Laczos-Haydock Recursio Bor: Feb 893 i Székesehérvár, Hugary- Died: 5 Jue 974 i Budapest, Hugary Corelius Laczos
From a abstract mathematical viewpoit, the method for puttig a symmetric matrix i three-diagoal form which is readily diagoalized was iveted by Laczos i 95. This is eough to calculate the Gree s fuctio of systems with a fiite umber of states, ad the algorithm is powerful ad coveiet eve for large matrices provided that they are sparse, that is, most elemets vaish. Haydock exteded the method ad developed applicatios to physical systems with may degrees of freedom.
Gree s Fuctio of ay System H= Hamiltoia, possibly represetig a complicated may-body system We seek G z u u, z E i ( causal) z H u ay ormalized state i Hilbert space We may always write : H u a u b u, u u, u u a u H u b u Hu u u b H a u H a u b u ad we ca proceed with 3 3
u H a u u u b H a b a u H u u H u a u b u b u u u u u, u u a u H u b u H a u b u sice the phase of b ca be chose at will, u u b H a u b u u H a u b u b 4 4
the ext step H u a u c u b u b3 u3 c u H u This might appear a borig way to get equatios of icreasig complexity, but here the clever idea first comes ito play, sice H u a u b u c u H u H u a u b u b3 u3 5 5
Geeral recursio relatios ivolve 3 sites H u a u b u b u a u H u b H a u b u u H a u b u b Hece, we have a geeral algorithm to fid the local Gree s fuctio of ay state u, mappig it to site of a chai of u that ca be geerated stepwise, with their levels a ad hoppigs to their earest eighbors. The chai is semi-ifiite, that is does ot ed, uless the system has a fiite Hilbert space or the choice of u was extremely clever. 6 6
Local Gree s Fuctio for a Chai ote: all sites are differet! b b b 3 b 4 Ay statioary quatum problem, with ay umber of degrees of freedom, ca be mapped ito a solvable liear chai oe-body tight-bidig model exactly. 7 7
ote: all sites are differet! b b b 3 b 4 u, u, u, u, site orbitals 3 a, a, a, a, site levels 3 b, b, b, hoppig it egrals 3 Hu a u b u b u We seek H a b b a b b a b3 b a b G z u u, z E i ( causal) z H G z Herglotz ( positive dos) 3 3 4 8 8
We eed G, the elemet H b E a b 3 b3 E a3 b4 ( E ) of E a b b E a b b E a 4 4 G ( E) D E D ( ) ( E) D E a b b E a b Det b E a b3 b3 E a3 b4 b E a 4 4 D E a b b E a b3 Det b3 E a3 b4 b E a b 4 4 5 b b b 3 b 4 Both are secular determiats represetig chais: recursio D starts with E-a b b b3 b 4 9 9
Expad by Laplace rule alog first lie: D E a b b E a b Det b E a b3 b3 E a3 b4 b E a 4 4 D E a b b E a b3 Det b3 E a3 b4 b E a b 4 4 5 b b E a b D ( E a ) D ( b ) Det b E a b3 b E a 3 3 b b E a b3 E a b3 E a b3 b3 E a3 b4 b3 E a3 b4 Det b 3 E a3 b4 bdet b4 E a4 b5 bdet b4 E a4 D ( E) ( E a ) D ( E) b D ( E)
G E D D E E D E ( Ea ) D b D E G E D ( E a ) D b D D ( E a) b D G E ( E a ) b G G E G E ( E a ) b G b ( Ea ) ( E a ) b G ad iteratig, cotiued fractio! G E Ea b Ea b3 Ea b4 E a3 E a4 b
G E Ea b Ea b3 Ea b4 E a3 E a4 b G E ( E a ) b G Simple case: For the uiform chai, with a ad b b, 4b E 4b E G E ( E) E b G E b ad oe gets back the semiellyptic level
Example: rectagular bad ( ) Ca we compute a cotiued fractio for this case? This is a iverse problem! 3 3
Momets The Laczos method is related to the momets method, which is itself worth cosiderig for its coceptual ad practical importace. Let > be ay state of the system, ad (ω) =< δ(ω H) >= /π ImG(ω) Gz ( ) z H I may models oe ca compute the momets d ( ) d ( H) H Relatio to Gree s fuctio: k H k G( z), z maxe k k k z k z Actually this gives a good descriptio outside the cotiuum, but a poor descriptio iside. ( H) Im Hi 4 4
The short time behavior of the evolutio operator elemet G(t) depeds o the first momets, that are ofte easier to compute; the log-time (t) is ofte eeded ad eve the asymptotic log-time tred is of iterest, but that iformatio must be sought elsewhere (see below). Lettig deote the umber of momets that we ca compute, we face the problem of usig them i the best way to represet the shape of the spectrum. Oe possibility would be choosig a fuctioal form which depeds o parameters ad imposig the values of momets. However, the Va Hove sigularities (whose positio ad ature is ot kow a priori) will prevet the uiform covergece of the procedure. A far more powerful method exists! From the kowledge of momets oe ca derived a fiite chai represetatio. 5 5
By the M + -site chai Hamiltioia H M of the form 3 3 3 Hu a u b u b u b b b 3 b 4 u, u, u, u, site orbitals a, a, a, a, site levels b, b, b, hoppig itegrals H a b b a b b a b3 b a b 3 3 4 p oe ca formally work out expressios for the kow H i terms of the ukow matrix elemets by matrix multiplicatio: H a, M H a b M p M
,,, the yield a, b k,, a very simply as k k follows: directly; oce a is foud, = a b b H a, M H a b M the from a ( a a ) b a 3 3 3 3 We ca cotiue i this way as log as we please: for every kow momet, equatio, ukow 7 7
rectagular bad ( ), odd d ( ) ( ), eve a, b 4 8 8
Trucated cotiued fractio + Loretzia broadeig i chages Im to Im a i a i ( a) The rectagular (E) with =.5 ad a broadeig δ =.5 (dashed) compared with the cotiued fractio approximatios usig =5 deomiators (left), = deomiators (cetre), =4 (right).icreasig the agreemet improves to some extet, but oe ever gets the right edges. Such a artificial procedure is too rough for may purposes; it is pathological, e.g. the secod ad all the eve momets of the approximate (E) diverge. 9 9
Trucated cotiued fractio + Loretzia broadeig Such a artificial procedure is too rough for may purposes; it is pathological, e.g. the secod ad all the eve momets of the approximate (E) diverge. Alterative: E ( E) ( x) dx histogram : fit by cotiuous fuctio, the d( E) (E)=. de Good 'almost everywhere', fails at Va Hove sigularities. Actually the theory of asymptotic Fourier trasforms shows that (t) for t depeds o sigularities! The Va-Hove sigularities are uiversal features, idepedet of the details of the Hamiltoia, oe ca gai them from a simpler model i the same uiversality class. Simpler model =Termiator
Cotiuum limit ad Va Hove sigularities: isert termiator tail G E b Ea b Ea b3 Ea b4 E a 3 te ( ) tail Ea te ( ) 4 The trucated cotiued fractio is the ratio of polyomials. To fid them efficietly, we eed Pade theory.
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Pade approximats The idea is: approximate f(x) by ratioal fuctios, rather tha polyomials as i MacLauri s method P[ m,, x] m p px px pmx qx qx qx Sice this depeds o m++ parameters oe ca impose that the followig series expasio holds: f ( x) P[ m,, x] c x c x m m m m That is, the approximat must have the first m++ coefficiets right. This ca be achieved by expadig P[m,,x] i power series ad solvig a system of equatios for the ukows p ad q.
Already with a few terms, Pade approximats are very ice 4
How to divide the cotiued fractio (i.e. Put it as a simple ratio) u, u, u, u, site orbitals 3 a, a, a, a, site levels 3 b, b, b, hoppig itegrals 3 Hu a u b u b u chai eigestates are f, [ ] m m m m m H a b b a b b a b3 b a b m 3 3 4 The -site chai Hamiltoia reads: H u a u b u b u. therefore, the the amplitudes f are give by the Schrodiger equatio: H f a m b m b m E f ( a E ) f b f, ( a E ) f b f b f,, f. m m ote: eed otherwise. We f all f
Defie the amplitudes ormalized to f P ( E ), P, P ( E ). f o The first equatio reads: ( a E ) ( a E ) b P ( E ) P ( E ). b The secod equatio reads: ( a E ) P ( E ) b P ( E ) b P ( E ) P ( E )... is a polyomial of degree. ( a E ) P ( E ) b P ( E ) b P ( E ) P =polyomial i E of degree. 6
Furthermore we ow show that P are orthogoal polyomials E, weighted over the DOS (E) * The wave fuctios are Orthoormal: f f but also * m f f m m It is coveiet to chage * de ( E) P ( E) P ( E), ( E) LDOS m m de E E.. to a eergy itegral usig * * f f de E E f f de E E f * f m m m m f P ( E ) f P ( E ) f 7
f f * m m f Combiig P ( E ) with de E E f f * * * m de E E fm f de E E f fpmp * * m m de E E P P de ( E) P ( E) P ( E), ( E) LDOS P =orthogoal polyomials i E 8 8
It ca be show (see Appedix A below) that itroducig orthogoal polyomials Q, Q, Q ( a E ) Q ( E ) b Q ( E ) b Q ( E ), b b Q ( E) D ( E) E a b b E a b D Det b E a b3 b3 E a3 b4 b4 E a4 b b b 3 b 4 that is, itroducig the chai with the first atom removed, b b 3 b 4 E a b b E a b3 D Det b3 E a3 b4 b4 E a4 b5 D E Q E G E Pade' approximat D E b P E 9 9
Missio: get termiator tail isert termiator tail ito fiite chai get tail t(e) from kow termiator ( E) isert t(e) i G G E Ea b Ea b3 Ea b4 Ea3 4 ( b E a te) Q E ( a E ) P ( E ) b P ( E ) b P ( E ), P G E where b P E ( a E ) Q ( E ) b Q ( E ) b Q ( E ), Q tail isertio: a a +b t ( E)
The termiator is a solvable model with kow Gree's fuctio ( E) idea: get t(e) ad apped to -site chai Gree's fuctio We evaluate the trucated termiator E : we ca fid the couterparts of a ad b that we call, from the Hamiltoia or from momets, Full termiator: if we stop at we get: E E E E i i i E E 3 E E E 3 E i 3 4 E 3 E t( E) 4 G 4 E 3 E Tail isertio: + t ( E) produces E E, which is kow 4
Startig from the trucated termiator: E E E E the tail isertio: + t ( E) produces, which is kow. We eed to fid t ( E), the tail! E From the r. r. ( E ) ( E ) ( E ) ( E ) ( E ) ( E ) ( E ) ( E ) ( E ) ( E ) ( E ) ( E ) oe fids: ( E ) ( E ) ( E ) ( E ) ( E ) ( E ) ( E ) ( E ) Settig, ( E ) ( E ) ( E ) ( E ) E ( E ) ( E ) ( E ) ( E ) ( E ) ( E ). 3 3
It is time to perform the tail isertio: + t ( ) produces E E E, which is kow. E ( E ( t ( E)) ) ( E ) ( E ) ( E ( t ( E)) ) ( E ) ( E ) Simplify: elimiate - stuff with r.r. E ( E ) ( E ) ( E ) ( E ) ( E ) ( E ) ( E ) ( E ) ( E ( t ( E))) ( E ) ( E ) ( E ) ( E ) ; ( E ( t ( E))) ( E ) ( E ) ( E ) ( E ) simplify 33
E ( E) The exact E is kow; ow we ca get the tail! t ( E ) t ( E) ( E ) ; ( E ) t ( E) ( E ) E E icidetally,compared with E tail is up ad dow For the same reasos, it holds that Formula for the tail Isertig the tail G E Q b P Q E bt E Q E G E b P ( E ) ( E) ( E ) ( ) ( ) ( ) ( E ) b t P ( E ) ad isertig t(e) we trasplat the tail ito the approximate cotiued fractio. 34 34
The Laczos-Haydock method is very good for oe-electro problems, also with complicated geometry. It is very useful for iteractig problems too, but there is a problem of error propagatio for log chais (>4 sites, say) The method is very useful for the precise determiatio of groud state ad a few excited states of iteractig systems- eigefuctios ad eigevalues. Two-sites chais project oto the groud states very efficietly
Simplest Recursio (two-sites Laczos) for groud state Trial state f E f H f V f ( H E ) f f H E V E f H f H i subspace: E V V E f Low eigestate better state f 36
Recall: Appedix A: proof of the Pade approximat for G D E a b b E a b Det b E a b3 b3 E a3 b4 b E a 4 4 Defie Δ = D ad Δ the shorteed chai with the first sites, up to site. E a b E a, Det, b E a E a b Det b E a b, 3 b E a E a b b E a b Det b E a b 3 b E a b 3 3 4... b E a 4 4 X These are the deomiators of G approximated with the trucated chais. 37
Det E a b 3 b E a b b E a b b E a X expad + (E) usig the last row: 38
E a b 3 ( E a ) Det b E a b b E a b b E a E a b 3 b Det b E a b b E a b b E a E a bdet E a b 4 3 b E a b 3 3 b E a b b X
E a bdet E a bdet E a b 4 3 b E a b 3 3 E a b 5 4 b E a b E a b 4 4 3 b b X b E a b 3 3 b E a X (E)=(E-a ) (E)-b (E) + - compare with the r.r. b P ( E ) ( E a ) P ( E ) b P ( E ) There is a clear similarity, that we exploit. We verify: (E)=(E-a ) (E)-b (E) solved by + - (E) b b b P ( E), (E) P 4
We verify: (E)=(E-a ) (E)-b (E) solved by + - (E) b b b P ( E), (E) P first r. r. : ( a E ) P ( E ) b P b P ( E ) multiply * b ( a E ) b P ( E ) b P b b P ( E ). b P, b b P give: ( a E ) ( E ) b ( E ) O. K. geeral r. r. : ( a E ) P ( E ) b P ( E ) b P ( E ), multiply * b b ( a E ) b b b P ( E ) b b b P ( E ) b b b b P ( E ) b yields ( a E ) ( E ) b ( E ) ( E ) (E) b b b P ( E) D ( E) for site chai 4 4
(E) b b b P ( E) D ( E) for -site chai D E G E we eed D E for site chai: D E but D E is just obtaied from D by removig site. D E a b b E a b Det b E a b3 b3 E a3 b4 b4 E a4 b b b 3 b 4 P, P ( a E ) P ( E ) b P ( E ) b P ( E ) (E) b b b P ( E) D ( E) 4
b b 3 b 4 E a b b E a b3 D Det b3 E a3 b4 b4 E a4 b5 Itroduce orthogoal polyomials Q, Q, Q ( a E ) Q ( E ) b Q ( E ) b Q ( E ), b b Q ( E) D ( E) G E D E Q E D E b P E Pade' approximat 43 43