich rve use to c*orlstrlict. ;t ..;t.at.rl

ticscribing fermions on a lat,tice of sites numbered 1 through .\I. Fermi st,atistics are in~posed rw: anticornmutation relations on the operator algebra: i.e. the operator ci, and its adjoint (-,, satisfy

Thus, when i f j or cr # 3. the order of two adjacent operators with indices ia and j3 can be reversed provided we introduce a factor of - 1. \$'hen i = j and a = 3. Eqs. (.A. 1.2) specialize to \Vith these properties in nlind, consider the result of r;] acting on Eq. (:\. 1.1 I. ~VPha1.r

= r' (.t,in,tnll . . . ".\lt".tf~! 1) n(r;3)n''p)

where P = nit + rl~ii n?~i n?~ +. .. + n!,) is the number of operator interchanges needed t ro movr cL-, from the front. to its properly ordered position in the operator cornposit.ion. Sirri ilarly.

ivhere P= n!;+rt!~+- n.;~+n~~+...+n,,- 1. Sow consider c,, and cjLl acting in succession on the same state:

That is. the operator ri,, = C!~C,, annihilates the state if n,, = 0 but returns it unchanged if n,, = 1. It remains to be seen whether n;, can take on values other than 0 or I. To settle that question. we first remark that the operator n;, is idempotent:

Then. given an eigenstate lo) = 1. . n;, . . .) of icia having eigenvalue n,,. Eq. (A.1.7)implies 1:irl;~lly. -inw lo) is arbitrary. it must be that n!,, E {(I, 1 }.

It is i~ppropriate.then, to treat the value n,, as the number of particles with ~pirl(1 ,tt site 1. With such an interpretation. n,, is the omupatton number operator. Eq. (3.I. I) is ;r state of defirlite particle number .\.. = ElC!, rc,,. and Eq. (A.1.;) is a stawrnrnt of thrl t 11riip In addition. Eq. (.A. 1.4) and Eq. (A.1.:) drsc-ribe r;, and I., , CL~rration and destruction npprators.

A.2 Real space representation

..\[I iwsembly of fermions moving on a lattice and interacting via a spin-independent. two- body potential can be described by the hamiltonian

ivhere Latin letters label sites and Greek letters label spin. t,, is the hopping integral

between sites i and j and LyiJtlJf is the matrix element for the interaction potential with respect to Wannier states lij) and li'j'). In order to ensure that the hanliltonian has real energy eigenvalues, those matrix elements must be chosen such that H = KO + is hermitian. First. notice that the kinetic energy term. H~,has an adjoint and the interact.ion term. HI, an atljoint

tvhrre r he final line in carh of Eqs. (.-1.2.2) and (-4.2.3)has beer) reached by r~lab~llinqrlw

-I~Pindices (1 + r' .lnd j i j'), Since the hernlitivity of fi fol1o~r.siror~~ the ir~,ii\.i,iual lwrnlitivity of fro and ifl, it muat be that t,, = t:, and I~,,,~~I= I~~,,,,.Sccon,l, in rtlt1 int.ersction term. the order of t.he operators rvithin each annihilation and creation pair car1 be reversed w prescribed by the anticommutation relation given in Eq. 1.4.1.2a). Such a rt~versal.follu~v~d by a simple relah~llingof the indices (i o j and 1' - j'). prrn~its11s to rccst the interaction term in an equivalent form:

It is clear by inspect.ion that the matrix elements of the interaction potential must cxhibit

the interchange symmetry u,,,~,t = ~~,,,l,t.This requirement (which follows immediately from the identities (ij)= Iji) and li'j') = Ij'i')) is a direct consequence of the indistinguishabi1it.y of particles of s single species. Finalk, in addition to these restrictions. it is common to

suppose that the interaction is local: i.e. o,,,~,l x b,,td,,I. The hermitivity of Fjo and the

locality of together ensure that local particle conservation is obeyed (see Appendix -4.6).

To summarize, our choice of hopping integral t,, and interaction potential ~~,,,t,t is re-

stricted to those which satisfy (i) t,, = t;, . (ii) cil,tjl = c:,,,, . (iii) r,,,tJ1 = c,,,t,I. and (iv) - - vIJ,~,t x d,,d,,t. [An equivalent statement of conditions (ii). (iii), and (iv) is that the inter-

action has the form vi,,,,~ = ~!"~~'6,,tdjj'where u!ocal = € W.] The simplest, possible 11 11 LPEC" (non-trivial) allowable model is the one-band Hubbard model, which specifies only nearest t if i and j arc n.n.'s = 0 othenvise

;~rl(_lan onsitc interaction

(i.t. C1-l = 1-5,,) for real coefficients t 0 and l-. 1; 1; > ..\pplying Eqs. (.-\.2..i) to the han~iltonian.the kinetic energy term becomes

ivhere (1;) indicates that each n.n. pair is summed over once. LVritten in terms ul the number operator ii,, = cluci, (using Eqs. (:1.1.3a) and Eg. (:\.1.7)), the interaction t.errn 1shich, for spin- 1/2 particles, is

Hence. the one-band Hubbard model for electrons on a lattice reads

A.3 Wavevector representation

A D-dimensional hy percubic lattice and its reciprocal are characterized by the basis vectors

(A.3. la) 1' where {C, = (c,. c:. . . . . cp) 1 e: = h',,) is the set of standard orthogonal unit vectors. In tilis cue. the Brillouin zone is ..;imply the c-artesian product

If the lattice is of linear size L. and hence consists of .\I = ( ~/n)~lattice sites. then the

I m11pLete) set

;dm consists of .\I vectors, with each E h' a D-tuple C = (kl.k2.. . . . kD\ characterizing .L Bloch state. t . \Ve can introduce annihilation and creation operators for Bloch states. rcL, and (:,- . v\a C.I the transformation equations

and the corn pleten~ssrelations

where 2, is the position vector of site j and the sums xEare over all E K. &. the kinetic energy part of the total hamiltonian. is diagonal in the wavevector Here we have introduced the dispersion relation

and expressed it in terms of the band-width LV = 4tD. Likewise. K,,the interaction part of the hsmiltonian, can be reexpressed as the sum of all possible scatterings between Bloch states of opposite spin.

Thus. the hamiltonian Eq. (-4.2.9)is equivalent to

A.4 Spin operator

The Pauli matrices form the basis for the spin space of spin-1/2 particles:

If we write the vector d = (az.oY. 0') then the spin operator at site i is given by ivtlere the sum takes c?,~over the values

Tl\tt follo~vin%are five ~lscfuls~~rnrnation ident.ities: . . = (1, a;,,-)

For esaniplc. the dot product of two spin operators can be evaluated using thew results:

If ive are comparing spins on different lattice sites (1 # j).this means that

where we have defined

but if we take i = j then Eq. (A.4.5)gives

That is, the magnitude of the local spin at the site i is an explicit function of the electron occupation there, :\ PPESDIS .4. B.-\SIC' FOR.lf.4 LIS,\f

The sum of this result over all sites.

Ilepcrrlds only on the total number of electrons and the nurnber of doubly occupied sites. C:'onlparing (--1.2.$):vit.h Eq. (--\.-I.10) ive see that t.he interaction term of the hamiltonian

- - .-Itotal spin operator .5 = XIh', has magnitude

or, via Eqs. (.-1.4..5) and (.A.-4.10).

A.5 Flux operator

Let us write the general han Itonian, Eq. (.A..L.l), as asurn H = C,C,K~,over the energy density operator ivllich rlieasures t hc spin o t:o~nponerltof the energ\. at site j. Uow iet 11s (iefine a jlrlr

The rtet jltlt operator is the sun1 over all local fluxes. nnci that it is ~iiagonalin the ivavevrctor representation.

111 going from the first to the second lines of Eq. (.\.5.6) we have taken advantage of ~IIP spatial homogen~ityof the lattice and represented by 0 the site that sits at the oriqin 6. In the last line ive have defined a new function Q having units of velocity. Recall that the dispersion relation is given bv

Clearly. Fc is just its gradient:

4 CG = grad cr = -i 1tOJ,qtr".%' . - Thus, we can immediately identify Cc as the group velocity of a particle with wavevector k and interpret the total flux operator as the net velocity of all the particles on the lattice. For the single band Hubbard model on a hypercubic lattice, the dispersion and group p, - 1 he hopping parameter t has units of energy and ta has units of vt~lucit.>,.i :\wor~iinqIy, t hc n1.t fiux operator is

= 3tn x(sinkln. sin kJn.. . . . sin kin1 iiFd.

-. Sote hat the expectation valuc i,~(l'lo)vanishes (since sine is an odd fl~rlc-tion)when- elver 10) = ni n,,(ri,)*c. represents a state tvhose ocrupation numbers pshibit an in-

to) .I version symmetry: . n)=).lloreover, for each state rhrrr? -t3sists l-ornplementary jtate 10') satisfying (i) ncu = ((31iii,Jo) = (0'1 n-,,l- lo') for all k.c~,~ii) oic1o) = --\dlr/d). and (iii) (01 li-(u)= (o'lh.lb). Consequently. the ~ns~nlbleavcraqc of the net flux is zero.

since we can partition the trace into a sum over those matrix elements which make no can- tribution and a sum over pairs of matrix elements which make non-vanishing contributions of equal magnitude but opposite sign. This tells us that, in the absence of an external field. there is no net momentum (mass flux = rnf1 and no net current (charge flux f= -ei'). A.6 Local number conservation

The equation of [notion (see liB.L.1 and 5B.1.2) for the number operator is n-rit.ten in the I'sing these to evaluate Eq. (-4.6.1) yields the equation of motion

This impiies t,hat the anti-hermitian parts of the energy density operator represent particle sources and sinks. For any local interaction potential. the right-hand-side of Eq. (X.6.3) vanishes and the equation of motion is just

Explicitly. That. is. !.he rate of rhange of t.he occupation of site j by spin n plectrorls is equal t.o i he rate at. u.hich such electrons arrive I j' -+ j)rriinus rho rate at ivhich they leave (1 7 1').I!t red timr ( T - it). lye %et

Notice that if ive sum over all sites j and spins n, Eq. ) becomes

-Thlls any hsrIliltonian of the fort11 given by Eq. (.-\.'.I) which has a local interaction pc~tent.iai

;~r~cl;L hcrrllitian kitletic crlcrgy tcrrn must. conserve t-otal particle number.

A.7 Density of levels

\!'P ~icfinet,he density of levels g(;) to be the number of energy levels with energy value1 -* that arp available to an assembly of non-interacting (I*= 0) electrons.

Sotice that with this definition, any summation that depends on Z only through the dispersion relation c~ can be represented as an integral weighted by the density of levels.

This form is particularly useful when one wishes to consider the limit of infinite lattice size in which the set of wavevectors K becomes dense in B and g(~jbecomes a continuous For .;iniplicity. we shall choose to measure cnergy in units of the band-~vicith \I*;~nri

It.n

unit length (1 = L. Then, rnaking use of the substitution r = - ros k and ds = sin k cik = s%n(k)vL - c:os2 kdk,from which it follo~vsthat

the density of levels can be written as

Therefore, the density of levels in one dimension is and. in tivo dimensions. is

is the (.ornpbte elliptic integai of t,he first kind [5tj].

Lcr 11s acid a subscript D to t.he the ,fensit! of l~v~lsto indicate t.he spatial tfimrlrlsion

of the system: 2 d( I-4u2} lyirh gl (dl = J- as a base case. Eq. (-i.i.10) constitutes a recursion relation for go. . . 1 his reveals several interesting properties of low dimensional lattices. First. gl (51 /'?)= x. S~corlcl.since

we find that Thus. the liensity of levels displays unbounded \.-an How sinq~laritiesin one rlim~esionat

condition on g~(;);Lsstircs us that c hese singularities arc inteqrable. Since gl. g.,. . . . an, ~*onvolutionsinvolvinq g~ and y?. it follotvs that t heae sinqulariti~sare integrated out. f hat isgo (L 1 is bollndd for 1) ;. '2. Third. giwn that

Finally. let 11s define a function that nieasures the density of states from rhe band rdge.

It is clear that. g:ge(;) vanishes for ;< 0 and 1. > 1 since there are no states outside !.he hnrrci. Inside the band (0 < J < 1) we have

in one dimension and. applying Eq. (.4.i.14) to in hiqher (iitnensions. -P he t1i.o factors in the integrand ul Eq. A.7. L 7) are both nou-ztlro 011ly u. hcn

(D+ 1)"J - x 0<1<1 and O< < 1. D

That is. LC- hen O < x < ( D -+- 1 ),,. Hence. we can explicitly restrict the integral to this rarl%e:

Then. just inside the band edge (for some infinitesimal 6)

-- dl: - - - r2J, fi,/ZT- 7rq

and

In general g2ge(t)- (&) Dd2, which we can prove as follows. Let us suppose that - D-1 *That is. y:''(t) = -rn(v t) (as hypoth~sized)with the coefficients ;D 4let~rrrlirled by the r~cursionrelation ;D+I = :l:DLD (-/I = l/'a) where we have defined

\\'e s~lmmsrizethe results for the first few values of D. Appendix B

Many-body formalism

B. 1 Green's function formalism

B.1.1 Time ordering operator

Given an operator .i.let us define rhe parameterized operator .-i(s)= ~".-k". [n \vhat

follows. we shall consider operators with red-tim~-i(it) and ternp~mtnn.-i( T) para1nPtt.r- izations bot h t and r iv~shall loosely call 'time' coordinates). Thev are (Jistinguished by their hchaviour t~nderhermitian conjugation. Since f;' is hermitian, we have

and

The time ordering operator T, orders its arguments according to the parameter s with largest values to the left and includes a factor of - 1 for each interchange of fermion opera- - - tdrs. For t3sample. given operators .-\. B, and C'. 1vr.c (-all use the Hmviside function to ~.ritr ~~spiicitiythe time ordering of tivo of thorn.

I hrc~of theni.

;tnd so on. By definition.

ivhere P is the number of permutations of fermion operators needed to rearrange the prod-

uct as qiven on the left side 1.0 agree with the order on the right side. That is to say. r he arguments of [,he time ordering operator can be treated as if they comniutp (bosonic operators) or anticornmute (fermionic operators).

8.1.2 Equations of motion

'1'1~equation of motion for an operator .i(~)is the differential equation The gweral case the equation of [notion for a time ordered product of n operators).

ran be proved by induction.

B. 1.3 Green's functions

Green's unctions (or response functions) are ensemble averages of various products of op erators. Four or five different kinds of Green's functions are commonly used. i1.e shall consider two: the temperature Green's function

(B.1.9)

and the retarded Green's function

(B.1.10) St at4 I1lorr precisely. r iiese are r he one-particle temperat urc and retarded Green's frrnr- or.Higher order ( ri-particle) Functions are defined in the obvious ivay:

\tee can show that the tcniperature function is a function of t.he time rliff~rrnce:, - 7:.

[\Ye have used the cyclic property of the trace in the fifth line.] In the same ivay. ive can

show that FR is a function of the time difference tl - t.1. It is convenient to make this explicit by writing F and F~ with a single functional dependence, i.e. F(q - r2)and ~~(t~- t,:). In general, each n-particle function depends on 2n - I time coordinates. This is a consequence of the time invariance of the hamiltonian, which allows us to take one of the 2n explicit coordinates as an arbitrary time zero (by simultaneously shifting all coordinates). ttepcnds only on th~three coordinates t = tI - t2.t' = t.: - t3. and t" = t7 - I.!. The temperature function F(r) has the additional property that it is (anti-jperiodic iirldcr 5hift,s 7 c; r + .I. For any T satisfying -.I < 7 < 0 we have

We can account for both the ferrnionic and bosonic cases it' we treat F(r)as having period

In contrast. F~ (t)is manifestly aperiodic; it has a sharp discontinuity at t = 0. B. 1.4 Fourier series for temperature functions

Suppose that Fi T)is a periodic function of 7 having a period of 2.j and satisfyins F(r +.)I= r Fj r 1. 5irlc-t. F is pcriodir, it may brl cspanded in a Fourier seri~s

and. as usnal. tve have used the :ign to keep track of the bosonic and fernlionicr (-;ws simultaneo~isly.For fermionic F, the prefactor is

( 1 if m is odd

and for bosonic F, it is

0 if nl is odd ;(I +prnr)= :(I + (-1)'9 1 if m is even

Hence we can write B. 1.5 Fourier respresentations

LP~.11s ilefinr a new complex-valued function F(:),represented in the upper half plane bv

[n the last line. we have simplified the notation for the matrix elements: (Q,J&P,) = 4, Putting the result of this integration into Eq. (B.1.25). we arrive at

\C.(,an now take Eq. iB.l.27) as the definition for F(:) in the entire complex plane. Sinw

1i is hermitian -- and its eigenvalues real - F(:) is analytic on C 'i 8.

F( :) car! also be expressed as a Stieljes transform,

The spectral function, defined by

is proportional to the magnitude of the jump in F(r)across the real axis (a branch cut in In idditior~,the sp~crralfunction satisfies various sun1 r~iles:e,q

= p C r-.JKm e,, .-I,, Thr latter equality fixes ihe high frequency behaviour of FI:I:

I'llat is. F(ZIhas the asymptotic form

rhc utility of the function F(z)is its close relation to both the Fourier transform. F~I:~. 01 the retarded Green's function and to the Fourier components, F(J,,),of the t.emperaturc Grwn's function. The retarded Green's function is represented by

~rsith3 Fourier transform F~(;)= F(,. + irl) derived from F(:) by evaluating it just abow rhe rpal asis (c.f. Eq. (B.1.23)). The small imaginary part (+irl) is required t.o pnsilrc the ronwrsence of the integral in Eq. (B.L.26).[The sign of the infinitesimal controls the time ordering: e.g. F;l(d)= F(; - irl) is the Fourier transform of the adcanced Creen'a function F" (t) = +r( [.it it).B(o)] ,)B(-t) .] The Fourier components of the temperature Crcrn's f~~nctionarc related to Fi z) hy

S(r1 ~ix- = F(&) . Lw', - If

B.1.6 Analytic continuation

The strategy for performing calculations is typically to work with the Fourier components of the temperature Green's functions (since sums over discrete frequencies are easier to execute numerically than integrations over continuous variables), to identify ~(iu,)= F(;,), and then to analytically continue the function F(:) to the whole complex plane. [Such a function

is guaranteed to be unique since F(d,) is known at an infinite set of points id,}, including

the point at infinity F(w, = xl) = 0.1 Once F(Z) is known. any kind of Green's function can be con~tructedfrom it. [n particular.

Of course. ;In! analytical rxpression for F(;, j will immediately yield f:I z) via t.hc identity

t~ut.in general. most computational schemes will only give us the numerical values of F(**,) at the IIatsubara frequencies and not its functional form. l'nder such circumstances the bcsr. can do is to find a fitting function capable of reproducing the potentially conlplirateti pole structure of F(z).

B.1.7 Moments of the spectral function

The moments of the spectral function are defined to be

Each moment provides information concerning the shape of the spectral function. If the function is zero outside some fixed interval, then it is uniquely determined by its set of moments Dl = {.Mr : r = 1.2. . . . ,111 [ST]. We have already shown in Sect. B. 1.5 that the analytic continuation of the Green's function to the upper half complex plane can be represented in terms of the spectral function, ..In c.spansion of Eq. [ R. 1. I1 in powers of 1 :' :.

~lenlonsrraresthat the elements of %J? are precisely the expansion coefficients of such a series. For concreteness, wc now suppose that F(:) can be represented as a contiruled fraction

Then. in keepins xith the asymptotic form given by Eq. (8.1.34). we may also rvrit~

where we have defined a function 9 called the self-energy. Reconciling Eqs.(B.l.-t:l) and . - iR.I.44) requires that Xi = (i.4. B],). z, = <, and

Finally. an asymptotic expansion of Eq. (B.1.44) gives

Appealing to the uniqueness of the taylor series. we can equate the two expansions ( Eqs. (8.1.A%) :\ PPE.L'LWS and { B. 1 .-f6)) term by term to zet the various moments of the spectral function:

T t~cfirst identification nierely expresses the riorrr~alizationof the spectral function. T hc. second says that the spectral tveighted average of the single particle energy is the sarrip ;~5 t tip non-interacting (?= 0) vaiue. Ttle third tells 11s that Xi is a measure of how rn11t:tl spclcrral ivcight (as compared ~vith the non-interacting case) has been shifted alvay from the energy zero,

Occupation functions

B.2.1 Basic occupation functions

The fermi function is defined by /[XI = (eJz t 1 ) and exhibits the following useful properties:

(B.2. la)

(B.2.1b)

(B.2.lc) (B.2. ld) The proofs are straiqhtfor~vard.Eqs. ( B.2. la) and ( B.2.l b) require onlv n minor rparranqe-

Eqs. (B.2. lc) and (B.2. Id) are immediate consequences of the definition of the ferrni md bose .\lstsubara frequencies: i.e. since e''''. = 1 and r'Jwn = - 1, we get

and

Likewise. the bose function is defined by b[r]= (edz - I)-' and possesses analogous 5[i;, + rj = - f[r] where elJ-a = - 1

The proofs arc. rnorc or less. identical to those in the fermi case: rh~t~vo ftlrlctions can be expressed onr in terrrls of the other i~sin~t hr follotvino t hrw

.-\gain the proofs are straight forward: 8.2.2 Extended occupation functions

The fermi case

Alternativ~ly,ive could start by taking

;is the definition lor f. The advantage is that this form leads rather naturally to a generalization of f which we might call the extended fermi function.

It has the property that

for any permutation o E sym(m). Provided [.hat x, = s,,-1. we can use a partial fraction ~iecornpositianto ~vrite

If u.e let x, 2 r,,, 1 then t.he last line of Eq. ( 8.2.18) becomes a part.ial derivative: i.e.

ivhich defines / recursively with Eq. (8.2.15) serving as the base case.

F71r any iPt (J,,I?. . . . . x,,,) of nz distinct elements we ran partition the arguments of J into groups of distinct values:

.\fr+l times -.W?+l times -.!f, +1 times

Soticc. that this form is cornplet,ely general since the function is symrnet,ric under interchai!ge of its arguments (e.g. f[r, g, :. r. z] = f[r.x.y. I.:] with .If, = 1. .\I, = 0, and .\I, = I). The total number of arguments is given by a sum over the multiplicities.

We now show that this (entirely general) form of f, given in Eq. (B.2.20). can be rct-rievcd from t. he partichlar case OF tiistinct arguments:

This constitutes a complete solution to the recursion formula of Eq. (B.2.19). The extended occupation functions are a useful calculational tool for computing llatsub- ;Ira frequency sums. They allow one to avoid complicated contour integrals, the approach usually taken. Let us start with a simple example.

1 - 2 tanh 3x/2

Sums which cannot be written in terms of extended ferrni functions are often functions of [.heir derivatives. For instance

~vtlichis equivalent to

The bose case

The identical steps can be followed to create an extended base function defined by

ivhich satisfies

and

" 1 p, b@ ..:.. q,.q... ..x2,..., .Tn? .... &,I = n-- [,x . . . 1 . (B.2.28) Jfi! l .!fl+l times -.Lf2+1 times JL+Lrimes 8.3.3 Partial occupation functions

The (iigmnla function is llefin~tdas

\vilere :) is the well known anayric continuation t.o the c.ornple.u plane of t.he factorial

The digamma function is the closed form solution to the infinite summation

Thus. a simple partial summation (fixed p) over bose frequencies can be eval~iated'as follows. Sow usinq fermi frequencies.

\Ye can niefine !he partial occupation functions

and their extended versions

which are evaluated recursively in a manner completely analogous to Eqs. (B.2.19) and (i3.2.27). B.2.4 Theta expansion

\v here the trvo-argument part iai occupation function b'[r . Y] is rclated to b'[x] by

provided that x t y. 'rbe 8 functions provide a closed-form r~presentationof t.he hiqh-

t'rcquency i~symptoticsof il broad class of llatsubara sums. In particular, the sum

can he separated into a finite sum over all 'low' frequencies

and an infinite sum over the remaining frequencies ~vhere,in Eq. ( 6.2.lO j. ive have llsed the fact t.hat the free susceptibility \' adrnits a Laureut cspansior~in the frequency variable

Propagators

8.3.1 Temperature and retarded Green's functions

.-Ipropagator is a particular kind of Green's function in which the ensemble average con- rains an ~qualnumber of annihilation and creation operators. :\s iisual. they romp in rhc tc?niperatu re

itrld retarded

varieties. After some manipulation Eq. (8.3.2)can be put in the form

i~t(jt; j't') = ( (c,, (it), c;,~(it1) })

panicle hole for times t > r'. Thus. the retarded Green's function has a simple intprpretation as the sum of the probability aniplitucies for t tie t.ivo processes where

i i) ari electron of spin c.1 is placed in the lattice at site jl, tirtie t1 and propagates t.o the site j where it is removed at time t > t':

ii) a hole of spin C, is cr~at.edat site j', time t' and propaqat.es to sire 1 nvherc it is filled at t.irne t.

Here 'propagation' is rnorc or less synonymous with evolution in time of the >!stern. It is ~~sr~allytuorc. c-onv~nientro ivork in the temperature formalism sinc~the time order- ins operator can bc made to act on any number of operators. Thus. the 2eneralization to higher order C; reen's functions can be accomplished while avoiding the corn piicated nesting of commutators seen in Eq. ( B. 1.12). For ~xample.the n-particle propagator is definc.d by

B.3.2 Fourier components of the temperature Green's function

The one-particle propagator can be ivritten in terrns of its Fourier cornpor~ent

.-lccording to Eq. (D.3.5) the equation of motion of the free propagator is given by

Substituting Eq. tB.3.5) into the left hand side gives

Then. since the right hand side can be written as \ye set (id,,- <,+ p)~:(k. &,I = 1 or. losing the unnecessary spin label.

B.3.3 Fourier components of the retarded Green's function

The retarded Green's function

can be represented in terms of its Fourier transform

1R &(k,d)- = pL(?,-? ,I G (k:- t, t') .\I -

B.3.4 Connecting temperature and retarded Green's functions

In Fourier transformed wavevector-frequency variables. the one-particle propagator is

Its spectral function has several useful properties. First. from its definition (given in Eq. (B.l.Xl)), it folloivs that the spectral function is real and

\ B.1.32), ~vcfind t.hat the spectral function sat.

- Following IB. 1.6, there exists a functiori G(k. I). wparately analytic in the i1ppt.r Io~vt1rhalf ~ron~plexplanes. such that

. . I he function i' has a spectral rrpresent.ation

rv hich implies that

and

In addition, notice that which follo~vsfrom the fact that .-\(c. ;) is real. Thew conjuqation syr~lrnetriesinliicat~ - that it is rnough to know the \.due of Gik.d,) at LIatsubara frequencies in the upper half plane. The rest can bc recovered by complex conjugation. .is it turns out, th~cah~~ of

.-\Ik. -31 is related to the jurnp in the imaginary part uf G(i.z) across the real axis. Following Eq. (B.l.:lO).iv~tind that

Since the frre propagator has the form

in the cornpiex plane. it looks like

just above the real axis. [Here, P denotes the Cauchy principal value.] Thus, via Eq. (8.3.2-1). n.e discover that the spc(:t.ral function for r he frre propagator.

is n &functiot~peak located at the single-part-icle enersy :;. Fourier transforming t.his rcsult hack t.o reai t.ime. ive qet

This describes s particle with wavevector C and energy <; propagating through the syst.ern .as a wave. The Heaviside factor indicates that the particle ivas introduced into the syst~m

;it t = 0. For the interacting case.

~vhichgives a spectral function

1 2;6(& -

;~icli.i[nall ir11;io;inary parts. the excitations are still Ion=-lived and behaw similarly to :'we p;trtic\rs rsrept for their finite iifetinle:

Soticc also that. since the spectral function is non-nqative for fcrmions. Eq. (B.;l.;lO)says

1 hat the imaginary part of the retarded self-energy must be negative. rhat same condition t.nsures that the r~aitime Green's function in Eq. (B.3.31) is tiecaving ivith increasing tirtre. Appendix C

Functional differentiation

C .1 Functional differentiation

Flcrc. LVP demonstrate how functional differentiation with respect to A fictitious tlstcrr~ai field (-an be used to generate a complete skeleton expansion for the self-t~nc~rg~and the tivo-particle correlator. For the sake of generality. our derivation assumes a local. spin- llependent irlteraction cIj(jr.j'r') from which one can recover the Hubbard result at thr ~ndof our c-alculation by setting

The two-body interaction has a corresponding action

Likewise. an external field v,?(jr. j'r') (non-local in time and space) has an action

J J deX(jj), ()) dr / dr'<:'( j'#, ,T) . 0 0 a 13'

The propagator G(vex) in the presence of such an external field can be written as a series t~sparisionin terms of 1:"" and t.he unpett.urbed propagators C;(rt" = 0). That is.

ivhert? t.he subscript 'con' denotes a restriction to fully connected diagrams. Thus, to second order in t,hc external field strength. we have

or. in the condensed notation.

From here. it is natural to define functional differentiation with respect to

&G.( 1: L'IP) vex =O = [c:Pt)cd2h( lkf con d~+;~('?'.2) (T (I/)]) In t-erms of the t~vo-particlecorrtliat.ion f~lnction.t. his becomes

-1little rr~anipulationallo~vs 11s to recast Eq. (C.l.,$)in a more provocativr form:

This implies that functional differentiation of G can be related to functional differentiation of its inverse (with the addition of two external legs) by

We know. however, that the propagator G(veX)is completely characterized by its proper .-\ PPESDIS C'. F 'SC'TlO.Y.4 L DIFFER E.YTI--\TIO,Y self-energy via the Dyson's equation

the secotld t.tarrn on the right-hand side of Eq. (C.1.12) nnst read

Putting the result for t,hc derivative of (3-I. viz.

6C;

into Eq. (C. 1.9) gives

Finally, comparing this result with Eq. (C.1.S) yields C.2 Skeleton expansion

In order to evaluate dS/'dtex,\re must assume a functional form for the self-energ). [Es- cutinlly. t his choice (let~rniinesthe hasis for t. he diagrammatic ~xpansion.]Let 11s suppose t.hi~t5 is a I'urlctional ot' C;'(I:") ivtrich dcp~ridson the two-body interaction I.. Then. A gcneralizcd chain rule.

ailow us to rc~vritedyi'd~:'~ in terms of a self-enersy-generated effective interactiori

and t.he two- article correlator (via Eq. (C'.1.(3)):

5u bstituting this result into Eq. (C.I. 17) gives an integral equation for the two-particle 1-orrelation function:

If we define a generalized two-particle interaction r according to t.hen tve can eliminate all occurences of the correlation function in Eq. I C.2.I). By hct.orin5 o~ittire external propagating lines. we are left ~vithan integral equation for T.

fiere. we rewrite the correlator t.errn using Eq. (C.2.5) to give

This allows us to express the self-energy in terms of T:

Recall that r goes as .-\ PPESDIS C'. F I'SC'TIO.Y.4 L DIFFERENTI.4TIO.Y so t.hat according to Eq. iC.2.9)

5tart.ing from the lowest order self-energy diagram. this quation uniquely deternlincs nil hiqher order self-energy contributions. Appendix D

Equation of motion met hod

D .1 Linear funct ionals

.\ Iinmr jrrnrtional ,I : (W-- R) + (W--+ R) acts according to

where .J(r.y) is the kernel of J. Composition of two linear functionals. J and E:. goes like

If ~LTextcnd the definition to allow .J : (R x R + R) + (R < R 4 R) to act on functions of two arguments. i.e.

(D.1.3)

then we can think of the kernel of K o J, the bracketed term in the last line of Eq. (D.1.2). as the action of K on the kernel of J,

\\'e may identify Go(jr: j'r') as the kernel of a fi~nctionalC;* which acts on functions exhibit- illy the same periodicity as GUljr: j'r') ao that. in analogy with Eq. (D.1.91. Eq. i D. l.lOs)

(-an be ~vritt.ens cu

rhat is. for any function / satisfying /(Jrj = -/(j. r + .1) we have

dr"&,,ltd;(r- r")f (j"rl')

Since j is arbitrary, we have verified that [Go]'' 3 CiU = id. LVhat is the effect of the reversed composition G' o [GO]''*I Acting on an arbitrary function (anti-periodic in its time coordinate, as before). it gives .An intrqxtion by parts yields the identity

since f(jOi = -/(j.j). Hence. Eq. (D.i.1-1) reacis

Finally. according to Eq. (D.l.lOb), the term in braces is just 0',,1d(r - rt) which implies

that (Goo [G']") f(jr) = j(jr) or Go o = id. Thus, we have demonstrated that [Go]-' serves as both the left and right inverse to Go. D .2 Evaluating the conimut at or of the grand canonical hamil- toliian with an annihilation operator

The qrand canonical harniitorlian can be partitioned into a sum of r.1~0operators. li = li,)+ I<,. ~vh~reICu contains a11 those terms which survive the I-- O limit.

mcl lCI contains those that do not.

[n order to tierive the equation of motion of the n-particle Green's function, ivc shall need to know the commutator of the grand canonical hamiltonian with a single annihi!ation operator. It is convenient to evaluate this conlmut,ztor in two steps:

As to the first term, the identity

implies that .-\PPE-YDIS D. EQI'.-lTIO.Y OF AIOTIOS AIETHOD

The spe(.ialization of Eq. (D,2.-1)to the case of j' = J" qives t.he identity

ivhich ice car1 11sc to evaluate the cornmutator

Following Eq. (D.2.3). the final resuit is

D.3 Equation of motion

From the def nition of the n-particle temperature Green's function t;~llo\vst he equation of motion llaking use of Eq. (D.2.9).the time ordered term can be rea.ritten as

C'ollect.inq all the n-part.icle G rerttls of' Eq. ( D.J.2) on the left-hand side produc~s

= ,,,,,n,, (1.2..... n. 1: 1l.2' .....n'. 1') + O'(1. l')G\ ,..,,(:!.:I .....n:2'.:11 ..... n')

ivhcre the operator [GU]'~is the inverse of the linear operator whose kernel is the free propagator GU = Glr*,o (as we showed in Appendix D. 1).

Here. and in what follows, we employ the condensed notation

The particular cases of greatest interest are the equation of motion for the oneparticle

I,(=) to in(licsre ;I surnnlation ovrr the p unique permutations of the function indicts t.vl~ererruntrib~itions from odd permutations pick up a factor of - I. (-sing a more explicit notation. Eqs. (D.4.lb)and (D.-1.1~)can be written ,u

Our prescription. then. is to terminate the hierarchy of equations by setting C', = O for I r r.That is, we choose to neglect correlations between r,,, or more particles. Before we work this out in detail, let us point out that - due to the spin symmetry of the

harniltonian - it is meaningful to write

Hartree-Fock approximation

Csing Eq. (D.-L.2),the equation of motion for the oneparticle temperature Green's function can be reexpressed in terms of the two particle correlations, or. (.ollectinq all the Green's functions on the left-hand side,

The spin symmetry and spatial homogeneity of the system ensure that the term G'( 1: 1') is particularly sinlple to evaluate:

It is not,hing more than half the clectron density. Hence. with the definition oi is new operator

satisfying

we can write Eq. (D.4.S) in the form

Therefore, when C2 = 0 we have G = eHF.the propagator in the Hartree-Fock ap proximation. This propagator differs from the free propagator only in the energy shift (see Eq. (D.4.Y)) which results from treating the interaction in the mean field approximation. Notice that in the limit of small interaction strengths (U + 0) and low number density (n+ 0). the Hartree-Fock propagator becomes indistinguishable from the free propagator. [In the first limit, the energy penalty for double occupancy of a site becomes vanishingly small. In the second limit, the probability of double occupancy becomes vanishingly small.] .-\ PPE.F'DIS D. EQI':\TIO.V OF AIOTIOS !.IETHOD

Beyond Hartree-Fock

The nest step in our systematic approximation scheme is to take C': O and C'] = 0. To hthgir~.the t \to-particle Green's function, Eq. ( D.:j.

By pxpanclirr5 the left-hand side. rve find that tivo terms can be evaluated with Eq. I D.4.51.

The resulting cancellation removes all &function terms and we are left with

The threeparticle contribution is obtained from Eq. (D.4.3). .\.laking the approximation that Cg = 0, we get Insertin5 t.his result into Eq. (D.-i.l:%), cancelling t,erms. and bringing ali C.',~(l2:1'2') r.0 the left yields

ti-hen a f A. Eq. ( D.4.15) describes correlations between particlcs of opposite spin.

and whet1 c~ = 3, it describes correlations between particles of identica 1 spin.

r7HI: 1 ( I)]- C++(12: 1'2') = -l-G( 1: 2')C'+-( 12: l+lf)+ LVG(1: 1')CV+-( 12: 1 +?I). ( D.4.17)

.-it this point, it is most 11sefu1to covert the equations of motion we have derived from differential to integral form. For example. the equation of motion for the single particle Green's function, Eq. (D.1.10), can be integrated to give

where a summation and integration

is implicit over every bold-face variable m. Notice that in the second line we have made .-\PPESDIS D. EQI'.-1TfO.y OF J1OTZO.y -1IETHOD

the identification

Y(1:3)G(3: 1') = K'+-(11; l'l+) ( D. t 20)

t.o $ve 11s the familiar Dyson equation in terms of the function 5. containing all proper self-enerqy corrections beyond the mean field approximation. For thc. tivo-particle correlations, the corresponding int.egra.1cq uations are

(3:2')

\Ve can exploit the fact that the integral equation for C++depends only on C'+- by sub stituting Eq. (D.-1.22)for C++ everywhere in Eq. (D.4.21). The resulting integral equatiorl for C',, is

Finally, what remains is to solve G. S, and C+- self-consistently. The self-consistency loop is illustrated below.

via (D.4.23) via(D.J.XI) via(D.1.18) via(D.1.23) via (D:1.20) -----+C+- -5 > G -C+- -... (D.4.24)

In practice. this is extremely difficult to do. .-\PPE.YDlS D. EQI'.ATIOS OF .\lOTIOS .\fETHOD D.5 Generalized interaction

It is lls~fulto write the correlation function C1,- in terms of a 5eneraiized interaction. I',

;ic.r.irlg bc~tivtvr~particles of opposite spin: Appendix E

MAPLE code

E.l Pade package

This code creates a SlhPLE package that can be invoked using the command with(pade) ;. The Padc inversion is performed by calling the procedure approx where its first argument is a list. of input points (rlsually hlatsuhara frequencies) and the second argument is a list of t.he corresponding values of the function of interest at those points. The procedure ret,urns a function of a single cornp1e.u variable.

> restart.; > pade := table(): > pade [approx] := proc(xx: : list ,bb : :list) > local x,b,n,p,q,i,sl,s2,v,z,first,second,row; > first := proc(v::vector) > local n,i;

> n : = linalg[vectdim] (v) ; > vector([seq(v[i] ,i=1. .n/2) 1) ; > end : > second := proc(v::vector) > local n,i; > a := linalg[vectdirn] (v) ; > linalg[vectorl ([seq(v[i] ,i=n/?+l. .n)l); > end : > row := proc(z::constant,n::posint,fz::con~tant) 215 local i,j ;

[l,seq(zei,i=l..n-1) ,seq(-zAj*fz,j=l..n)] ;

end : x := vector(xx);

b : = vector(bb1 ; n := linalg[vectdiml(x); if type(n,even) then

v : = linalg[linsolve] (linalg [matrix] (Cseq(row(x[il ,n/Z,b[i]) ,i=l..dl) ,b); p := first(v1;

q := second(v1;

n : = linalg [vectdim] (p) ; sl := [l ,seq(zei,i=l. .n-111 ;

s2 := [seq(zAi,i=l. .n)]; unapply((lina1g [innerprodl (linalg [vector] (sl) ,p) / (l+linalg [innerprod] (linalg [vector] (s2), q) ) ,z) else ERROR('vector must consist of an even number of entries') fi

end : save(pade,cat(libname,'/pade.m'));

restart ;

E.2 Hubbard2d package

The HubbardZd package creates new hI.\PLE array data types that can be indexed by lvavevector and Matsubara frequency. It also provides various routines lor calculating and plotting quantities of interest in the T-matrix approximation of the attractive Hubbard model (two dimensions only).

> restart: > HubbardZd := table(): *** Initialization Routines *** > ~ubbard2d[init] := proc 0 global 't~pe/BZvector','index/GO~method','index/femi,method', 'index/bose,method','index/BZ~method','index/0Z~method2', 'index/int,BZ-method','&=','&==';

' type/BZvectorC : = list (integer) ; 'index/GO,method' := proc(indices,array) local i,j;

i : = ~ubbardZd[ind] (OD(!, indices) 1 ; j := op(2,indices); if nargs=2 then if j>O then

array[i,(j+l)/21 ; else

conjugate(arrayCi, (-j+1)/21) ; fi; else array[i,(j+l)/2] := op(args[3]); fi;

end : ' index/f ermi-method ' := proc (indices,array) global fMax; local i,j ; i : = ~ubbard2dcindJ(op(1, indices) ) ;

j : = op(2, indices) ; if abs(j)>fMax then 0; elif nargs=2 then if j>O then arrayci, (j+l)/21 ; else conjugate(arrayCi, (-j+1)/21); fi; else array [i, (j+l) /2] := op(args [3] 1 ; fi; end :

global bMax; local i,j; i : = ~ubbard2d[ind](op(1, indices) ) ;

if abs(j)>bMax then

0; elif nargs=2 then if j>=O then array [i , j/2+1] ; else

fi; else array [i ,j /2+1] : = opcargs (3. . nargsl ; fi; end :

global fMax; local i; i : = ~ubbard2d[ind] (op (1,indices) ) ; if nargs=2 then array Cil ; else

array ti] := op (args [3] ) ; fi; end :

global fMax; local i, j ; i := HubbardZd [indl (op(1, indices) ) ; .-\ PPE.VDIS E. .\I.APL E CODE

j : = Hubbard2d rind] Cop (2,indices) ) ; if narqs=2 then

array [itj] ; else

array [itj 3 : = op (args [31) ; fi;

end :

'index/int-BZ-method' := proc(indices,array) global fMax;

local i,j ; i := op(1,indices);

j : = Hubbard2d [ind] (op (2,indices) ) ; if nargs=2 then

array [i, j] ; else

array [i , j I : = op (args C31) ; fi;

and : '%=' := proc(left,right) global Digits; if evalf(abs(1eft-right))

print ( ' 2D Hubbard model calculations ' ) ; print ('Kevin Beach' ) ;

print ( ' You must specify the physical parameters ' ) ; print('1attice size L, interaction strength U, chemical potential mu, .-\PPE.VD1.Y E. .ll.-\PLE CODE

temperature T, frequency cutoff FC,');

> print ( 'and set the global variables' ) ;

> print ( ' INPUT-PATH , OUTPUT-PATH , DUMP-PATH , VERBOSE. ' ) ; > print ('To begin, execute the setup0 procedure. '1 ; > end:

> ~ubbard2dLinit-BZ] := proco > global BZ,N; > local i,j ; > 02 := [seq(seq([i, j] , i-1-N. .N) j=1-N. .N)] ; > Hubbard2d [out] ( ' Br ioullin zone vectors initialized' ) ; > end: > HubbardZd [init-BZ81 := prod) > global BZ8,rnult,kMax1N; > local i,j,tmpl,tmp2; > mult := array(BZ,method,l..kMax); > BZ8 := [seq(seq([i,j], j=O. .i) ,i=O..N)] ; > mult[[O,O]] := 1; > mult[[N,N]] := 1; > mult[[N,O]] := 2; > for j from 1 to N-1 do > mult[[j ,011 := 4; > od; > for j from 1 to N-1 do > mult Kj,jl] := 4; > od; > for j from 1 to N-1 do > rnult[[~,j]] := 4; > od; > for i from 2 to N-1 do > for j from 1 to i-1 do > mult[[i, j]] := 8; > od ; .-\PPESD1.Y E. .\I.-\ PL E ("ODE

> od ; > end: > ~ubbard2d[init-C] : = proc()

> global BZ8,bFreqPos ,C, kMax ,bMax ; > local q,v; > C := array(bose,method,l..kMax, l..bMax/2+1); > for Q in BZ8 do > for v in bFreqPos do > C[Q,v] := 0; > od ; > od; > ~ubbard2d[out] ( 'C Matrix initialized. ' ) ; > end: > Hubbard2d [init-GO] := proc (1 > global BZ8,bMax,fMax,fFreqPosx2,kMax,Mat,xi,G0,D~P~PATH; > local k,w; > GO := array(GO,method,l..kMax, l..(fMax+l+bMax)/2); > for k in BZ8 do > for u in fFreqPosx2 do > ~0[k,u] := (I*Mat[u] - xi[k])^(-1); > od ; > od ;

> Hubbard'Zd[out] ('Free Propagator Matrix initialized. ' ) ; > save GO, ".DUMP,PATH,'/'.'CO.m'; > end: > Hubbard2d [init-C] : = proc() > global BZ8,fFreqPosRestr,xi,CO,C,DG,lowerMat,upperMat; > local k,u; > G := array(fermi-method,l..kMax, louerMat..upperMat);

> DG := array (f ermi-method ,1 . .kMax , lowernat. .upperMat ) ; > for k in 028 do > for v in fFreqPosRestr do > ~[k,w] := GO[k,u] ; .-\PPE-VDIS E. Jl.-\PLE CODE

> DG[k,u] := 0; > od; > od ;

> ~ubbard2d[out] ( ' Propagator Matrix init ial ized . ' ; > end: > Hubbard2d [init-Matsubara] : = proc () global Mat,bMax,fMax ,T,fFreq,fFreqPos .fFreqPosRestr ,fFreaPosx2. bFreq,bFreqPos,bFreqStrictPos,1ov~rMat,upperMat; local i; Mat := array(-fMax-bMax..fMax+bMax); for i from -fMax-bMax to fMax+bMax do Mat [i] := evalf (Pi*i*T) ;

od ;

fFreq := [seq(2*i+l, i=-(fMax+1)/2.. (EMU-1)/2)] ; fFreqPos := [seq(2*i+l,i=O..(fMax-1)/2)]; f FreqPosRestr := fFreqPos [lowerMat . .uppeflat] ; fFreqPosx2 := [seq(?*i+l,i=O..(fMax-l)/2+bMax/2)]; bFreq : = [seq(2*i, i=-bMax/2. .bMax/2)] ; bFreqPos := [seq(2*i,i=O..bMax/2)]; bFreqStrictPos := [seq(2*iJi=l..bMax/2)]; ~ubbard2d[out] ( 'Matrix of Matsubara frequencies initialized.' ; > end: > Hubbard2d(init,nO] : = proc() > global nO,M,T,f ,mult ,BZ8; > local k,tmp; > tmp := 0; > for k in BZ8 do

> tmp := tmp + 2*mult [k] *f [k] /M; > od; > no := tmp: > Hubbard2dCoutf ('Free Number density initialized. ' ) ; > end:

> Hubbard2d[init~upndo] := proc() .APPE.YDIN E. .i1.-1PLE CODE

> global BZ8,bFreqStrictPos,XO,U,T,M,rnult,nupndoO,nupndo; > local Q,v,tmp; > tmp := 0; > for Q in BZ8 do

> trnp : = tmp + mult [Q] *(-U*Chi2(Q)

+ T*U-2*(XO[Q,O] '3/(1+U*XO[Q ,011) ; > for v in DFreqStrictP~sdo

> tmp := tmp + ~*T*U^~*~U~~[~]*R~(XO[~,V]-~/(~+U*Y.O~Q,V])); > od ;

> od ;

> nupndo := nupndo0;

> ~ubbard2d[out] ('Density-density correlation calculated. ' ) ; > end:

> HubbardZd Cinit-Theta] : = proc(order : :nomegint) > global T,bMax,Theta; > local l ,tmp; > Theta : = array(2. . (order+l)) ; > for 1 from 2 to (order+l) do > if 1=2 then > tmp := unapply((Psi(bMax/2+1-z) - Psi(bMax/2+1-a)>/(a-z) , (z,a)) ; > else > tmp := DL21 (tmp)/(l-2) ; > fi;

> ThetaCl] := unapply(eva1f (T* (tmp (z/ (2*Pi*T*I) ,O)+(-1) -1 *tmp(-z/(2*Pi*T*I) ,O)1 /(2*Pi*T*I)^l) ,z) ; > od: > Hubbard2d[out] ('Theta functions precalculated. ' ) ; > save Theta, ".DUMP,PATH.'/'.'Theta.m'; > end: > Hubbard2d [init-Sigma01 : = proc () > global BZ8 ,fFreqPosRestr ,Sigma,Sigma0,loverMat ,; > local k,u; > Sigma0 := array(fermi-method,l..kMax, lowerMat..upperMat); > for k in BZ8 do > for u in fFreqPosRestr do > SigmaO[k,w] := 0; > od ;

> od ;

> HubbardZd[out] ( 'Sigma Matrix initialized. ' ) ; > end: > Hubbard2d [init-Sigma] : = proc 0 > global BZ8,fFreqPosRestr,Signta,Sigma0,louerMat,upperMat; > local k,w; > Sigma := array(fermi-method,l..kMax, lowerMat..upperMat); > for k in BZ8 do > for v in fFreqPosRestr do > Sigma[k,u] := SigmaO[k,v] ; > od ; > od ;

> HubbardZdCout] ('Sigma Matrix initialized. ' ) ; > end: > HubbardZdCinit-xx] := proc(0rder::nonnegint) > global BZ ,BZ8, f ,M ,T,bMax ,kMax ,n0,U ,xi, xx ; > local l,ll,k,Q,tmpl,tmp2,xDX,Y,v; > x := array(int,BZ,method,2..order,l..kMax); > xx := array(int,BZ,method,l..order,l..kMax); > for Q in BZ8 do > for 1 from 2 to order do > x(1,Ql := 0; > od; > for k in BZ do > tmpl : = f [k] +f [q-k] -1 ; > tmp2 := (xi [kl +xi [Q-k] > ; > for 1 from 2 to order do od; od ;

Y := sort (taylor(~/(~+~*~),v=inf inity , (order+l) 1) ;

for Q in 028 do

tmpl := [op(Y) [2. .nops(Y)]] ; for II from 2 to order do tmpl := subs(y.ll=x[ll,Q],tmpl); od ; for 1 from 1 to order do xx[l,~] := subs(v=l,trnpl[lj); od ;

Hubbard2dCoutl ('xx coefficients precalculated. ' ) ; save XX, ".DUMP,PATH.'/'.'xx.m' > end:

> Hubbard2d[init_XO] : = proc() > global BZ8,bFreqStrictPos ,XO ,DUMP-PATH ; > local Q ,v ,X00 ; > XO := array(bose,method,l..kMax, l..bMax/2+1); > for Q in BZ8 do > XOO : = return,ChiOR(Q) ; > for v in bFreqStrictPos do > XO[Q,V] := ~0O(I*Mat[v]); > od ;

> XO [q, O] : = Hubbard2d [ChiO] (Q ,O) ; > od ; > HubbardZd[out]('Free Pair Susceptibility Initialized.'); > save XO, ".DUMP,PATH.'/'.'XO.rn'; > end: > HubbardZdCinit-X] := proc () > global BiX,bFreqPos,X,XO; > local Q ,v; > X := array(bose,method,l. .kMax, 1. .bMax/2+1); > for 4 in BZ8 do > for v in bFreqPos do > X[Q,vl := XO[Q,vl; > od: > od;

> HubbardZd[out] ('Pair Susceptibility Initialized. ' ) ; > end: > Hubbard2d [init-xi] := proco > global BZ8,E,fJfp,fpp,fppp,kMax,xi,mu,T,U; > local k; > E := array(BZ,method,l..kMax); > xi := array(BZ,method,l..kMax); > f : = array (BZ-method , 1 . . kMax) ; > fp := array(BZ,rnethod, 1. . kMax) ;

> f pp : = array (BZ-method, 1 . . kMax) ; > fppp := array(BZ,method,l..kMax); > for k in BZ8 do > E [k] : = Hubbard2d [dispersion] (k) ; > xi[k] := E[k]-mu;

> f [k] : = Hubbard2d [f ermi] (xi [k] ) ; > fpck] := -f[k]*(l-f[k])/T; > fpplkl := (fCkl*(l-f[kl)'2-f[k]'2*(1-fCkl))/~'2;

> fppp [k] := (-f [kl* (1-f [k] ) '3+4*f [k] '2*(1-f [k] ) -2 -f [kI -3* (1-f Ck] ) /T-3 ; > od;

> Hubbard2dCoutI ('Single particle energies initializad. ' ) ; > Hubbard2d [out] ('Occupation numbers initialized. ' ) ; > end: *** Calculation Routines *** > global 828; > local k;

> iiubbard2d[out] ('Completed k = . . . ' ) ; > for k in BZ8 do

> Hubbard2d [talc-Sigma0,selectl (k,order) ; > od ;

> Hubbard2d[out](' SiqmaO Matrix initialized. ' ) ; > end: > ~ubbard2d[calc~SigmaO~select]:= proc(k::BZvector,order::nonnegint) > global BZ8,fMw,bMax,bFreq,fFreqPosRestr,SigmaO,T,U,nO,M,Mat,

XO ,GO,DUMP,PATH ,xx ,Theta; local 1,g,v,Q,v,tmp; for v in fFreqPosRestr do tmp := 0; for Q in 02 do for v in bFreq do

tmp : = trnp - Ua2*T*X0(Q ,v] *GO [a-k ,v-w] / ( ~+U*XO[Q , v] ) /M ; od ;

g : = I*Mat [w] +xi [Q-k] ; for 1 from 2 to order do trnp : = tmp - Ua2*xx [l-1 ,Q] *Theta[l] (g) /H; od ; od; Sigma0 [k ,w] := trnp ; save SigmaO, ".DUMP-PATH.'/'.'S~~~~O.III'; od; > end: > HubbardZdCcalc-SigmaO-select-add-order] := proc(k::BZvector,lov::no~egint,high::noanegint) > global BZ8,fMax,bMax,bFreq,fFreqPosRestr,SigmaO,T,U,M,Mat,XO,GO, DUMP-PATH ,xx ,Theta; > local l,g,w,Q,tmp; > for v in fFreqPosRestr do tmp := 0; for Q in BZ do

g : = I*Mat [w] +xi [Q-k] ; far 1 from low to high do

tmp := trnp - Ua2*xx [1-1 ,Q]*Theta[l] (g) /M ; od ;

od ; SigmaOCk.~] = SigmaO[k,v] + tmp; save SigmaO, ".DUMP-PATH.'Jr.'SigmaO.m';

od ; > end: t** Return Routines *** > ~ubbard2d[return,ChioR] := proc(~::BZvector) > global BZ,f ,xi,M;

> unapply(add((f [k]+f [Q-k]-l)/(z-xi[k]- xi[^-k] ) , k=BZ)/M,z) ; > end: > ~ubbard2d[return-Chi0R-old] : = proc (4: : BZvector) > global BZ,f,xi,M; > local k,tmp; > tmp := 0; > for k in 02 do

> tmp := tmp + (f [k] +f [Q-k] -1) /(z-xi Ckl -xi [a-k] ) ; > od ;

> and: > Hubbard2d [return-GOR] : = proc(k: : BZvector) > global xi; > unapply(l/ (z-xi [k] 1 ,z);

> end: > ~ubbard2d[return_Sigmal2] := proc(k: :BZvector) .-\ PPE.YD1.Y E. Jl;\PL E CODE

global EZ,f ,xi,M,T,U; local kk,kkk,Q,Ei,Ej,Ri,Rj,Si,Sj,tmp; tmp := 0; for kk in BZ do for Q in BZ do

tmp : = tmp + U-2* (f [kk] *f [Q-kk] +( 1-f [kk] -f m-kkl ) *f [Q-k] ) / (z+xi[q-k] -xi [kk] -xi @-kkl) /Ma2: for kkk in BZ do

Ei := xi[kk]+xi[Q-kk] ;

Ej : = xi [kkk] +xi [Q-kkk] ; Ri : = f [kkj +f [Q-kk] -1 ;

Rj : = f [kkk] +f m-kkk] -1 ; Si : = f [kk] *f [Q-kk] ;

S j := f [kkk] *f [Q-kkk] ; if Ei&=Ej then

tmp := tmp + UA3*(Si*(Rj-Sj)/(z+xi[Q-kl-Ei)/T + Rj *(Ri*f [Q-kj-Si) / (z+xi[Q-k] -Ei>-2) /M'3 ; else

tmp := tmp + um3*((Rj*(FLi*f[Q-k]-Si) /(z+xi [Q-kl -Ei) - Ri*(Rj*f [Q-k]-Sj)/(z+xi [Q-k]-Ej)) /(El-EJ 1 )/MA3; > fi; > od ; > od ; > od; > unapply (tap ,z) ; > end: > Hubbard2d[return_ChiOGOR] := proc(k::BZvector) > global BZ,f,xi,M,T; > unapply (-add (add( (f [kk] *f [Q-kk] + (1-f Ckk] -f [Q-kk] *f [Q-kl) / (z+xi[Q-k] -xi [kk] -xi Okk] ,Q4Z) , kk=BZ) /Ma2,z) ; .-I\ PPE,YDl.Y E. lf.4 Pl. E CODE

> end: > Hubbard2d [return,~hi0~0~~ofd] := proc(k: ve vector) > global BZ,f ,xi,H,T; > local kk,Qltmp; > tmp := 0; > for kk in BZ do > for Q in BZ do > cmp := tmp + (

/ (z+xi (Q-k] -xi [kk] -xi [Q-kk] ) ; > od; > od ; > unapply(-tmp/Ma2 ,d; > end: > Hubbard2d[return,Chi2GOR] := proc(k::BZvector) global BZ,f ,xi ,M,T; local kk,kkk,Q,Ei,Ej,Ri,Rj,Si,Sj,tmp; tmp := 0; for kk in BZ do for kkk in BZ do for Q in BZ do

Ei := xi [kk] +xi [Q-kk] ; Ej := xi[kkk]+xi[q-kkk] ; Ri := f [kk] +f [Q-kk] -1 ; Rj := f [kkk] +f [a-kkk] - 1 ; Si : = f [kkl *f [Q-kkl ;

S j : = f [kkk] *f [Q-kkk] ; if Ei&=Ej then tmp := tmp + ~i*(~j-Sj)/(z+xi [Q-k] -Ei) /T + ~j * (~i*f[Q-k] -~i)/ (z+xi [Q-k] -Ei) -2 ; else tmp := tmp + (Rj * (Ri*f [Q-k] -Si) / (z+xi[Q-k] -Ei) - Ri* (Rj *f [Q-k] -S j ) /(z+xi[Q-k] -Ej 1) / (Ei-Ej ; .-\ PPE.TDl-Y E. Af.4 PL E CODE

> fi; > od ; > od ; > od ; > unapply(tmp/M"3 ,z); > end:

*** updzte Ro1jti;es **t

> ~ubbard2d[update-C] : = proc () > global BZ8,bFreqPos1C,X,U; > local Q ,v; > for Q in BZ8 do > for v in bFreqPos do

> c[Q,v] := -U*(X[~,V])^~/(~+U*XC~,V]); > od ; > od ;

> Hubbard2d [out] ( ' C Matrix recalculated. ' ) ; > end: > Hubbard2d [update-G] : = proc () > global BZ8,fFreqPosRestr,xilGOlG,DG,Sigma; > local k,v; > for k in BZ8 do > for w in fFreqPosRestr do > G[k.u] := (I*Mat[w] - xi[k] - Sigma[k,u]) > DC[k,u] := G[k,w] - GO[k,u]; > od; > od ; > HubbardZd [out] ('Propagators recalculated. ' ; > end: > Hubbard2d [update-n] := proc () > global n,M,T,f,mult,fFreqPosRestr,BZ8,DC; > local i,k,u,tmp; > tmp := 0; > for k in 028 do > tmp : = tmp + 2*mult [k] *f [k] /M; > for w in fFreqPosRestr do

> tmp : = tmp + 4*T*mult [k] *~e(DG [k,u] ) /M ; > od ; > od ; > n := tmp:

> Hubbard2dCoutj ( ' Number density recalculated. ' 1 : > end:

> Hubbard2d [updat e-nupndo] : = proc ( ) > global BZ8,bFreqStrictPos,XOIXlUlTlMlmult,nupndoO,nupndo; > local Q , v ,tmp ; > tmp := 0; > for Q in 828 do

> tmp : = tmp - mult [Q] *T*U* ( X [Q ,0]'2/ (l+U*X [Q ,0]) - XO~9,0~'2/(1+U*XO[Q,O]) ); > for v in bFreqStrictPos do

> tmp := tmp - 2*T*U*mult [Q] *Re( X[Q,v] -2/(1+U*~[Q ,v] ) - XO~Q,vf~2/(l+U*XOCQlvl~); > od ; > od ; > nupndo := nupndoO + tmp/M; > end: > Hubbard2d [update-Sigma] : = proco > global n0,BZ8,BZ,fFreqPos,X,XOlG,C0,Sigma,Si~ping,SC; > local k,u,Q,v,tmp; > if SC=-1 then

> HubbardZd[out] ('This command can only be executed in SC node‘) ; > else > for k in BZ8 do > for v in fFreqPos do > if SC=2 or SC=4 or SC=5 then > for Q in BZ do > for v in Hubbard2d [bConvFreq] (u) do tmp := tmp + x[~,v]*G[Cj-k,v-v] /(r+u*x[~,vI1 - XO [Q ,v] *GO [Q-k , v-v] /(~+u*xo[Q,v]); od ;

od ; else for Q in BZ do for v in Hubbard2dCbConvFreql (4 do

trnp : = tmp + X [Q ,v] *GO (4-k, v-v] /(1+U*XCQ,vl) - XO[Q,V]*GO[Q-k,v-v] /(l+U*XOCQ,vI > ; > od ; > od ; > fi; > SigmaCk .w] : = Sigma0 [k,w3 - U'Z*T*tmp/M; > od ; > od ;

> ~ubbard2d[out] i'Self -energy recalculated. ' ) ; > fi; > end: > Hubbard2d [update-Sigma-f ast] : = proc() > global n0,BZ8,BZ,fFrrqPos,X,XO,C,GO,Sigma,SigmaO,U,T,M,damping; > local k,w,Q,v,tmp; > if SC=-1 then

> HubbardZd[out] ('This command can only be executed in SC mode') ; > else > for k in 828 do > for w in fFreqPos do > if SC=2 or SC=4 or SC=S then > for Q in BZ do

> for v in HubbardZd[bFreqPosPlus] (w) do .-\ PPESDIS E. AI.4 PLE CODE

> tmp := tmp + X[Q ,v] *G[Q-k,v-w] /(l+U*XCQ,vl> - XO [Q ,v] *GO [Q-k ,v-w]

/(i+U*XOCQ,vl> ; od ; od; slss for Q in BZ do for v in Hubbard2d[bConvFreq] (u) do

tmp : = tmp + X [Q ,v] *GO [Q-k ,v-w] /(r+v*x[o,v]) - XO [Q ,v] *GO [Q- k ,v-u]

/(l+u*xoCq,vl> ; > od ; > od ; > f i ; > Sigmack ,wl := Sigma0 [k,w] - Ua2*T*tmp/M; > od ; > od ;

> Hubbard2d[out] ( 'Self -energy recalculated. ' ) ; > fi; > end: > ~ubbard2d[update-X] := proc (1 > global BZ8,bFreqPos,Mat,U,XO,X,SC; > local Q,v,Xl,pp; > if SC=-1 then > HubbardZd[out]('This command can only be executed in SC mode'); > else > if SC = 1 or SC = 4 then > pp := 1; > elif SC = 3 or SC = 5 then > pp := 2; > else .-\PPESDlS E. JI.4 PLE C'ODE

> > r'i ; > for Q in BZ8 do > for v in bFraqPos do

> X 1 : = X0 [Q , v] +pp*Hubbard2d Whi] (9, v) ; > if SC = 3 or SC = 5 then

> XI := XI + Hubbard2d[~2~hi](~,v): > fi; > X[Q,v] := XI;

> od ; > od ;

> HubbardZd[out] ('Pair Susceptibility recalculated.' ) ; > fi; > end: *** View Routines *** > ~ubbard2dLview-Spectralll] : = proc(br) > global Gbar , N ,mu,n ,U; > local a,b,c,ChemPot; > ChemPot := mu+n*U/2;

> a := CURVES ( [ [ChemPot ,-0.251 , [cheflot ,51 1) ; > b := plot({seq(-Im(Cbar[i ,i] (z+Trbr)/Pi), i=O. .N)>,z=-2.-2,y=-1. -5, t itle='Spectral Functions along the [I. 11 direction' , label~=['~','~(~)~],co1or=~black',thickne~s=l,axe~=80XED,

labelfom= [SYMBOL, 101 ) ;

> c : = TEXT ( [ChemPot ,-0.251 , m' ,ALIGNBELOW) ; > display([a,b,c] ,font=[sYMBOL]); > end: > ~ubbard2d[view~Spectra~lO]:= proc(br) > global Gbar,N,mu,n,U; > local a,b ,c ,ChemPot ; > ChemPot := mu+n*U/2; > a := CURVES ( [[Chem~ot ,-0.251 , [ChemPot ,53j ) ; > b := plot ({seq(-Im(Cbar(i ,0](z+~*br)/Pi) ,i=O. .N)) ,z=-2. -2,y=-1. -5, title='Spactral Functions along the [1,0] direction', Labels= ['u' , ' A(W) '1 ,color='black' ,thickness=l , axes=BOXED, labelf ont= [SYMBOL, 101 ) ;

> c : = TEXT ( [ChemPot, -0.251 , m' ,ALIGNBELOW) ; > display( [a,b , c] ,font= [SYMBOL] ) ; > end: > ~ubbard2d[vieu,X~pectrall]: = proc(br) > global Xbar,N,mu,n,U; > local a,b,c,ChemPot; > ChemPot := mu+n*U/2; > a : = CURVES ( [ [~hem~ot, -0.251 , KhemPot ,51 1) ; > b := plot ({seq(~m(~bar[i ,ifi+I*br)/Pi), i=O. .N)>,z=-2..2,y=-8. -8, title='X Spectral Functions along the [I,!] direction', label~=[~w~,'~(w)~],color='black',thickne~s=l,axes=80XED,

labelf ont= [SYMBOL,101 ) ;

> c : = TEXT( [ChemPot ,-0.251 , 'm' ,ALIGNBELOW) ; > display ( [a, b , c] ,font= [SYMBOL] ) ; > end:~ubbard2d[view~~~pectrall0] := proc (br) > global Xbar,N,mu,n,U; > local a,b,c,ChemPot; > ChemPot := mu+n*U/2;

> a := CURVES ( [ [ChemPot , -0.251 , [ChemPot ,511 ) ; > b := plot((seq(Im(~bar[i,~] (z+I*br)/Pi) ,i.O..-2. 2,-8 .8, title='X Spectral Functions along the [l ,O] direction' , labels= ['w ' , 'A(u) '1 ,color='black' ,thic-es=BOXED, labelf ont= [SYMBOL,101 ) ;

> c := TEXT( [Chem~ot,-0.251 , 'm' ,ALIGNBELOW) ; > display( [a, b,c] ,font=[~YllBOL] ) ; > end: > ~ubbard2d[viev_DOS] := proc(br1 > global DOS,mu,n,U; > local a,b,c,ChemPot; > ChemPot := mu+n*U/2; > a : = CURVES( [[~hem~ot, -0,253,CChernPot,31 11 ; > b := plct (DOS(z+I*br) ,z=-2. .2,y=-1.-3 ,title='Density of Staresc, labels=['~','N(w)'] ,~ol~~r='black',thic~es=BOXED,

labelf cnt= [SYMBOL, 101 ) ;

> c := TEXT([C~~~PO~,-O.~~],'~',ALIGNBELOW);

> dispf ay ( [a, b ,c] ,font= [SYMBOL] ) ; > end:

> Hubbard2d [view-XDOS] : = proc (br) > global XDOS,mu,n,U; > local a,b,c,ChemPot; > ChernPot := mu+n*U/2; > a : = CURVES ( [[ChemPot ,-0.253 , [ChemPot ,3]1 ) ; > b := plot(XDOS(z+I*br),z=-2..2,y=-1..3,title='Density of States', labe1~=['~','~~(w)'],colour~'black',thic~ess~l,~es~BOXED,

labelf ont= [SYMBOL, 101 ; > c := TEXT([ChemPot,-0.251 ,'~',ALIGNBELOW); > display([a,b,c] ,font=[SYMBOL]); > end: > HubbardZd[vieu-sumrules] := proc(br,range) > global Cbar,DOS,mult; > local tmp ,tmpl ,tmp2, tmp3, k ; > if eval(Gbar) = 'Cbar' then > print('G must be continued to the complex plane first.'); > else > tmp := 0: tmpl :=0:tmp2 :=0:tmp3 :=0: > for k in BZ8 do

> tmp : = evalf (int (-Im(Gbar [k[]] (z+I*br)/~i),z=-range. .range)) : > if k= [O ,O1 then > tmpl := tmp: > fi; > tmp2 := tmp2 + mult [k] *tmp ;

> tmp3 : = tmp3 + mult [k] *evalf (int (-Im(Gbar[ka] (z+br*I) /Pi/(exp(z/T)+l)) ,z=-range. .range)) ; .A PPESD1.L' E. .~IAPL E CODE

> od:

> print ('Int (A( [O ,O] ,omega) ,omega=-infinity.. infinity) ' = trnp1) ; > print('Int(N(omega),omega=-infinity..infnty' = 2*tmpZ/M);

> print ( ' 1nt (N(omega)/(ee (beta*omega)+l) , omega=-inf inity . . infinity) ' = 2*tmp3/M) ; > fi; > and:

> Hubbard2d[view_number] : = proc (1

global X ,U,n ,nO ,T ,M,mu ,nupndo ; local i,l,fid,outfile,header,hartree;

1~ubbard2d[updat e-nupndo] () ; 1 := nargs ; header := false;

hartree : = U* (no-1) 12 ; for i from 1 to 1 do if type(args(i] ,relation) then if lhs(args [i] )='FILEi then

outf ile := rhs(args [i]) ; if not type(outf ile,string) then

print ('FILE argument must be a valid string. ' ) ; outfile := 'outfile'; fi; fi; elif args [i] =' HEADER' then header := true; elif args [i] =' SCHARTREE' then hartree := U* (n-1)/2; fi;

if not outfile='outf ile' then ~ubbardZd(out](cat('vriting to file: ',outfile)); fid := fopen(outfile,~PPEND); if header then fprintf (f id, ' T mu mu'

n n(up)n(do) XU0,OI ,O)\nO ; gfprintf (f id,' T mu

n n(up)n(do) X( CO,Ol ,OhL; fi; #fprintf(fid,'%l2,8f %12.8f %12.8f 112.8f %12.8f\n1,8*T,

8*(mu+hartree) ,n,nOa2/4+ nupndo, x[[O .o] ,o] 18) ; fprintf(fid1'%l2.8f X12.8f %12.8f %12.8f X12.8f %12.8f\nC,

T.mu,(mu+hartree) ,n,nOA2/4+ nupndo,X[[O,O] ,011; f close(f id) ; else print('+) = evalf(n,lO)); print ( ' = evalf (nOa2/4 + aupndo ,lo) . evalf (na2/4+ nupndo,10) ) ; print('X((0,OI ,0)) = evalf (X[[O,O] ,01 ,10)*'W^(-I)' , evalf (X[[0,0] ,0]/8,lO)*'t-(-1)'); print('l+~X([~,O] ,0)' = evalf (~+U*X[[O,O] ,0] ,lo)) ; print('mu' = evalf((mu+n*U/2*TILDE) ,lO)*'W' , evalf (8* (mu+n*U/2*TILDE) ,101 * ' t ' ) ; f i ;

end : *** Miscellaneous Routines *** > Hubbard2d [El] : = proc (k: : BZvector) > global xi; > xi [kl ; > end: > HubbardZdCEZ] := proc(k::0Zvector,Q::BZvector) > global xi; > xi [k] + xi CQ-k] ; > end: > HubbardZdCR?] : = proc(k: : BZvector ,Q: :BZvector) > global f; > f [k3 + f [Q-kl - 1; > end: > Hubbard2dCR31 := proc(k::BZvector,kk::BZvactor,Q::BZ~ector~ > global f;

> (f [kk] + f [Q-kk] - I)*f [Q-k] - f (kk] *f [Q-kk] ; > end: > ~ubbard2d[return-X] : = proc (Q : : BZvector) > -global BZ,M; > unapply(add(~ubbard2d[~2](k,~)/(z-~ubbard2d[E2] (k,Q> > , ~=BZ)/M,z) ; > end: > Hubbard~dCreturn-XG] := proc(k::BZvector) > global BZ,M; > unapply (add(add(~ubbard2d[R3] (k ,kk ,Q)/ (z+~ubbard2d[El] (Q-k) - Hubbard2d[~2] (kk ,q) ) ,Q=BZ),kk=BZ) /M42,z); > end: > Hubbardldcadd-error-sigma] := proc(rnag::nonneg,seed::posint) > global BZ8 ,fFreqPosRestr,Sigma; > local k,w,tmp; > tmp : = Hubbard2d[return_random] (lOolOj ; > readlib (randomize) (seed) ; > for k in 828 do > for v in fFreqPosRestr do > Sigma[k,u] := (l+mag*tmp())*Sigma[k,d ; > od ; > ad; > end: > Hubbard2d [analytic-cont inuat ion-G] : = proc( j : : even) > global C,Gbar,DOS,fMax,mult,BZ8,M,Mat,numMat,x~a; > local k,x,i,b; > if jC2 then > print ( 'Too few points to execute the Pade approximant ' ) ; > elif jhmmMat then > print ( ' Not enough Matsubara frequencies ' ) ; > else > Gbar:='Gbar'; > DOS := x->O; > x := [seq(I*Mat[Z*i-l] ,i=L.j)]; > for k in BZ8 do > b[ku] := [seq(Sigma[k,Z*i-l] ,ill.. j)l ;

> Gbar [k[]] : = unapply(l/ (2-xi tk] -apprdx ,btk 0 1 ) (z)),z) ; > DOS := una~~l~(~~~(z)+mult[k]*~bar[k[~~(z),z): > od;

> DOS : = unapply(-2*1m(DO~(z) ) /M/Pi ,z) ; > fi; > print ( ' The propagator has been analytically continued to the complex plane. ' ) ; > end: > ~ubbard2d[analyt ic-cont inuat ion,~,select : = proc (j : : even,k : : BZvector) > global G,Cbar,DOS,fMax,mult,BZ8,M,Mat,numMat,xi,Sigma; > local x,i,b; > if j<2 then > print('T00 few points to execute the Pade approximant'); > elif j>numMat then > print ( 'Not enough Matsubara frequencies' ; > else > x := [seq(I*Mat[Z*i-l] ,i=l..j)] ; > b[k[]] := [seq(Sigma[k,2*i-11 ,i=l.. j)l ; > Gbar[k[]] := unapply(l/(~-xi[k]-a~~rox(~,b~k[]~)(~)),~); > fi; > printlrThe propagator has been analytically continued to the complex plane. ' ; > end: > ~ubbard2d[analyt ic-cont inuat ion-~igma~selectl:= proc(j::even,k::BZvector) > global Sigmabar ,DOS,fMax,mult ,028 ,M,Mat,numMat ,xi ,Sigma; > local x,i,b; > if j<2 then > print('Too few points to execute the Pade approximant'); > elif j>numMat then > print('Nor enough Matsubara frequencies'); > else > x := [seq(I*Mat [2*i-l] , i=l. . j I] ; > b[k[]] := [seq(Sigma[k,Z*i-I] ,i=l..j)] ; > ~igrnabarCkC11 := approx(x.bCkC13) : > fi; > print('The self-energy has been analytically continued to the complex plane. ' ) ; > end: > Hubbard2d [analyt ic-cont inuat ion-X] : = proc (j : : even) > global X,Xbar,XDOS1bMax,mult,BZ8,M,Mat,xi; > local Q,x,i,b; > if j<2 than > print('Too feu points to execute the Pade approximant'); > elif j>(bMax/2+1) then > print ( ' Not enough Matsubara frequencies ' ) ; > else > Xbar:='Xbarl; > XDOS := x->O ; > x := [seq(I*Mat[2*i], i=O..j-I)] ; > for Q in BZ8 do > b[Qn] := [seq(X[~,2*i] ,i=O..j-111 ; > ~bar[QU] := approx(x,b[Pnl) ;

> XDOS : = unapply(XD0S (2)+mult [PI *Xbar C9 1 (2) ,d; > od ; > XDOS := unapply(2*Im(XDOS(z) ) /M/Pi ,z) ; > fi; > print('The propagator has been analytically continued to the complex plane.'); > end: > Hubbard2d [f ConvFreq] : = proc(v: : integer) > global fMax; > local i ,vv ; > if type (v, even) then > vv : = abs (v) ; > alse > vv := abs(v)+l; > fi; > [seq(2*i+l,i=(vv-fMax-1)/2..(vv+fMax-l)/2)]; > end: > ~ubbard~d[b~onvFreq]:= proc(v::integer) > global bMax; > local i,vv; > if type(v ,even) then > vv := abs(v); > else > vv := abs(v)+l; > fi; > [seq(2*i, i=(vv-bMax) /2. . (vv+bMax)/2) 1 ; > end: > ~ubbard2d[b~req~trict~osPlus]:= proc(v::integer) > global bMax; > local i ,vv; > if type(v ,even) then

> vv : = abs (v) ; > else > vv := abs(v)+l; > fi;

> [seq(2*i, i=l. . (vv+bMax)/2)] ; > end: > Hubbard2d[converge] := proc(1: : posint) > global SC; > local st; > if SC=-1 then > ~ubbard~d[out]('This command can only be executed in SC mode') ; > else > SC := readstat('1 - X(COG)G0, 2 - x(cOGO)G, 3 - x(cG)GO, 4 - X(COG)G, 5 - X(GC)G: '1 ; > st := time0;

> while ~ubbard2d[converge-step] () >lo-(-1) do od; > print('elapsed time' = time() - st); > fi; > end: > ~ubbard2d[converge-step] : = proc (1 local tmp; if SC=-1 then Hubbard2d[out]('This command can only be axecuted in SC mode'); else

Hubbard2d [update-n] () ; if U<>O then Hubbard2d[update,~] () ;

Hubbardld[update-Sigma] () ;

tHubbard2d update,^] () ; fi;

tmp := Hubbard2d [convergence-check] () ; ~ubbard2d update,^] (1 ;

tmp ; fi; > end: > Hubbardld [convergence-check] : = proc (1 > global BZ8 ,fFreqPos ,G ,GO ,Sigma; > local k,u,tmp,tmp2; > if SC=-1 then > ~ubbard2d[out]('This command can only be executed in SC mode ' 1 ; > else > tmp := 0; > tmp2 := 0; > for k in BZ8 do > for v in fFreqPos do

> tmp : = tmp + evalf (abs(G [k,w] '(-1)

+ Sigma[k,v]-GO[k,u]'(-1))) ; tmp2 := tmp2 + 1; od ; od ;

Hubbard2d [out] (cat ( 'Convergence measure : ' , convert Ctmp/tmp2 ,name) 1 1 ; > tmp/tmp2; > fi; > end: > ~ubbard2d[ChiO] := proc(Q::BZvector,v::even) global BZ,f,fp,fpp,fppp,xi,M,Mat,T; local k, trnp ,x ; tmp := 0; if v=O then for k in BZ do x : = xi [k] +xi [Q-k] ; if x&=0 then tmp := tmp - fp[k] + fpp[k] /2*x - fpppck] /3*xm2; else

tmp := tmp + (f [k] +f [g-k] -1 1/ (I*Mat (v] -xi [k] -xi [Q-k] ) ; fi; od; else

tmp : = add((f [k] +f [Q-k] -1) /(I*Mat [v] -xi [k] -xi [Q-k] > ,k=BZ) ; fi; tmp/M ; end : Hubbard2d [Chi21 := proc(Q : : BZvector) global BZyf,M,T; local k,kk,tmp,x,y; trnp := 0; for k in BZ do for kk in BZ do

x : = Hubbard2d [E2] (k,Q) ;

y : = Hubbard2dLE21 (kk ,Q) ;

tmp := trnp + (Hubbard2d [R2] (kk ,Q)*f [k] *f [Q-k] - ~ubbard2d[R2] (k ,a) *f [kk] *f [Q-kk] ) / (x-y) ; > fi; > od ; > od; > tmp/Ma2; > end: > HubbardZd[DChiJ := proc(Q::BZvector,v::even) > global kMax,T,M; > local k,u;

> T*add(add(GO [k, u] *DG[q-k, v-w] ,u=Hubbard2d[f ConvFr~ql(v) ) ,k=BZ) /M; end: > Hubbard2dED2Chil := proc(Q::BZvector,v::even) > global kMax,T,M; > local k,w; > T*add(add(DC [k, w] *DG [Q-k, v-uh=Hubbard2d[f conv~raq](v) ,k=BZ) /M; > end: > ~ubbard2d[dispersion] := proc(k: :BZvector) > global N;

> evalf (-(cos(k[l] *Pi/N)+cos(kC2] tPi/N) )/4) ; > end: > Hubbard2d [f ermi] : = proc(z) > global T; > l/(exp(z/T)+l) ; > end: > Hubbard2dLbosel : = proc(z! > global T; > l/(exp(z/T)-1) ; > end:

> Hubbard2d[ind] : = proc (k: : BZvector) option remember; global N: local ml,m2,trnp; rnl := abs(k[1]); m2 := abs(k[2]); while ml>N do rnl := m1 - 2*N od; while m2>N do m2 := m2 - 2*N od; if mlml then trnp := ml; rnl := m2; m2 := tmp fi; l+m2+ml*(m1+1)/2; > end:

> Hubbard2d [out] : = proc(x) > global VERBOSE; > if VERBOSE = 1 then > print (x) ; > Pi; > end: > HubbardZd [setup] := proc0 > global L,M,N,U,mu,T,n,nO,FC,bMax,fMax,kMax,k,mult, xi,f ,fp,fpp,fppp,Mat ,louerMat,upperMat,nMat, damping, DUMP-PATH ,INPUT-PATH , OUTPUT-PATH ; > forget (HubbardZd [ind] ) ; > if DUMP,PATH='DUMP,PATH' then DUMP-PATH := " fi; if INPUT,PATH='INPUT-PATH' then INPUT-PATH := " fi; if OUTPUT,PATH='OUTPUT-PATH' then OUTPUT-PATH := " fi; if type(L,odd) then ERROR('Lattice size must be an even number.') fi; if LFC then Max := fMax - I; else bMax := fMax + 1 ; fi; if loverbfat = '1owerKat' or upperMat = 'upperMat' then loverMat := 1;

upperMat : = (fMax+ 1 ) /2 ; fi; numMat := upperMat-lowerMat+l ; if 1ouerMatCl then ERROR('1overMat cannot be less than 10 fi; if upperMat>(fMax+l) /2 then

ERROR( cat ( 'upperMat cannot be greater than ' ,convert( (fMax+l) /2,string) ) ) f i; if lowerMat>upperMat then ERROR('1overMat cannot be greater than upperMat.0 fi; kMax := (N+l) *(N+2) /2;

A := 0; damping := 0; > Hubbard2d [init-021 (1 ; > Hubbard2d [init,BZ8] () ; > Hubbard2d [init-Matsubara] 0 ;

> Hubbard2d [init-xi] () ; > Hubbard2d [init-no] 0 ; > end: > save(Hubbard2d,cat(libname,'/Hubbard2d.m')); > restart; References

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