Methods of QFT in Condensed Matter Physics WiSe 2014/15 Lecturer Dr. Oleg Yevtushenko, Tutor Dennis Schimmel ([email protected]) http://homepages.physik.uni-muenchen.de/~oleg.yevtushenko/qft-cm-WiSe-14-15/ OutLine-QFT-CM-WiSe-14-15.htm Problem Set 3 “Random Phase Approximation” Let us consider a 3d system of interacting fermions (and treat spin only as a degeneracy) with the Hamiltonian R 3 † 1 ∇2 − µ + U (~r, τ ))Ψ(~r, τ ) (1) H= d rΨ (~r, τ )(− 2m R 3 3 0 † + d rd r Ψ (~r, τ )Ψ† (~r0 , τ )V (~r − ~r0 )Ψ(~r0 , τ )Ψ(~r, τ ). (2) Here U is weak external potential and V is the Coulomb interaction. Exercise 1. “Fermionic Coherent states” Coherent states are introduced via ˆb|bi = b|bi. (3) Recall that for fermions b is a Grassmann number (and for bosons b is a complex number). (a) Show that for fermions we have |bi = exp(−bˆb† )|0i, where |0i is the vacuum. (b) Using (a), compute the overlap between a Fock-state with n particles and a coherent state |bi. What is the approriate way to change coordinates from Fock-states to coherent states? P (c) Prove that T r(A) = n hn|A|ni can be rewritten as an integral over coherent states. Exercise 2. “Ring Graph Expansion” For this exercise, consider the hamiltonian (1), and set U = 0. In the lecture you saw that in the case of high density the dominant contributions to the free energy due to interactions stem from diagrams where “bubbles” are arranged one after the other in a ring. Show that the contribution of a ring graph with n bubbles comes with a factor of 1/n, that is T X n (n) FRP A = − Π , (4) 2n q q with Πnq := X 2T V (q) Gp Gp+q , L3 p where Gp is the electron Green’s function and Π is the polarization operator. 1 (5) Exercise 3. “Calculating Π” The polarization operator plays a crucial role in various phenomena, for example in response functions. In this exercise we will demonstrate how interactions lead to screening (via polarization). Let us introduce effective quantities Uef f and Vef f via Uef f = V U ; Vef f = . 1−VΠ 1−VΠ (6) These are the screened external potential and screened interaction. Since the full computation of Π is lengthy, we will restrict ourselves to some approximations: We consider the limit |q| pF , ω EF and T → 0. (a) Show from (5) that Π(~q, ωm ) = 2 X 1 1 L3 β iΩn − k i(Ωn + ωm ) − k+q (7) ~k,Ωn (b) Calculate the Matsubara sum (over frequencies Ωn ) in the expression for Π. (c) To calculate the sum over ~k, change to the continuum approximation and introduce spherical coordinates. (d) Expand the numerator and denominator of the integrand in q. (e) Perform all the integrals. You should find in the considered limits: s s+1 Π ≈ 2ν0 ( log( ) − 1), 2 s−1 where s = ω vF q (8) and ν0 is the density of states at the Fermi energy. Exercise 4. “The spectrum of excitations” In this exercise we wish to consider the retarded polarization operator. (a) The dispersion relation of collective excitations is given by the poles of the effective interaction Vef f . Show that the poles lie at V (~q)Π(~q, ω) = 1, V (~q) = 4πe2 . q2 (9) (b) What is the form of the dispersion relation for large and small q? Sketch the dispersion relation of the collective excitations and compare this with the dispersion of quasi-particle excitations, ωq = vF q. (c) There is also some regime where there is no viable way of introducing collective excitations. This regime is characterized by the absence of any poles in the screened interaction Vef f . A simple way of excluding poles is to look at the imaginary part. Show that Z 2mpF π ω ImΠ(~q, ω) = − d3 k 0 θ(|~q0 + ~k 0 | − 1)θ(1 − k 0 )δ( − ~k 0 ~q0 − (q 0 )2 /2). (10) 3 (2π) vF pF 2 (d) What conditions do we have to impose for the imaginary part to be non-zero? (e) Sketch the region of ImΠ 6= 0. (f) Show that the conditions for ImΠ 6= 0 are consistent with those for the creation of real (as opposed to virtual) electron-hole pairs. 3