Computation of physical properties

In this subsection we look at how the dierent physical properties of the Kondo model can be calculated. The physical properties of a system are properties that can be mea-sured and whose values describe the state of the system. We will show how to calculate static thermodynamic quantities such as susceptibility, heat capacity, and entropy. Beside computing the thermodynamic quantities, we will also show how to calculate more elusive dynamical quantities such as the spectral function, which is a quantity that makes it very easy to interpret experimental results.

5.3.1 Thermodynamic properties

As mentioned above we look at the static thermodynamic quantities like the entropy, spe-cic heat, and charge susceptibility of the Kondo model. These quantities can easily be calculated once we have the energy spectrum of the system. With the energy spectrum we can calculate the partition function Z. As we know from statistical physics, once we have the partition function we can calculate all other thermodynamic properties of a given system. The truncation of the Hamiltonian does not aect the calculation of thermody-namic properties as long as the energies thrown away are of values much higher than k_BT above the ground state. This is because theses energies are suppressed exponentially. In

general the contribution of the impurity to the thermodynamic quantities is what we are interested in and this can be derived from the impurity free energy F_imp =−βln(Z/Z_cb), whereβ = (k_BT)⁻¹ andZ_cb is the partition function of the non-interacting system, where cb stands for the conduction band. The partition function is dened as follows

Z_N(T) = Tre^−βHN =X

i

e^−βEiN (5.6)

where N is the iteration number in the NRG scheme.

Entropy

The entropy can be obtained in thermodynamics from the relationship E = F +T S where E is the energy, F the free energy, T the temperature, and S the entropy. From this expression one can get that

S

k_B = E−F

k_BT =βE−βF =βE+ lnZ, (5.7)

where the energy is dened as

E =hHi= Tr(He^−βH)

Tre^−βH . (5.8)

Specic heat

One can obtain the specic heat from the second derivative of the free energy. The specic heat is a very interesting quantity experimentally, the specic heat due to the impurity can be calculated in the NRG method using the following equation

Cimp =Ctotal−Ccb (5.9)

C_total denotes the specic heat of the composite system (impurity and conduction band) whileC_cbis that of the conduction band without the impurity. C_totalandC_cbare calculated as follows

C_l(T) kB

= β² Z

X

i

E_i²e^−βEi −(X

i

E_ie^−βEi)²

!

(5.10) where the subscriptlcould either betotalorcbandEiare the eigenenergies of the system.

Susceptibility

The magnetic susceptibility can be calculated using the formula, χ_imp(T) =β(gµ_B)² 1

Z X

i

S_z,i² e^−βEi− 1 Z_cb

X

i

S_z,cb,i² e^−βEi

!

(5.11) g is the electronic gyromagnetic factor, S_z the z-component of the total spin, and µ_B is the Bohr magneton. Z is the partition function of the combined system, impurity and conduction electrons. The contribution of the impurity to the susceptibility of the system is obtained in the preceding equation. The rst term represents the combined system while the second one is just that of the conduction electron.

5.3.2 Dynamical properties

The importance of the dynamical properties cannot be overemphasized, considering that in many experiments transport quantities are measured instead of thermodynamic quan-tities. The calculation of the spectral function was the rst chief extension of the NRG in calculating dynamical properties [55, 56]. Some examples of dynamical properties are local single-particle spectral function, dynamical spin susceptibility, and charge susceptibility.

One can also use the dynamical properties to distinguish between inelastic and elastic contributions to the cross-section of scattering. The above quantities can be calculated both at zero and at non-zero temperatures. As examples for the dynamical properties in this section we will propose the spectral function and the dynamical spin susceptibility.

Spectral function

The spectral function is the most experimentally interesting quantity. It is the energy resolution for a particle in a given quantum state. The spectral function gives an indication of how well the excitation created by adding a particle in a quantum state can be described by a free non-interacting particle. The spectral function for non-interacting free electrons is usually a delta function whereas that for interacting systems diers from a delta function but may still be a peak function. The impurity spectral function is dened as follows

A_σ(ω, T) = −1

πImG_σ(ω, T). (5.12) One can immediately see from the preceding equation that, the spectral function is dened in terms of the imaginary part of the Green's function G. ω andT are the frequency and temperature respectively, while G_σ(ω, T) denotes the retarded impurity Green's function dened as

G(t) =−iθ(t)h[d(t), d^†(0)]+i, (5.13) where θ(t) is the Heaviside function which is 0 for all variables less than zero and 1, when otherwise, while t is the time. d^†_σ(0) and d_σ(t) are the ordinary fermionic opera-tors in which an electron is created at time zero and destroyed at time t, respectively.

d_σ(t) = e^iHtˆ d_σ(0)e^−iHtˆ in the time evolution picture which is the Heisenberg representa-tion, whereHˆ is the Hamiltonian operator. For nite systems and arbitrary temperatures it is convenient to write the retarded Green's function in the Lehmann's representation as follows

A_N(ω, T) = 1 Z_N(T)

X

n,m

|hn|d^†_σ|mi|²(e^−βEn+e^−βEm)δ(ω−(E_m−E_n)) (5.14) The spectral function atT = 0is easy to calculate as compared to its non-zero temperature counterpart and is dened as

A_N(ω, T = 0) = 1 Z_N(0)

X

n

|hn|d^†_σ|0i|²δ(ω−E_n+E₀) +|h0|f_σ^†|ni|²δ(ω+E_n−E₀) (5.15) E₀ is the ground state energy whereas E_n is the energy of then−th excited level. In the zero temperature case all transitions are from the ground state |0i to the excited states.

The data we obtain from the above formula is discrete because it is just a set of positions and δ-function weights. To compare this result with continuous experimental spectra

results, we need to broaden them [57]. This broadening can be done using functions like the Gaussian or the logarithmic Gaussian distributions which have widths comparable to that of the conduction bandwidth. We do not give any details of this broadening in thesis since no direct use is made of it. It is only mentioned here in order to give a complete picture of what can be obtained using the NRG technique.

Spin susceptibility

This is the response of a system to an applied magnetic eld. The spin susceptibility is given in linear response theory as follows

χ(t) = −ihψ₀|T Sˆ _imp^z (t)S_imp^z (0)|ψ₀i (5.16) whereTˆis the time ordering operator. Like in the case of the spectral function it can also be expressed in the Lehmann representation. The expression in the simple case of T = 0 is dened as

χ(ω) = πX

n

|hn|S_imp^z |0i|²δ(ω−(E_n−E₀)) (5.17) The spin susceptibility calculated above is then broadened so it can be compared to experimental results.