Question

As I was trying to understand slave rotor mean field theory, I have faced a problem while solving the Coulomb staircase example in which I had to calculate the expectation value of some quantity at T→0. For an atomic system with two orbitals giving rise to a spin/orbital index of N = 4, we have the average occupancy ⟨ˆQ⟩ ⟨Q^⟩ ⟨ˆQ⟩ = 1ZatN ∑Q = 0 (NQ) Q e−EQ / T where Q represents the fermionic number and the partition function of electron ensemble in atomic orbitals Z at is Zat = N∑Q = 0 (NQ) e−EQ / T given that the energy is EQ = ϵ0Q + U2 (Q−N2) 2 Numerically, I can obtain the so-called Coulomb staircase as T→0, which gives the dependence of average occupancy on the atomic level position ϵ0. My question is that, how can I obtain an analytical solution? Is there an approximation that I can do one −EQ / T term that give rise to analytical solution? Just out of curiosity, I have tried to do TS expansion on the exponential coefficient e − EQ / T goes to zero for T = 0.

Answer 1

At T→0 K, states are filled only up to the Fermi energy, simplifying the exponential term in the occupancy equation. However, finding an analytical solution for the average occupancy at absolute zero is challenging because series expansion methods do not generally converge.

Step-by-step explanation:

The question revolves around the analytical solution to the average occupancy of an atomic system with two orbitals in slave rotor mean field theory.

At absolute zero temperature (T→0), the Fermi factor and Pauli's exclusion principle dictate that states are filled up to the Fermi energy level, and at T→0 K, the exponential term describing occupancy — e−EQ/T — simplifies significantly.

Since the exponential term goes to zero for energy levels above the Fermi energy, one can determine the occupancy electron states below this energy which are completely filled.

Furthermore, analytical solutions around T→0 K can typically be approached via a series expansion; however, such an expansion does not converge as the temperature approaches absolute zero, indicating that approximation methods for analytical solutions must be considered with caution.

The Coulomb staircase provides a numerical method to examine this behavior, although the quest for an analytical solution at such a low temperature remains challenging due to the nature of the exponential term in the partition function.