Mapping Single-orbital Hubbard model The Hubbard model describes the onsite interaction between electrons of opposite spin by a single parameter, U. The Hubbard Hamiltonian may take the following form: : 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 \sigma, c_i^{\dagger},c_i denote the
creation and annihilation operators of an electron on a localized orbital on site i, and 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 d-bands are non-degenerate), • the orbitals are so localized that only nearest-neighbor hopping t is taken into account
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 a_{p\sigma} and 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 \sigma, is: :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{U n_{\uparrow} n_{\downarrow}-\mu \left(n_{\uparrow}+n_{\downarrow}\right)}_{H_{\text{loc}}} where • H_{\text{bath}} describes the non-correlated electronic levels \epsilon_p of the bath • H_{\text{loc}} describes the impurity, where two electrons interact with the energetical cost U • H_{\text{mix}} describes the hybridization (or coupling) between the impurity and the bath through hybridization terms V_p^{\sigma} The Matsubara Green's function of this model, defined by G_{\text{imp}}(\tau) = - \langle T c(\tau) c^{\dagger}(0)\rangle , is entirely determined by the parameters U,\mu and the so-called hybridization function \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 \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: :(\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 G(i\omega_n) for a given hybridization function \Delta(i\omega_n) and 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
Self-consistency equations The self-consistency condition requires the impurity Green's function G_\mathrm{imp}(\tau) to coincide with the local lattice Green's function G_{ii}(\tau) = -\langle T c_i(\tau)c_i^{\dagger}(0)\rangle : : 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)} = G_\mathrm{loc}(i\omega_n) where \Sigma(k,i\omega_n) denotes the lattice self-energy.
DMFT approximation: locality of 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: : \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.
DMFT loop 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 U, \mu and temperature T: • Start with a guess for \Sigma(k,i\omega_n) (typically, \Sigma(k,i\omega_n)=0) • Make the DMFT approximation: \Sigma(k,i\omega_n) \approx \Sigma_\mathrm{imp}(i\omega_n) • Compute the local Green's function G_\mathrm{loc}(i\omega_n) • Compute the dynamical mean field \Delta(i\omega) = i\omega_n + \mu - G^{-1}_\mathrm{loc}(i\omega_n) - \Sigma_\mathrm{imp}(i\omega_n) • Solve the AIM for a new impurity Green's function G_\mathrm{imp}(i\omega_n), extract its self-energy: \Sigma_\mathrm{imp}(i\omega_n) = (\mathcal{G}_0)^{-1}(i\omega_n) - (G_\mathrm{imp})^{-1}(i\omega_n) • Go back to step 2 until convergence, namely when G_\mathrm{imp}^n = G_\mathrm{imp}^{n+1}. ==Applications==