1 Time reversal. 1.1 Without spin. Time-dependent Schrödinger equation: 2m + V (r) ψ (r, t) (7) Local time-reversal transformation, T :
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1 1 Time reversal 1.1 Without spin Time-dependent Schrödinger equation: i t ψ (r, t = Local time-reversal transformation, T : Transformed Schrödinger equation d (f T (t dt ] 2m + V (r ψ (r, t (1 t 1 < t 2 <... < t n T t 1 > T t 2 >... > T t n (2 T (t 2 t 1 = (t 2 t 1 (3 f (T t + T dt f (T t = = dt i t ψ (r, T t = T = T 1 (4 2m + V (r f (T t dt f (T t dt df (T t = dt (5 ] ψ (r, T t (6 On the other hand, i t ψ (r, t = ] 2m + V (r ψ (r, t (7 ψ (r, T t = ψ (r, t = C ψ (r, t (8 Properties: C is anti-hermitian, and anti-linear, C 2 = 1, C 1 = C (9 ψ Cϕ = ϕ Cψ = Cψ ϕ (10 C (c 1 ϕ 1 + c 2 ϕ 2 = c 1Cϕ 1 + c 2Cϕ 2. (11 However, the transformation C preserves the norm of the wavefunctions, Relationship to operators: ( C (pψ =C i ψ Cψ Cψ = ψ ψ. (12 C (rψ = r (Cψ = Cr = rc (13 = Cψ = p (Cψ = Cp = pc (14 i CL = C (r p = rc p = (r p C = LC (15 1
2 1.2 With spin Hamilton operator Pauli-Schrödinger equation i t ψ (r, t = Time-reversed magnetic field: = H = p2 2m + V (r + µ (L + 2S (16 2m + V (r + µ ] (L + 2S ψ (r, t (17 Time-reversed Pauli-Schrödinger equation i t ψ (r, T t = 2m + V (r + µ ] (L + 2S ψ (r, T t (18 = 2m + V (r µ ] (L + 2S ψ (r, T t (19 On the other hand: i t ψ (r, t = = 2m + V (r + µ ] (L + 2S ψ (r, t (20 2m + V (r µ ] (L 2S ψ (r, t (21 It is then tempting to suppose that This equation is obviously satisfied if ψ (r, T t = L ψ (r, t = LC ψ (r, t (22 il t ψ (r, t = 2m + V (r µ ] (L + 2S Lψ (r, t (23 i t ψ (r, t = 2m + V (r µ ( L + 2L 1 SL ] ψ (r, t (24 Let s introduce the simplified notation: T := LC L 1 SL= S = C S C = S LC = LC S (25 T S = ST. (26 It is easy to prove that T = σ y C (27 is a satisfactory choice (in many text-boos T = iσ y C is chosen. Proof of Eq. (26: σ x = ( = σ x σy = ( 0 i i 0 = σ y σz = ( = σ z (28 2
3 T 1 σ x T = ( σ y C σ x (σ y C = σ y σ x σ y = σ x (29 T 1 σ y T = ( σ y C σ y (σ y C = σ y (30 T 1 σ z T = ( σ y C σ z (σ y C = σ y σ z σ y = σ z (31 Properties: T 1 = Cσ y = σ yc = σ y C = T (32 (33 T 2 = 1 (34 From the relationship, ψ T ϕ = ψ σ y Cϕ = σ y ψ Cϕ = ( σy rs ψs Cϕ r = ϕ r Cσy rs ψ s = ϕ Cσ y ψ = ϕ T ψ, (35 it follows that ψ T ψ = ψ T ψ = 0, (36 i.e. ψ and T ψ are orthogonal and, also, T is norm-conserving, T ψ T ψ = ψ T 2 ψ = ψ ψ. (37 The operator of spin-orbit coupling, 4m 2 c 2 ( V p σ, commutes with T : T 1 ( V p σt = ( T 1 ( V p T ( T 1 σt = ( V ( p ( σ = ( V p σ. ( Kramers degeneracy Let us consider an eigenfunction, ψ (r 1 s 1,..., r N s N of the N-electron Hamiltonian, Hψ = Eψ (39 where T 1 HT = H. (40 The time-reversed wavefunction, T ψ, is then also eigenfunction of H with the same eigenvalue, T 1 HT ψ = Eψ = H (T ψ = E (T ψ. (41 The representation of T is T = σ y (1... σ (N y C = ( 1 N Cσ y (1... σ (N y = ( 1 N T 1 = T 2 = ( 1 N, (42 T + = T 1 = ( 1 N T, (43 since for any = 1,..., N T S ( = S ( T. (44 Furthermore, ψ T ψ = ψ σ (1 y... σ y (N Cψ = ( 1 N ψ C σ (1 y... σ y (N ψ = Eq. (10 ( 1N σ y (1... σ y (N ψ C ψ = ( 1 N ψ σ 1 y... σ N y C ψ = ( 1 N ψ T ψ (45 Corollary: For odd number of electrons ψ and T ψ are orthogonal, therefore, the eigenstates of the system are at least two-fold degenerate. 3
4 1.4 Kramers degeneracy of loch-states We consider the Hamiltonian derived from the Dirac equation up to first order of 1/c 2 : H = p2 2m + V (r p4 8m 3 c m 2 V (r + c2 4m 2 ( V p σ (46 c2 This one-electron Hamiltonian is invariant w.r.t. time-reversal, T 1 HT = H. (47 From the previous section it follows that the eigenstates are at least two-fold degenerate: and T ψ is orthogonal to ψ. What is T ψ? A loch-eigenfunction is defined as Hψ = εψ (48 H (T ψ = ε (T ψ (49 H = (p + 2 2m + V (r It is straightforward to show that ψ (r = e ir u (r (50 (p + 4 8m 3 c m 2 V (r + c2 4m 2 ( V (p + σ c2 (51 H u = ε u (52 thus, T 1 H T = H (53 T 1 H u = ε T 1 u (54 H ( T 1 u = ε ( T 1 u (55 ε = ε (56 and the two degenerate wavefunctions are: ( ψ (r = e ir u (r u (r and ψ (1 (r = e ir ( iu (r iu (r ( Space inversion Let s consider the case when also space inversion applies: V (I r = V ( r = V (r (58 I H I = H (59 This also implies that ε = ε with the corresponding wavefunction for, ( ψ (2 (r = u ( r e ir. (60 u ( r 4
5 In case of both time-reversal and inversion symmetry, the two eigenfunctions for with the same energy ε (= ε are orthogonal: ψ (1+ (r ψ(2 (r d3 r = i [u (r u ( r u (r u ( r] d 3 r = 0 (61 Corollary: The loch-states of a nonmagnetic centro-symmetric crystal are at least twofold degenerate. 1.6 Sorting out by spin-expectation value In general, the eigenfunctions ψ (µ (µ = 1, 2 are not eigenfunctions of the spin-operator S z for any prechosen quantization axis z. This is only the case in the absence of spin-orbit coupling. Nevertheless, it is possible to construct the orthonormal linear combinations, c 1, c 2 C, c c 2 2 = 1, such that ψ (+/ ψ (+ = c 1 ψ (1 + c 2ψ (2 (62 ψ ( = c 2ψ (1 + c 1ψ (2 (63 σ x ψ (+/ = ψ (+/ σ y ψ (+/ = 0 (64 and ψ (+/ σ z ψ (+/ = ±P (65 0 P 0 (66 Thus we can sort out the two degenerate states by the spin-character, P. 5
6 FIG. 2: and structure of Pt from the fully relativistic (red and the relativistic with the spin-orbit coupling scaled to zero (blac calculation. FIG. 3: Calculated fully relativistic band structure of bcc Fe. The small inset shows a comparison to the calculation with the spin-orbit coupling scaled to zero (x=0. The spin-orbit interaction leads to avoided crossings. 6
7 FIG. 4: Calculated relativistic Fermi surface of Cu (upper left, Au (upper right and Pt (lower left: 9th band, lower right: 11th band, and the expectation values of ˆβσ z for the Ψ + states are indicated as color code. Note the different scale for Cu and Au in comparison to Pt. 7
8 FIG. 5: Calculated relativistic Fermi surface for the bands 7-10 of bcc Fe. The expectation values of the ˆβσ z operator are given as color code. 8
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