MIT OpenCourseWare http ocw mit edu 8 334 Statistical Mechanics II Statistical Physics of Fields Spring 2008 For information about citing these materials or our Terms of Use visit http ocw mit edu terms IV Perturbative Renormalization Group IV A Expectation Values in the Gaussian Model Can we treat the Landau Ginzburg Hamiltonian as a perturbation to the Gaussian model In particular for zero magnetic eld we shall examine H H0 U Z Z L 2 2 t 2 K 2 d x m m m u dd xm4 IV 1 2 2 2 d The unperturbed Gaussian Hamiltonian can be decomposed into independent Fourier modes as Z 1 X t Kq 2 Lq 4 dd q t Kq 2 Lq 4 2 H0 m q m q 2 V q 2 2 d 2 IV 2 The interaction mixes up the normal modes and Z dd xm x 4 Z Z d d q1 dd q2 dd q3 dd q4 ix q1 q2 q3 q4 d e m q1 m q2 m q3 m q4 u d x 2 4d U u IV 3 where summation over and is implicit The integral over x sets q1 q2 q3 q4 0 and U u Z dd q1 dd q2 dd q3 m q1 m q2 m q3 m q1 q2 q3 2 3d IV 4 From the variance of the Gaussian weights the two point expectation values in a nite sized system with discretized modes are easily obtained as hm q m q i0 q q V t Kq 2 Lq 4 IV 5 In the limit of in nite size the spectrum becomes continuous and eq IV 5 goes over to hm q m q i0 2 d d q q t Kq 2 Lq 4 53 IV 6 The subscript 0 is used to indicate that the expectation values are taken with respect to the unperturbed Gaussian Hamiltonian Expectation values involving any product of m s can be obtained starting from the identity X X ai aj hmi mj i0 IV 7 exp ai mi exp 2 i i j 0 which is valid for any set of Gaussian distributed variables mi This is easily seen by completing the square Expanding both sides of the equation in powers of ai leads to ai aj ai aj ak ai aj ak al hmi mj i0 hmi mj mk i0 hmi mj mk mk i0 2 6 24 ai aj ak al ai aj hmi mj i0 hmi mj i0 hmk ml i0 hmi mk i0 hmj ml i0 hmi mk i0 hmj ml i0 1 2 24 IV 8 Matching powers of ai on the two sides of the above equation gives 0 for odd Y mi IV 9 i 1 sum over all pairwise contractions for even 0 1 ai hmi i0 This result is known as Wick s theorem and for example hmi mj mk ml i0 hmi mj i0 hmk ml i0 hmi mk i0 hmj ml i0 hmi mk i0 hmj ml i0 IV B Expectation values in Perturbation Theory In the presence of an interaction U the expectation value of any operator O is computed perturbatively as follows R R Dm Oe H0 U Dm e H0 O 1 U U 2 2 hOi R R e H0 U Dm e H0 1 U U 2 2 Dm IV 10 Z0 hOi0 hOUi0 hOU 2 i0 2 Z0 1 hUi0 hU 2 i0 2 Inverting the denominator by an expansion in powers of U gives 1 1 2 2 2 hOi hOi0 hOUi0 hOU i0 1 hUi0 hUi0 hU i0 2 2 1 hOi0 hOUi0 hOi0 hUi0 hOU 2 i0 2hOUi0 hUi0 2hOi0 hUi20 hOi0 hU 2 i0 2 n X 1 c hOU n i0 n n 0 IV 11 54 The connected averages are de ned as the combination of unperturbed expectation values appearing at various orders in the expansion Their signi cance will become apparent in diagrammatic representations and from the following example Let us calculate the two point correlation function of the Landau Ginzburg model to rst order in the parameter u In view of their expected irrelevance we shall ignore higher order interactions and also only keep the lowest order Gaussian terms Substituting eq IV 4 into eq IV 11 yields Z d d q1 dd q2 dd q3 hm q m q i hm q m q i0 u 2 3d IV 12 hm q m q m q m q m q m q q q i i 1 i 2 j 3 j 1 2 3 0 hm q m q i0 hmi q1 mi q2 mj q3 mj q1 q2 q3 i0 O u2 To calculate hOUi0 we need the unperturbed expectation value of the product of six m s This can be evaluated using eq IV 9 as the sum of all pair wise contractions 15 in all Three contractions are obtained by rst pairing m to m and then the remaining four m s in U Clearly these contractions cancel exactly with corresponding ones in hOi0 hUi0 The only surviving terms involve contractions that connect O to U This cancellation persists at all orders and hOU n ic0 contains only terms in which all n 1 operators are connected by contractions The remaining 12 pairings in hOUi0 fall into two classes i 4 pairings involve contracting m and m to m s with the same index e g hm q mi q1 i0 hm q mi q2 i0 hmj q3 mj q1 q2 q3 i0 IV 13 i i jj 2 3d d q q1 d q q2 d q1 q2 2 2 2 t Kq t Kq t Kq3 where we have used eq IV 6 After summing over i and j and integrating over q1 q2 and q3 such terms make a contribution Z d d q3 1 n 2 d d q q IV 14 4u 2 2 d t Kq 2 t Kq32 ii 8 pairings involve contracting m and m to m s with di erent indices e g hm q mi q1 i0 hm q mj q3 i0 hmi q2 mj q1 q2 q3 i0 IV 15 i j ij 2 3d d q q1 d q q3 d q1 q3 t Kq 2 t Kq 2 t Kq22 Summing over all indices and integrating over the momenta leads to an overall contribution of Z d 2 d d q q d q2 1 8u IV 16 t Kq 2 2 2 d t Kq22 Adding up both contributions we obtain Z 4u n 2 dd k 1 2 d d q q 2 1 O u hm q m q i t Kq 2 t Kq 2 2 d t Kk 2 IV 17 55 IV C Diagrammatic Representation of Perturbation Theory The calculations become more involved at higher orders in perturbation theory A diagrammatic representation can be introduced to help keep track of all possible contrac Q tions To calculate the point expectation value h i 1 m i qi i at pth order in u proceed according to the following rules 1 Draw external …