Linear Response Theory: The connection between QFT and experiments

Transcription

1 Phys540.nb 39 3 Lnear Response Theory: The connecon beween QFT and expermens 3.1. Basc conceps and deas Q: ow do we measure he conducvy of a meal? A: we frs nroduce a weak elecrc feld E, and hen measure he elecrc curren J. In general, J s a funcon of E, and hen we can make a Taylor expanson for hs funcon J(E) = J(E = 0) + J ' (E = 0) E + 1 2! J '' (E = 0) E2 + (3.1) Symmery ells us ha when E = 0, J = 0, so he frs erm n he equaon above s zero. And herefore, he leadng order erm s he lnear erm of E. If we gnore all hgher order erms beyond he leadng order one, we ge a lnear relaon beween J and E J = σ E (3.2) A he end of he day, we fnd ha as long as E s weak enough, J shall be proporonal o E and he coeffcen s he conducvy. Because J s a lnear funcon of E a small E, he weak E lm s also known as he lnear response regme. The phlosophy descrbed above apples o many expermenal echnques. Very ypcally, n an expermen we frs nroduce a small perurbaon (a me ) o he sysem and hen we wa and see how he sysem response o hs perurbaon a a laer me (' > ). If he perurbaon s week enough, he response of he sysem s a lnear funcon of he perurbaon, and we say ha he expermen s done n he lnear response regme. In he lnear response regme, we can compue he slope of he lnear funcon (.e. he measuremen of he expermen) usng quanum feld heory. Ths ype of calculaon s known as he lnear response heory Lnear Response Theory he densy marx and s me-evoluon If we wan o measure he expecaon of he quanum operaor (say B ), he quanum sascal mechancs ells us ha a emperaure T = 1/β, hs s wha we wll ge (see Sec. 1.1 for deals) B = n n B exp-β - μ N n n n exp-β - μ N n = Tr B exp-β - μ N Tr exp-β - μ N Anoher way o wre he same formula s o defne he so-called densy marx (3.3)

2 40 Phys540.nb ρ 1 = Tr exp-β - μ N exp-β - μ N = exp-β - μ N - Ω (3.4) The prefacor 1 Tr exp-β -μ N s a number, whch s jus a renormalzaon facor. We can absorb no he exponenal ρ = exp-β - μ N - Ω (3.5) where Ω = - 1 β ln Tr exp-β - μ N (3.6) and Ω s known as he grand poenal n sascal physcs. Usng ρ, we fnd ha B = Tr B ρ Now, we consder a sysem descrbed by he amlonan and we perurbaon hs sysem, so ha he new amlonan s oal = + V() A (3.7) (3.8) The perurbaon ncludes wo facor, a erman quanum operaor A and s coeffcen (a real number) V(). V() sasfes he condon ha a =, s zero, V( ) 0 (3.9) In oher words, frs we have a sysem descrbed by he amlonan, hen we urn on a perurbaon V() A. ere we use he esenberg pcure, so ha he quanum saes do no change wh me and all he me-evoluons come from quanum operaors. B () = n n B () ρ () n (3.10) The me-evoluon of he quanum operaor B () s sraghforward. I jus follow he me-evoluon of oal. We can defne he me-evoluon operaor U oal(, ) = Texp - ħ + V(') A d ' (3.11) And hen, we know ha B () = U oal(, ) B S U oal(, ) (3.12) where B S s he same operaor n he Schrödnger pcure. As shown above, he densy marx ρ s deermned by he amlonan. ere, we have wo amlonan, he full amlonan oal = + V() A and he unperurbed amlonan. Whch one shall we use for ρ? The answer s a b couner nuve. Insead of he full amlonan, here, we should use he unperurbed par. Ths s because he densy marx comes from sascal mechansm. A me =, before we urn on he perurbaon, he sysem reaches hermal equlbrum for he unperurbed amlonan. Then, afer we urn on he perurbaon V(') A, he sysem doesn have enough me o reach a new hermal equlbrum (for he new amlonan oal = + V() A ). Insead, when we consder he sascal wegh for each quanum sae, we should use he wegh for he orgnal amlonan, nsead of he unperurbed one oal. There are wo drec consequence for hs. (1) ρ, = 0 and (2) he me-evoluon of ρ s deermned by he unperurbed amlonan, nsead of oal,.e., ρ () = U (, ) ρ S U (, ) (3.13) where U (, ) = Texp - ħ d ' (3.14) In summary, here we emphasze agan ha quanum mechancs and quanum sascal physcs have wo oally dfferen me scales. For quanum mechancs, he dynamcs of a quanum operaor changes nsananeous. If we perurb a amlonan, a quanum operaor knows

3 Phys540.nb 41 mmedaely and s me-evoluon s deermned by he full amlonan oal = + V() A, mmedaely afer we urn on he perurbaon. On he oher hand, he me scale n quanum sascal physcs s much slower. When we perurb a quanum sysem, wll ake he sysem a long me o reach new equlbrum. Therefore, shorly afer we perurb he sysem, we sll use he unperurbed amlonan o compue hermal average me-evoluon of B () Now, we know ha B () = Tr U oal(, ) B S U oal(, ) U (, ) ρ S U (, ) (3.15) If he perurbaon V() A s weak enough, we can use he approxmaon e x = 1 + x + Ox 2 and hus U oal(, ) = Texp - ħ + V(') A d ' = Texp - ħ d ' + Texp - ħ d ' - ħ d'' V('') A + OV 2 (3.16) Because he me-evoluon operaor requres he me orderng (operaors wh smaller should be placed on he rgh sde), we fnd ha '' d ' - d '' exp - d ' V('') A exp - d ' + OV 2 = U oal(, ) = Texp - ħ ħ ħ '' U (, ) - ħ d '' U (, '') V('') A U ('', ) + OV 2 ħ (3.17) In he las sep, we used he fac ha Texp- ħ d ' s jus he me evoluon operaor for he unperurbed amlonan. Smlarly, U oal(, ) = U (, ) + ħ As a resul, d '' U ('', ) V('') A U (, '') + OV 2 (3.18) B () = Tr U (, ) B S U (, ) U (, ) ρ S U (, ) + d'' V('') Tr U ħ d'' V('') Tr U ħ = Tr B () ρ S() + ħ ('', ) A U (, '') B S U (, ) U (, ) ρ S U (, ) - (, ) B S U (, '') A U ('', ) U (, ) ρ S U (, ) d '' V('') Tr U ('', ) A U (, '') B S ρ S U (, ) - ħ d'' V('') Tr U (, ) B S U (, '') A U ('', ) ρ S + OV 2 (3.19) ere, we used he facs ha (U )-1 = (U ) and ρ, = 0. Because ρ, = 0, ρ, U = ρ, (U ) = 0 and hus we can swch he order for ρ and he me-evoluon operaors. The frs erm n he formula above s he expecaon value of B () for he unperurbed amlonan. B () = B () + ħ d '' V('') Tr U ('', ) A U ('', ) U ('', ) U (, '') B S U (, ) ρ S - ħ d'' V('') Tr U (, ) B S U (, '') U ('', ) U ('', ) A U ('', ) ρ S + OV 2 = B () + ħ d '' V('') Tr A ('') U (, ) B S U (, ) ρ S - d'' V('') Tr U (, ) B S U (, ) A ('') ρ S + OV 2 (3.20) = B () + ħ d '' V('') Tr A ('') B () ρ S - d'' V('') Tr B () A ('') ρ S + OV 2 = B () + ħ d ' V(') A (') B () - B () A (') + OV 2

4 42 Phys540.nb = B () + ħ d ' V(') A ('), B () + OV 2 ere, all he expacaon values are compued for he unperurbed amlonan,. Very ypcally, B () = 0,.e. he quany ha we measure n an expermenal s zero before we urn on he perurbaon, and herefore B () = ħ d ' V(') A ('), B () + OV 2 (3.21) To he leadng order, he measurable quany B () s proporonal o he perurbaon V('), and hus hs s known as he lnear response heory. To compue he non-lnear response, one needs o go o hgher order n he above expanson, whch wll no be consdered n hs lecure. In he lnear response regme, he measuremen s proporonal o he perurbaon. The coeffcen of hs lnear relaon s A ('), B (), whch only depends on he unperurbed amlonan. In oher words, he lnear-response measuremen measures he nrnsc properes of he sysem Lnear Response Theory and he Suscepbly generalzed suscepbly In general, he perurbaon and observaon dscussed n he prevous secon can also have spaal dependence,.e., he A and B operaor can depends on he real space coordnae. There, he more general lnear response heory akes he followng form (he devaon s he same and we gnore hgher order non-lnear erms). B x,, = ħ d x ' d ' Vx ', ' A x ', ', B x, = d + x ' d' ħ θ( - ') A x ', ', B x, Vx ', ' (3.22) ere, n he las sep, we change he upper bound for he me negral from o +. In he same me, we added an sep funcon θ( - ') θ(x) = 1 x > 0 0 x < 0 (3.23) so he formula remans he same. From now on, we wll drop he sub-ndex for expecaon values. In oher words, all he expecaon values compued n hs secon are for he orgnal amlonan, whou he small perurbaon. We can defne he generalzed suscepbly χx, ; x ', ' = ħ θ( - ') A x ', ', B x, (3.24) Please pay specal aenon o hs sep funcon θ( - '). We need here, because he me negral requres ' <. As wll be dscussed below, hs mples causly. I s easy o show ha B (x, ) = d x ' d ' χx, ; x ', ' Vx ', ' (3.25) Please noce ha he me negral now s from o +. In oher words, we can descrbe our perurbaon (npu) as a funcon Vx ', '. Ths funcon measures he srengh of our perurbaon a me ' and a he locaon x '. Then we measure he oupu, whch s also a funcon of space and me B (x, ),.e. our measurable quany measure a he me and he poson x. The generalzed suscepbly ells us he connecon beween he npu and he oupu. For any gven npu, he negral above ells us mmedaely wha oupu we shall expec. NOTE: In he defne of he generalzed suscepbly, a sep funcon of me arses. Ths sep funcon s acually expeced and mus be here, because mples causaly (f he sep funcon s no here, our heory mus be wrong, because volae causaly). The sep funcon ells us ha he response wll be nonzero, only f > '. Noce ha he npu (he perurbaon) s appled a ' and we are measurng he resul a. Ths sep funcon acually mples ha for measuremen a me, only npus before hs me pon conrbue o he measuremen, whch s known as causaly.

5 Phys540.nb The frequency space Now, we conver he lnear response heory obaned above no he frequency space. There are wo movaons for hs: I smplfes he formula, as wll be shown below Real expermens are ofen down hs way. One sends n an sgnal (perurbaon) a frequency ω and hen measure he response a he same frequency ω. For ω = 0, s known as a DC measuremen. For ω 0 bu small, s ypcally known a AC measuremen. For hgh ω, he name vares dependng on he echnques, IR, opcal, UV, X-ray, ec. ere, we assume ranslaonal symmery along he me axs (whch s rue for mos condensed maer sysems), χx, ; x ', ' = χx, + Δ; x ', ' + Δ (3.26) In oher words, we assume ha χx, ; x ', ' only depends on he dfference beween me and '. Defne Bx, ω = d e ω B x, (3.27) Vx, ω = d e ω Vx, (3.28) χx, x ', ω = d e ω (-') χx, ; x ', ' The nverse ransformaon s easy o oban B x dω, = 2 π e- ω Bx, ω Vx dω, = 2 π e- ω Vx, χx, ; d ω x ', ' = 2 π e- ω (-') χx, x ', ω (3.29) (3.30) (3.31) (3.32) Usng he lnear response heory obaned above, we fnd ha Bx, ω = d e ω B x, = d x ' d e ω d ' χx, ; x ', ' Vx ', ' = d x ' d e ω d ' d ω' 2 π e- ω' (-') χx, dω'' x ', ω' 2 π e- ω'' ' Vx ', ω'' = d dω' d ω'' x ' 2 π 2 π d e( ω-ω') d ' e (ω'-ω'') ' χx, x ', ω' Vx ', ω'' = d x ' dω' 2 π d ω'' 2 π 2 π δ(ω - ω') 2 π δ(ω' - ω'') χx, x ', ω' Vx ', ω'' = d x ' χx, x ', ω Vx ', ω (3.33) We fnd ha f he npu s a frequency ω, he oupu mus have exacly he same frequency whn he lnear response heory (non-lnear response may show up a dfferen frequences, bu hey wll be very weak a small V). The formula ha we obaned here s known as he Kubo formula, whch s he key relaon n he lnear response heory. Bx, ω = d x ' χx, x ', ω Vx ', ω (3.34) The frequency-momenum space One can furher conver he formula no he momenum space.

6 44 Phys540.nb Bk, ω = d e - k x + ω B x, (3.35) Vk, ω = d e - k x + ω Vx, (3.36) χk, ω = d e - k x -x '+ ω (-') χx, ; x ', ' (3.37) ere, we assume he ranslaon symmery,.e., χ x, x ', ω only depends on he dfference beween x and x '. The procedure would be essenally he same, so ha we wll no repea here. A he end of he day, one fnds ha Bk, ω = χk, ω Vk, ω (3.38).e., he oupu and npu has exacly he same frequency and wavevecor and her srenghs are proporonal o each oher. The coeffcen here s he generalzed suscepbly a he same frequency and wavevecor. For a wde range of expermens, we are measuremen some ype of χk, ω Suscepbly from he Quanum Feld Theory Q: ow do we compue he suscepbly usng he quanum feld heory? The rearded Green s funcon and he advanced Green s funcon χx, ; x ', ' = ħ θ( - ') A x ', ', B x, = - ħ θ( - ') B x,, A x ', ' = - ħ θ( - ') B x, A x ', ' - A x ', ' B x, (3.39) χx, ; x ', ' = - ħ θ( - ') B x,, A x ', ' (3.40) Ths formula s very smlar o he Green s funcon ha we defned n prevous chapers. In fac, hs quany s one ype of Green s funcon, known as he rearded Green s funcon. G R x, ; x ', ' = - ħ θ( - ') B x,, A x ', ' (3.41) Smlarly, one can defne advanced Green s funcon as G A x, ; x ', ' = ħ θ(' - ) B x,, A x ', ' (3.42) The name of he rearded Green s funcon comes from he fac ha f we apply a perurbaon A x ', ' a me ', only measuremen a laer me > ' wll be affeced. The advanced Green s funcon s he oppose: only < ' obans nonzero conrbuons. For many expermens, A and B are acually he same quanum operaor. For example, f we apply a volage and hen measure he change n densy, he perurbaon o he amlonan s V ρ, where V s he volage and ρ s he charge densy operaor. The measuremen ρ s he same quanum operaor. ere, we assume hey are he same operaor and we wll label hem by O. Furhermore, we assume ha O s a bosonc operaor. From now on, we wll se ħ = 1 (same as wha we dd n he prevous chapers when we compue he Green s funcons) G R x, ; x ', ' = - θ( - ') O x,, O x ', ' (3.43) G A x, ; x ', ' = θ(' - ) O x,, O x ', ' (3.44) The relaon beween he rearded Green s funcon and he specral funcon As shown n he prevous chapers, we can defne oher ype of Green s funcons, e.g., G > x, ; x ', ' = - O x, O x ', ' (3.45)