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Bethe's Contributions to Solid State Theory: A Tribute H R Krishnamurthy

T h is a r tic le a tte m p ts to a c q u a in t th e r e a d e r w ith th e se m in a l c o n tr ib u tio n s m a d e b y H a n s A lb r e c h t B e th e to th e a re a o f so lid sta te th e o r y . It sh o u ld b e r e a d in c o n ju c tio n w ith th e o th e r tr ib u te s to B e th e 's w o r k th a t a p p e a r e d in th e O c to b e r 2 0 0 5 is su e o f R e so n a n ce . 1 . In tr o d u c tio n H a n s B eth e a ctu a lly sta rted h is p h y sics ca reer a s a `S o lid S ta te T h eo rist'! H e jo in ed th e U n iv ersity o f M u n ich in 1 9 2 6 a s a p ro t¶e g¶e o f A rn o ld S o m m erfeld . H e q u ick ly m a stered th e n ew ly fo u n d ed q u a n tu m m ech a n ics (b eca u se, a s h e la ter reca lled , h e w a s n o t b u rd en ed b y k n o w in g th e recip es o f th e ò ld ' q u a n tu m th eo ry !) a n d sta rted a p p ly in g it to cu ttin g ed g e p ro b lem s in so lid sta te a n d a to m ic p h y sics. H e o b ta in ed h is d o cto ra te in th eo retica l p h y sics in 1 9 2 8 . F o r h is th esis resea rch h e stu d ied th e p a ssa g e o f electro n s th ro u g h cry sta ls. T h is su b ject w a s o f g rea t in terest a t th a t tim e sin ce q u a n tu m th eo ry h a d su rp risin g ly p red icted th a t electro n s, fo rm erly th o u g h t to b e p a rticles, co u ld , lik e lig h t, a lso b eh av e a s w av es, a n d D av isso n a n d G erm er h a d ju st th en p erfo rm ed th eir p io n eerin g ex p erim en ts w h ich v eri¯ ed th is b y d em o n stra tin g th a t cry sta ls d i® ra cted electro n s in m u ch th e sa m e w ay a s X -ray s. B eth e d ev elo p ed th e d eta iled th eo ry to ex p la in th ese ex p erim en ts. T h is w o rk led to h is ¯ rst p u b lish ed p a p er [1 ], o n th e `T h eo ry o f d i® ra ctio n o f electro n s b y cry sta ls'. D u rin g th e v ery n ex t y ea r h e cem en ted h is p la ce in th e h isto ry o f so lid sta te th eo ry w ith h is sem in a l w o rk en titled `S p littin g o f term s in cry sta ls' [2 ], o n e o f th e ¯ rst

RESONANCE  November 2005

H R Krishnamurthy is at the Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore. He is also an honorary professor at the JNCASR, Bangalore. His research interests are primarily in the area of quantum many body theory, especially strongly correlated electron systems.

Keywords Crystal field theory, Bethe’s Ansatz, Bethe–Peierls method.

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GENERAL ARTICLE

Of the first 18 papers of Bethe, 11 are in the area of solid state physics. These helped set up the foundations of the modern Quantum Theory of Solids.

a p p lica tio n s o f g ro u p th eo ry to q u a n tu m m ech a n ics (see th e n ex t sectio n fo r a m o re d eta iled a cco u n t o f th is w o rk ). O v era ll, a lth o u g h o n ly 1 4 o u t o f h is 3 0 0 o r so p a p ers a re in th e a rea o f so lid sta te p h y sics, th ey co m p rised th e m a jo rity o f h is in itia l p a p ers (7 o u t o f th e ¯ rst 1 1 , 1 1 o u t o f th e ¯ rst 1 8 , th e rest b ein g in w h a t o n e m ig h t ca ll à to m ic a n d m o lecu la r p h y sics') a n d h elp ed to set u p th e fo u n d a tio n s o f th e m o d ern Q u a n tu m T h eo ry o f S o lid s. In w h a t fo llow s, I sk etch b rie° y th o se o f B eth e's co n trib u tio n s to so lid sta te th eo ry w h ich a re readily available in E n g lish tra n sla tio n s [3 ]. 2 . C r y sta l F ie ld th e o r y

The Hamiltonian of an isolated atom/ ion is invariant with respect to the continuous group of rotations in 3 dimensions.

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A n a to m o r io n p la ced in va cu u m is in a n en v iro n m en t w h ich is b eliev ed to b e ìso trop ic'; i.e., a ll d irectio n s in sp a ce a re eq u iva len t. C o rresp o n d in g ly, its H a m ilto n ia n (i.e., its en erg y ex p ressed a s a fu n ctio n o f th e co o rd in a tes a n d m o m en ta o f its electro n s) is sy m m etric o r in va ria n t w ith resp ect to ro ta tio n s (eg ., o f th e co o rd in a te a x es). M a th em a tica lly, in th e q u a n tu m m ech a n ics o f electro n s in a n a to m / io n , th e set o f a ll p o ssib le th ree-d im en sio n a l ro ta tio n s is rep resen ted b y a `co n tin u o u s sy m m etry g ro u p ' [4 ] u n d er w h o se a ctio n th e H a m ilto n ia n is in va ria n t. T h is h a s th e co n seq u en ce th a t th e n et sp in a n g u la r m o m en tu m (S ) a n d o rb ita l a n g u la r m o m en tu m (L ) o f a ll th e electro n s, a n d th eir z co m p o n en ts (u n less sp in -o rb it in tera ctio n s a re im p o rta n t, in w h ich ca se th e to ta l a n g u la r m o m en tu m (J ) a n d its z co m p o n en t) a re `g o o d q u a n tu m n u m b ers' th a t ca n b e u sed to la b el th e a to m ic en erg y eig en sta tes. W h en th e q u a n tu m n u m b ers a re n o n -zero , th e en erg y eig en sta tes ex h ib it ch a ra cteristic d eg en era cies. T h e sim p lest su ch ex a m p le sh ow s u p in th e q u a n tu m m ech a n ics o f th e h y d ro g en a to m . J u st fro m th e fa ct th a t th e p o ten tia l seen b y th e electro n m ov in g a ro u n d a ¯ x ed n u cleu s is sp h erica lly sy m m etric o r iso tro p ic, o n e ca n sh ow th a t th e electro n ic orb ita ls o r q u a n tu m sta tes


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ca n b e ch a ra cterized b y a set o f fo u r q u a n tu m n u m b ers. T h ese a re th e p rin cip a l q u a n tu m n u m b er n (w h ich ta k es va lu es 1 ,2 .....) w h ich d eterm in es th e `size' o f th e o rb ita l, th e a zim u th a l (o r o rb ita l a n g u la r m o m en tu m ) q u a n tu m n u m b er ` (w h ich ta k es in teg er va lu es fro m 0 to n ¡ 1 ) w h ich d eterm in es th e m a g n itu d e o f th e o rb ita l a n g u la r m o m en tu m , th e `m a g n etic' q u a n tu m n u m b er m (w h ich ta k es va lu es fro m ¡ ` to + ` ) w h ich d eterm in es th e o rien ta tio n o f th e o rb ita l o r th e z-co m p o n en t o f th e o rb ita l a n g u la r m o m en tu m , a n d th e sp in q u a n tu m n u m b er s (w h ich ta k es va lu es § 1 = 2 ) w h ich d eterm in es th e z-co m p o n en t o f th e sp in a n g u la r m o m en tu m . F u rth erm o re, o n e ca n sh ow th a t th e en erg y m u st b e in d ep en d en t o f m a n d s a n d ca n d ep en d o n ly o n n a n d `. H en ce ea ch en erg y lev el E (n ;`) m u st h av e a d eg en era cy o f 2 (2 ` + 1 ). [In th e h y d ro g en a to m , th e en erg ies, g iv en b y th e B o h r fo rm u la E n = ¡ m e 4 = (8 ² 20 h 2 n 2 ), a re a lso in d ep en d en t o f `, lea d in g to a h ig h er d eg en era cy, o f 2 n 2 , b u t th is is d u e to a n a d d itio n a l sp ecia l sy m m etry o f th e h y d ro g en a to m p ro b lem , a n d is n o t g en eric.] In ca se o f a to m s w ith h ig h er a to m ic n u m b er, o n e m u st in clu d e th e e® ects o f electro n -electro n in tera ctio n s [4 , 5 ]. If th is is d o n e a p p rox im a tely, in th e fo rm o f a n av era g e, self-co n sisten t p o ten tia l seen b y th e electro n s, w h ich is still ta k en to b e sp h erica lly sy m m etric, th en th e sa m e d eg en era cy stru ctu re a s o u tlin ed a b ov e fo r th e h y d ro g en a to m em erg es, b u t w ith th e en erg ies d ep en d en t o n ` a s w ell. T h e lev els m u st n ow b e ¯ lled w ith th e req u isite n u m b er o f electro n s (eq u a l to th e a to m ic n u m b er) resp ectin g th e P a u li-ex clu sio n -p rin cip le (a cco rd in g to w h ich n o tw o electro n s ca n o ccu p y th e sa m e o rb ita l, i.e., h av e a ll th e sa m e q u a n tu m n u m b ers). T h e ex isten ce o f th e d eg en era cies m en tio n ed a b ov e th en lea d s [5 ] to th e so ca lled `sh ell stru ctu re' o f a to m s w h ich is resp o n sib le fo r th e sy stem a tic a n d p erio d ic a ltern a tio n in th e p h y sica l a n d ch em ica l p ro p erties o f th e elem en ts a s b ro u g h t o u t b y th e p erio d ic ta b le. W h en th e electro n


The symmetry under rotations in 3d is responsible for the ‘shell structure’ of atoms.

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n u m b ers (eq u a l to th e a to m ic n u m b ers) h av e th e sp ecia l va lu es o f 2 , 1 0 , 1 8 , 3 6 , 5 4 , a n d 8 6 , co rresp o n d in g to th e n o b le g a ses H e, N e, A r, K r, X e, a n d R n resp ectiv ely, a ll th e d eg en era te lev els co rresp o n d in g to a set o f f n ;`g a re co m p letely ¯ lled , co rresp o n d in g to `clo sed sh ells', w h ich a re en erg etica lly v ery sta b le a n d h en ce in ert. F o r o th er a to m s, o n e g ets p a rtia lly ¯ lled sh ells, w h en ce th e rem a in d er o f th e electro n -electro n in tera ctio n e® ects fo rces th e electro n s in th e o u term o st, `va len ce' sh ell, to co -o p era tiv ely a rra n g e th eir m o tio n a cco rd in g to th e H u n d 's ru les; i.e., in su ch a w ay a s to ¯ rst m a x im ize th eir to ta l sp in S , a n d th en (co n sisten t w ith th is) th eir to ta l o rb ita l a n g u la r m o m en tu m L , (a n d , w h en sp in -o rb it e® ects a re im p o rta n t, th eir to ta l a n g u la r m o m en tu m J ). A cco rd in g ly, th e g ro u n d sta te w ill h av e a d eg en era cy o f (2 S + 1 )(2 L + 1 ) (red u ced to (2 J + 1 ) if sp in -o rb it in tera ctio n s a re in clu d ed ). T h is stru ctu re is resp o n sib le fo r a ll th e w o n d erfu l va riety o f ch em ica l, m a g n etic, o p tica l, a n d o th er p ro p erties o f d ifferen t a to m s a n d io n s in ìso tro p ic' (i.e, in g a seo u s, a n d , fo r m a n y p u rp o ses, in liq u id -lik e) en v iro n m en ts [5 ].

An atom or ion in a crystalline environment is symmetric only under the point group of that crystal.

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In h is m o n u m en ta l w o rk o n th e `S p littin g o f term s in cry sta ls' m en tio n ed a b ov e [2 , 3 ], B eth e ra ised , a n d a n sw ered co m p letely, w ith h is ch a ra cteristic th o ro u g h n ess, th e q u estio n a s to in w h a t w ay th e a b ov e fea tu res ch a n g e w h en th e sa m e a to m / io n is in a cry sta llin e en v iro n m en t [6 ]. T h e ch a n g e is n ecessita ted b y th e p resen ce o f th e cry sta llin e electric ¯ eld (C E F ), a risin g fro m th e electro sta tic fo rces o n th e electro n s o f th e sp eci¯ c a to m / io n u n d er co n sid era tio n d u e to th e o th er io n s su rro u n d in g it. T h e sy m m etry (g ro u p ) u n d er w h ich th e a to m ic/ io n ic H a m ilto n ia n w h ich in clu d es th e C E F is in va ria n t is th e sm a ller, d iscrete, p o in t (g ro u p ) sy m m etry o f th e cry sta l, o n e o f 3 2 p o ssib ilities (th e `C ry sta llo g ra p h ic P o in t G ro u p s') [4 , 6 ] d ep en d in g o n th e cry sta l ty p e. U sin g h is k n ow led g e o f th e th eo ry o f rep resen ta tio n s o f th e cry sta llo g ra p h ic p o in t g ro u p s [4 ], B eth e co m p letely so rted


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o u t th is p ro b lem , cov erin g th e ca ses o f sm a ll, in term ed ia te a n d la rg e C E F . T h e w o rk h a s b een ex ten siv ely u sed ev er sin ce, a n d is o n e o f th e co rn ersto n es o f m o d ern so lid sta te th eo ry [4 ]. It is o f d irect ex p erim en ta l releva n ce in u n d ersta n d in g th e sp ectra (esp ecia lly electro n p a ra m a g n etic reso n a n ce [7 ]) o f io n s a n d im p u rities in cry sta ls, a n d is a sta rtin g p o in t fo r th e m o d ellin g o f m a n y cry sta llin e so lid s. A s a n illu stra tiv e ex a m p le, co n sid er a n a to m / io n in a cu b ic (sim p le-cu b ic(S C ), fa ce cen tered cu b ic(F C C ), o r b o d y cen tered cu b ic(B C C )) cry sta l. T h en th e C E F h a s th e p o in t g ro u p sy m m etry o f a cu b e, i.e., it is in va ria n t u n d er a ll th e sy m m etry o p era tio n s th a t ta k e a cu b e in to itself, eg ., ro ta tio n s b y m u ltip les o f 9 0 o a ro u n d th e x , y o r z a x es. A s a co n seq u en ce, a s B eth e sh ow ed , th e 5 `3 d -o rb ita ls' (co rresp o n d in g to n = 3 a n d ` = 2 ) w h ich w o u ld b e `d eg en era te' in a n iso la ted io n a s ex p la in ed a b ov e, g et sp lit in to o n e g ro u p o f th ree lev els, referred to a s t2 g o rb ita ls, (co rresp o n d in g to th e o rb ita ls la b elled d x y , d y z a n d d z x a cco rd in g to th eir a n g u la r d ep en d en ce) a n d a n o th er g ro u p o f tw o lev els, referred to a s e g o rb ita ls, (co rresp o n d in g to th e o rb ita ls la b elled d x 2 ¡y 2 a n d d 3 z 2 ¡r 2 a cco rd in g to th eir a n g u la r d ep en d en ce). In a cry sta l o f ev en low er sy m m etry, eg ., a tetra g o n a l cry sta l, th ese w o u ld g et sp lit fu rth er. T h ese `sp lit lev els' h av e n ow to b e ¯ lled w ith electro n s a cco rd in g to (a m o d i¯ ed v ersio n o f th e) `H u n d 's ru les', a n d th e co n seq u en t electro n ic co n ¯ g u ra tio n s p lay a cru cia l ro le in d eterm in in g th e p ro p erties o f cry sta ls co n ta in in g tra n sitio n m eta l io n s in w h ich th e 3 d o rb ita ls a re p a rtia lly ¯ lled . A p a rt fro m th e elem en ta l tra n sitio n m eta l cry sta ls, in sta n ces o f su ch cry sta ls in clu d e fa m o u s recen t ex a m p les lik e th e h ig h -T c cu p ra te su p erco n d u cto rs, a n d th e `m a n g a n ites' w h ich sh ow `co lo ssa l m a g n eto -resista n ce'. F igu re 1 d ep icts th e a b ov e in th e co n tex t o f th e d o p ed m a n g a n ite m a teria l L a 1 ¡x C a x M n O 3 , w h ere fo r x = 0 , a ll th e M n io n s h av e a ch a rg e o f + 3 , w ith fo u r d electro n s, a n d


In a cubic crystal, the 5 ‘3d orbitals’ are split into one group of 3 levels and another group of 2 levels.

Crystal field effects are essential for understanding properties of crystals, especially those containing atoms/ions with partially filled shells.

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Figure 1. The phase diagram of La1-xCaxMnO3 in the doping x and temperature T plane (courtesy M Uehara, K H Kim and S W Cheong). Various kinds of anti-ferromagnetic insulator(AF), paramagnetic insulator(PI), ferromagnetic insulator(FI), ferromagnetic metal(FM) and charge/orbitally ordered insulator(CO/O O*) are the phases shown. O and O* are orthorombic (Jahn–Teller distorted) and orthorombic(octahedron rotated) structural phases. The figure also illustrates the splitting of the 3d levels of Mn due to the CEF and how these are occupied for x=0 and x=1.

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fo r x = 1 , a ll th e M n io n s h av e a ch a rg e o f + 4 , w ith th ree d electro n s. F o r in term ed ia te va lu es o f x , th e M n io n s ° u ctu a te b etw een th e tw o co n ¯ g u ra tio n s, a n d th is is resp o n sib le fo r th e ex tra o rd in a rily co m p lica ted p h a se d ia g ra m th a t is a lso sh ow n in F igu re 1 . T h e cry sta l ¯ eld e® ects th a t B eth e d rew o u r a tten tio n to n ea rly 7 5 y ea rs a g o h av e to b e in clu d ed fro m th e o u tset in co n stru ctin g m o d els fo r u n d ersta n d in g th e p h a se d ia g ra m a n d th e p ro p erties o f th e p h a ses [8 ]. 3 . P a ss a g e o f F a st P a r tic le s th r o u g h M a tte r A y ea r a fter B eth e's w o rk o n cry sta l ¯ eld th eo ry ca m e th e sem in a l p a p er `T h eo ry o f th e p a ssa g e o f fa st co rp u scu la r ray s th ro u g h m a tter' [9 ] In th is p a p er B eth e stu d ied th e p ro b lem o f h ow ch a rg ed p a rticles trav ellin g a t h ig h sp eed s lo se en erg y a s th ey p a ss th ro u g h m a tter. T h is w a s a n im p o rta n t p ro b lem a t th e tim e b eca u se th e rela tio n sh ip b etw een th e en erg y a n d th e `ra n g e' o f th e p a rticles (i.e. th e av era g e d ista n ce trav elled b y th em b efo re th ey a re sto p p ed ) w a s a n im p o rta n t to o l fo r m ea su rin g th e en erg y o f th e p a rticles. B eth e sim p li¯ ed co n sid era b ly th e q u a n tu m th eo ry o f a to m ic co llisio n s p ro p o sed


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b y M a x B o rn in 1 9 2 6 , a n d u sed it to ca lcu la te th e cro ss sectio n s fo r a ll th e d i® eren t sca tterin g m ech a n ism s ex p erien ced b y th e ch a rg ed p a rticles (electro n s, p ro to n s, ® p a rticles,...) a s th ey co llid e w ith th e electro n s a n d n u clei co m p risin g th e m a tter, su ch a s ela stic co llisio n s, in ela stic co llisio n s, io n iza tio n p ro cesses, etc, a n d o b ta in ed , in h is ow n m o d est w o rd s, `sa tisfa cto ry to g o o d ' a g reem en t w ith ex p erim en t. S o m e id ea o f th e im p o rta n ce o f th is w o rk ca n b e g lea n ed fro m th e a m a zin g sta tistics th a t th is a rticle w a s cited [1 0 ] 5 4 5 tim es d u rin g th e p erio d 1 9 6 1 -1 9 7 5 , a n d 9 1 tim es in 1 9 7 5 -7 6 , fo rty -¯ v e y ea rs a fter its p u b lica tio n !

Bethe calculated the cross sections for all the different scattering mechanisms experienced by charged particles moving through matter.

4 . T h e B e th e A n sa tz In th e v ery n ex t y ea r, w h ile v isitin g E n rico F erm i a t R o m e, B eth e w ro te th e fa m o u s p a p er Ò n th e th eo ry o f m eta ls I: E ig en va lu es a n d eig en fu n ctio n s o f th e lin ea r a to m ic ch a in ' [1 1 ] w h ere h e p ro p o sed th e celeb ra ted `B eth e's A n sa tz'. A lth o u g h h is p a p er is p h ra sed in term s o f th e H eitler{ L o n d o n th eo ry o f ex ch a n g e p ro cesses a p p lied to a ch a in o f a to m s, in term s o f m o d ern term in o lo g y B eth e o b ta in ed in h is p a p er all th e 2 N eig en sta tes a n d eig en va lu es o f th e sp in -1 / 2 n ea rest n eig h b o u r H eisen b erg H a m ilto n ia n o n a `rin g ' (i.e., a ch a in w ith p erio d ic b o u n d a ry co n d itio n s) o f N sites (w ith co u p lin g J ) ex a ctly a n a ly tica lly fo r a rb itra ry N ! T h is w a s a n a sto u n d in g a cco m p lish m en t th a t w a s w ay a h ea d o f h is tim es, a n d fo rm s th e fo u n d a tio n fo r th e b u lk o f th e ex ten siv e resea rch w o rk o n èx a ctly so lu b le 1 -d m o d els o f q u a n tu m m a n y -b o d y th eo ry ' th a t h a s sin ce fo llow ed . It is d i± cu lt to a p p recia te th is w o rk w ith o u t so m e ex p o su re to th e q u a n tu m m ech a n ics o f sp in -1 / 2 o p era to rs, so so m e m o re d eta ils a b o u t th is w o rk a re g iv en sep a ra tely, in B ox 1 . 5 . M o d e r n T h e o r y o f M e ta ls T w o y ea rs a fter th e `B eth e A n sa tz', in 1 9 3 3 , ca m e tw o


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B o x 1 . S o m e D e ta ils a b o u t th e B e th e A n sa tz The Heisenberg Hamiltonian for an N-site ring of spin-1/2 atoms can be written as H =¡J

S~ n ¢ S~ n +

XN

1

=¡J

n= 1

XN

( S nz S nz + 1 + S nx S nx + 1 + S ny S ny + 1 )

n= 1

=¡J

XN

[S nz S nz + 1 +

n= 1

1 + ¡ (S S + S n¡ S n+ + 1 ) ] ; 2 n n+ 1

´ S~ 1 . P z The z-component of the total spin operator, S z ´ n S n commutes with H , and the values it takes (from N = 2 to ¡ N = 2 in steps of 1 ) can hence be used as a `good quantum number' for labelling its eigenstates. The state in which all spins are up, which we will denote as j-i ´ j " ;" ;::: " i has S z = N = 2, and is easily veri¯ed to be an eigenstate of H with eigenvalue E - = ¡ N J = 4. If J is positive, corresponding to the ferromagnetic case, this state is the lowest energy state; if J is negative, corresponding to the antiferromagnetic case, it is the highest energy state. where S~ n is a spin-1/2 operator at site n , and S~ N

+ 1

States with smaller values of S z are obtainable by starting from this reference state and °ipping spins down, say D of them, corresponding to S z = N = 2¡ D , at an arbitrary subset of sites f m 1 ;m 2 ;:::m D g . We will denote this state, which can be obtained for example by letting D spin lowering operators act on j-i, as jm 1 ;m 2 ;:::m D i ´ S m¡ 1 S m¡ 2 :::S m¡ D j-i. There are as many distinct such states as the binomial coe±cient N

C

D

´

N ! : D !( N ¡ D ) !

The eigenstates corresponding to D spins down should hence be expressible as the linear combination X m

jm 1 ;m 2 ;:::m

D

iÃ ( m 1 ;m 2 ;:::m

D

):

1 ;m 2 ;:::m D

It is useful to think of the down spins as `particles' with a hard-core (so that no more than one of them can be placed at a site) , whence Ã ( m 1 ;m 2 ;:::m D ) is their `wave-function' on the discrete lattice on which these particles move. It is not hard to see that the (x-x, y-y) or transverse part of the Heisenberg Hamiltonian makes the particles move to the left or to the right with an amplitude of J = 2. Whereas the z-z part of H gives a site energy of J to each particle, and an interaction energy of ¡ J (i.e., attractive in the ferromagnetic case and repulsive in the anti-ferromagnetic case) when a pair of particles come onto adj acent sites. Box 1. continued...

62


GENERAL

ARTICLE

When D =1, i. e, there is only one down spin, there are no interaction e®ects and it is not hard to see that Ã ( m ) satis¯es the `SchrÄo dinger equation' E Ã ( m ) = E - Ã ( m ) + J [¡ Ã ( m ) +

1 ( Ã ( m ¡ 1) + Ã ( m + 1) ) ] : 2

This has the solution Ã ( m ) / e x p ( ik m ) , i. e. , the eigenfunctions are `plane waves' with wave-vector k . The periodic boundary conditions, requiring that Ã ( m ) = Ã ( m + N ) imply that k can only take the values k = 2 ¼ º =N where º is an integer. The corresponding energy eigenvalues are E k = E - + ( J ¡ J co s ( k ) ) : When D =2, i. e. , there are two down-spin particles, they move in the form of the same independent plane waves when they are far away. But because of the interactions that become operative when they come on near neighbour sites, they can exchange their momenta or wave-vectors, and also have a scattering phase-shift in their wave-functions. Indeed, Bethe showed that in this case the solution for the wave-function can be written in the form Ã ( m 1 ;m 2 ) / [ e i(k 1 m

1+

1 2+ 2

k 2m

Á 1 ;2 )

+ e i(k 2 m

1+

k1m

1 2¡ 2

Á 1 ;2 )

]; m

1

< m 2;

with 2 co t( Á 1 ;2 = 2) = co t( k 1 = 2) ¡ co t( k 2 = 2) determining a `two-particle phase shift' as a function of the wave-vectors k 1 and k 2 of the particles, and the energy eigenvalue being given by E ( k 1 ;k 2 ) = E - + ( J ¡ J co s ( k 1 ) ) + ( J ¡ J co s ( k 2 ) ) : The requirement of periodic boundary conditions, namely Ã ( m 1 ;m 2 ) = Ã ( m 2 ;m 1 + N ) now implies that the wave-vectors are determined in terms of two integers º 1 and º 2 as N k 1 ¡ Á 1 ;2 = 2 ¼ º 1 ; N k 2 ¡ Á 1 ;2 = 2 ¼ º 2 :

Bethe showed that this pair of coupled equations for the wave-vectors k 1 and k 2 has ( N ¡ 1) ( N ¡ 2) = 2 real solutions, corresponding to ` scattering states', for 0 · º 1 · ( º 2 ¡ 2) · N ¡ 3 and ( N ¡ 1) 'complex conj ugate pair' solutions, corresponding to ` bound states' (or ànti-bound states' depending on the sign of J ) , for º 1 = º 2 ¡ 1 or for º 1 = º 2 , accounting for all of the total of N ( N ¡ 1) = 2 eigenstates corresponding to D = 2. Next came the great leap; the `Bethe's Ansatz', that for all D ¸ 2, the solution for the wave-function (say for m 1 < m 2 < m 3 ::: < m D ) can be found in the form X i[P D k P j m j + 1 P D ¡ 1 P D Á 0] 2 j= 1 j= 1 j 0= j+ 1 P j;P j ; Ã ( m 1 ;m 2 ;:::;m D ) / e P

Box 1. continued...


63


involving the same `two-particle phase shift functions' as de¯ned above (i. e. ,2 co t( Á j;j 0 = 2) = co t( k j = 2) ¡ co t( k j 0 = 2) ) . Here P denotes an index that identi¯es and runs over the D ! permutations of the numbers 1 ; 2 :::D , and P j denotes the number that replaces j under the permutation P . After guessing the form of the solution, Bethe showed that it does satisfy the SchrÄo dinger equation for Ã ( m 1 ;m 2 ;:::m D ) , the energy eigen-values being now given by XD E fkjg = E - + ( J ¡ J co s ( k j ) ) : j= 1

The periodic boundary condition now determines the wave-vectors k 1 ;:::k D in terms of D integers º 1 ;:::º D via the set of D coupled equations: X N kj ¡ Á j;j 0 = 2 ¼ º j j 06= j

and the solutions for these (including scattering states that correspond to real values of k and bound states that correspond to complex k ) exactly account for all the eigenstates with D down spins. The Bethe Ansatz wavefunction and its rami¯cations have many beautiful mathematical properties, which have been explored in great detail in the enormous amount of research in the area of ` exactly integrable quantum many body problems' that followed in the wake of Bethe's work. For example the wave function can be shown to correspond to a description of an arbitrary scattering as a succession of pair-wise scatterings, leading to the `factorizability of the S matrix' into a product of two-body S matrices [13] . In the thermodynamic limit ( N ;D ! 1 ) , it embodies an in¯nite number of conserved quantities. An incredible variety of problems in condensed matter physics and quantum ¯eld theory in one (space) dimension [13, 14, 17] such as gases of Fermi or Bose particles with delta function interactions, the Hubbard model, etc. , as well as models of magnetic impurities in metals [16] , have been shown to be soluble using the Bethe Ansatz. It is also profoundly related to exactly solved models of statistical mechanics [12] , and to the ideas of quantum inverse scattering and conformal ¯eld theory [15] . For more information, see references [12-17] .

celeb ra ted m o n o g ra p h s in th e H an dbu ch. der P hysik 2 4 : th e ¯ rst o n th e `Q u a n tu m M ech a n ics o f 1 a n d 2 -electro n p ro b lem s' b y B eth e a lo n e, a n d th e seco n d en titled È lectro n th eo ry o f m eta ls', co -a u th o red w ith A S o m m erfeld . E x cep t fo r th e ¯ rst ch a p ter o f a b o u t 3 5 p a g es w h ere S o m m erfeld ela b o ra ted h is `free electro n th eo ry o f m eta ls' (see ch a p ter 2 o f [6 ]) th e rest o f th e 3 0 0 p a g e a rticle w a s w ritten b y B eth e. It rev iew ed th e p a p ers a lrea d y p u b lish ed in th e ¯ eld , su ch a s th e sem in a l w o rk b y B lo ch w h ich sh ow ed th a t electro n s m ov in g in th e p e-

64


GENERAL

ARTICLE

rio d ic p o ten tia l a risin g fro m a cry sta l la ttice o f io n s essen tia lly co n tin u ed to p ro p a g a te lik e free electro n s b u t w ith a n a ltered en erg y -m o m en tu m rela tio n ; th e w o rk o f B rillo u in o n th e recip ro ca l la ttice, etc. In a d d itio n , it set o u t a h o st o f n ew id ea s, lay in g th e fo u n d a tio n s o f th e m o d ern th eo ry o f so lid s, a n d v ery so o n a ch iev ed th e sta tu s o f a cla ssic. A d elig h tfu l in terv iew o f B eth e reg a rd in g th is w o rk b y N D M erm in w a s p u b lish ed in P hysics T oday in J u n e 2 0 0 4 a n d w a s rep ro d u ced in th e p rev io u s issu e o f R eson an ce 1 .

Reson ance , Vol.10 , No.10, pp.96-104, 2005. 1

6 . T h e B e th e { P e ie r ls M e th o d In 1 9 3 5 , w h ile v isitin g th e H H W ills P h y sica l L a b o ra to ry a t th e U n iv ersity o f B risto l, B eth e's a tten tio n w a s d raw n b y W L B ra g g to th e p ro b lem o f th e o rd erd iso rd er tra n sitio n in b in a ry a lloy s su ch a s ¯ -b ra ss. H ere, in a b ip a rtite cry sta l la ttice ( i.e, a la ttice w h ich is m a d e u p o f tw o su b la ttices `1 ' a n d `2 ' w h o se sites a re ea ch o th er's n ea rest n eig h b o u rs, eg ., a B C C la ttice w h ich is m a d e u p o f tw o S C su b la ttices) ea ch site/ cell is ra n d o m ly o ccu p ied w ith eq u a l p ro b a b ility b y eith er o f tw o ty p es o f a to m s A a n d B (C u a n d Z n in ca se o f b ra ss) a t h ig h tem p era tu res. B u t b elow a sh a rp tra n sitio n tem p era tu re, a su p er-la ttice w ith a d o u b led u n it cell fo rm s, w ith o n e su b la ttice (say `1 ') p referen tia lly o ccu p ied b y A a to m s, a n d th e o th er su b la ttice (`2 ') p referen tia lly o ccu p ied b y B a to m s. A t th e sim p lest lev el o f m o d ellin g o f su ch a lloy s, th e en erg etics o f a n a rb itra ry co n ¯ g u ra tio n o f A a n d B a to m s ca n b e w ritten in term s o f a va ria b le ¾ j , w h ich ta k es th e va lu e + 1 if th e site j is o ccu p ied b y a n A a to m , a n d th e va lu e ¡ 1 if it is o ccu p ied b y a B a to m , a s th e su m E =

1 V 2

j;j 0>

X < j;j 0>

¾ j ¾ j 0;

such as  -brass, below a sharp transition

(1 )

w h ere < in d ica tes th a t th e su m is o n ly ov er p a irs o f sites w h ich a re n ea rest n eig h b o u rs. A cco rd in g to th e


In binary alloys

temperature, a super-lattice with a doubled unit cell forms.

65


Bragg and Williams published the simplest ‘mean-field theory of the orderdisorder transition in binary alloys.

ru les o f sta tistica l m ech a n ics, th e p ro b a b ility o f o ccu rren ce o f d i® eren t co n ¯ g u ra tio n s o f A a n d B a to m s is p ro p o rtio n a l to th e B o ltzm a n w eig h t e x p (¡ E = k B T ) w h ere T is th e a b so lu te tem p era tu re o f th e sy stem a n d k B is th e B o ltzm a n co n sta n t. T h is lea d s to a co m p etitio n b etw een en erg y a n d en tro p y w h ich is resp o n sib le fo r th e tra n sitio n . A t h ig h tem p era tu res, th e en tro p y o f th e m a n y p o ssib le (2 N if N is th e n u m b er o f la ttice sites) co n ¯ g u ra tio n s d o m in a tes a n d S j ´ < ¾ j > = 0 (fo r a 5 0 % a lloy ), co rresp o n d in g to th e eq u a l p ro b a b ility o f o ccu p a n cy o f sites b y A o r B a to m s. B u t a t low tem p era tu res, th e low est en erg y co n ¯ g u ra tio n s in w h ich A a to m s a re su rro u n d ed b y B a to m s a n d v ice-v ersa d o m in a te (fo r p o sitiv e V ) lea d in g to S j = § S 6= 0 a ltern a tely o n su b la ttices `1 ' a n d `2 '. B ra g g a n d W illia m s h a d ju st th en p u b lish ed [1 8 ] th e sim p lest `m ea n -¯ eld th eo ry ' o f th is tra n sitio n , in a n a lo g y w ith W eiss's m o lecu la r-¯ eld th eo ry o f ferro m a g n etism , w h ich a ssu m es th a t th e sp o n ta n eo u sly g en era ted selfco n sisten t m o lecu la r `¯ eld ' th a t p ro d u ces th e ò rd erin g ' in a g iv en u n it cell is d eterm in ed co m p letely a n d u n iq u ely b y th e àv era g e o rd er S ' in th e cry sta l a s a w h o le. A cco rd in g ly, th e en erg etics o f th e o ccu p a n cy o f th e sites o f su b la ttices `1 ' a n d `2 ' a re sim p ly a p p rox im a ted b y E

1

= ¡ z V ¾ 1 S ;E

2

= zV ¾ 2S

(2 )

w ith z b ein g th e co -o rd in a tio n n u m b er (i.e., th e n u m b er o f n ea rest n eig h b o u rs) o f th e la ttice (6 fo r th e S C la ttice, 8 fo r th e B C C la ttice, etc). A ca lcu la tio n o f S u sin g th is en erg etics lea d s to th e sim p le self co n sisten t eq u a tio n S = ta n h (

zV S ) kB T

(3 )

w h ich h a s a n o n zero so lu tio n fo r S w h ich ¯ rst a p p ea rs

66


GENERAL

ARTICLE

b elow a critica l tem p era tu re g iv en b y k B T c = z V a n d g row s to th e m a x im u m va lu e o f 1 a s T ! 0 . H o w ev er th is a p p rox im a tio n h a s th e u n sa tisfa cto ry fea tu re th a t fo r all T > T c , th e en erg y, sp eci¯ c h ea t, etc a re a ll zero , a n d th e en tro p y is th e m a x im u m p o ssib le, n a m ely N k B ln (2 )!

Bethe developed a

B eth e's ch a llen g e w a s to g o b ey o n d th is sim p le B ra g g { W illia m s(B W ) th eo ry, a n d in clu d e th e e® ects o f `° u ctu a tio n co rrectio n s' to th e a b ov e. T h is in v estig a tio n led to h is p a p er en titled `S ta tistica l th eo ry o f su p er-la ttices' [1 9 ] In th is p a p er B eth e in tro d u ced th e im p o rta n t co n cep ts o f `sh o rt ra n g e o rd er', ch a ra cterized fo r ex a m p le b y th e n ea rest n eig h b o u r co rrela tio n C ´ < ¾ j ¾ j 0 > (referred to a s ò rd er o f n eig h b ou rs' in th e p a p er) a s d istin ct fro m th e `lo n g ra n g e o rd er' (referred to a s `lo n g d ista n ce o rd er' in th e p a p er) ch a ra cterized b y S (in th e B W th eo ry, C = S 2 ) a n d sh ow ed th a t o n ly th e la tter d isa p p ea rs a b ov e th e critica l tem p era tu re. T h e p a p er is a sto u n d in g in th a t it co n ta in ed th e g erm s o f m a n y o f th e id ea s th a t w ere rein v en ted in la ter d ev elo p m en ts cu lm in a tin g in th e m o d ern th eo ry o f critica l p h en o m en a . F u rth erm o re, B eth e in tro d u ced a series o f in g en io u s a p p rox im a tio n s su ch th a t, in th e lim it o f la rg e co o rd in a tio n n u m b er, z ! 1 k eep in g z V ¯ x ed , th e B W th eo ry reem erg ed , b u t fo r a n y ¯ n ite z , th e resu lts sy stem a tica lly im p rov ed u p o n th e B W th eo ry. In p a rticu la r, sen sib le resu lts w ere o b ta in ed fo r p h y sica l q u a n tities fo r T > T c a s w ell, a s illu stra ted in F igu re 2 , ta k en fro m th e a b ov e p a p er. T h e a p p rox im a tio n s w ere fu rth er d ev elo p ed in fo llow -u p w o rk b y P eierls [2 0 ], a n d a re referred to a s th e `B eth e{ P eierls M eth o d ' o r th e `B eth e{ P eierls A p p rox im a tio n ' a n d h av e b een w id ely u sed ev er sin ce. T h is w o rk w a s a lso th e p recu rso r to la ter w o rk b y K ik u ch i, S u zu k i a n d o th ers o n self-co n sisten t clu ster a p p rox im a tio n s fo r th e sta tistica l m ech a n ics o f la ttice m o d els.

Bragg–Williams


theory of orderdisorder transitions that included ‘fluctuation corrections’ to the theory.

67


(A)

(B)

(C) Figure 2. Results for the ‘order of neighbours’ (A), long distance order (B) and the specific heat (C) obtained by Bethe . The curves labelled (a)- (e) are obtained respectively using :(a) the linear chain approximation (equation (4) of [19]); (b) the second approximation with no long-distance order (§ 5 of [19]); (c) the approximation of Bragg and Williams; (d) and (e) the first and second approximation including order at long distances (§§ 6 and 8 of [19]).

7 . C o n c lu d in g C o m m e n ts A ro u n d 1 9 3 4 B eth e's in terests h a d a lrea d y sta rted sh iftin g tow a rd s n u clea r p h y sics. T h e a b ov e p a p er, a n d a n o th er p a p er o n th e sa m e su b ject, en titled Ò rd er a n d d iso rd er in a lloy s', p u b lish ed in 1 9 3 8 , w ere th e la st o f h is co n trib u tio n s to so lid sta te th eo ry. S o lid sta te th eo ry 's lo ss, h ow ev er, w a s to th e g a in o f m a n y o th er ¯ eld s o f p h y sics, a s ela b o ra ted b y th e o th er trib u tes p u b lish ed in th e p rev io u s issu e o f R eson an ce to o n e o f th e m o st ex tra o rd in a ry a n d g ifted m in d s o f th e la st cen tu ry.

68


GENERAL

ARTICLE

Suggested Reading [1]

H A Bethe, Theory of Diffraction of Electrons by Crystals, Ann. Physik, Vol. 87, pp.55-129, 1928.

[2]

H A Bethe, Splitting of terms in Crystals, Ann. Physik, Vol.3, pp.133-208, 1929.

[3]

e.g. see Selected Works of H A Bethe, World Scientific, Vol.18, 1997, series in 20th Century Physics.

[4]

e.g. see M Tinkham, Group Theory and Quantum Mechanics, McGrawHill, 1964. e.g. see J C Slater , Quantum Theory of Atomic Structure, Vols I and II, McGraw-Hill, 1960.

[5] [6]

e.g. see N Ashcroft and N D Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976.

[7]

e.g. see A Abragam and B Bleany, Electron Paramagnetic Resonance of Transition Ions, Clarendon press, Oxford 1970.

[8]

e.g. see M B Salamon and M Jaime, Rev. Mod. Phys. , Vol.73, p.583, 2001 and T V Ramakrishnan, H R Krishnamurthy, S R Hassan and Pai G Venketeswera, in Colossal Magnetoresistive Manganites, ed.T Chatterji, Kluwer Academic Publishers, Dordrecht, Netherlands, Feb. 2004, condmat/0308396, 2003.

[9]

H A Bethe, Theory of the passage of fast corpuscular rays through matter, Ann. Physik, Vol. 5, pp.325-400, 1930.

[10]

E Garfield, Citation Classics. 3. Articles from the Physical Sciences Published in the 1930's, Current Contents, Vol.44, p.5-9, 1976.

[11]

H A Bethe, On the theory of metals I :Eigenvalues and eigenfunctions of the linear atomic chain, Zeits. Physik Vol.71, pp. 205-226, 1931. R J Baxter, Exactly Solved Models of Statistical Mechanics, Academic Press, 1982.

[12] [13]

H B Thacker, Rev. Mod. Phys. , Vol.53, p.253, 1982.

[14]

B S Shastry, S S Jha and V Singh (eds), Exactly Solved Problems in Condensed Matter and Relativistic Field Theory, Springer Verlag, 1985.

[15]

V E Korepin, A G Izergin and N M Bogoliubov, Quantum Inverse Scattering Method and Corrrelation Functions, Cambridge University Press, 1993.

[16]

N Andrei, K Furuya and J H Lowenstein, Rev. Mod. Phys. Vol.55, p.331, 1983. A M Tsvelick and P B Wiegmann, Adv. Phys., Vol.32, p.453, 1983. N Andrei, Lectures at the Trieste Summer School 1992, condmatt/9408101

[17]

M Takahashi, Thermodyanamics of One-dimensional Solvable Models, Cambridge University Press, 1999.

[18]

W L Bragg and E J Williams, Proc. Roy. Soc. London Ser., Vol.A145, p.699, 1934.

[19]

H A Bethe, Statistical Theory of Super-lattices, Proc. Roy. Soc. London Ser. , Vol.A150, pp.552-575, 1935.

[20]

R Peierls, Proc. Roy. Soc., Vol.154, p.207, 1936.


Address for Correspondence H R Krishnamurthy Centre For Condensed Matter Theory Department of Physics Indian Institute of Science Bangalore 560 012, India Email: [email protected]

69