Spectra Made Simple Lab

Up to four interacting quantum dots, each attached to its own lead and all repelling each other. The exact many-body problem lives in a Fock space that grows as 4M. Steady-state density functional theory (i-DFT) gets the spectral function of every dot from an effectively non-interacting Kohn–Sham system instead: one self-consistent equation per dot for the occupations, then one extra equation per frequency. That is enough to capture Coulomb blockade and, at low temperature, the Kondo resonance.

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Number of dots M
Exact many-body problem
i-DFT

What you are seeing

  • Coulomb blockade. Lowering the gate fills the dots one electron at a time. Every extra electron costs its on-site repulsion Ul or the inter-dot repulsion Ulm, so the occupations climb in steps and the spectral map splits into bright addition and removal lines. The Kohn–Sham system reproduces these steps through the sharp features (derivative discontinuities) of the Hxc potential.
  • Spectral functions from i-DFT. Each point of the map is obtained by attaching a weak probe to the dot at bias ω and solving one equation for the current it would draw. The exchange–correlation bias turns the single Kohn–Sham peak into the full many-body spectrum, including the Hubbard side bands.
  • Kondo. Switch on the Kondo correction and cool down (the Kondo presets use kBT = 10−5γ): a sharp resonance appears at the Fermi level for dots with an odd occupation. The GCE + γ reference is built from isolated dots and cannot show the Kondo effect; in the paper the i-DFT Kondo spectra are benchmarked against numerical renormalization group calculations.
  • Cost. The GCE reference diagonalizes a Hamiltonian in a Fock space of 4M states, and for dots coupled to leads the many-body problem is far harder still. i-DFT needs M coupled equations for the occupations and then one independent equation per frequency, about the cost of a mean-field calculation.

Model: M dots with on-site interactions Ul and a common inter-dot interaction Ulm, no inter-dot hopping, each coupled to its own reservoir with broadening γ (wide-band limit), common gate v, zero bias. i-DFT: Hxc potential built from single-impurity (SIAM) and constant-interaction (CIM) terms, xc bias from its current dependence, optional Kondo correction, functional widths w0 = w1 = 0.16. The functionals are designed for regime I, Ulm ≤ Ul. Reference (GCE + γ): grand-canonical ensemble of the isolated dots (4M states), with every transition broadened by a Lorentzian of width γ, shown for the spectra; it is accurate when kBT ≳ γ. Occupations are shown from i-DFT only: a simple broadened isolated-dot reference is not reliable at low temperature, where the ground state of the isolated dots switches abruptly. Energies in units of γ; spectral functions summed over spin. Reference: N. Sobrino & S. Kurth, Spectral and transmission properties of multiple correlated quantum dots made simple. Code and data: github.com/Nahualcsc/idft-multiple-quantum-dots, doi:10.5281/zenodo.20154932.