Single-orbital Hubbard model

The Hubbard model ^[7] describes the onsite interaction between electrons of opposite spin by a single parameter, ${\displaystyle U}$. The Hubbard Hamiltonian may take the following form:

${\displaystyle H_{\text{Hubbard}}=t\sum _{\langle ij\rangle \sigma }c_{i\sigma }^{\dagger }c_{j\sigma }+U\sum _{i}n_{i\uparrow }n_{i\downarrow }}$

where, on suppressing the spin 1/2 indices ${\displaystyle \sigma }$, ${\displaystyle c_{i}^{\dagger },c_{i}}$ denote the creation and annihilation operators of an electron on a localized orbital on site ${\displaystyle i}$, and ${\displaystyle n_{i}=c_{i}^{\dagger }c_{i}}$.

The following assumptions have been made:

• only one orbital contributes to the electronic properties (as might be the case of copper atoms in superconducting cuprates, whose ${\displaystyle d}$-bands are non-degenerate),
• the orbitals are so localized that only nearest-neighbor hopping ${\displaystyle t}$ is taken into account

The auxiliary problem: the Anderson impurity model

The Hubbard model is in general intractable under usual perturbation expansion techniques. DMFT maps this lattice model onto the so-called Anderson impurity model (AIM). This model describes the interaction of one site (the impurity) with a "bath" of electronic levels (described by the annihilation and creation operators ${\displaystyle a_{p\sigma }}$ and ${\displaystyle a_{p\sigma }^{\dagger }}$) through a hybridization function. The Anderson model corresponding to our single-site model is a single-orbital Anderson impurity model, whose hamiltonian formulation, on suppressing some spin 1/2 indices ${\displaystyle \sigma }$, is:

${\displaystyle H_{\text{AIM}}=\underbrace {\sum _{p}\epsilon _{p}a_{p}^{\dagger }a_{p}} _{H_{\text{bath}}}+\underbrace {\sum _{p\sigma }\left(V_{p}^{\sigma }c_{\sigma }^{\dagger }a_{p\sigma }+h.c.\right)} _{H_{\text{mix}}}+\underbrace {Un_{\uparrow }n_{\downarrow }-\mu \left(n_{\uparrow }+n_{\downarrow }\right)} _{H_{\text{loc}}}}$

where

• ${\displaystyle H_{\text{bath}}}$ describes the non-correlated electronic levels ${\displaystyle \epsilon _{p}}$ of the bath
• ${\displaystyle H_{\text{loc}}}$ describes the impurity, where two electrons interact with the energetical cost ${\displaystyle U}$
• ${\displaystyle H_{\text{mix}}}$ describes the hybridization (or coupling) between the impurity and the bath through hybridization terms ${\displaystyle V_{p}^{\sigma }}$

The Matsubara Green's function of this model, defined by ${\displaystyle G_{\text{imp}}(\tau )=-\langle Tc(\tau )c^{\dagger }(0)\rangle }$, is entirely determined by the parameters ${\displaystyle U,\mu }$ and the so-called hybridization function ${\displaystyle \Delta _{\sigma }(i\omega _{n})=\sum _{p}{\frac {|V_{p}^{\sigma }|^{2}}{i\omega _{n}-\epsilon _{p}}}}$, which is the imaginary-time Fourier-transform of ${\displaystyle \Delta _{\sigma }(\tau )}$.

This hybridization function describes the dynamics of electrons hopping in and out of the bath. It should reproduce the lattice dynamics such that the impurity Green's function is the same as the local lattice Green's function. It is related to the non-interacting Green's function by the relation:

${\displaystyle ({\mathcal {G}}_{0})^{-1}(i\omega _{n})=i\omega _{n}+\mu -\Delta (i\omega _{n})}$ (1)

Solving the Anderson impurity model consists in computing observables such as the interacting Green's function ${\displaystyle G(i\omega _{n})}$ for a given hybridization function ${\displaystyle \Delta (i\omega _{n})}$ and ${\displaystyle U,\mu }$. It is a difficult but not intractable problem. There exists a number of ways to solve the AIM, such as

• Numerical renormalization group
• Exact diagonalization
• Iterative perturbation theory
• Non-crossing approximation
• Continuous-time quantum Monte Carlo algorithms