Summary of Mattuck Chapters 16 and 17

Transcription

1 Summary of Mattuck Chapters 16 and 17 Tomas Petersson Växjö university

2 1 Phonons form a Many-Body Viewpoint Hamiltonian for coupled Einstein phonons Definition of Einstein phonon propagator Evaluation of propagator by exact summation of graphs Question of convergence 2 Quantum Field Theory of Phase Transitions in Fermi Systems Qualitative theory of phase transitions Anomalous propagators and the breakdown of the perturbation series in the condensed phase The generalized matrix propagator Application to ferromagnetic phase in system with δ-function interaction Summary

3 Hamiltonian for coupled Einstein phonons Hamiltonian for coupled Einstein phonons where Linear chain of N atoms. Interatomic distance d. Harmonic coupling constant 1 2 mω2 0 N [ p 2 H = l 2m + 1 ] 2 mω2 0ul 2 1 N 2 mω2 0 u l u l+1 u l = l 1 1 ) (b l + b 2mω l 0 Neglecting end effect. p l = 1 i l=1 mω0 2 ( ) b l b l

4 Hamiltonian for coupled Einstein phonons Make a Fourier transform of b l to b k and the Hamiltonian can be written ( H = ω 0 b k b k + 1 ) ω ( ) ( ) 0 cos kd b k + b 2 2 k b k + b k k = H 0 + H 1 k>0 Where H 0 is the Hamiltonian for a set of oscillators through the whole lattice with a common frequency, called bare Einstein phonons. The perturbing term H 1 is responsible for the k-dependent frequency. Problem: Solution: H 0 H 1 so ordinary perturbation theory can t be used. Canonical transformation.

5 Hamiltonian for coupled Einstein phonons Canonical transformation φ k = b k + b k H = ω 0 2 k>0 π k = i ( ) b k b k [ ] π k π k + (1 cos kd) φ k φ k 2 Introducing boson operators a k, a k ω0 ) (a k + a ω k = b k +b k k ωk ( ) a ω k a k = b k b k 0 Leading to the simple Hamiltonian and dispersion law H = ( ω k a k a k + 1 ) ω k = ω 0 1 cos kd 2 k

6 Definition of Einstein phonon propagator Definition of Einstein phonon propagator Simplest propagator to understand for the Einstein phonon is G(k, t) = i ψ o T {b k (t)b k (0)} ψ 0 where T is the time-ordering operator and b k (t) is the operator b k i the Heisenberg picture.

7 Definition of Einstein phonon propagator In the case of zero interaction, G(k, t) assumes the free propagator form G 0 (k, t) = iθ t e iω 0t Fourier transform G 0 (k, ω) = 1 ω ω 0 + iδ G 0 (k, t) = iθ te iω 0t G 0 (k, ω) = 1 ω + ω 0 iδ

8 Definition of Einstein phonon propagator An alternative propagator is where the φ k s are D(k, t) = i ψ 0 T {φ k (t)φ k (0)} ψ 0 φ k = b k + b k φ k = b k + b k which will generate the free propagator D 0 (k, ω) = 2ω 0 ω 2 ω iδω 0 D 0 includes propagating both forward and backwards in time.

9 Evaluation of propagator by exact summation of graphs Evaluation of propagator by exact summation of graphs The perturbing part of the first Hamiltonian becomes in the interaction picture (the case where H 0 H 1 ) H 1 (t) = [ ] ] V p b p (t) + b p [b (t) p (t) + b p(t) p>0 V p = ω 0 cos pd 2 The four products can be represented graphically b p b p b p b p b p b p b p b p

10 Evaluation of propagator by exact summation of graphs Diagram rules Factor of ig 0 (k, ω) for the forward propagating (+k) lines. Factor of ig0 (k, ω) for the backward propagating ( k) lines. Factor of iv k for each interaction wiggle.

11 Evaluation of propagator by exact summation of graphs We expand the Einstein phonon propagator as ig(k, ω) = Define an irreducible self-energy diagram which can t be drawn as two parts connected by a positive k-line. Sum all reducible self-energy diagrams and create a Dyson s equation. = + = + + ; +...

12 Evaluation of propagator by exact summation of graphs The Dyson equation becomes which gives ig(k, ω) = ig 0 + (ig 0 ) The poles of G(k, ω) occur at ( i ) (ig) G(k, ω) = ω + ω ( cos kd) ω 2 ω0 2 (1 cos kd) ω = ω 0 1 cos kd which is identical with the phonon dispersion law.

13 Evaluation of propagator by exact summation of graphs The perturbing part of the second Hamiltonian, where the canonical transformation was made, becomes in the interaction picture H 1 (t) = p>0 V p φ p (t)φ p(t) We expand this alternative Einstein phonon propagator id(k, ω) = We use the diagram rules Factor of id 0 (k, ω) for each line. Factor of iv k for each wiggle.

14 Evaluation of propagator by exact summation of graphs The simple expansion will result in id(k, ω) = id 0 1 D 0 V k 2ω 0 D(k, ω) = ω 2 ω0 2 (1 cos kd) The poles of D(k, ω) produces (again) the phonon dispersion law.

15 Question of convergence Question of convergence The summations made in the examples put a restriction on where ω is valid. One can arrive at the same propagator D(k, ω) using a different technique without having any restriction on ω at all. To my knowledge there is no way to know when you can disregard the convergence criteria and when you can t.

16 Quantum Field Theory of Phase Transitions in Fermi Systems Phase transition Sudden change from a normal state to a condensed state Short-range order long-range order Perfect symmetry broken symmetry Parameters that determine phase transitions T - temperature ρ - density or λ - coupling constant

17 Phase transition characteristic First-order transitions will have a discontinuity at the transition point for the long-range order parameter. Second-order transitions have continuously changing long-range order parameter, but a discontinuity in its first derivative at the transition point.... and so on.

18 Qualitative theory of phase transitions Qualitative theory of phase transitions In a condensed system we have a long-range order without an external field. an internal field causing long-range order. interaction between the particles of the system causing this internal field. In a normal phase, the field is limited and causes only short-range order.

19 Qualitative theory of phase transitions A qualitative description of the internal field F internal long-range field O long-range order parameter λ interparticle interaction strenght T temperature O will be a function of F, λ and T O = O λt (F ) The field will in turn be a function of O F = F λt (O) Combined O = O λt (F λt (O)) which one must find by self-consistent method.

20 Anomalous propagators and the breakdown of the perturbation series in the condensed phase Anomalous propagators and the breakdown of the perturbation series in the condensed phase Normal perturbation series breaks down in any condensed phase. Ferromagnetic Normal Phase or Phase In a ferromagnetic phase there is a possibility that the particle spin may be flipped.

21 Anomalous propagators and the breakdown of the perturbation series in the condensed phase We therefore add an additionally anomalous propagators which flip a spin G fer (k, k, t t) = i ψ o T {c k (t )c k (t)} ψ 0 G fer (k, k, t t) = i ψ o T {c k (t )c k (t)} ψ 0 which are zero in the normal phase. Anomalous propagation exists in all condensed phases. They are not taken into account in the normal expansion. Causes the breakdown of the normal perturbation expansion in the condensed system.

22 The generalized matrix propagator The generalized matrix propagator Construct a perturbation theory valid for the condensed state Modify the self-consistent self-energy expansion. Replace normal propagators by matrix propagators. Diagonal elements will be the normal propagators. Off-diagonal elements will be the anomalous propagators. G fer (k, t t) = [ Gfer (k, k, t t) G fer (k, k, t t) G fer (k, k, t t) G fer (k, k, t t) ]

23 The generalized matrix propagator It is possible to write G fer in the same form as the ordinary normal propagator by introducing the spinor operators ( ) ck (t) ( ) γ k (t) = γ c k (t) k (t) = c k (t), c k (t) which leads to G fer (k, t t) = i ψ o T {γ k (t )γ k (t)} ψ 0 The Hamiltonian can now be written H = k (ɛ k µ)γ k γ k + 1 V klmn (γ 2 l γ n)(γ k γ m) + constant klmn

24 The generalized matrix propagator Both G and H have the same form in terms of the γ k s as the ordinary G and H have in terms of the c k s. Use the same perturbation expansion as before, just change ordinary propagators to matrix propagators. The Dyson equation thus becomes G fer (k, ω) = 1 G 1 0 fer (k, ω) (Gfer ) which just is the self-consistent equation for the long-order parameter described earlier.

25 Application to ferromagnetic phase in system with δ-function interaction Application to ferromagnetic phase in system with δ-function interaction Calculate the magnetization in Hartree-Fock approximation with the developed formalism H = kσ ɛ k c kσ c kσ + V klmn σσ clσ c kσ c mσc nσ The matrix propagator is = 1 1 +

26 Application to ferromagnetic phase in system with δ-function interaction We assume that the internal magnetization field is in the z direction so that G = G = 0. Applying the diagram dictionary from the book (Table 17.3) yields where and N σ = i G σσ (l, ω) = d 3 l dω (2π) 3 2π G σσ(l, ω)e iω0 1 ω (ɛ l µ) VN σ + iωδ

27 Application to ferromagnetic phase in system with δ-function interaction We express the magnetization by and we can eventually arrive at M = N N = N N N + N N NV = 1 2ɛ F 2M ] [(1 + M) (1 M) 3

28 Application to ferromagnetic phase in system with δ-function interaction Magnetization as a function of NV 2ɛ F (a) Paramagnetic Region (b) Unsaturated Ferromagnetic (c) Saturated Ferromagnetic Depends on the Fermi-energy and the coupling. The magnetization changes continuously at the critical point, while the derivative doesn t - this is a second-order phase transition.

29 Summary Summary We can use the same perturbation expansion for condensed phase as for normal phase if we replace ordinary propagators by matrix propagators. let the diagonal elements be the normal propagators. let the off-diagonal elements be the anomalous propagators.