Resonant inelastic x-ray scattering studies of elementary excitations Ament, L.J.P.

excitations

Ament, L.J.P.

Citation

Ament, L. J. P. (2010, November 11). Resonant inelastic x-ray scattering studies of elementary excitations. Casimir PhD Series. Retrieved from https://hdl.handle.net/1887/16138

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A p p e n d i x A

QED

This appendix contains some details of the derivation of the higher-order electron- photon coupling Hamiltonian.

A.1 H

to second order

Below the components of HΨ= ΩHαΩ⁻¹+ i~(∂^tΩ)Ω⁻¹ are calculated. First, ΩHαΩ⁻¹= Hα+1

2

h ˜D², Hα

i

= Hα+1 2

h ˜D², −eφi

. (A.1)

where in the last step only terms up to O(m⁻²) are kept, and where (2mc)²h ˜D², φi

= σⁱσ^j(DiDjφ − φDiDj) = σⁱσ^j(Di(∂jφ) + DiφDj− φDiDj)

= σⁱσ^j(∂i(∂jφ) + (∂jφ)Di+ (∂iφ)Dj) = ∂_i²φ + 2(∂iφ)Di

= −ρ/0+2i

~

(∇φ) · (p + eA). (A.2)

Second,

i~∂^tΩ = i~

2(2mc)²∂t(p + eA)²+ e~σ · B

= ie~

2(2mc)²[(∂tA) · (p + eA) + (p + eA) · (∂tA) + ~σ · (∂^tB)]

= ie~

2(2mc)²[2(∂_tA) · (p + eA) + ~σ · (∂tB)] (A.3)

Putting things together, one finds HΨ= Hα−1

2 e~²ρ (2mc)²0

− ie~

(2mc)²E · (p + eA) + ie~²

2(2mc)²σ · (∂tB)

= (p + eA)²

2m + e~

2mσ · B − eφ + e~

(2mc)²σ · E × (p + eA) + 1 2

e~²ρ (2mc)²0

− ie~

2(2mc)²~σ · (∂tB). (A.4)

Note that the E × p term is not Hermitian for dynamic fields, because E and p do not commute in that case. The ‘imaginary term’ in H_Ψ solves this issue: it can be rewritten as

− ie~

2(2mc)²~σ^k(∂_tB^k) = ie~²

2(2mc)²^ijkσ^k(∂_iE^j) = ie~²

2(2mc)²^ijkσ^k(∂_iE^j− E^j∂_i)

= − e~

2(2mc)²^ijkσ^k(−i~∂ⁱE^j+ Eⁱ(−i~)∂^j)

= − e~

2(2mc)²σ · (p × E + E × p) (A.5)

With this substitution, HΨ assumes the form of Eq. (2.22).

A.2 H

to third order

This section contains the derivation of HΨto order O(m⁻³).

Starting point is Eq. (2.7):

β(x) = ˜Dα(x) + ˜D0Dα(x) + ˜˜ D₀²Dα(x).˜ (A.6) Substitution in Eq. (2.5) gives the equation for α(x):

D˜0α(x) + ˜D

1 + ˜D0+ ˜D₀² ˜Dα(x) = 0. (A.7) Every ˜D₀ is commuted to the right:

0 =



D˜0+ ˜D²+ ˜D 1 + ˜D0



D ˜˜D0+ ie~

(2mc)²cσⁱEⁱ



α(x)

=



D˜0+ ˜D²+ ˜D



D ˜˜D0+ ie~

(2mc)²cσⁱEⁱ



+ ˜D



D ˜˜D₀+ ie~

(2mc)²cσⁱEⁱ



D˜₀+ ˜D₀ ie~

(2mc)²cσⁱEⁱ



α(x) (A.8) Using

h ˜D₀, Eⁱi

= −i~

2mc(∂₀Eⁱ), (A.9)

A.2 H_Ψ to third order 193

one obtains

− ˜D0α(x) =

 D˜²

1 + ˜D0+ ˜D₀²

+ ˜D ie~

(2mc)²cσⁱEⁱ

1 + 2 ˜D0



+ ˜De~²σⁱ(∂₀Eⁱ) (2mc)³c



α(x). (A.10)

D˜0α(x) is replaced by the second order Schr¨odinger equation, and only terms that contribute to the Schr¨odinger equation at order O(m⁻³) are considered:

− ˜D₀α(x) =

 D˜²

1 − ˜D²

+ ˜D ie~

(2mc)²cσⁱEⁱ+ ˜De~²σⁱ(∂0Eⁱ) (2mc)³c



α(x). (A.11) The normalization condition becomes

Z d³x



α(x)^†α(x) +n

1 + ˜D₀+ ˜D²₀ ˜Dα(x)o†n

1 + ˜D₀+ ˜D²₀ ˜Dα(x)o

= 1.

(A.12) Up to order O(m⁻³), this is

1 = Z

d³x



α(x)^†+ α(x)^†D˜²+n ˜D0Dα(x)˜ o†

D + α(x)˜ ^†D ˜˜D0D˜

 α(x)

= Z

d³x

"

α(x)^†+ α(x)^†D˜²+



D ˜˜D₀+ ie~

(2mc)²cσⁱEⁱ

 α(x)

†

D˜

+α(x)^†D˜



D ˜˜D0+ ie~

(2mc)²cσⁱEⁱ



α(x)

= Z

d³x α(x)^†

"

1 + ˜D²+

 ie~

(2mc)²cσⁱEⁱ

^†

D + ˜˜ D ie~

(2mc)²cσⁱEⁱ

# α(x)

= Z

d³x α(x)^†



1 + ˜D²+ e~

(2mc)²c

h ˜D, iσⁱEⁱi

α(x), (A.13)

where the commutator h ˜D, iσⁱEⁱi

= ~

2mc σⁱσ^j(∂iE^j) − 2i^ijkσ^kEⁱDj

(A.14) is hermitian. The renormalization operator becomes

Ω(x) = 1 + 1 2



D˜²+ e~

(2mc)²c

h ˜D, iσⁱEⁱi

= Ω(x)^†, (A.15) so that to order O(m⁻³)

Ω(x)⁻¹= 1 −1 2



D˜²+ e~

(2mc)²c

h ˜D, iσⁱEⁱi

. (A.16)

With this renormalization operator, one can obtain the Schr¨odinger equation for the normalized wave function Ψ(x). Its Hamiltonian HΨ consists of the parts

i~(∂^tΩ)Ω⁻¹= ie~

2(2mc)²{2(∂tA) · (p + eA) + ~σ · (∂^tB)}

+ e 2c

 −i~

2mc

³

σⁱσ^j(∂_t∂_iE^j) − 2i^ijkσ^k(∂_tEⁱD_j)

(A.17) and

ΩHαΩ⁻¹= 2mc²



− eφ

2mc² + ˜D²− ˜D⁴+ ˜D ie~

(2mc)²cσⁱEⁱ+ ˜De~²σⁱ(∂0Eⁱ) (2mc)³c



+ 1 2



D˜²+ e~

(2mc)²c

h ˜D, iσⁱEⁱi

, −eφ + 2mc²D˜²



= −eφ + (p + eA)²

2m + e~

2mσ · B − 1 2mc²

 (p + eA)²

2m + e~

2mσ · B

²

− e −i~

2mc

²

σ^jσⁱD_jEⁱ− e −i~

2mc

³

σ^jσⁱD_j(∂₀Eⁱ)

− −i~

2mc

²e

2 ∂²_iφ + 2(∂_iφ)D_i
− e²~ 2(2mc)²c

hh ˜D, iσⁱEⁱi , φi

, (A.18) where the commutator is

hh ˜D, iσⁱEⁱi , φi

= −h

φ, ˜DiσⁱEⁱi +h

φ, iσⁱEⁱD˜i

= −h φ, ˜Di

iσⁱEⁱ− ˜Dφ, iσⁱEⁱ + φ, iσⁱEⁱD + iσ˜ ⁱEⁱh φ, ˜Di

= −h φ, ˜Di

iσⁱEⁱ+ iσⁱEⁱh φ, ˜Di

= ~

2mc σ^jσⁱ− σⁱσ^j
Eⁱ(∂jφ)

= ~

2mci2^jikσ^kEⁱ(∂jφ) = −i~

2mc2 σ · E × (∇φ), (A.19) so that

ΩH_αΩ⁻¹= −eφ + (p + eA)²

2m + e~

2mσ · B − 1 2mc²

 (p + eA)²

2m + e~

2mσ · B

²

− e −i~

2mc

2

σ^jσⁱD_jEⁱ− e −i~

2mc

3

σ^jσⁱD_j(∂₀Eⁱ)

− −i~

2mc

²e

2 ∂²_iφ + 2(∂_iφ)D_i
+ ie²~²

(2mc)³cσ · E × (∇φ). (A.20) Putting the O(m⁻³) term of HΨtogether, one obtains

− 1 2mc²

 (p + eA)²

2m + e~

2mσ · B

²

− e −i~

2mc

3

σ^jσⁱDj(∂0Eⁱ) +

A.2 H_Ψ to third order 195

+ ie²~²

(2mc)³cσ · E × (∇φ) + e 2c

 −i~

2mc

³

σⁱσ^j(∂_t∂_iE^j) − 2i^ijkσ^k(∂_tEⁱD_j)

= 1

(2mc)³



−c (p + eA)²+ e~σ · B
2

+ie²~²

c σ · E × (∇φ)

−e

2c(−i~)³σ^jσⁱ(∂j∂tEⁱ) + (∂tEⁱ)Dj+ Dj(∂tEⁱ) +e(−i~)³

2c



σⁱσ^j(∂t∂iE^j) − 2i^ijkσ^k



(∂tEⁱ)Dj+ie

~

Eⁱ(∂tA^j)



= 1

(2mc)³



−c (p + eA)²+ e~σ · B
²

+e(−i~)³

2c −2i^ijkσ^k(∂_tEⁱ)D_j

−e

2c(−i~)³σ^jσⁱ(∂tEⁱ)Dj+ Dj(∂tEⁱ) −ie²~²

c σ · E × E



= 1

(2mc)³

h−c (p + eA)²+ e~σ · B
²

−e (−i~)³

2c (∂tEⁱ)Di+ Di(∂tEⁱ) − i^ijkσ^k(∂t∂jEⁱ)

#

, (A.21)

giving Eq. (2.23).

A p p e n d i x B

YTiO ³

This appendix contains some of the lengthy expressions involved in calculating the RIXS spectra of YTiO₃(section 5.4).

B.1 RIXS – Single site processes

The angular momentum ˆl and quadrupole operators ˆQ, ˆT in Eqs. (5.43−5.45) are defined as follows:

ˆlx= i(c^†b − b^†c) (B.1)

ˆly= i(a^†c − c^†a) (B.2)

ˆl_z= i(b^†a − a^†b) (B.3)

Qˆx= ˆl²_x− ˆl_y²= nb− na (B.4) Qˆ_z= 1

√3(ˆl_x²+ ˆl²_y− 2ˆl²_z) = 1

√3(2n_c− n_a− n_b) (B.5) Tˆx= ˆlyˆlz+ ˆlzˆly= −(b^†c + c^†b) (B.6) Tˆ_y= ˆl_xˆl_z+ ˆl_zˆl_x= −(c^†a + a^†c) (B.7) Tˆz= ˆlxˆly+ ˆlyˆlx= −(a^†b + b^†a) (B.8)

which are normalized by Tr ˆΓ²

= 2. The corresponding matrices Γd⁰d in Eq. (5.41) are

Γ^Qx =1 2





−1 0 0

0 1 0

0 0 0



, Γ^Qz = ₂^√1₃





−1 0 0

0 −1 0

0 0 2



,

Γ^Tx = −1 2





0 0 0 0 0 1 0 1 0



, Γ^Ty = −¹₂





0 0 1 0 0 0 1 0 0



,

Γ^Tz = −1 2





0 1 0 1 0 0 0 0 0



, Γ^lx= ¹₂





0 0 0

0 0 i

0 −i 0



,

Γ^ly = 1 2





0 0 −i

0 0 0

i 0 0



, Γ^lz =¹₂





0 i 0

−i 0 0

0 0 0



 (B.9)

with the indices d, d⁰ = (yz, zx, xy) (or for polarization dependence: α, β = (x, y, z)).

B.2 Multiplet factors

For the multiplet effect factors in Eq. (5.46), we have

M_d^A0^1gd = r2

3(hd⁰| ˆx |mi hm| ˆx |di + hd⁰| ˆy |mi hm| ˆy |di + hd⁰| ˆz |mi hm| ˆz |di) (B.10) M_d^Q0d^x= (hd⁰| ˆy |mi hm| ˆy |di − hd⁰| ˆx |mi hm| ˆx |di) (B.11) M_d^Q0d^z = 1

√3(2 hd⁰| ˆz |mi hm| ˆz |di − hd⁰| ˆx |mi hm| ˆx |di − hd⁰| ˆy |mi hm| ˆy |di) (B.12) M_d^T0^xd= − (hd⁰| ˆy |mi hm| ˆz |di + hd⁰| ˆz |mi hm| ˆy |di) (B.13) M_d^T0^yd= − (hd⁰| ˆz |mi hm| ˆx |di + hd⁰| ˆx |mi hm| ˆz |di) (B.14) M_d^T0^zd= − (hd⁰| ˆx |mi hm| ˆy |di + hd⁰| ˆy |mi hm| ˆx |di) (B.15) M_d^lx0d= −i (hd⁰| ˆy |mi hm| ˆz |di − hd⁰| ˆz |mi hm| ˆy |di) (B.16) M_d^ly0d= −i (hd⁰| ˆz |mi hm| ˆx |di − hd⁰| ˆx |mi hm| ˆz |di) (B.17) M_d^lz0d= −i (hd⁰| ˆx |mi hm| ˆy |di − hd⁰| ˆy |mi hm| ˆx |di) (B.18) Note that the position operators act on the core electrons, not the t_2g ones. Both the core and t_2g electrons are implied in the states |di , |d⁰i.

B.3 The operators ˆΓ in terms of orbitons 199

B.3 The operators ˆ Γ in terms of orbitons

In terms of the orbiton operators, we obtain the one-orbiton creation part of Γˆq=P

ie^iq·riΓˆi to be

ˆl⁽¹⁾_x,q−q

1 =i |c₀| 2

rN 3

hn(1 −√

3)u_q+ (1 +√ 3)v_qo

(sh θ_1,q+ ch θ_1,q)α^†_1,−q +n

(−1 −√

3)u_q+ (1 −√ 3)v_qo

(sh θ_2,q+ ch θ_2,q)α^†_2,−qi

(B.19) ˆl_y,q−q⁽¹⁾ ₂ =i |c0|

2 rN

3 hn

(1 +√

3)u_q+ (1 −√ 3)v_qo

(sh θ_1,q+ ch θ_1,q)α^†_1,−q +n

(−1 +√

3)u_q+ (1 +√ 3)v_qo

(sh θ_2,q+ ch θ_2,q)α^†_2,−qi

(B.20) ˆl⁽¹⁾_z,q−q3 = −i |c0|

rN 3 h

(u_q+ v_q)(sh θ_1,q+ ch θ_1,q)α^†_1,−q + (v_q− u_q)(sh θ_2,q+ ch θ_2,q)α^†_2,−qi

(B.21) Tˆ_x,q−q⁽¹⁾ ₁ =|c0|

6

√ Nhn

(1 +√

3)u_q+ (−1 +√ 3)v_qo

(ch θ_1,q− sh θ_1,q)α^†_1,−q +n

(1 −√

3)u_q+ (1 +√ 3)v_qo

(ch θ_2,q− sh θ_2,q)α_2,−q^† i

(B.22) Tˆ_y,q−q⁽¹⁾

2 =|c0| 6

√ Nhn

(1 −√

3)u_q+ (−1 −√ 3)v_qo

(ch θ_1,q− sh θ_1,q)α^†_1,−q +n

(1 +√

3)u_q+ (1 −√ 3)v_qo

(ch θ_2,q− sh θ_2,q)α_2,−q^† i

(B.23) Tˆ_z,q−q⁽¹⁾ ₃ = −|c0|

3

√ Nh

(u_q− v_q)(ch θ_1,q− sh θ_1,q)α^†_1,−q + (u_q+ v_q)(ch θ_2,q− sh θ_2,q)α^†_2,−qi

(B.24) Qˆ⁽¹⁾_x,q = |c0|

rN 3

h−(uq+ vq)(ch θ1,q− sh θ1,q)α^†_1,−q + (u_q− v_q)(ch θ_2,q− sh θ_2,q)α^†_2,−qi

(B.25) Qˆ⁽¹⁾_z,q= − |c0|

rN 3

h

(uq− vq)(ch θ1,q− sh θ1,q)α^†_1,−q + (uq+ vq)(ch θ2,q− sh θ2,q)α^†_2,−qi

(B.26) with q₁ = (π, 0, π), q₂ = (π, π, 0), q₃ = (0, π, π). The expressions for the two-orbiton creation part of ˆΓq=P

ie^iq·riΓˆi are ˆl⁽²⁾_x,q= i

√3 X

k

h

(vu⁰− uv⁰)ch θ₁sh θ⁰₁α^†_1,kα^†_1,−k−q

1−q +

+ (vu⁰− uv⁰)ch θ2sh θ⁰₂α^†_2,kα^†_2,−k−q

1−q

+ (uu⁰+ vv⁰)(ch θ₁sh θ₂⁰ − sh θ₁ch θ₂⁰)α_1,k^† α^†_2,−k−q

1−q

i

(B.27) with u, v, θ1, θ2= uk, vk, θ1,k, θ2,k and primed quantities u⁰, v⁰, θ₁⁰, θ⁰₂= uk+q₁+q, v_k+q1+q, θ_1,k+q1+q, θ_2,k+q1+q. Further, ˆl⁽²⁾y,qand ˆl⁽²⁾z,qhave the same form as ˆlx,q⁽²⁾

but with q₁ replaced by q₂and q₃ respectively. Next, Tˆ_x,q⁽²⁾ =X

k



−(uu⁰+ vv⁰) +uu⁰− vv⁰

√3 +uv⁰+ vu⁰ 3



ch θ₁sh θ⁰₁α^†_1,kα^†_1,−k−q

1−q

+



−(uu⁰+ vv⁰) −uu⁰− vv⁰

√3 −uv⁰+ vu⁰ 3



ch θ₂sh θ⁰₂α^†_2,kα^†_2,−k−q

1−q

+



−(uv⁰− vu⁰) −uv⁰+ vu⁰

√3 −uu⁰− vv⁰ 3



(ch θ₁sh θ₂⁰ + sh θ₁ch θ₂⁰)×

α_1,k^† α^†_2,−k−q

1−q

i

(B.28) Tˆ_y,q⁽²⁾=X

k



−(uu⁰+ vv⁰) −uu⁰− vv⁰

√3 +uv⁰+ vu⁰ 3



ch θ1sh θ⁰₁α^†_1,kα^†_1,−k−q

2−q

+



−(uu⁰+ vv⁰) +uu⁰− vv⁰

√3 −uv⁰+ vu⁰ 3



ch θ₂sh θ⁰₂α^†_2,kα^†_2,−k−q

2−q

+



−(uv⁰− vu⁰) +uv⁰+ vu⁰

√3 −uu⁰− vv⁰ 3



(ch θ1sh θ₂⁰ + sh θ1ch θ₂⁰)×

α_1,k^† α^†_2,−k−q

2−q

i (B.29)

where in the expression for ˆTy,q⁽²⁾ we replaced q₁ by q₂: u⁰, v⁰, θ₁⁰, θ⁰₂ = u_k+q2+q, v_k+q2+q, θ_1,k+q2+q, θ_2,k+q2+q.

Tˆ_z,q⁽²⁾=X

k



−2

3(uv⁰+ vu⁰) − (uu⁰+ vv⁰)



ch θ1sh θ₁⁰ α^†_1,kα^†_1,−k−q

3−q

+ 2

3(uv⁰+ vu⁰) − (uu⁰+ vv⁰)



ch θ2sh θ₂⁰ α^†_2,kα^†_2,−k−q

3−q

+ 2

3(uu⁰− vv⁰) − (uv⁰− vu⁰)



(ch θ1sh θ⁰₂+ sh θ1ch θ⁰₂)α^†_1,kα^†_2,−k−q

3−q

 (B.30) where we replaced q₁ by q₃: u⁰, v⁰, θ₁⁰, θ₂⁰ = u_k+q3+q, v_k+q3+q, θ_1,k+q3+q, θ_2,k+q3+q. Finally,

Qˆ⁽²⁾_x,q= − 1

√3 X

k

h−(uu⁰− vv⁰)ch θ1sh θ₁⁰ α^†_1,kα^†_1,−k−q

+ (uu⁰− vv⁰)ch θ2sh θ⁰₂α^†_2,kα^†_2,−k−q+

B.4 RIXS – two-site processes with superexchange model 201

+ (uv⁰+ vu⁰)(ch θ1sh θ₂⁰ + sh θ1ch θ₂⁰)α^†_1,kα^†_2,−k−qi

(B.31) Qˆ⁽²⁾_z,q= 1

√3 X

k

h

(uv⁰+ vu⁰)ch θ₁sh θ⁰₁α^†_1,kα^†_1,−k−q

− (uv⁰+ vu⁰)ch θ₂sh θ⁰₂α^†_2,kα^†_2,−k−q

− (uu⁰− vv⁰)(ch θ1sh θ₂⁰ + sh θ1ch θ₂⁰)α^†_1,kα^†_2,−k−qi

(B.32) where in both equations we replaced q₁by 0: u⁰, v⁰, θ₁⁰, θ₂⁰ = u_k+q, v_k+q, θ_1,k+q, θ_2,k+q.

B.4 RIXS – two-site processes with superex- change model

Functions f₁₁, f₂₂and f₁₂in Eqs. (5.79−5.80) are:

f11(k, q) = [−γ3,q(uv⁰+ u⁰v) − γ2,q(uu⁰− vv⁰) − (1 + γ1,q)(uu⁰+ vv⁰)] × (ch θ₁sh θ⁰₁+ sh θ₁ch θ⁰₁)

+ 2 [γ₁⁰(uu⁰+ vv⁰) + γ₂⁰(uu⁰− vv⁰) + γ₃⁰(uv⁰+ u⁰v)] ×

(sh θ₁sh θ₁⁰ + ch θ₁ch θ⁰₁) (B.33) f22(k, q) = [γ3,q(uv⁰+ u⁰v) + γ2,q(uu⁰− vv⁰) − (1 + γ1,q)(uu⁰+ vv⁰)] ×

(ch θ2sh θ⁰₂+ sh θ2ch θ⁰₂)

+ 2 [γ₁⁰(uu⁰+ vv⁰) − γ₂⁰(uu⁰− vv⁰) − γ₃⁰(uv⁰+ u⁰v)] ×

(sh θ2sh θ₂⁰ + ch θ2ch θ⁰₂) (B.34) f₁₂(k, q) = 2 [γ_3,q(uu⁰− vv⁰) + γ_2,q(uv⁰+ u⁰v) − (1 + γ_1,q)(uv⁰− u⁰v)] ×

(ch θ1sh θ⁰₂+ sh θ1ch θ⁰₂)

+ 4 [γ₁⁰(uv⁰− u⁰v) + γ₂⁰(uv⁰+ u⁰v) − γ₃⁰(uu⁰− vv⁰)] ×

(sh θ1sh θ₂⁰ + ch θ1ch θ⁰₂) (B.35) where we shortened notation by writing θ_1/2 = θ_1/2,k, θ⁰_1/2 = θ_1/2,k+q, u(⁰) = uk(+q), v(⁰) = vk(+q)and γ_i⁰= γi,k+q.

A p p e n d i x C

Phonon RIXS

This appendix contains the lengthy derivations involved in calculating the Ein- stein phonon RIXS spectra of chapter 7. In the following, we evaluate the scat- tering amplitude (7.8). We use (see Mahan [62], section 4.3.2)

hn|e^{ω0M(b†−b)}

n⁰ = e^−g/2hn| e^ω0Mb†e^−ω0Mb n⁰

= e^−g/2

n

X

k=0 n⁰

X

l=0

hn−k|(M/ω0)^k k!

 n!

(n − k)!

¹₂ (−M/ω0)^l l!

 n⁰! (n⁰− l)!

¹2

n⁰−l

= e^−g/2

n

X

k=0 n⁰

X

l=0

(M/ω0)^k k!

 n!n⁰! (n − k)!(n⁰− l)!

¹2 (−M/ω0)^l

l! δ_n0−l,n−k

=







e^−g/2Pn⁰ l=0

(M/ω0)^{l−n0 +n} (l−n⁰+n)!

h n!n⁰! (n−(l−n⁰+n))!(n⁰−l)!

i¹_{2 (−M/ω}

0)^l

l! for n > n⁰ e^−g/2Pn

k=0

(M/ω0)^k k!

h n!n⁰! (n−k)!(n⁰−(n⁰−n+k))!

i¹_{2 (−M/ω}

0)^{n0 −n+k}

(n⁰−n+k)! for n ≤ n⁰

=







e^−g/2Pn⁰ l=0

(−1)^l(M/ω₀)^{2l−n0 +n} l!(l−n⁰+n)!

√n!n⁰!

(n⁰−l)! for n > n⁰ e^−g/2Pn

l=0

(−1)^{l+n0 −n}(M/ω₀)^{2l+n0 −n} l!(l+n⁰−n)!

√ n!n⁰!

(n−l)! for n ≤ n⁰

(C.1)

with g = m²/ω²₀. For simplicity, we assume the system is initially in its ground state: n⁰= 0. In that case, the above expression simplifies to

hn| e^{ω0M(b†−b)}|0i = e^−g/2(M/ω0)ⁿ

√n! . (C.2)

These expressions are inserted in the Kramers-Heisenberg equation (7.12):

Ff g= T_el(⁰, )X

i

e^iq·RiF_i (C.3)

with

F_i= e^−g

n⁰_i

X

n=0 n

X

l=0

(−1)^l+n0i−n(M/ω₀)^2l+n0i l!(l + n⁰_i− n)!

pn⁰_i! (n − l)!

1 z + (g − n)ω0

+ e^−g

∞

X

n=n⁰_i+1 n⁰_i

X

l=0

(−1)^l(M/ω₀)^2l−n0i+2n l!(l − n⁰_i+ n)!

pn⁰_i! (n⁰_i− l)!

1 z + (g − n)ω0

. (C.4)

A p p e n d i x D

Magnetic spectral weight at the Γ point in 2D cuprates

This appendix contains the calculational details from Sec. 4.5.5, where the leading magnetic contribution to the q = 0 RIXS spectra at the Cu L edge is established.

D.1 H ¯

to fourth order in t/U

In this section we evaluate Eq. (4.74) in the presence of a core hole.

The first term of Eq. (4.74) changes the Hamiltonian to H¯_eff⁽⁴⁾= H₀−4t²

U X

i

p_ip^†_iX

δ

S_i· S_i+δ+ . . . (D.1)

where p^†_i creates a core electron, δ points to nearest neighbors, and the dots indicate the corrections to H₀ due to the four hop terms.

To handle the four hop terms systematically, we categorize all terms according to the ‘connections’ between sites. With a ‘connection’ is meant that one or more hops occur between the sites in question. For example, only the sites i and j, and j and k are connected if we select the following four hoppings from the V ’s:

t_ijX

σ

(c^†_iσc_jσ+ c^†_jσc_iσ) × ... × t_jkX

σ⁰

(c^†_jσ0c_kσ0+ c_jσ^† 0c_kσ0) × ... × t_ij(...) × ... × t_jk(...), (D.2)

but also processes with a different hopping order like tijtijtjktjkand tjktijtijtjk, etc., have the same connections and are thus categorized together. We will there- fore represent the category comprising all these processes by t²_ijt²_jk. The order of the t’s is not important, only the number of times a certain hop occurs is important for indicating the category.

We now establish the different categories. The most general string of t’s is t_ijt_klt_mnt_pq. A lot of these processes change the occupancy of one of the sites.

These terms vanish because of the P0’s at the beginning and end of every four hop term. This imposes a restriction on the indices i, j, k, l, m, n, p, q: if an index appears an odd number of times, the occupancy is changed and the process does not contribute to H_eff⁽⁴⁾. We are left with the following categories: (a) t⁴_ij, (b) t²_ijt²_kl(i, j 6= k, l), (c) t²_ijt²_jk(i 6= k), and (d) tijtjktkltli (i, j, k, l form a square). If there is no core hole present at any of the sites involved, we can just copy-paste the results from Ref. [82]. Below we analyze the processes where there is a core hole present.

Processes in category (a) do not flip any spins. The doublon (the doubly occupied site) hops 4 times between site i and j, where i is the core hole site.

We obtain the (a) contribution:

H_eff⁽⁴⁾ = · · · + p_ip^†_i t⁴ U_c³P₀X

σ

c^†_iσc_jσc^†_jσc_iσP₀c^†_iσc_jσc^†_jσc_iσP₀, (D.3)

where the dots indicate other fourth order terms. At the core hole site, c^†_iσc_iσ= 1, and we use the projected spins as defined in Eq. (2.14) of Ref. [82] to get

H_eff⁽⁴⁾= · · · + p_ip^†_i t⁴ U_c³

 {1

2(1 − σ_j^z)}²+ {1

2(1 + σ^z_j)}²



= · · · + p_ip^†_i t⁴

U_c³¹1. (D.4) The corresponding part of H0 is

H0= · · · −16t⁴

U³ Si· Sj (D.5)

which should be replaced by Eq. (D.4). Dropping the constant term, this gives H¯_eff⁽⁴⁾= H0+X

i

p_ip^†_iX

δ

16t⁴

U³ Si· Si+δ+ . . . (D.6) Processes in category (b) do not appear in the fourth order expansion of the half-filled Hubbard model. If we add one doubly occupied site (namely, the core hole site i), there still is no contribution to ¯H_eff⁽⁴⁾. The only matrix elements for which this is not a priori clear, involve configurations pictured in Fig. D.1.

If the spins at k and l are parallel, the corresponding matrix element is obvi- ously zero. Working out the other matrix elements (12 pathways for interchanging anti-parallel k and l, and 12 pathways for leaving anti-parallel k and l invariant),

D.1 ¯H_eff⁽⁴⁾ to fourth order in t/U 207

i j

k l

Figure D.1: The processes of category (b) involve t²_ijt²_kl, connecting i to j and k to l.

i j k j i k

Figure D.2: The processes of category (c) are subdivided into two cases: the core hole site is connected to one other site (left), or the core hole site is connected to two other sites (right).

it turns out that all pathways interfere to give 0, just as in the half-filled case.

Therefore, H₀is not modified by category (b) processes.

Processes in category (c) connect three sites i, j and k. They can be sub- divided into two cases: in the first one, the core hole site i is connected to one other site, and the second one, the core hole site i is connected to two other sites.

Starting with the first case, we have for the following matrix elements:

↑j↑k→↑j↑k and ↓j↓k→↓j↓k: H¯_eff⁽⁴⁾= · · · − p_ip^†_i t⁴ U_c³

 1

2¹1 + 2S_j^zS_k^z



, (D.7)

↑j↓k→↑j↓k and ↓j↑k→↓j↑k: H¯_eff⁽⁴⁾= · · · − p_ip^†_i t⁴ U_c³

 1

2¹1 − 2S_j^zS_k^z



, (D.8)

↑j↓k→↓j↑k and ↓j↑k→↑j↓k: H¯_eff⁽⁴⁾= · · · + 0. (D.9) Adding these contributions and dropping the constants yields

H¯_eff⁽⁴⁾= · · · + 0. (D.10) The second case gives

↑j↑k→↑j↑k and ↓j↓k→↓j↓k: H¯_eff⁽⁴⁾= · · · + p_ip^†_i2t⁴ U_c³

 1

2¹1 + 2S_j^zS_k^z



, (D.11)

↑_j↓_k→↑_j↓_k and ↓_j↑_k→↓_j↑_k: H¯_eff⁽⁴⁾= · · · − p_ip^†_i

 2t⁴

U U_c² + 4t⁴ U_c²(2Uc+ U )

−2t⁴ U_c³

  1

2¹1 − 2S_j^zS_k^z



, (D.12)

↑_j↓_k→↓_j↑_k and ↓_j↑_k→↑_j↓_k: H¯_eff⁽⁴⁾= · · · + p_ip^†_i

 2t⁴

U U_c² + 4t⁴ U_c²(2U_c+ U )



× S_j⁺S_k⁻+ S_j⁻S_k⁺
. (D.13) Adding the contributions and dropping the constants yields

H¯_eff⁽⁴⁾= · · · + p_ip^†_i

 4t⁴

U U_c² + 8t⁴ U_c²(2Uc+ U )



Sj· Sk. (D.14)

The corresponding part of H0(including the second neighbor terms from category (d))¹is

H₀= · · · +X

j6=k

4t²_ijt²_ik

U³ S_j· Sk (D.15)

and it is replaced by H¯_eff⁽⁴⁾= H₀+X

i

p_ip^†_iX

j6=k

 4t⁴

U U_c²+ 8t⁴

U_c²(2U_c+ U )−4t⁴ U³



S_j· S_k+ . . . (D.16)

where the sum over j 6= k is over all pairs of neighbors j, k of i.

Finally, the processes of category (d) are

↑j di

↑k ↑l

→ ↑j di

↑k ↑l

+ flipped : H¯_eff⁽⁴⁾= · · · − p_ip^†_i2t⁴

U_c³

 1

4¹1 + S_j^zS^z_k+ S_k^zS_l^z+ S_j^zS_l^z



, (D.17)

↑j di

↓k ↑l

→ ↑j di

↓k ↑l

+ flipped : H¯_eff⁽⁴⁾= · · · + p_ip^†_i 2t⁴

U U_c²

 1

4¹1 − S_j^zS_k^z− S_k^zS_l^z+ S_j^zS^z_l



, (D.18)

↓j d_i

↑k ↑l

→ ↓j d_i

↑k ↑l

+ flip + j ↔ l : H¯_eff⁽⁴⁾= · · · + p_ip^†_i 2t⁴

U U_c²

 1

2¹1 − 2S_j^zS_l^z



, (D.19)

↓j di

↑k ↑l

→ ↑j di

↓k ↑l

+ flip + j ↔ l : H¯_eff⁽⁴⁾= · · · − p_ip^†_i t⁴

U_c³ + t⁴ U U_c²



S_j⁺S_k⁻+ S_j⁻S_k⁺

+S_k⁺S_l⁻+ S_k⁻S_l⁺
, (D.20)

↓_j d_i

↑_k ↑_l → ↑_j d_i

↑_k ↓_l + flip + j ↔ l : H¯_eff⁽⁴⁾= · · · − p_ip^†_i t⁴

U_c³ + t⁴ U U_c²



S_j⁺S_l⁻+ S_j⁻S_l⁺
. (D.21)

where we have labeled the sites as shown in Fig. D.3, with dithe doubly occupied core hole site.

1We do this because, first, it simplifies notation: the second and third neighbor terms in the sum get the same prefactor. Second, Coldea also does this, and following him makes it easy to compare to his mean field result for the ring exchange terms.

D.1 ¯H_eff⁽⁴⁾ to fourth order in t/U 209

i j

k l

Figure D.3: The processes of cat- egory (d) (t_ijt_jkt_klt_li) describe ring exchange.

Dropping constants, the final result for processes of category (d) is H_eff⁽⁴⁾= · · · − X

i,squares at i

p_ip^†_i

 2t⁴ U U_c² +2t⁴

U_c³



(S_j· S_k+ S_k· S_l+ S_j· S_l) . (D.22)

The corresponding terms in H0 (minus the second neighbor terms, which were absorbed in the correction to H0due to category (c)) are

H0= · · · −8t⁴ U³

X

hi,ji

Si· Sj+80t⁴ U³

X

squares

[(Si· Sj)(Sk· Sl) + (Si· Sl)(Sj· Sk)

−(S_i· S_k)(S_j· S_l)] . (D.23) The mean field result is easily obtained [82, 83]:

H0= · · · −48t⁴ U³

X

hi,ji

Si· Sj−20t⁴ U³

X

i,k

Si· Sk, (D.24)

where the sum over i, k is over all pairs of next nearest neighbors. Then, at the mean field, the intermediate state Hamiltonian for processes of category (d) becomes

H¯_eff⁽⁴⁾ = H0+X

i

p_ip^†_i X

squares at i

 24t⁴

U³ (Si· Sj+ Sj· Sk+ Sk· Sl+ Sl· Si) +20t⁴

U³ (Si· Sk+ Sj· Sl)

−

 2t⁴ U U_c² +2t⁴

U_c³



(S_j· Sk+ S_k· Sl+ S_j· Sl)



+ . . . (D.25) Adding all contributions from all categories, we obtain

H¯_eff⁽⁴⁾ = H₀+X

i

p_ip^†_i

 X

j

 16t⁴ U³ −4t²

U



S_i· Sj+X

j6=k

4t⁴

U U_c² + 8t⁴ U_c²(2Uc + U )

−4t⁴ U³



S_j· S_k+ X

squares

 24t⁴

U³ (S_i· S_j+ S_j· S_k+ S_k· S_l+ S_l· S_i) +20t⁴

U³ (S_i· S_k+ S_j· S_l) −

 2t⁴ U U_c² +2t⁴

U_c³



(S_j· S_k+ S_k· S_l +Sj· Sl)



. (D.26)