Con epts of Theoreti al Solid State 1 Physi s ψ 0 gψ 0 g(x)ψ 0 λ Alexander Altland and Ben Simons 1 Copyright (C) 2001: Permission is granted to anyone to make verbatim opies of this do - ument provided that the opyright noti e and this permission noti e are preserved. ii Chapter 1 Colle tive Ex itations: From Parti les to Fields The goal of this se tion is to introdu e some fundamental on epts of lassi al (cid:12)eld theory in the framework of a simple model of latti e dynami s. In doing so, we will be ome a - quainted with the notion of a ontinuum a tion, elementary ex itations, olle tive modes, symmetries and universality | on epts whi h will pervade the rest of the ourse. One ofthe moreremarkablefa ts about ondensed matterphysi s isthat phenomenol- ogy of phantasti omplexity is born out of a Hamiltonianthat does not look parti ularly impressive. Indeed, it is not in the least diÆ ult to write down mi ros opi \ ondensed matter Hamiltonians" of reasonable generality. E.g. a prototypi al metal might be de- s ribed by H = He +Hi +Hei; X 2 X pi He = + Vee(ri (cid:0)rj); 2m i i6=j X 2 X P I Hi = + Vii(RI (cid:0)RJ); 2M I I6=J X Hei = Vei(RI (cid:0)ri); (1.1) iI where, ri (RI) are the oordinates of i;I = 1;:::;N valen e ele trons (ion ores) and He;Hi, and Hei des ribe the dynami s of ele trons, ions and the intera tion of ele trons and ions, respe tively. (For details of the notation, see Fig. 1.1.) Of ourse, the Hamil- tonian (9.30) an be made \more realisti ", e.g. by remembering that ele trons and ions arry spin, adding disorder or introdu ing host latti es with multi-atomi unit- ells, however for developing our present line of thought the prototype H will do (cid:12)ne. The fa t that an inno uously looking Hamiltonian like (9.30) is apable of generating a vast panopti um of metalli phenomenology an be read in reverse order: one will normally not be able to make theoreti al progress by approa hing the problem in an \ab 1 2 CHAPTER 1. COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS r i R I Figure 1.1: One-dimensional artoon of a (metalli ) solid. Positively harged ions lo ated at positions RI are sourrounded by a ondu tion ele tron loud (ele tron oordinates denoted by ri). Both ele trons and ions are free to move, as des ribed by the kineti P P 2 2 energy terms pi=(2m) and Pi=(2M) of Eq. (9.30), respe tively. While, the motion oftheionsismassively onstrainedbythelatti epotentialVii (indi atedbysolidlines)the dynami s of the ele trons is a(cid:11)e ted by their mutual intera tion (Vee) and the intera tion with the ore ions (Vei). initio"manner,i.e. anapproa hthattreatsallmi ros opi onstituentsasequallyrelevant degrees of freedom. But how then, an su essful analyti al approa hes be developed? Mu h ofthe answer to this question liesin anumber ofbasi prin iplesinherent to generi ondensed matter systems 1. Stru tural redu ibility of the problem. Whi h simply means that not all om- pounds of the Hamiltonian (9.30) need to be treated simultaneously. E.g. when the interest is foremostly in the vibrational motion of the ion latti e, the dynami s of the ele tron system an often be negle ted or, at least, be treated in a simplisti manner. Similarly, mu h of the dynami s of the ele trons is independent of the ion latti e, et . 2. Inthemajorityof ondensed matterappli ationsoneisnotsomu hinterestedinthe full pro(cid:12)le of a given system but rather in its energeti ally low lying dynami s. This is partly motivated by pra ti al aspe ts (In daily life, iron is normaly en outered at room temperature and not at its melting point.), partly by the tenden y of large systems to behave \universal" at low temperatures. Universality means that systems di(cid:11)ering in mi ros opi detail (e.g. di(cid:11)erent types of intera tion potentials, ion spe ies et .) exhibit identi al olle tive behaviour. As a physi ist, one will normallyseek forunifyingprin iplesin olle tivephenomenaratherthantodes ribe the spe ialties of individual spe ies. Hen e the fundamental importan e of the universality prini iple. However, universality is equally important in the pra ti e of ondensed matter theory. It implies, e.g., that at low temperatures, details of the fun tional form of mi ros opi intera tion potentials are of se ondary importan e, i.e. that one may employ simple model Hamiltonians. 1.1. CLASSICAL HARMONIC CHAIN: PHONONS 3 3. For most systems of interest, the number of degrees of freedom, N, is formidably 23 large, e.g. N = O(10 ). However, ontrary to the (cid:12)rst impression, the magnitude of this (cid:12)gure is rather an advantage. The reason is that in addressing ondensed matterproblemswemaymakeuseofthe on epts of statisti sandthat(pre isely 1 due to the largeness of N) statisti al errors tend to be negligibly small . 4. Finally, ondensed matter systems typi ally possess a number of intrinsi symme- tries. E.g. our prototype Hamiltonianabove is invariant under simultaneous trans- lationand rotationof all oordinates whi h expresses the globalGalileiinvarian eof the system (a ontinuous set of symmetries). Spin rotation invarian e ( ontinuous) and time reversal invarian e (dis rete) are other examples of frequently en ountered symmetries. The general importan e of symmetries needs no stressing: symmetries entail onservation laws and onservation laws simplify any problem. Yet in on- densed matter physi s, symmetries are \even more" important. The point is that a onserved observable isgenerallytiedtoanenergeti allylow-lyingex itation. Inthe universal low temperature regimes we will typi ally be interested in, it is pre isely the dynami s of these low level ex itations that governs the gross behaviour of the system. In subsequent se tion, the sequen e \symmetry ! onservation law ! low- lying ex itations" will be en ountered time and again. At any rate, identi(cid:12) ation of the fundamental symmetries will typi ally be step no.1 in the analysis of a solid state system. Employing a hain of harmoni ally bound atoms as an example, we next attempt to illustrate how su h prin iples an be applied to onstru t \e(cid:11)e tive low energy" models of solid state systems. I.e. models that are universal, en apsulate the essential low energy dynami s, and anberelatedtoexperimentallyobservabledata. Wewillalsoobserve that 2 the low energy dynami s of large systems naturally relates to on epts of (cid:12)eld theory; In a way, this and the next few hapters represent a (cid:12)rst introdu tion to the use of (cid:12)eld theoreti al methods in solid state physi s. 1.1 Classi al Harmoni Chain: Phonons Coming ba k to our prototype Hamiltonian (9.30), let us fo us on dynami al behaviour of the positively harged ore ions onstituting the host latti e. For the moment, let us negle t the fa tthat atomsare quantum obje ts, i.e. treatthe ionsas lassi al. Tofurther simplify the problem, we onsider an atomi hain rather than a generi d-dimensional solid. I.e. the positions of the ions are given by a sequen e of points with average spa ing a. Relyingonthe redu tion prin iple(1.) we next arguethat tounderstand the behaviour 1 Theimportan eofthispointisillustratedbytheempiri alobservationthatthemostresistivesystem lassesinphysi als ien esareofmedium(andnotlarge)s ale. E.g. metalli lusters,mediumsizenu lei 1(cid:0)2 or large atoms onsist of O(10 ) fundamental onstituents. Su h problems are well beyond the rea h of few body quantum me hani swhilenot yet a essibleto reliablestatisti almodelling. Often the only viablepath to approa hingsystems of this type is massiveuse of phenomenology. 2 Inthis oursewewillfo usonthedynami albehaviouroflargesystems,asopposedtostati stru tural properties. E.g., we will not address questions related to the formation of de(cid:12)nite rystallographi stru tures in solidstate systems. 4 CHAPTER 1. COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS of the ions the dynami s of the ondu tion ele troni se tor of is of se ondary importan e, i.e. we set He = Hei = 0. At striktly zero temperature, the system of ions will be frozen out, i.e. the one- dimensional ion oordinates RI (cid:17) R(cid:22)I = Ia de(cid:12)ne a regularly spa ed array. Any deviation from a perfe tly regular on(cid:12)guration has to be payed for by a prize in potential energy. For low enough temperatures (prini iple 3.), this energy will be approximately quadrati in the small deviation from the equilibrium position (the dashed line in Fig. 1.1.) The 3 redu ed low energy Hamiltonian of our system then reads XN (cid:18) 2 (cid:19) PI ks 2 H = + (RI (cid:0)RI+1 (cid:0)a) ; (1.2) 2M 2 I=1 where the oeÆ ient ks determines the steepness of the latti e potential. Noti e that H an be interpreted as the Hamiltonian of N parti les of mass M elasti ally onne ted by springs with spring onstant ks (see Fig. 1.2). x φ n-1 n k s m (n-1)a na (n+1)a Figure 1.2: Toy model of a one-dimensional solid: A hain of elasti ally bound massive point parti les. hange x $ R 1.1.1 Lagrangian Formulation and Equations of Motion What are the elementary low energy ex itations of th is system? To answer this question we might, in prini ple, solve Hamiltons equations of motion; this is possible be ause H is quadrati in all its oordinates. However, we must keep in mind that few of the problems en ountered in general solid state physi s enjoy this property. Further, it seems unlikely that the low energy dynami s of a ma ros opi ally large hain { whi h we know from our experien e will be governed by large s ale wave type ex itations { is adequately des ribed in terms of an \atomisti " language; the relevant degrees of freedom will be of di(cid:11)erent type. Ratherwhatweshoulddoismakemu hmoreex essive usefromourbasi prin iples 1.-4. Notably, we have so far neither payed attention to the intrinsi symmetry of the problem nor to the fa t that N is large. 3 Sir William Rowan Hamilton 1805- 1865; a mathemati ian redited with the dis overy of quaternions, the (cid:12)rst non- ommutative algebra to be stud- ied. He also invented important new methods in Me hani s. 1.1. CLASSICAL HARMONIC CHAIN: PHONONS 5 Now omes a very imporant point: To redu e a mi ros opi ally formulated model down to an e(cid:11)e tive low energy model, the Hamiltonian is often not a very onvenient starting point. It is usually more eÆ ient to start out from an a tion. As usual, the 4 Lagrangian a tion of our system is de(cid:12)ned as Z T S = L(R;R_)dt; 0 where (R;R_) (cid:17) fRI;R_Ig symboli ally represents the set of all oordinates and their time derivatives. The Lagrangian L related to the Hamiltonian (1.2) is given by XN (cid:18) 2 (cid:19) PI ks 2 L = T (cid:0)U = (cid:0) (RI (cid:0)RI+1 (cid:0)a) ; (1.3) 2M 2 I=1 where T and U stand for kineti and potential energy, respe tively. . Exer ise. Re apitulate the onne tion between Hamiltonian and Lagrangian in N- parti le lassi al me hani s. For onvenien e we assume thatour atomi hain has the topologyof a ring, i.e. adopt periodi boundary onditions RN+1 = R1. Further, anti ipating that the e(cid:11)e t of latti e vibrationsonthesolidisweak (i.e. long-rangeatomi orderismaintained)weassumethat the deviation from the equilibrium position is small (jRI(t)(cid:0)R(cid:22)Ij (cid:28) a), i.e. the integrity of the solid is maintained. With RI(t) = R(cid:22)I + (cid:30)I(t) ((cid:30)I+1 = (cid:30)1) the Lagrangian (1.3) simpli(cid:12)es to N (cid:18) (cid:19) X M _2 ks 2 L = (cid:30)I (cid:0) ((cid:30)I+1 (cid:0)(cid:30)I) : 2 2 I=1 To make further progress, we now use that we are not on erned with the behaviour of our system on `atomi ' s ales. (In any ase, for su h purposes a modelling like the one above would be mu h too primitive!) Rather, we are interested in experimentally observable behaviour that manifests itself on ma ros opi length s ales (prin iple 2.). For example, one might wish to study the spe i(cid:12) heat of the solid in the limit of in- 23 (cid:12)nitely many atoms (or at least a ma ros opi ally large number, O(10 )). Under these onditions, mi ros opi models an usually be substantially simpli(cid:12)ed (prin iple 3.). In parti ular it is often permissible to subje t a dis rete latti e model to a so- alled ontin- uum limit, i.e. to negle t the dis reteness of the mi ros opi entities of the system and to des ribe it in terms of e(cid:11)e tive ontinuum degrees of freedom. 4 Joseph-LouisLagrange 1736-1813;Lagrangewas a mathemati ian whoex elledinall(cid:12)eldsofanalysis,numbertheory,analyti al,and elestial me hani s. In 1788 he published M(cid:19)e anique analytique, whi hsummarisedalltheworkdoneinthe(cid:12)eldofme hani ssin e the time of Newton and is notable for its use of the theory of di(cid:11)erentialequations. Inithetransformedme hani sintoabran h of mathemati alanalysis. 6 CHAPTER 1. COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS φ n Continuum Limitφ(x) (n-1)a na (n+1)a Figure1.3: Continuumlimitoftheharmoni hain. For larity,the (horizontal)distortion of the point parti les has been plotted against the verti al. Inthe present ase, takinga ontinuumlimitamountstodes ribingthe latti e(cid:13)u tua- tions (cid:30)I in terms of smooth fun tions of a ontinuous variable x (Fig. 1.3). Clearly su h a des riptionmakes sense onlyifrelative(cid:13)u tuationsonatomi s ales areweak. (Otherwise the smoothness ondition would be violated.) However, if this ondition is met { as it will be for suÆ iently large values of the sti(cid:11)ness onstant ks { the ontinuum des ription is mu h more powerful than the dis rete en oding in terms of the 've tor' f(cid:30)Ig. All steps we need to take to go from the Lagrangian to on rete physi al predi tions will be mu h easier to formulate. Introdu ing ontinuum degrees of freedom (cid:30)(x), and applying a (cid:12)rst order Taylor 5 expansion, we de(cid:12)ne (cid:12) (cid:12) XN Z L 1=2 (cid:12) 3=2 (cid:12) 1 (cid:30)I ! a (cid:30)(x)(cid:12) ; (cid:30)I+1 (cid:0)(cid:30)I ! a x(cid:30)(x)(cid:12) ; (cid:0)! dx; x=Ia x=Ia a 0 I=1 1=2 whereL = Na. Notethat,asde(cid:12)ned,thefun tions(cid:30)(x;t)havedimensionality[Length℄ . Expressed in terms of the new degrees of freedom, the ontinuum limit of the Lagrangian then reads Z L 2 _ _ m _2 ksa 2 L[(cid:30)℄ = dx L((cid:30);x(cid:30);(cid:30)); L((cid:30);x(cid:30);(cid:30)) = (cid:30) (cid:0) (x(cid:30)) ; (1.4) 0 2 2 where the Lagrangian density L has dimensionality [energy℄=[length℄ and we have des- ignated the parti le mass by the more ommon symbol m (cid:17) M. Similarly, the lassi al a tion assumes the ontinuum form Z Z Z L _ S[(cid:30)℄ = dt L[(cid:30)℄ = dt dx L((cid:30);x(cid:30);(cid:30)): (1.5) 0 We have thus su eeded in abandoning the N-point parti le des ription in favour of one involving ontinuous degrees of freedom, a ( lassi al) (cid:12)eld. The dynami s of the latter is spe i(cid:12)ed by the fun tionals L and S whi h represent the ontinuum generalisations of the dis rete lassi al Lagrangian and a tion, respe tively. . Info. The ontinuum variable (cid:30) is our (cid:12)rst en ounter with a (cid:12)eld. Before pro eeding with our example, let us pause to make some preliminary remarks on the general de(cid:12)nition of 5 Indeed, for reasons that will be ome lear, higher order ontributions to the Taylor expansion are immaterialin the long-range ontinuum limit. 1.1. CLASSICAL HARMONIC CHAIN: PHONONS 7 theseobje t s. Thiswillhelptopla ethesubsequentdis ussionoftheatomi hainintoalarger ontext. Mathemati ally speaking, a (cid:12)eld is a mapping (cid:30) : M ! T; z 7!(cid:30)(t); froma ertainmanifoldM, often alledthe'basemanifold',intoatarget or(cid:12)eldmanifoldT, see 2 Fig. 1.4. Inourpresentexample,M =[0;L℄(cid:2)[0;T℄ (cid:26)R istheprodu tofintervalsinspa eand time, andT =R istherealnumbers. Ingeneralappli ations,thebasemanifoldwillbea(subset of) some d-dimensional spa e-like manifold R multiplied by a time-like interval: M (cid:26) R(cid:2)R. (E.g. in our present example, R 'S1 is isomorphi to the unit ir leS1.) Sometimes, espe ially in problems relating to statisti al me hani s, M (cid:26) R is just spa elike. However, we are always free to assumethatlo ally M isisomorphi to some subssetofd+1- ord-dimensionalrealve tor spa e. In ontrast, the target manifold an be just any (di(cid:11)erentiable) manifold. From real or omplex numbers, over ve torspa es and groupsto the fan iest obje ts of mathemati al physi s. Inapplied(cid:12)eldtheory, (cid:12)eldsdo notappearas (cid:12)nalobje ts butratheras inputto fun tionals (see Fig. 1.4) Mathemati ally, a fun tional S : (cid:30) 7! S[(cid:30)℄ 2 R is a mapping that that takes a (cid:12)eld as its argument and maps it into the real numbers. The fun tional pro(cid:12)le S[(cid:30)℄ essentially determinesthe hara ter ofa (cid:12)eldtheory. Noti e that the argumentof afun tionalis ommonly indi ated in angular bra kets [:℄. T φ S φ S[ ] M Figure 1.4: S hemati vizualization of a (cid:12)eld: a mapping (cid:30) from a base manifold M into a target spa e T (here the real numbers, but T an be more ompli ated). A fun tional assigns to ea h (cid:30) a real number S[(cid:30)℄. The grid embedded into M indi ates that (cid:12)elds in ondensed matter physi s arise as ontinuum limits of dis rete mappings. While these formulations may appear unne essarily abstra t, remembering the dry math- emati al ba kbone of the theory often helps to avoid onfusion. At any rate, it takes some time and pra ti e to get used to the on ept of (cid:12)elds and fun tionals. Con eptual diÆ ulties 8 CHAPTER 1. COLLECTIVE EXCITATIONS: FROM PARTICLES TO FIELDS in handling these obje ts an be over ome by remembering that any (cid:12)eld in ondensed matter physi s arises as the limitof a dis rete mapping. E.g. in our example, the (cid:12)eld(cid:30)(x) obtained as N ontinuumapproximationofthedis reteve tor f(cid:30)Ig2R ; thefun tionalL[(cid:30)℄isthe ontinuum N limitof the fun tion L:R !R, et . While in pra ti al al ulations (cid:12)elds are usuallyeasier to handlethan their dis rete analogs, it issometimes easier to thinkabout problemsof (cid:12)eldtheory in a dis rete language. Within the disr rete pi ture, the mathemati al aparatus of (cid:12)eld theory redu es to (cid:12)nite dimensional al ulus. ||||||||||||||{ Although Eq. (1.4) ontains the full information about the model, we have not yet learned mu h about its a tual behaviour. To extra t on rete physi al information from Eq. (1.4) we need to derive equations of motion. At (cid:12)rst sight, it may not be entirely lear what is meant by the term `equations of motion' in the ontext of an in(cid:12)nite di- mensional model: The equations of motion relevant for the present problem obtain as generalization of the onventional Lagrange equations of N-parti le lassi al me hani s to a model with in(cid:12)nitely many degrees of freedom. To derive these equations we need to generalize Hamilton's extremal prin iple, i.e. the route from an a tion to the asso iated equations of motion, to in(cid:12)nite dimensions. As a warmup, let us brie(cid:13)y re apitulate how the extremal prin iple worked for a system with one degree of freedom: Suppose the dynami s of a lassi al point parti le with oordinate x(t) is des ribed by R the lassi al Lagrangian L(x;x_), and a tion S[x℄ = dtL(x;x_). Hamilton's extremal prin iple states that the on(cid:12)gurations x(t) that are a tually realised are those that extremise the a tion ÆS[x℄ = 0. This means that for any smooth urve t 7! y(t), 1 lim (S[x+(cid:15)y℄(cid:0)S[x℄) = 0: (1.6) (cid:15)!0 (cid:15) I.e. to (cid:12)rst order in (cid:15) the a tion has to remain invariant. Applying this ondition, one (cid:12)nds that it is ful(cid:12)lled if and only if x solves Lagrange's equation of motion d (x_L)(cid:0)xL = 0: (1.7) dt . Exer ise. Re apitulate the derivation of (1.7) from the lassi al a tion. In Eq. (1.5) we are dealing with a system of in(cid:12)nitely many degrees of freedom (cid:30)(x;t). Yet Hamilton's prin iple is general and we may see what happens if (1.5) is subje ted to an extremal prin iple analogous to Eq. (1.6). To do so, we substitute (cid:30)(x;t) ! (cid:30)(x;t)+(cid:15)(cid:17)(x;t) into Eq. (1.5) and demand vanishing of the (cid:12)rst order ontribution to an expansion in (cid:15) (see Fig. 1.5). When applied to the spe i(cid:12) Lagrangian (1.4), substituting the `varied' (cid:12)eld leads to Z Z L (cid:16) (cid:17) _ 2 2 S[(cid:30)+(cid:15)(cid:17)℄ = S[(cid:30)℄+(cid:15) dt dx m(cid:30)(cid:17)_ (cid:0)ksa x(cid:30)x(cid:17) +O((cid:15) ): 0