Dynamical mean-field theory

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 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

The self-consistency condition requires the impurity Green's function ${\displaystyle G_{\mathrm {imp} }(\tau )}$ to coincide with the local lattice Green's function ${\displaystyle G_{ii}(\tau )=-\langle Tc_{i}(\tau )c_{i}^{\dagger }(0)\rangle }$:

${\displaystyle G_{\mathrm {imp} }(i\omega _{n})=G_{ii}(i\omega _{n})=\sum _{k}{\frac {1}{i\omega _{n}+\mu -\epsilon (k)-\Sigma (k,i\omega _{n})}}}$

where ${\displaystyle \Sigma (k,i\omega _{n})}$ denotes the lattice self-energy.

The only DMFT approximations (apart from the approximation that can be made in order to solve the Anderson model) consists in neglecting the spatial fluctuations of the lattice self-energy, by equating it to the impurity self-energy:

${\displaystyle \Sigma (k,i\omega _{n})\approx \Sigma _{imp}(i\omega _{n})}$

This approximation becomes exact in the limit of lattices with infinite coordination, that is when the number of neighbors of each site is infinite. Indeed, one can show that in the diagrammatic expansion of the lattice self-energy, only local diagrams survive when one goes into the infinite coordination limit.

Thus, as in classical mean-field theories, DMFT is supposed to get more accurate as the dimensionality (and thus the number of neighbors) increases. Put differently, for low dimensions, spatial fluctuations will render the DMFT approximation less reliable.

Spatial fluctuations also become relevant in the vicinity of phase transitions. Here, DMFT and classical mean-field theories result in mean-field critical exponents, the pronounced changes before the phase transition are not reflected in the DMFT self-energy.

In order to find the local lattice Green's function, one has to determine the hybridization function such that the corresponding impurity Green's function will coincide with the sought-after local lattice Green's function. The most widespread way of solving this problem is by using a forward recursion method, namely, for a given ${\displaystyle U}$, ${\displaystyle \mu }$ and temperature ${\displaystyle T}$:

1. Start with a guess for ${\displaystyle \Sigma (k,i\omega _{n})}$ (typically, ${\displaystyle \Sigma (k,i\omega _{n})=0}$)
2. Make the DMFT approximation: ${\displaystyle \Sigma (k,i\omega _{n})\approx \Sigma _{\mathrm {imp} }(i\omega _{n})}$
3. Compute the local Green's function ${\displaystyle G_{\mathrm {loc} }(i\omega _{n})}$
4. Compute the dynamical mean field ${\displaystyle \Delta (i\omega )=i\omega _{n}+\mu -G_{\mathrm {loc} }^{-1}(i\omega _{n})-\Sigma _{\mathrm {imp} }(i\omega _{n})}$
5. Solve the AIM for a new impurity Green's function ${\displaystyle G_{\mathrm {imp} }(i\omega _{n})}$, extract its self-energy: ${\displaystyle \Sigma _{\mathrm {imp} }(i\omega _{n})=({\mathcal {G}}_{0})^{-1}(i\omega _{n})-(G_{\mathrm {imp} })^{-1}(i\omega _{n})}$
6. Go back to step 2 until convergence, namely when ${\displaystyle G_{\mathrm {imp} }^{n}=G_{\mathrm {imp} }^{n+1}}$.

DMFT can be extended to Hubbard models with multiple orbitals, namely with electron-electron interactions of the form ${\displaystyle U_{\alpha \beta }n_{\alpha }n_{\beta }}$ where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ denote different orbitals. The combination with density functional theory (DFT+DMFT)[4][8] then allows for a realistic calculation of correlated materials.[9]

Extended DMFT yields a local impurity self energy for non-local interactions and hence allows us to apply DMFT for more general models such as the t-J model.

In order to improve on the DMFT approximation, the Hubbard model can be mapped on a multi-site impurity (cluster) problem, which allows one to add some spatial dependence to the impurity self-energy. Clusters contain 4 to 8 sites at low temperature and up to 100 sites at high temperature.

Spatial dependencies of the self energy beyond DMFT, including long-range correlations in the vicinity of a phase transition, can be obtained also through diagrammatic extensions of DMFT[10] using a combination of analytical and numerical techniques. The starting point of the dynamical vertex approximation[11] and of the dual fermion approach is the local two-particle vertex.

DMFT has been employed to study non-equilibrium transport and optical excitations.[12] Here, the reliable calculation of the AIM's Green function out of equilibrium remains a big challenge.