SELF-CONSISTENT EQUATIONS A1137 where uA(n) and uo(n)are, respectively, the chemical So far everything is formal and exact. We now write potentials of an interacting and a noninteracting homo- in the spirit of the previous sections geneous gas of density n, and g and go are the respective G-1(r,,,, [nD=G-1(r, r; [n]+Gxo-1(r,r; [n].(4.7) densities of states. 10 It follows immediately that the low-temperature heat The second term we approximate as for a slowl capacity is given by Cr=YT s, which giv Y=3+k Ng, (u)+ n(r)(g(un(n))-go(o(n))dr x[n]=(1/20)2(/V)×g(), (4.9) We shall not present a treatment, analogous to and x(n), xo(n)are, respectively, the susceptibilities for ec IIB, in which exchange effects are included exactly. uniform systems with and without interactions he development is straightforward but leads to nown divergence in the low-temperature specific heat. APPENDIX I: GRADIENT EXPANSION OF THE DENSITY IV. SPIN SUSCEPTIBILITY In this Appendix we show that for a system of slowly To obtain a theory of the spin susceptibility of an varying density our procedure gives the density electron gas, we first extend the theory of HK to include correct to order V 2 inclusive. When dealing with such the effects of spin interaction with an external magnetic a system we may proceed in two entirely equivalent direction and write the magnetic-moment density as Eqs.(2. and can solve the self-consistent equations, field. The result is that if we take the field in the s ways: (1) (2.9), for n(r), and(2)we can go bac m(r)=-(1/20)0y+*(r)1(t)-(x)4()10),(4.1)totheunderlyingvaiationalprinciple(25,make gradient expansion and determine n(r) directly. We the ground-state energy can be written in the form shall here follow the second route to estimate the errors (r) E, m =(o(r)n(r)-H(rm(r)dr From(2.5)and the expansion (2.12)of T.[n], we obtain where G is a universal functional of n and m, and the where u is the chemical potential [cf. HK, Eq.(68)] correct m(r), n(r)make a minimum Note however that because of our approximation of For small m we expand G in the form keeping only the first term in(2.11), some other contri outions of order V 2 are missing in(A1.1) G=GD]+ G(r,r; D ])m(r)m()dr dr+.;(4.3), To solve(A1. 1), let us write the external charge ity as the linear term vanishes for a paramagnetic system in ns()≡f(r/ro), which m=0 when H=0. From the stationary property where fo-co(slow spatial variation), and try the order and that n(r)=m(r)+n1(r), (A1.3) where ()+/Gr;[n])m(ar=0,(44) no(r)=fo(r/ro) 14) exactly neutralizes the external charge and n is assumed where n is the zero-field density. We now formally to approach zero as ro=o0. Neglecting, for the moment, invert this equation, which gives the terms of order V2 in(A1. 1)and substituting (A1.3)into(Al. 1), we obtain n(r)=/G-I(r,r; [n]H(r)dr' dr'tun(no)+ni(r)uA'(no)+o(n1? For a uniform field this gives for the susceptibility Now define y aH m(r)dr=G(r,r En]dr dr.(4.6) r and write 10 J. M. Luttinger, Phys. Rev. 119, 1153(1960) n1(r)≡f1(R), (A1.7)SELF —Co N S1STENT EQ UATIONS So fal every thmg» fo~al » We now write ~ s sectjolls, G —i rr';[n cond pproxim ate va ryi gg which gives x[n]=x,[n]+— [x(n(r)) x n —xp(n(r))] r dr, (4.8) C,,=yv-, (3.16) where respective y the chemical „( ) r and Po(n) ' ' teract»g h where phi teracting and a n he respective tentials of an s of density ~ d and go are t￾It follows immediately t a low Perature heat capacity is given by =-'ee Xg Q)+ «)(g6 n(. ~(n))—ap(t p(n))) « We shall no p treatment, analogous to b 1 d t Sec. known divergenence in the 1ow- e where = (1/2 )'(~/~)&&g, ( ), (4.9) res ective ly, the susceptibilities for unl orm sy stems wiith and wit hou t interactions. APPENDIX I:. GRADIEEINT EXPANSIONN OF IV. SPIN SUSCEPTIBILITY s in susceptibility of an as, we 6rstexten d the h fHK 1 d *()o ()-s * ~ theh groun-und-state energy can be written ln t e ().()-~() ()&d G G[n]+— 2 t e anisl e o pa rama agn etic system in th h h W lC eL=— 0f (4.2) we fin, or order and tha t I d I —H(r)+ G(r,r', [n])m(r')dr =0, p (4.4) n is the zero-field density. ensit . We now formally inver t this equation, wh h ' m r = G—'(r,r', [n])H(r')dr'. (4.5) ~ ~ e ' ' For a uniform Geld this gives for the susceptibihty ' ' or 8 x[n]=— G '(r,r'; m(r) dr= G [n]) dr dr . 4.6) — r,r; h p. Rpv. 119, 1153 (1960). 'o J. M. Luttinger, phys, Rcp, @Vi d'+GL () ()], ( 2 r—r'i universal functional of e an m, orrect m(r), ( 42) For small m we expan [n])m(r)m(r') dr dr'; + (4.3) 1 's endix we showw that or a 's en ' w f system of slow y ives the density In thi pp y o p corre e the self-consistent q d (2.9), for n(r), an 2) ob er i ' ' al rinciple and determine e r)di 1. W th dm t too estimate the error From (2.5) an d thee e pansion (2.12 o, n, otential [cf. HK, Eq.. (68)]. is the chemical po e ( k pngoonl y the first term in .11),, o t er co utlon t' ns of order 1 let us wri e To solvee A1. ), 'te the externa c (A1.2) density as d t the ' where ro~~— (slow spatia l variation), v an ry + ansatz where n(r) =np(r)+ni(r), (A1.3) Now de6ne and write R=r/rp, ni(r) =fi(R) . (A1.6) (A1.'/) no(r) =fp(r/«) (A1.4) exactly e y neu tralizes the external er charge annd e& is assumed e lecting, for the momen, terms of order V~' in (A1.1 (A1.3) A1.1), bt i in'to )+ .().'(.)+o( '). fr—r'/ (A1.5)