Linear Response Theory: The connection between QFT and experiments

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Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and hen measure he elecric curren J. In general, J is a funcion of E, and hen we can make a Taylor expansion for his funcion 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 firs erm in he equaion above is zero. And herefore, he leading order erm is he linear erm of E. If we ignore all higher order erms beyond he leading order one, we ge a linear relaion beween J and E J =se (3.2) A he end of he day, we find ha as long as E is weak enough, J shall be proporional o E and he coefficien is he conduciviy. Because J is a linear funcion of E a small E, he weak E limi is also known as he linear response regime. The philosophy described above applies o many experimenal echniques. Very ypically, in an experimen we firs inroduce a small perurbaion (a ime ) o he sysem and hen we wai and see how he sysem response o his perurbaion a a laer ime ( ' > ). If he perurbaion is week enough, he response of he sysem is a linear funcion of he perurbaion, and we say ha he experimen is done in he linear response regime. In he linear response regime, we can compue he slope of he linear funcion (i.e. he measuremen of he experimen) using quanum field heory. This ype of calculaion is known as he linear response heory. 3.2. Linear Response Theory 3.2.1. he densiy marix and is ime-evoluion If we wan o measure he expecaion of he quanum operaor (say B` ), he quanum saisical mechanics ells us ha a emperaure T = 1 b, his is wha we will ge (see Sec. 1.1 for deails) n n B` exp -b H` -mǹ n Tr B` exp -b H` -mǹ B = = n n exp -b H` -mǹ n Tr exp -b H` -mǹ (3.3) Anoher way o wrie he same formula is o define he so-called densiy marix 1 r` = exp -b H` -mǹ = exp -b H` -mǹ -W Tr exp -b H` -mǹ (3.4) The prefacor 1 Tr exp -b H` -m Ǹ is a number, which is jus a renormalizaion facor. We can absorb i ino he exponenial

40 Phys540.nb r` = exp -b H` -mǹ -W (3.5) where W=- 1 b ln Tr exp -b H` -mǹ (3.6) and W is known as he grand poenial in saisical physics. Using r`, we find ha B = Tr B` r` (3.7) Now, we consider a sysem described by he Hamilonian H and we perurbaion his sysem, so ha he new Hamilonian is H` oal = H` + V À (3.8) The perurbaion includes wo facor, a Hermiian quanum operaor À and is coefficien (a real number) V. V saisfies he condiion ha a =, i is zero, V Ø ö0 (3.9) In oher words, firs we have a sysem described by he Hamilonian H, hen we urn on a perurbaion V À. Here we use he Heisenberg picure, so ha he quanum saes do no change wih ime and all he ime-evoluions come from quanum operaors. B` = n n B` r` n (3.10) The ime-evoluion of he quanum operaor B` is sraighforward. I jus follow he ime-evoluion of H` oal. We can define he ime-evoluion operaor Ù oal, = T exp - Â Ñ H` + V ' À d' (3.11) And hen, we know ha B` = Ù oal, B` S Ù oal, (3.12) where B` S is he same operaor in he Schrödinger picure. As shown above, he densiy marix r` is deermined by he Hamilonian. Here, we have wo Hamilonian, he full Hamilonian H` oal = H` + V À and he unperurbed Hamilonian H`. Which one shall we use for r`? The answer is a bi couner inuiive. Insead of he full Hamilonian, here, we should use he unperurbed par H`. This is because he densiy marix comes from saisical mechanism. A ime =, before we urn on he perurbaion, he sysem reaches hermal equilibrium for he unperurbed Hamilonian H`. Then, afer we urn on he perurbaion V ' À, he sysem doesn have enough ime o reach a new hermal equilibrium (for he new Hamilonian H` oal = H` + V À). Insead, when we consider he saisical weigh for each quanum sae, we should use he weigh for he original Hamilonian H`, insead of he unperurbed one H` oal. There are wo direc consequence for his. (1) r`, H` = 0 and (2) he ime-evoluion of r` is deermined by he unperurbed Hamilonian H`, insead of H` oal, i.e., r` = Ù H`, r` S Ù H`, (3.13) where Ù H`, = T exp - Â Ñ H` d' (3.14) In summary, here we emphasize again ha quanum mechanics and quanum saisical physics have wo oally differen ime scales. For quanum mechanics, he dynamics of a quanum operaor changes insananeous. If we perurb a Hamilonian, a quanum operaor knows i immediaely and is ime-evoluion is deermined by he full Hamilonian H` oal = H` + V À, immediaely afer we urn on he perurbaion. On he oher hand, he ime scale in quanum saisical physics is much slower. When we perurb a quanum sysem, i will ake he sysem a long ime o reach new equilibrium. Therefore, shorly afer we perurb he sysem, we sill use he unperurbed Hamilonian o compue hermal average.

Phys540.nb 41 3.2.2. ime-evoluion of B` Now, we know ha B` = Tr Ù oal, B` S Ù oal, Ù H`, r` S Ù H`, (3.15) If he perurbaion V À is weak enough, we can use he approximaion x = 1 + x + O x 2 and hus Ù oal, = T exp - Â Ñ H` + V ' À d' = T exp - Â Ñ H` d' + T exp - Â Ñ Hd' - Â Ñ '' V '' À + O V 2 (3.16) Because he ime-evoluion operaor requires he ime ordering (operaors wih smaller should be placed on he righ side), we find ha Ù oal, = T exp - Â Ñ H` d' - Â Ñ '' exp - Â Ñ Hd' V '' À exp - Â '' Ñ Hd' + O V 2 = Ù H`, - Â Ñ '' Ù H`, '' V '' À Ù H` '', + O V 2 '' (3.17) In he las sep, we used he fac ha T exp - Â Ñ H` d' is jus he ime evoluion operaor for he unperurbed Hamilonian H`. Similarly, Ù oal, = Ù H`, + Â Ñ '' Ù H` '', V '' À Ù H`, '' + O V 2 (3.18) As a resul, B` = Tr Ù H`, B` S Ù H`, Ù H`, r` S Ù, + H` Â Ñ '' V '' Tr Ù H` '', À Ù H`, '' B` S Ù H`, Ù H`, r` S Ù, - H` Â Ñ '' V '' Tr Ù H`, B` S Ù H`, '' À Ù H` '', Ù H`, r` S Ù, H` = Tr B` r` S + Â H` Ñ '' V '' Tr Ù H` '', À Ù H`, '' B` S r` S Ù, - H` (3.19) Â Ñ '' V '' Tr Ù H`, B` S Ù H`, '' À Ù H` '', r` S + O V 2 Here, we used he facs ha Ù H` -1 = Ù H` and r`, H` = 0. Because r`, H` = 0, r`, Ù H` = r`, Ù H` = 0 and hus we can swich he order for r` and he ime-evoluion operaors. The firs erm in he formula above is he expecaion value of B` for he unperurbed Hamilonian. B` = B` H` + Â Ñ '' V '' Tr Ù H` '', À Ù H` '', Ù H` '', Ù H`, '' B` S Ù H`, r` S - Â Ñ '' V '' Tr Ù H`, B` S Ù H`, '' Ù H` '', Ù H` '', À Ù H` '', r` S + O V 2 = B` H` + Â Ñ '' V '' Tr À '' Ù H`, B` S Ù H`, r` S - H` Â '' V '' Tr Ù H`, B` S Ù H`, À '' r` S H` + O V 2 (3.20) = B` H` + Â Ñ '' V '' Tr À '' B` r` S H` -Â '' V '' Tr B` À '' r` S + O V H` 2 = B` H` + Â Ñ ' V ' À ' B` - B` À ' H` + O V 2

42 Phys540.nb = B` H` + Â Ñ ' V ' À ', B` H` + O V 2 Here, all he expacaion values are compued for he unperurbed Hamilonian H`, H`. Very ypically, B` H` = 0, i.e. he quaniy ha we measure in an experimenal is zero before we urn on he perurbaion, and herefore B` = Â Ñ ' V ' À ', B` H` + O V 2 (3.21) To he leading order, he measurable quaniy B` is proporional o he perurbaion V ', and hus his is known as he linear response heory. To compue he non-linear response, one needs o go o higher order in he above expansion, which will no be considered in his lecure. In he linear response regime, he measuremen is proporional o he perurbaion. The coefficien of his linear relaion is À ', B` H`, which only depends on he unperurbed Hamilonian. In oher words, he linear-response measuremen measures he inrinsic properies of he sysem. 3.3. Linear Response Theory and he Suscepibiliy 3.3.1. generalized suscepibiliy In general, he perurbaion and observaion discussed in he previous secion can also have spaial dependence, i.e., he À and B` operaor can depends on he real space coordinae. There, he more general linear response heory akes he following form (he deviaion is he same and we ignore higher order non-linear erms). B` x Ø,, = Â Ñ Ø x ' ' V x Ø ', ' À x Ø ', ', B` x Ø, = H` Ø Â x ' + ' Ñ q - ' À xø ', ', B` x Ø, V x Ø ', ' H` Here, in he las sep, we change he upper bound for he ime inegral from o +. In he same ime, we added an sep funcion q - ' q x = 1 x > 0 0 x < 0 so he formula remains he same. From now on, we will drop he sub-index H` for expecaion values H`. In oher words, all he expecaion values compued in his secion are for he original Hamilonian H`, wihou he small perurbaion. We can define he generalized suscepibiliy (3.22) (3.23) c x Ø, ; x Ø ', ' = Â Ñ q - ' À xø ', ', B` x Ø, (3.24) Please pay special aenion o his sep funcion q - '. We need i here, because he ime inegral requires ' <. As will be discussed below, his implies causiliy. I is easy o show ha B` x, = x Ø ' ' c x Ø, ; x Ø ', ' V x Ø ', ' (3.25) Please noice ha he ime inegral now is from o +. In oher words, we can describe our perurbaion (inpu) as a funcion V x Ø ', '. This funcion measures he srengh of our perurbaion a ime ' and a he locaion Ø x '. Then we measure he oupu, which is also a funcion of space and ime B` x,, i.e. our measurable quaniy measure a he ime and he posiion Ø x. The generalized suscepibiliy ells us he connecion beween he inpu and he oupu. For any given inpu, he inegral above ells us immediaely wha oupu we shall expec. NOTE: In he definie of he generalized suscepibiliy, a sep funcion of ime arises. This sep funcion is acually expeced and mus be here, because i implies causaliy (if he sep funcion is no here, our heory mus be wrong, because i violae causaliy). The sep funcion ells us ha he response will be nonzero, only if > '. Noice ha he inpu (he perurbaion) is applied a ' and we are measuring he resul a. This sep funcion acually implies ha for measuremen a ime, only inpus before his ime poin conribue o he measuremen, which is known as causaliy.

Phys540.nb 43 3.3.2. The frequency space Now, we conver he linear response heory obained above ino he frequency space. There are wo moivaions for his: I simplifies he formula, as will be shown below Real experimens are ofen down his way. One sends in an signal (perurbaion) a frequency w and hen measure he response a he same frequency w. For w=0, i is known as a DC measuremen. For w 0 bu small, i is ypically known a AC measuremen. For high w, he name varies depending on he echniques, IR, opical, UV, X-ray, ec. Here, we assume ranslaional symmery along he ime axis (which is rue for mos condensed maer sysems), c x Ø, ; x Ø ', ' = c x Ø, +D; x Ø ', ' +D (3.26) In oher words, we assume ha c x Ø, ; x Ø ', ' only depends on he difference beween ime and '. Define B x Ø, w = Âw B` x Ø, (3.27) V x Ø, w = Âw V x Ø, (3.28) c x Ø, x Ø ', w = Âw -' c x Ø, ; x Ø ', ' (3.29) The inverse ransformaion is easy o obain B` x Ø w, = 2 p - w B x Ø, w V x Ø w, = 2 p - w V x Ø, c x Ø, ; Ø w x ', ' = 2 p - w -' c x Ø, Ø x ', w Using he linear response heory obained above, we find ha (3.30) (3.31) (3.32) B x Ø, w = Âw B` x Ø, = x Ø ' Âw ' c x Ø, ; x Ø ', ' V x Ø ', ' = Ø w' x ' Âw ' 2 p - w' -' c x Ø, Ø w'' x ', w' 2 p - w'' ' V x Ø ', w'' = Ø w' w'' x ' 2 p 2 p  w-w' '  w'-w'' ' c x Ø, Ø x ', w' V x Ø ', w'' = x Ø ' w' 2 p w'' 2 p 2 pd w-w' 2 pd w' -w'' c xø, x Ø ', w' V x Ø ', w'' (3.33) = x Ø ' c x Ø, x Ø ', w V x Ø ', w We find ha if he inpu is a frequency w, he oupu mus have exacly he same frequency wihin he linear response heory (non-linear response may show up a differen frequencies, bu hey will be very weak a small V). The formula ha we obained here is known as he Kubo formula, which is he key relaion in he linear response heory. B x Ø, w = x Ø ' c x Ø, x Ø ', w V x Ø ', w (3.34) 3.3.3. The frequency-momenum space One can furher conver he formula ino he momenum space.

44 Phys540.nb B k Ø, w = -Â k Øÿx Ø +Â w B` x Ø, (3.35) V k Ø, w = -Â k Øÿx Ø +Â w V x Ø, (3.36) c k Ø, w = -Â k Øÿ x Ø -x Ø ' +Â w -' c x Ø, ; x Ø ', ' (3.37) Here, we assume he ranslaion symmery, i.e., x,x ', only depends on he difference beween x and x '. The procedure would be essenially he same, so ha we will no repea i here. A he end of he day, one finds ha B k Ø, w =c k Ø, w V k Ø, w (3.38) i.e., he oupu and inpu has exacly he same frequency and wavevecor and heir srenghs are proporional o each oher. The coefficien here is he generalized suscepibiliy a he same frequency and wavevecor. For a wide range of experimens, we are measuremen some ype of c k Ø, w. 3.4. Suscepibiliy from he Quanum Field Theory Q: How do we compue he suscepibiliy using he quanum field heory? 3.4.1. The rearded Green s funcion and he advanced Green s funcion c x Ø, ; Ø x ', ' = Â Ñ q - ' À xø ', ', B` x Ø, =- Â Ñ q - ' B` x Ø,, À x Ø ', ' =- Â Ñ q - ' B` x Ø, À x Ø ', ' - À x Ø ', ' B` x Ø, c x Ø, ; Ø x ', ' =- Â Ñ q - ' B` x Ø,, À x Ø ', ' (3.39) (3.40) This formula is very similar o he Green s funcion ha we defined in previous chapers. In fac, his quaniy is one ype of Green s funcion, known as he rearded Green s funcion. G R x Ø, ; x Ø ', ' =- Â Ñ q - ' B` x Ø,, À x Ø ', ' (3.41) Similarly, one can define advanced Green s funcion as G A x Ø, ; x Ø ', ' = Â Ñ q ' - B` x Ø,, À x Ø ', ' (3.42) The name of he rearded Green s funcion comes from he fac ha if we apply a perurbaion À x Ø ', ' a ime ', only measuremen a laer ime > ' will be affeced. The advanced Green s funcion is he opposie: only < ' obains nonzero conribuions. For many experimens, À and B` are acually he same quanum operaor. For example, if we apply a volage and hen measure he change in densiy, he perurbaion o he Hamilonian is V r`, where V is he volage and r` is he charge densiy operaor. The measuremen r` is he same quanum operaor. Here, we assume hey are he same operaor and we will label hem by Ò. Furhermore, we assume ha Ò is a bosonic operaor. From now on, we will se Ñ = 1 (same as wha we did in he previous chapers when we compue he Green s funcions) G R x Ø, ; x Ø ', ' =-Âq - ' Ò x Ø,, Ò x Ø ', ' G A x Ø, ; x Ø ', ' =Âq ' - Ò x Ø,, Ò x Ø ', ' (3.43) (3.44) 3.4.2. The relaion beween he rearded Green s funcion and he specral funcion As shown in he previous chapers, we can define oher ype of Green s funcions, e.g.,

Phys540.nb 45 G > x Ø, ; x Ø ', ' =-Â Ò x Ø, Ò x Ø ', ' G < x Ø, ; x Ø ', ' =-Â Ò x Ø ', ' Ò x Ø, (3.45) (3.46) I is easy o realize ha G R x Ø, ; x Ø ', ' =-Âq - ' Ò x Ø,, Ò x Ø ', ' =-Âq - ' Ò x Ø,, Ò x Ø ', ' - Ò x Ø ', ' Ò x Ø,, =q - ' G > x Ø, ; x Ø ', ' - G < x Ø, ; x Ø ', ' (3.47) Now we go o he frequency space. And remember, as shown in Eq. (1.117), due o hisorical reasons, when we conver G > and G < o he frequency space, we need an exra facor of (-Â). There is nohing fundamenal here. I is jus a convenion. G R x Ø, x Ø ', w = Âw -' G R x Ø, ; x Ø ', ' = Âw -' q - ' G > x Ø, ; x Ø ', ' - G < x Ø, ; x Ø ', ' = Âw -' q - ' -Â w' -Â w' -' G > x Ø, x Ø ', w' - -Â w' -Â w' -' G < x Ø, x Ø ', w' =-Â w' Â w-w' -' q - ' G > x Ø, x Ø ', w' - -Â w' Â w-w' -' q - ' G < x Ø, x Ø ', w' (3.48) =-Â w' G > x Ø, x Ø ', w' - G < x Ø, x Ø ', w' ' + Â w-w' -' Remember ha he definiion of he specral funcion is A x, x', w = G > x, x', w - G < x, x', w (3.49) And herefore, G R x Ø, x Ø ', w =-Â w' A x, x', w' ' + Â w-w' -' (3.50) Now we need o ake care of he inegral + Â w-w' -' = ' =+ (3.51) 1 1 Â w-w' Â w-w' -' =' = 1 1 Â w-w' Lim Ø+ Â w-w' -' - Lim Ø' Â w-w' -' = 1 1 Â w-w' Lim Ø+ Â w-w' -' - 1 To ensure ha he limi Lim Ø+ Â w-w' -' exiss, we need o add a small posiive imaginary par o w, i.e., wøw+âd where d>0 and d<<1. There, Lim Ø+ Â w-w' -' Ø Lim Ø+ Â w-w' -' -d -' = 0 (3.52) And hus G R x Ø, Ø Â A x, x', w' x ', w+âd =-Â w' A x, x', w' = w' w+âd-w' w+âd-w' If we go o he momenum space, he formula reads A k Ø, w' G R k Ø, w+âd = w' w+âd-w' (3.53) (3.54) or ofen A k Ø, w' G R k Ø, w = w' w-w' +Âd (3.55) As we discussed in previous chapers, in quanum field heory (he Green s funcion approach), he key quaniy ha we compue is he specral funcion A k Ø, w'. Wih his quaniy, we can easily ge he rearded Green s funcion and hus he suscepibiliy

46 Phys540.nb 3.4.3. The relaion beween he rearded Green s funcion and he ime-ordered Green s funcion We know ha he ime-ordered Green s funcion can be wrien as A k Ø, w' G k Ø, Âw n = w' Âw n -w' (3.56) where Âw n represens he Masubara frequencies. Therefore, i is easy o noe ha o obain he rearded Green s funcion, we jus need o subsiue he Masubara frequency for he ime-ordered Green s funcion ino w+âd. G R k Ø, w = G k Ø, Âw n Øw+Âd (3.57) Summary, in QFT, we compue he ime-ordered Green s funcions, which is much easier o compue han oher Green s funcions. However, for real experimenal measuremens, we need o deal wih oher Green s funcions (e.g. he rearded Green s funcion for linear-response measuremens). Bu he good news is, as soon as we know he ime-order Green s funcions, oher ones can be obained easily.

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