7. Response functions

Transcription

1 7. Response functions 7.1 Random Phase Approximation The truncation of the equation of motion for the Greens function with the Hartree Foc mean field approximation transforms the Hamiltonian into an effective single particle operator; this too simple treatment of the influence of interactions leads to a lac of realism. A better approach than the truncation of the equation of motion hierarchy is the approximate evaluation also of the higher order equations of motion. This allows to better account for the interaction as the full Hamiltonian enters the equation of motion at each step. 1 The so called random phase approximation (RPA) involves replacement of operator products by averages as in the mean field approximation (neglect of the fluctuation of averages). For the single particle Greens function G σ (ω) = c σ ; c σ (7.1) the result is again the Hartree-Foc result. The magnetic susceptibility is an important quantity describing the response of a material to an external magnetic field H(t) = H e (ω+iδ)t : M α (t) = M α β dt M α (t); M β (t ) H β e i(ω+iδ)t (7.) with cartesian directions α, β. As the Greens function is homogeneous in time we have M α (t) = M α β = M α β dt M α ; M β (t t) H β e i(ω+iδ)(t t) e i(ω+iδ)t dt M α ; M β (t ) e i(ω+iδ)t H β (t) 1 This chapter is based on K. El, W. Gasser, Die Methode der Greenschen Funtionen in der Festörperphysi, Aademie-Verlag Berlin 1979, and W. Nolting, Grundurs Theoretische Physi 7, Viel-Teilchen-Theorie, Springer 9. 99

2 (7.3) with a field independent part of the magnetization M α which is important for ferromagnetic systems and the second part describing the magnetization induced by an external magnetic field. This part is proportional to H(t); the prefactor represents the magnetic susceptibility χ: M α (t) = M α + β χ αβ (ω)h β (t) (7.4) Comparison yields χ αβ (ω) = dt e i(ω+iδ)t M α ; M β (t) M α ; M β (ω iδ) (7.5) Thus, χ αβ (ω) is given by the Fourier transform of the Greens function. We can now replace the magnetic moment operator M i at site i by a spin operator S i to obtain χ µν ij (ω) = Mµ i ; Mν j (ω) = g µ B S µ i ; Sν j (ω) (7.6) with gyromagnetic factor g and Bohr magneton µ B. Of particular interest are the longitudinal susceptibility χ zz ij (ω) = g µ B S z i ; Sz j (ω) (7.7) and the transversal susceptibility χ + ij (ω) = g µ B S + i ; S j (ω) where S± i = S x i ± is y i (7.8) The operators S z i, S+ i and S i can again be replaced by creation and annihilation operators: S z i = 1 (n i n i ), S + i = c i c i, S i = c i c i (7.9) which leads to and χ zz ij (ω) = 1 4 g µ B(δ σσ 1) n iσ ; n jσ (ω) (7.1) χ + ij (ω) = g µ B c i c i ; c i c i (ω) (7.11) 1

3 lining the susceptibilities to special two particle Greens functions. We will now apply the random phase approximation to the calculation of the transversal magnetic susceptibility within the Hubbard model H = σ (ε µ)c c + U c c σ σ N q c 1 c 1 + q ; (7.1) 1q the random phase approximation will be valid in the limit of wea interactions. In space we have for the susceptibility χ + ( q, ω) = g µ B c c + q ; c (ω) c (7.13) q The equation of motion for this two-particle Greens function is ω c c + q ; c c q (ω) = ( n n + q ) δ, + q + + ( ε + q ) ε c c + q ; c c (ω)+ q + U (c c N 1 c 1 q 1 c + q q 1 c + q1 c 1 c 1 + q 1 c + q ); c c (ω) q 1, q 1 (7.14) According to the principles of the random phase approximation the excess operators in the higher order Greens functions are replaced by averages: c c 1 c 1 q 1 c + q+ q 1 n 1 δ q 1, c c + q n δ 1 q 1 c q1 c q q 1 (7.15) Here, conservation of momentum and spin was used: Also c σ c σ = n σ δ δ σσ (7.16) c + q1 c 1 c 1 + q 1 c + q n 1 δ q 1, c c + q n + q δ 1, + q c + q1 c + q+ q 1 (7.17) This yields ( ω ω ( q) ) c c + q ; c (ω) = ( n c q ) n + q [ δ, U ] c + q N c ; + q c c (ω) q 11 (7.18)

4 with the Stoner single particle excitation spectrum ω σσ ( q) = ε + q ε + U(n σ n σ) (7.19) Now we divide Eq. (7.18) by ( ω ω ( q) ) and sum over : c c + q ; c (ω) = n n + q c q ω ω ( q) n n + q ω ω ( q) U N After renaming summation indices this means c c + q ; c c q (ω) = δ, + q c c ; + q c (ω) c (7.) q n n + q ω ω ( q) 1 + U N δ, + q n n + q ω ω ( q) (7.1) with the susceptivility of the Stoner model χ ( q, ω) = g µ B n n + q ω ω ( q) (7.) This yields for the transverse susceptibility χ + ( q, ω) = χ ( q, ω) 1 U Ng µ Bχ ( q, ω) (7.3) The denominator in this RPA expression for the susceptibility can become small for certain q and ω values so that the susceptibility becomes big. Noninteracting susceptibility The spin susceptibility χ 1 can be expressed 3 in terms of the Matsubara spin-spin correlation function (χ 1 ) s t = 1 β dτe T iωτ τss (q, τ) S t ( q, ) 3 This part was wored out by Michaela Altmeyer; than you! 3 S. Graser, T.A. Maier, P.J. Hirschfeld, D.J. Scalapino, New J. Phys. 11, 516 (9). 1

5 with the Matsubara frequency ω, the imaginary time τ and the spin operators for the different orbitals s S s (q) = 1 d sα( + q) σ αβ d sβ (), (7.4),αβ where α and β are spin indices. The charge susceptibility χ can be calculated in a very similar manner, (χ ) s t = β dτe iωτ T τ n s (q, t)n t ( q, ) (7.5) with n s (q) = 1 d N sα( + q)d sβ ()δ α,β. (7.6),αβ In the noninteracting case ( 3 i=1 Tr(σ i ) = 6), both are equivalent and can be written in a more general way as st (q, iω) = 1 N β dτe iωτ Tτ d pα(, τ)d qα ( + q, τ) αβ d sβ (, )d tβ ( q, ). (7.7) The imaginary-time ordered expectation value can be evaluated by means of Wic s theorem. st (q, iω) = 1 N β + 1 N β dτe iωτ dτe iωτ αβ αβ G pt α ( + q, τ)g sq α (, τ)δ α,β δ,+q G pq α (, τ)g st β (, )δ q,. (7.8) The second term leads to an unphysical delta function δ q, and will therefore be neglected in the following considerations, while the first term can be further analyzed by writing the imaginary time Green s functions as Fourier transforms of Matsubara Green s functions (the summation over the spin index α leads to a factor of ), G pt (, τ) = 1 e iωnτ G pt (, iω n ). (7.9) β ω n 13

6 Insertion into equation (7.8) yields st (q, iω) = 1 β dτe iωτ N β ω n,ω m e iωnτ G pt (+q, iω n )e iωmτ G sq (, iω m ), (7.3) so that we can now perform the integration over imaginary time. st (q, iω) = 1 N β eiβ(ω ωn+ωm) 1 i(ω ω ω n,ω n + ω m ) Gpt (+q, iω n )G sq (, iω m ). m (7.31) Due to the fact that the frequencies are Fermionic Matsubara frequencies, the fraction is zero when ω ω n + ω m ; however, when ω = ω n + ω m we have to consider the limit e iβ(ω ω n+ω m ) 1 i(ω ω n + ω m ) δ ω n,ω+ω m = 1 + iβ(ω ω n + ω m ) 1 δ ωn,ω+ω i(ω ω n + ω m ) m = βδ ωn,ω+ω m, (7.3) so that st (q, iω) = G pt ( + q, iω n )G sq (, iω m )δ ωn,ω+ω βn m ω n,ω m = G pt ( + q, iω + iω m )G sq (, iω m ). βn ω m Maing use of the spectral representation of the Green s function G pt (, iω n ) = µ (7.33) a s µ()a p µ () iω n E µ (), (7.34) where the E µ denote the eigenvalues of the tight binding Hamiltonian and a s µ are the s components of the corresponding eigenvectors, this can be written as st (q, iω) = βn ω m µν a s µ()a p µ () iω m E µ () 14 a q ν( + q)a t ν ( + q) iω + iω m E ν ( + q).

7 Evaluation of the sum over Matsubara frequencies leads to (7.35) st (q, iω) = a s µ()a p µ ()a q ν( + q)a t ν ( + q) [ f(eν ( + q)) f(e µ ()) ] N iω + E ν ( + q) E µ (),µν (7.36) where N is the product of the number of considered points and the number of bands in the model hamiltonian. The static homogeneous spin susceptibility can be calculated quite easily from χ S (q) = 1 (χ 1 ) pp ss (q, ω = ) sp = 1 N sp,µν ν ( + q) [ f(eν ( + q)) f(e µ ()) ]. E ν ( + q) E µ () a s µ()a p µ ()a p ν( + q)a s (7.37) The susceptibility (7.36) should be properly normalized, taing into account the number of possible spin orientations, the number of considered points and the number of bands in the model Hamiltonian ( N n). The computational implementation needs an extra consideration. When the energies in the denominator are the same we have a unphysical singularity, which can be avoided using the rule of l Hopital, so that χ S (q) = 1 (χ 1 ) pp ss (q, ω = ) sp = β N sp,µν [ a s µ()a p µ ()a p ν( + q)a s ν ( + q) 1 e β(e ν(+q) µ) + e β(e ν(+q) µ) e β(e µ() µ) + e β(e µ() µ) + (7.38) Random Phase Approximation (RPA) The random phase approximated susceptibilities can be obtained from a Dyson-type equation. The charge susceptibility can be calculated from (χ RPA ) pq st = χpq st (χ RPA ) pq uv(u c ) uv wzχ wz st, (7.39) uvwz 15 ].

8 while the spin susceptibility has a plus sign instead, (χ RPA 1 ) pq st = χpq st + (χ RPA ) pq uv(u s ) uv wzχ wz st. (7.4) uvwz 16