Fermi-Golden-Rule calculations

The NRG is perfectly suited to deal with local quantum events such as absorption or

emission of a generalized quantum impurity system in contact with non-interacting

reser-voirs.

30,38,45,61,69

If the rate of absorption is weak, such that the system has sufficient time to equilibrate on average, then the resulting absorption spectra are described by the Fermi-Golden rule,

⁵⁴

A(ω) = 2π X

i,f

ρ

^I_i

(T ) · |h f | C ˆ | i i|

²

· δ(ω − E

_if

), (4.27) where i and f describe complete basis sets for initial and final system, respectively. The system starts in the thermal equilibrium of the initial system. The operator ˆ C describes the absorption event at the impurity system, i.e. corresponds to the term in the Hamil-tonian that couples to the light field. The transition amplitudes between initial and final Hamiltonian are fully described by the matrix elements C

_{f i}

≡ h f | C ˆ | i i . In Eq. (4.27), the frequency ω shows threshold behavior, with the frequency threshold given by the difference in the ground state energies of initial and final Hamiltonian, ω

_thr

≡ ∆E

g

≡ E

_g^F

− E

_g^I

.

The only difference between emission and absorption spectra is the reversed role of initial and final system, while also having ˆ C → C ˆ

^†

. Specifically, the emission process starts in the thermal equilibrium of the final Hamiltonian, with subsequent transition matrix elements to the initial system. This also implies that emission spectra have their contributions at negative frequencies, i.e. frequencies smaller than the threshold frequency yet blurred by temperature, indicating the emission of a photon.

While absorption or emission spectra are already defined in frequency domain, they can nevertheless be translated into the time domain through Fourier transform,

A(t) ≡

Z dω

2π e

^−iωt

A(ω) = X

i,f

ρ

^I_i

(T ) h i | C ˆ

^†

| f i e

^−iEft

h f | C ˆ | i i e

^iEit

=

e

^iHˆIt

C ˆ

^†

e

^−iHˆFt

| {z }

≡C(t)ˆ

· C ˆ

I

T

. (4.28)

Thus absorption spectra can also be interpreted similar to correlation functions: at time t = 0 an absorption event occurs (application of ˆ C, which for example rises an electron from a low lying level into some higher level that participates in the dynamics). This alters the Hamiltonian, such that the subsequent time evolution is governed by the final Hamiltonian. At some time t > 0 then, the absorption event relaxes back to the original configuration (application of ˆ C

^†

), such that A(t) finally describes the overlap amplitude of the resulting state with the original state with no absorption. While the “mixed” time evolution of ˆ C(t) in Eq. (4.28) appears somewhat artificial, it can be easily rewritten in terms of a regular time-dependent Heisenberg operator with a single Hamiltonian. By explicitly including a further static degree (e.g. a low lying hole from which the electron was lifted through the absorption event, or the photon itself), this switches ˆ H

^I

to ˆ H

^F

, i.e. between two dynamically disconnected sectors in Hilbert space [compare discussion of type-1 and type-2 quenches in M¨ under et al. (2011)].

Within the complete NRG eigenbasis, Eq. (4.27) becomes A(ω) = 2π X

ss⁰∈KK/ F

n

h s

⁰

| A ˆ | s i

^In

· R

^X_n^sI

(T ) ·

^In

h s | A ˆ

^†

| s

⁰

i

^Fn

× δ(ω − E

_ss0

), (4.29)

4. MPS diagrammatics for the numerical renormalization group

iteration n

collecting spectral data in single sweep having (i,f)

 {KK}

cond-mat/0607497

f

i

^ C ^ C

y

e^¡¯En~sD0 Z

Figure 4.5: MPS diagram for the calculation of absorption or emission spectra using Fermi-Golden-rule (FGR-NRG) mediated by the operator ˆ C. The two center legs (horizontal black lines) refer to the state space of the initial Hamiltonian, while the outer legs (horizontal dark gray lines) refer to the state space of the final Hamiltonian. Note that the matrix elements of ˆ C are mixed matrix elements between eigenstates of initial and final Hamiltonian.

with X

s

∈ { K, D } the state space sector of state s. The MPS diagram for Eq. (4.29) to be evaluated is shown in Fig. 4.5. Its structure is completely analogous to the calculation of generic correlation functions in Fig. 4.3, except that similar to the quantum quench earlier, here again the basis sets from two different Hamiltonians come into play.

³⁰

In contrast to the quantum quench situation in Fig. 4.4, however, no overlap matrices emerge. Instead, all matrix elements of the local operator ˆ C

^†

are calculated in a mixed basis between initial and final eigenstates. The double sum over Wilson shells (one from the outer trace Fig. 4.5 in the complete basis of the final Hamiltonian, and one in the construction of the FDM) is again reduced to a single sum over Wilson shells with the constraint (i, f) ≡ (s, s

⁰

) ∈ / KK.

The reduced density matrices R

^X,I_n

are constructed w.r.t. the initial Hamiltonian, but exactly correspond to the ones introduced in the FDM context in Eq. (4.20) otherwise.

Technical remarks

Absorption or emission spectra in the presence of Anderson orthogonality or strongly

cor-related low-energy physics typically exhibit sharply peaked features close to the threshold

frequency with clear physical interpretation. While in principle, a single Hamiltonian with

dynamically disconnected Hilbert space sectors may be used, this is ill-suited for an NRG

simulation. Using a single NRG run, this can only resolve the low-energy of the full

Hamil-tonian, i.e. of the initial system as it is assumed to lie lower. Consequently, the sharp

features at the threshold frequency will have to be smoothened by an energy window

com-parable to ω

_thr

= ∆E

g

in order to suppress discretization artifacts. This problem is fully

collecting data in a single sweep having (X

₁

=X

₂

,X

₃

,X

₄

)  {KKK}

s₃ s₂

iteration n s₄

^ B ^ C ^ D

s₃

s₄ s₁

X₃ X₄

±

_X1X2

e^¡¯En

0

~sD

Z

Figure 4.6: MPS diagram for the evaluation of a three-point correlation functions as in Eq. (4.31).

circumvented only by using two separate NRG runs, one for the initial and and one for

the final Hamiltonian. With the NRG spectra typically collected in logarithmically spaced

bins, having two NRG runs then, it is important that the data is collected in terms of the

frequencies ν ≡ ω − ω

_thr

taken relative to the threshold frequency ω

_thr

.

Im Dokument Tensor networks and the numerical renormalization group (Seite 56-59)