8.512 Theory of Solids II Spring 2009

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1 MIT OpeCourseWare Theory of Solids II Sprig 2009 For iformatio about citig these materials or our Terms of Use, visit:

2 Lecture 1: Liear Respose Theory Last semester i 8.511, we discussed liear respose theory i the cotext of charge screeig ad the free fermio polarizatio fuctio. This theory ca be exteded to a much wider rage of areas, however, ad is a very useful tool i solid state physics. We ll begi this semester by goig back ad studyig liear respose theory agai with a more formal approach, ad the returig to this like supercoductivity a bit later. 1.1 Respose Fuctios ad the Iteractio Represetatio I solid state physics, we ordiarily thik about may body systems, with somethig o the order of particles. With so may particles, it is usually impossible to eve thik about a wave fuctio for the whole system. As a result, it is ofte more useful for us to thik i terms of the macroscopic observable behaviors of systems rather tha their particular microscopic states. Oe example of such a macroscopic property is the magetic susceptibility χ H M H, which is a measure of the respose of the et magetizatio M of a system to a applied magetic field H ( r, t). This is the type of behavior we will be thikig about: we ca mathematically probe the system with some perturbig exteral probe or field (e.g. H ( r, t)), ad try to predict what the system s respose will be i terms of the expectatio values of some observable quatities. Let Ĥ be the full may body Hamiltoia for some isolated system that we are iterested i. We spet most of thikig about how to solve for the behavior of a system govered by Ĥ. As iterestig as that behavior may be, we will ow cosider that to be a solved problem. That is, we will assume the existece of a set of eigekets { } that diagoalize H ˆ with associated eigevalues (eergies) E. I additio to Ĥ, we ow tur o a exteral probe potetial Vˆ, such that the total Hamiltoia H T ot satisfies: Hˆ T ot = H ˆ + Vˆ (1.1) I particular, we are iterested i probe potetials that arise from the couplig of some exteral scalar or vector field to some sort of desity i the sample. For example, the exteral field ca be a electric potetial U ( r, t), which couples to the electroic charge desity ρˆ( r) such that ˆV = d rˆρ ( r) U ( r, t) (1.2) V where the electro desity operator ˆρ ( r) is give by N ˆρ ( r) = δ ( r r i ) (1.3) i=1 1

3 Respose Fuctios ad the Iteractio Represetatio 2 I first quatized laguage, with r i beig the positio of electro i the N electro system. I secod quatized otatio, recall ρˆ( r) = Ψ ( r) Ψ ( r) (1.4) where Ψ ( r) ad Ψ ( r) are the electro field creatio ad aihilatio operators, respectively. The mometum space versio of the electro desity operator, ˆρ (q ), is related to ρˆ( r) through the Fourier trasforms: i ρˆ( r) = e q r ρˆ( q) (1.5) q i r Ψ ( r) = e k c (1.6) k k such that ρˆ(q ) = e r = c c (1.8) k i q r k q k (1.7) Equatio (1.7) is the first quatized form of ˆρ ( q), ad equatio (1.8) is the secod quatized form with c the creatio operator for a electro with mometum 1 k q ad c the destructio k q operator for a electro with mometum k. Returig to equatio (1.1), we d like to thik about Vˆ as a perturbatio o the exteral field free system Hamiltoia H. ˆ This leads us aturally to cosider H ˆ as the uperturbed Hamiltoia withi the iteractio picture represetatio. Recall that this H ˆ is a very complicated beast with all of the electro electro repulsios icluded, but for our purposes we just take as a give that there are a set of eigestates ad eergies that diagoalize this Hamiltoia. Recall the formulatio of the iteractio represetatio: i h φ (t) = ( H ˆ + Vˆ ) φ (t) (1.9) t We ca uwid the atural time depedece due to H ˆ from the state ket φ (t) to form a iteractio represetatio state ket φ (t) I by Ht φ (t) I = e i ˆ φ (t) (1.10) φ (t) Ht I = e i ˆ φ (t) (1.11) Note that i the absece of Vˆ, these iteractio picture state kets are actually the Heiseberg picture state kets of the system. Also, we have ow officially set h = 1. After substitutig (1.11) ito (1.9), we obtai Ht ˆ Ht i h = e i ˆ V e i ˆ φ (t) t (1.12) = Vˆ I φ (t) (1.13) k 1 k ad q are actually wavevectors, which differ from mometa by a factor of h. Whe i doubt, assume h = 1.

4 Respose Fuctios ad the Iteractio Represetatio 3 where we have set Ht ˆ Ht VˆI = e i ˆ V e i ˆ (1.14) Thus the iteractio picture state ket evolves simply accordig to the dyamics govered solely by the iteractio picture perturbig potetial VˆI. More geerally, we ca write ay observable (operator) i the iteractio picture as ˆ Ht ˆ Ht A I = e i ˆ Ae i ˆ (1.15) We ca itegrate equatio (1.12) with respect to t to get φ (t) = φ dt Vˆ I (t ) φ (t 0 i ) (1.16) At first it seems like we have ot doe much to beefit ourselves, sice all we have doe is to covert the ordiary Schrodiger equatio, a PDE, ito a itegral equatio. However, if VˆI is small, the we ca iterate equatio (1.16): φ (t) φ 0 i dt VˆI (t ) φ 0 + (1.17) The essece of liear respose theory is that we focus ourselves o cases where VˆI is sufficietly weak that the perturbatio series represeted by equatio (1.17) has essetially coverged after icludig just the first o trivial term listed above. This term is liear i VˆI. Throughout this discussio, we will be workig at T = 0, so φ 0 is simply the groud state of the o perturbed total system Hamiltoia H. ˆ Note that we have take our iitial time, i.e. the lower limit of itegratio i equatio (1.16), to be. This is because we wat to imagie turig o the probig potetial Vˆ adiabatically, that is so slowly that the system tracks the groud state for all fiite times. If we were to tur o the probe sharply, the system would exhibit complicated rigig behavior that we are ot iterested i. We ow retur to our model experimet for studyig the properties of our system. After applyig some probe via the exteral potetial Vˆ, we wat to measure the respose of some observable of the system A. ˆ We characterize this respose through the expectatio value of A, ˆ Â : A = φ (t) Aˆ φ (t) (1.18) = φ (t) Aˆ I φ (t) (1.19) The key ow is to substitute i the approximatio for φ (t) give by equatio (1.17) ito equatio (1.19). Sice we have oly kept terms up to liear order i VˆI, we must be careful oly to keep terms to this order. After performig this substitutio, we arrive at A ˆ φ A dt e ηt φ ˆ ˆ 0 [A I ( r, t), V I (t 0 ˆ φ 0 i )] φ 0 (1.20) The mysterious factor e ηt comes from our adiabatic switchig o of the potetial. This esures that the system evolves smoothly from t = to t. Evetually, we will sed η 0. Sice we are iterested i positive times t close to 0 whe compared with, we do t eed to worry about the e ηt messig aythig up.

5 Respose Fuctios 4 The other mysterious piece of equatio (1.20) is the appearace of the commutator [Aˆ I ( r, t), VˆI (t )]. These two terms simply come from the two possible terms liear i VˆI arisig from the substitutios φ (t) φ dt VˆI (t 0 i ) φ 0 ad φ (t) φ 0 + i dt Vˆ I (t ) φ 0 Note that the itegratio is with respect to t, sice it comes from the expressio for φ (t) which ivolves a itegratio of VˆI with respect to t. The observable ˆ A is also a fuctio of space ad time, but there is o reaso to itegrate over it at this poit. This is oe way to remember what to itegrate over if you forget some day. 1.2 Respose Fuctios What we re really iterested i, however, is ot A ˆ itself, but the chage i A ˆ relative to the uperturbed state: δ A ˆ = δ A( r, ˆ t) φ 0 δ A ˆ φ 0 (1.21) = lim 0 ieηt dt e η(t t) φ 0 [A ˆ I ( r, t), V ˆ I (t )] φ 0 (1.22) η Now is whe we will specialize to the specific type of probe potetial describe i the previous example. For cocreteess, we cosider the potetial of equatio (1.2): Vˆ = d rρˆ( r) U ( r, t) V U( r, t) commutes with the Hamiltoia, so the iteractio picture represetatio of Vˆ is give by ( ) ( ) Ht Ht VˆI = e i ˆ dr ρˆ r U r, t e i ˆ (1.23) V Ht = dr e i ˆ ρ(r ˆ )e i Ht ˆ U V ( ) r, t (1.24) ( ) = dr ρˆi(r ) U r, t (1.25) V Substitutig this expressio for VˆI back ito equatio (1.22), we obtai: A( r, t) = lim ie ηt dt d r e η(t t) δ ˆ φ 0 [A ˆ I ( r, t), ρˆi( r, t )] φ 0 U( r, t ) (1.26) η 0 V We defie the respose fuctio χ as the kerel of this expressio for δ A( ˆ r, t) : δ ˆ A( r, t) = dt d r χ( r, r, t t ) U( r, t ) (1.27)

6 Electro Desity Respose to a Applied Electric Potetial 5 χ is a fuctio of (t t ) oly, sice Ĥ is idepedet of time. The iterpretatio of equatio (1.27) is that if we shake the system with a exteral potetial U(r, t ), the the respose of the system i terms of some observable Â at the poit r ad time t is modulated by the respose fuctio χ( r, r, t t ). Thus from comparig this defiitio with equatio (1.26), we see that χ( r, r, t t ) (1.28) i φ 0 [A I ( r, t), ρˆi( r, t )] φ 0 e η(t t) ˆ θ(t t ) Note that i equatio (1.27) we exteded the limits of itegratio from to for coveiece, ad thus have added the Heaviside step fuctio θ(t t ) to our defiitio of χ( r, r, t t ). Recall that θ(t) = 0 for t < 0 ad θ(t) = 1 for t > 0. This esures causality i our defiitio of χ, sice the system should ot be able to respod to the perturbatio before it happes. Notice also that based o this defiitio, the respose fuctio is purely a fuctio of the system s uperturbed Hamiltoia H; ˆ U does ot appear aywhere i the expressio. Thus ivestigatios of χ ca reveal iformatio about the systems Hamiltoia. I this defiitio, the electro desity ˆρ I ( r, t ) appears because we specialized to the case of a applied exteral electric potetial that couples to the system s charge desity. For a probe that couples to some other desity, such as magetizatio desity ˆm(r, t ), we ca simply replace ρˆi( r, t ) by ˆm I (r, t ) i defi itio (1.28). 1.3 Electro Desity Respose to a Applied Electric Potetial I this sectio, we will specialize further to the case where we observe the respose of the electro desity to a applied potetial that couples to the desity. Thus we are pickig A ˆ = ρˆ. We begi by takig the Fourier trasform of equatio (1.28) with respect to time: t = t t 0 χ( r, r, ω) = i dt e (iω η)t φ 0 [ρˆi( r, 0), ρˆi( r, t )] φ 0 (1.29) Recall that we have a complete set of eigestates of Ĥ: Ĥ = E = 1ˆ Isertig this complete set of states ito the commutator [ ˆρ I ( r, 0), ˆρ I ( r, t )] = [ ˆρ I ( r, 0), ˆρ I ( r, t )] (1.30) ad otig that ad ˆρ I = e i Ĥt ˆρe i Ĥt (1.31) e i Ĥt = e ie t (1.32)

7 we obtai Electro Desity Respose to a Applied Electric Potetial 6 0 i(e E 0 )t (iω η)t χ( r, r, ω) = i dt φ 0 ˆ ρ( r) ρˆ( r ) φ 0 e (1.33) φ 0 ρˆ( r ) ρˆ( r) φ 0 e i(e E0 )t (iω η)t (1.34) All of the time depedece has ow bee brought up ito the expoetials, so it is trivial to perform the itegratio over time. This yield the spectral represetatio of χ( r, r, ω): [ ] ρˆ( r ) φ 0 φ 0 ρˆ( r ) ˆ r) φ 0 χ( r, r, ω) = φ 0 ρˆ( r) ρ( (1.35) ω (E E 0 ) + iη ω + (E E 0 ) + iη If there is traslatioal ivariace i the sample, the the respose fuctio χ( r, r, ω) should be simply a fuctio of the differece r r. I this case, the spatial Fourier trasform is simple: 1 χ( q, ω) = d r d r e i q ( r r ) χ( r r, ω) (1.36) V [ φ ] 0 ρˆ(q ) ρˆ( q) φ 0 φ 0 ρ( = ˆ q) ρˆ(q ) φ 0 (1.37) ω (E E 0 ) + iη ω + (E E 0 ) + iη where ρ( ˆ q) ρˆq is give by equatios (1.7) ad (1.8) i first quatized or secod quatized otatio, respectively. Sice the electro desity ρ( ˆ r) is a real fuctio, we have the importat relatio ρˆ = ρˆ which is a simple cosequece of the ature of the Fourier trasform. This implies that q q (1.38) 2 φ 0 ρˆ(q ) ρˆ( q ) φ 0 = ρˆ ( q) φ 0 (1.39) Usig this alog with the relatio { } 1 Im = πδ(x) (1.40) x + iη we arrive at the ext importat result: Im{χ( q, ω)} = π { ρˆ (q ) φ 0 2 δ (ω (E E 0 )) (1.41) ρ( ˆ q) φ 0 2 δ (ω + (E E 0 ))} (1.42) Why are we iterested i the imagiary part of χ? The imagiary part of χ gives us iformatio about dissipatio, i.e. the absorptio ad loss of eergy as a result of the iteractio with the probe. We will ofte use the otatio χ ( q, ω) = Im{χ( q, ω)} We ca plot χ ( q, ω) as a fuctio of ω for fixed q (see plot). The locatio of the peaks tells us about the types of excitatios beig produced. As we will see shortly, it actually turs out that kowledge of χ ( q, ω) is all we eed; the real part of χ( q, ω), deoted χ ( q, ω), ca be recostructed from χ ( q, ω) aloe.

8 Saity Check: Free Fermios Saity Check: Free Fermios To covice ourselves that of this formalism is really workig, we will try it out o the case of free fermios, which we studied last semester i Now, χ( q, ω) is simply Π 0 ( q, ω). The groud state for free fermios is just the simple spherical Fermi sea, filled up to exactly to the Fermi eergy. The excited states are of the form k = hole at k, e at k + q (1.43) These sigle particle hole excitatios are the oly types of excitatios possible i this case, sice the exteral field U couples to the desity ρˆq = c k c. The matrix elemets we eed k q k are simple to calculate as well, sice all that is required is a filled iitial state below the Fermi sea (with wave vector k ad a ope state above the Fermi sea with wave vector k + q to jump ito. Thus k ρˆ q φ 0 = (1 f k+ q)f k (1.44) where f is 1 if the state with mometum k k is occupied i the groud state, ad 0 if it is empty. Substitutig this i, we get [ ] (1 f k+ q)f k (1 f k q)f k Π 0 ( q, ω) = ω (ɛ q ɛ k+ k ) + iη (1.45) k ) + iη ω + (ɛ q ɛ Lettig k k+ k q = k k = k + q we ca switch the dummy summatio variables o the secod term ad combie both terms ito oe: Π 0 ( q, ω) = = (1 f k+ q )f k (1 f k )f k+ q ω (ɛ (1.46) k+ k ) + iη q ɛ k f k f k+ q ω (ɛ (1.47) k+ k ) + iη q ɛ k (1.48) This is exactly the Lidhardt formula that we derived i The Correlatio Fuctio S( r, t) Let s switch gears ow ad talk about aother object that we will see is related to the respose fuctio. We defie the correlatio fuctio S( r, t) = φ 0 ρˆh ( r, t)ˆρ H (0, 0) φ 0 (1.49) S( r, t) described fluctuatios of the electro desity across the sample i space ad time. Due to the traslatioal ivariace of the sample, we arbitrarily set oe of argumets to (r, t ) = (0, 0)

9 The Correlatio Fuctio S( r, t) 8 ad observe the desity correlatio with aother poit ( r, t). What we wat to show ext is that there is a relatioship betwee dissipatio ad fluctuatios. Fourier trasformig S( r, t) i space yields S( q, t) = φ 0 ρˆh( q, t) ρ ˆ H ( q, 0) φ 0 (1.50) = ρˆ φ 0 2 e i(e E0 )t q (1.51) where the secod lie follows by isertig a complete set of states betwee the desity operators ad actig the e ±i Ĥt operators o the eigestates to the left ad the right. Notice that this is very similar with what we did earlier o our way to derivig the form of the respose fuctio. Now we take the Fourier trasform i time: S( q, ω) = d r dt e iωt e i S( q, t) (1.52) = 2π ρˆ 2 q φ 0 δ (ω (E E 0 )) (1.53) This expressio for S( q, ω) is idetical to the first (absorptive) term i the expressio for the imagiary part of the respose fuctio χ ( q, ω). This ca be restated as the Zero Temperature Fluctuatio Dissipatio Theorem: 1 χ ( q, ω) = (S( q, ω) S( q, ω)) (1.54) 2 This shows that the eergy absorbed i a probig experimet of the type described i this lecture is directly related to desity fluctuatios across the system. Although so far we have derived everythig at T = 0, the Fluctuatio Dissipatio Theorem ca be exteded to fiite temperatures as well. Usig the thermal average ( ) H Tr e β ˆ A ˆ A ˆ T = ( ) (1.55) Tr e β H ˆ the derivatio ca be redoe to arrive at the fiite temperature Fluctuatio Dissipatio Theorem: where B (ω) is the Boltzma statistical factor 1 ( ) χ ( q, ω) = e βω 1 S( q, ω) 2 (1.56) S( q, ω) = 2 ( B (ω) + 1) χ ( q, ω) (1.57) 1 B (ω) = e βω 1 (1.58) By playig with these relatios, we ca further derive the followig two results: S( q, ω) = e βω S( q, ω) (1.59) χ ( q, ω) = χ ( q, ω) (1.60) Equatio (1.59) is simply a statemet of the law of detailed balace.

10 Measurig S( q, ω) Measurig S( q, ω) It is possible to measure S( q, ω) directly through scatterig experimets. Depedig o the particle desity of iterest, the scatterig ca be performed usig electros, X rays, eutros, etc. This process is govered by the iteractio Hamiltoia Ĥ it = v( r i R) (1.61) i i( r i R) q = e vq (1.62) q = v ρˆ e i q R q (1.63) q where R is the positio of the scatterig electro ad { r i } are the sample electros coordiates. For ow, we imagie probig the electro desity by sedig i high eergy ( kev) electros. These electros iteract with the electros i the sample through the Coulomb iteractio. Thus To esure sigle scatterig, we eed to work i the regime of weak couplig. Thus we ca apply the first order Bor Approximatio ad Fermi s Golde Rule to obtai the scatterig rate 2π W i f = f Hˆ 2 it i δ(e i E f ) (1.66) h We take the iitial ad fial states of our scatterig probe to be plae waves k i ad k f, respectively. The the iitial ad fial states of the system are q 4πe 2 v = (1.64) q If we wated to perform eutro scatterig, the the { r i } would be the sample s uclear coord iates, ad the iteractio would be the cotact potetial q 2 2πb v = δ( r) (1.65) r M i = φ 0 k i (1.67) f = k f (1.68) Let Q = k i k f so that ω > 0 whe eergy is lost to the system ad Q is the mometum trasfer to the system. The 2 2 k i ) R ρˆ R e i( e i q R φ 0 d k f W i f = 2π v q q δ (ω (E E 0 )) (1.69) 2 h = 0 agai. q ω = E k i E k f

11 Measurig S( q, ω) = vq 2π ρˆ φ δ(e f E i ) (1.70) Q 0 2 = v S( Q Q, ω) (1.71) Thus the scatterig rate for scatterig with a mometum trasfer Q ad eergy loss hω is related to the correlatio fuctio S( Q, ω) very simply through a scalig by the square of the Q th Fourier compoet of the iteractio potetial.

Lecture 25 (Dec. 6, 2017)

Lecture 25 (Dec. 6, 2017) Lecture 5 8.31 Quatum Theory I, Fall 017 106 Lecture 5 (Dec. 6, 017) 5.1 Degeerate Perturbatio Theory Previously, whe discussig perturbatio theory, we restricted ourselves to the case where the uperturbed