HOME

TheInfoList



OR:

Dynamical mean-field theory (DMFT) is a method to determine the electronic structure of
strongly correlated materials Strongly correlated materials are a wide class of compounds that include insulators and electronic materials, and show unusual (often technologically useful) electronic and magnetic properties, such as metal-insulator transitions, heavy fermio ...
. In such materials, the approximation of independent electrons, which is used in
density functional theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
and usual
band structure In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
calculations, breaks down. Dynamical mean-field theory, a non-perturbative treatment of local interactions between electrons, bridges the gap between the nearly free electron gas limit and the atomic limit of
condensed-matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases which arise from electromagnetic forces between atoms. More ge ...
. DMFT consists in mapping a
many-body The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
lattice problem to a many-body ''local'' problem, called an impurity model. While the lattice problem is in general intractable, the impurity model is usually solvable through various schemes. The mapping in itself does not constitute an approximation. The only approximation made in ordinary DMFT schemes is to assume the lattice
self-energy In quantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines self-energy \Sigma, and represents the contribution to the particle's energy, or effective mass, due to interactions between ...
to be a momentum-independent (local) quantity. This approximation becomes exact in the limit of lattices with an infinite coordination. One of DMFT's main successes is to describe the
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
between a metal and a
Mott insulator Mott insulators are a class of materials that are expected to conduct electricity according to conventional band theories, but turn out to be insulators (particularly at low temperatures). These insulators fail to be correctly described by band ...
when the strength of
electronic correlation Electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons. Ato ...
s is increased. It has been successfully applied to real materials, in combination with the
local density approximation Local-density approximations (LDA) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space (and no ...
of density functional theory.


Relation to mean-field theory

The DMFT treatment of lattice quantum models is similar to the
mean-field theory In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of ...
(MFT) treatment of classical models such as the
Ising model The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
. In the Ising model, the lattice problem is mapped onto an effective single site problem, whose magnetization is to reproduce the lattice magnetization through an effective "mean-field". This condition is called the self-consistency condition. It stipulates that the single-site observables should reproduce the lattice "local" observables by means of an effective field. While the N-site Ising Hamiltonian is hard to solve analytically (to date, analytical solutions exist only for the 1D and 2D case), the single-site problem is easily solved. Likewise, DMFT maps a lattice problem (''e.g.'' the
Hubbard model The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems. It is particularly useful in solid-state physics. The model is named for John Hubbard. The Hubbard model states that each el ...
) onto a single-site problem. In DMFT, the local observable is the local
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential ...
. Thus, the self-consistency condition for DMFT is for the impurity Green's function to reproduce the lattice local Green's function through an effective mean-field which, in DMFT, is the hybridization function \Delta(\tau) of the impurity model. DMFT owes its name to the fact that the mean-field \Delta(\tau) is time-dependent, or dynamical. This also points to the major difference between the Ising MFT and DMFT: Ising MFT maps the N-spin problem into a single-site, single-spin problem. DMFT maps the lattice problem onto a single-site problem, but the latter fundamentally remains a N-body problem which captures the temporal fluctuations due to electron-electron correlations.


Description of DMFT for the Hubbard model


The DMFT 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_=t \sum_ c_^c_ + U\sum_n_ n_ where, on suppressing the spin 1/2 indices \sigma, c_i^,c_i denote the creation and annihilation operators of an electron on a localized orbital on site i, and n_i=c_i^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


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 The Anderson impurity model, named after Philip Warren Anderson, is a Hamiltonian that is used to describe magnetic impurities embedded in metals. It is often applied to the description of Kondo effect-type problems, such as heavy fermion sys ...
(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_ and a_^) 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_=\underbrace_ + \underbrace_+\underbrace_ where * H_ describes the non-correlated electronic levels \epsilon_p of the bath * H_ describes the impurity, where two electrons interact with the energetical cost U * H_ describes the hybridization (or coupling) between the impurity and the bath through hybridization terms V_p^ The Matsubara Green's function of this model, defined by G_(\tau) = - \langle T c(\tau) c^(0)\rangle , is entirely determined by the parameters U,\mu and the so-called hybridization function \Delta_\sigma(i\omega_n) = \sum_\frac, which is the imaginary-time Fourier-transform of \Delta_(\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_0)^(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 The numerical renormalization group (NRG) is a technique devised by Kenneth Wilson to solve certain many-body problems where quantum impurity physics plays a key role. History The numerical renormalization group is an inherently non-perturbative ...
*
Exact diagonalization Exact diagonalization (ED) is a numerical technique used in physics to determine the eigenstates and energy eigenvalues of a quantum Hamiltonian. In this technique, a Hamiltonian for a discrete, finite system is expressed in matrix form and diag ...
*
Iterative perturbation theory Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
* Non-crossing approximation *
Continuous-time quantum Monte Carlo In computational solid state physics, Continuous-time quantum Monte Carlo (CT-QMC) is a family of stochastic algorithms for solving the Anderson impurity model at finite temperature. These methods first expand the full partition function as a se ...
algorithms


Self-consistency equations

The self-consistency condition requires the impurity Green's function G_\mathrm(\tau) to coincide with the local lattice Green's function G_(\tau) = -\langle T c_i(\tau)c_i^(0)\rangle : : G_\mathrm(i\omega_n) = G_(i\omega_n) = \sum_k \frac 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 In quantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines self-energy \Sigma, and represents the contribution to the particle's energy, or effective mass, due to interactions between ...
, by equating it to the impurity self-energy: : \Sigma(k,i\omega_n) \approx \Sigma_(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 In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
. Here, DMFT and classical mean-field theories result in mean-field
critical exponents Critical or Critically may refer to: *Critical, or critical but stable, medical states **Critical, or intensive care medicine * Critical juncture, a discontinuous change studied in the social sciences. * Critical Software, a company specializing i ...
, the pronounced changes before the phase transition are not reflected in the DMFT self-energy.


The 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(i\omega_n) # Compute the local Green's function G_\mathrm(i\omega_n) # Compute the dynamical mean field \Delta(i\omega) = i\omega_n + \mu - G^_\mathrm(i\omega_n) - \Sigma_\mathrm(i\omega_n) # Solve the AIM for a new impurity Green's function G_\mathrm(i\omega_n), extract its self-energy: \Sigma_\mathrm(i\omega_n) = (\mathcal_0)^(i\omega_n) - (G_\mathrm)^(i\omega_n) # Go back to step 2 until convergence, namely when G_\mathrm^n = G_\mathrm^.


Applications

The local lattice Green's function and other impurity observables can be used to calculate a number of physical quantities as a function of correlations U, bandwidth, filling (chemical potential \mu), and temperature T: * the
spectral function The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
(which gives the band structure) * the
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
* the double occupancy of a site * response functions (compressibility, optical conductivity, specific heat) In particular, the drop of the double-occupancy as U increases is a signature of the Mott transition.


Extensions of DMFT

DMFT has several extensions, extending the above formalism to multi-orbital, multi-site problems, long-range correlations and non-equilibrium.


Multi-orbital extension

DMFT can be extended to Hubbard models with multiple orbitals, namely with electron-electron interactions of the form U_ n_n_ where \alpha and \beta denote different orbitals. The combination with
density functional theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
(DFT+DMFT) then allows for a realistic calculation of correlated materials.


Extended DMFT

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 solid-state physics, the ''t''-''J'' model is a model first derived in 1977 from the Hubbard model by Józef Spałek to explain antiferromagnetic properties of the Mott insulators and taking into account experimental results about the streng ...
.


Cluster DMFT

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.


Diagrammatic extensions

Spatial dependencies of the self energy beyond DMFT, including long-range correlations in the vicinity of a
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
, can be obtained also through diagrammatic extensions of DMFT using a combination of analytical and numerical techniques. The starting point of the dynamical vertex approximation and of the dual fermion approach is the local two-particle vertex.


Non-equilibrium

DMFT has been employed to study non-equilibrium transport and optical excitations.{{Cite journal , last1=Aoki , first1=Hideo , last2=Tsuji , first2=Naoto , last3=Eckstein , first3=Martin , last4=Kollar , first4=Marcus , last5=Oka , first5=Takashi , last6=Werner , first6=Philipp , date=2014-06-24 , title=Nonequilibrium dynamical mean-field theory and its applications , url=https://link.aps.org/doi/10.1103/RevModPhys.86.779 , journal=Reviews of Modern Physics , language=en , volume=86 , issue=2 , pages=779–837 , doi=10.1103/RevModPhys.86.779 , arxiv=1310.5329 , bibcode=2014RvMP...86..779A , s2cid=119213862 , issn=0034-6861 Here, the reliable calculation of the AIM's Green function out of equilibrium remains a big challenge.


References and notes


See also

*
Strongly correlated material Strongly correlated materials are a wide class of compounds that include insulators and electronic materials, and show unusual (often technologically useful) electronic and magnetic properties, such as metal-insulator transitions, heavy fermion ...


External links


Strongly Correlated Materials: Insights From Dynamical Mean-Field Theory
G. Kotliar and D. Vollhardt
Lecture notes on the LDA+DMFT approach to strongly correlated materials
Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.)
Lecture notes DMFT at 25: Infinite Dimensions
Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.)
Lecture notes DMFT – From Infinite Dimensions to Real Materials
Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.) Correlated electrons Materials science Quantum mechanics Electronic structure methods