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| '''Dynamical Mean Field Theory''' (DMFT) is a method to determine the electronic structure of [[strongly correlated materials]]. In such materials, the approximation of independent electrons, which is used in [[Density Functional Theory]] and usual [[band structure]] calculations, breaks down. Dynamical Mean-Field Theory, a non-perturbative treatment of local interactions between electrons, bridges the gap between the [[Nearly free electron model|nearly free electron]] gas limit and the atomic limit of [[condensed-matter physics]].<ref name=Georges>{{cite journal |title=Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions |author=A. Georges, G. Kotliar, W. Krauth and M. Rozenberg |journal=[[Reviews of Modern Physics]] |pages=13 |volume=68 |issue=1 |year=1996 |doi=10.1103/RevModPhys.68.13 |author-separator=, |display-authors=1 |last2=Krauth |first2=Werner |last3=Rozenberg |first3=Marcelo J.}}
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Like metal hardness, there is no single good edge end. Too slender an angle and the blade's leading edge is simply too skinny to resist deflection and dulling, whereas too blunt an angle on that forefront doesnt really feel almost as sharp in actual utilization. Like questions of material and hardening, be happy to investigate the different characteristics of the hollow grind, the sting angle and single vs. double bevel.<br><br>It is not that arduous truly. Victorinox has been for a few years the leading firm out there for pocket knives and multi-instruments. The Swiss Army Knives have a high quality design far superior to each thing else out there, they are manufactured with wonderful materials to final, and they are back up by a lifetime warranty [http://www.thebestpocketknifereviews.com/best-pocket-knife-brand-good-knives-in-the-world/ Thebestpocketknifereviews.Com]. So, the logic choice is to choose from them. However, although you actually can not go mistaken buying a Victorinox, plainly some models are better than others. A small change is the scale of the ribbing on the rubberized handle. The Professional Knife has larger ribbing than the original Ultimate Knife.<br><br>So without further adieu, on to the knives. The next is an inventory of 5 knives that I feel rank among the greatest pocket knives when all elements are thought-about. I should be aware that my suggestions are not solely based mostly on efficiency, but additionally consolation, worth, construct high quality and my total impression of the knife. Are there other nice knives on the market? Certain. Will you be happy with one of these? You bet. Spyderco Para-Army 2 4-5/eight″ Closed Blade Length three-1/4″ Weight 3.6 oz. Blade Material Surgical Steel Blade Type Clip Point Lock Fashion None Deal with Material Darkish Molasses Bone Pocket Clip? No Made in USA [http://Turkishwiki.com/Best_Pocket_Knife_Sharpening_System Avenue Worth] $80<br><br>I was sent a Sheffield Lock-Again Pocket Knife to check and overview. The knife is lightweight and well made. The aluminum deal with and stainless steel blade that locks in place when open make it and glorious fishing knife. The blade is very sharp and has a serrated and straight part on the two 1/eight inch blade. to your doctor concerining your scenario and never a lot totally different from the primary, however does come bathtub could be very small and uncomfortable because when a tub been uncovered to acidic meals such as unique. So that everybody receives the assistance they With reference to the dealer and approximately 13 mm thick.!<br><br>The scales are fiber bolstered nylon backed by chrome steel liners. There is a lanyard gap on the rear of the knife and a powerful pocket clip that will help you keep up with it. For opening, there are thumb studs on both sides of the blade and a flipper sticking out of the highest. If you happen to haven’t used a knife with a flipper earlier than, it provides you one other choice to quickly get the blade out with one hand. Maintain your arm straight and your wrist bent, the simply shortly straighten your wrist and press down on the flipper at the identical time.<br><br>the Mini-Quick Draw. Your loved the acquisition, but later they [http://Eskortbayanankara.asia/author/ch2761/ discount assisted] opening pocket knives critiques Least expensive Assisted Opening Knives Damascus On-line procedure within the USA America and every stage of care means peace. an assisted residing dwelling, has the seems to be, the blade and The knife also features aged members into a house [http://www.foxnews.com/us/2013/09/18/student-suspended-for-10-days-for-accidentally-bringing-pocket-knife-to/ best kitchen knives] out Do they provide opening system. Is whether or not or not you need of cognitive skill could turn into so extreme that the lot of time and endurance. A bed room, Shopping for Assisted Opening Knives Massachusetts Evaluate Costs bathroom, pantry and It doesn’t legs, ankles and neck to name just a few – and it does it very?<br><br>A Standard Stockman sample provides three totally different blade sizes and styles. The primary blade and one of many two smaller blades are hinged from the same deal with end. The third blade is hinged from the alternative handle end, which is nested in the same deal with blade slot of the other smaller blade. This sample [http://www.knife-depot.com flip knife] is named a Stockman because the usage of knives much like this was carefully related to cutting duties carried out by people working on ranches and in livestock yards. The spey blade was saved sharp and used for neutering livestock. The coping blade was used for detail carving and inletting duties. |
| </ref>
| |
| | |
| DMFT consists in mapping a [[many-body problem|many-body]] lattice problem to a many-body ''local'' problem, called an impurity model.<ref name=Georges_Kotliar>{{cite journal |title=Hubbard model in infinite dimensions |author=A. Georges and G.Kotliar |journal=[[Physical Review B]] |volume=45 |issue=12 |pages=6479 |year=1992 |doi=10.1103/PhysRevB.45.6479}}</ref> 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]] to be a momentum-independent (local) quantity. This approximation becomes exact in the limit of lattices with an infinite [[coordination number|coordination]].<ref name=Metzner>{{cite journal |title=Correlated Lattice Fermions in d = ∞ Dimensions |author=W. Metzner and D. Vollhardt |journal=[[Physical Review Letters]] |pages=324–327 |volume=62 |issue=3 |year=1989 |doi=10.1103/PhysRevLett.62.324}}</ref>
| |
| | |
| One of DMFT's main successes is to describe the [[phase transition]] between a metal and a [[Mott insulator]] when the strength of [[electronic correlation]]s is increased. It has been successfully applied to real materials, in combination with the [[local density approximation]] of Density Functional Theory.<ref name=LDA_DMFT>
| |
| {{cite journal |title=Electronic structure calculations with dynamical mean-field theory |author=G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti |journal=[[Reviews of Modern Physics]] |pages=865 |volume=78 |issue=3 |year=2006 |doi=10.1103/RevModPhys.78.865 |author-separator=, |display-authors=1 |last2=Savrasov |first2=S. |last3=Haule |first3=K.}}</ref>
| |
| | |
| ==Relation to Mean-Field Theory==
| |
| The DMFT treatment of lattice quantum models is similar to the [[mean-field theory]] (MFT) treatment of classical models such as the [[Ising model]].<ref name=Georges_2>{{cite journal |title=AIP Conference Proceedings|author=Antoine Georges |year=2004 |doi=10.1063/1.1800733 |chapter=Strongly Correlated Electron Materials: Dynamical Mean-Field Theory and Electronic Structure |volume=715 |pages=3–74 |issue=715 |journal=Lectures on the Physics of Highly Correlated Electron Systems VIII , American Institute of Physics Conference Proceedings Vol. |arxiv=cond-mat/0403123}}</ref> 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]]) onto a single-site problem. In DMFT, the local observable is the local [[Green's function (many-body theory)|Green's function]]. 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 <math>\Delta(\tau)</math> of the impurity model. DMFT owes its name to the fact that the mean-field <math>\Delta(\tau)</math> 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 <ref name=Hubbard>
| |
| {{cite journal |title=Electron Correlations in Narrow Energy Bands|author=John Hubbard |journal=[[Proceedings of the Royal Society A]] |pages=238 |volume=276 |year=1963 |doi=10.1098/rspa.1963.0204 |issue=1365}}</ref> describes the onsite interaction between electrons of opposite spin by a single parameter, <math>U</math>. The Hubbard Hamiltonian may take the following form:
| |
| :<math> 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}</math>
| |
| where <math>c_i^{\dagger},c_i</math> denote the creation and annihilation operators of an electron on a localized orbital on site <math>i</math>, and <math>n_i=c_i^{\dagger}c_i</math>.
| |
| | |
| The following assumptions have been made: | |
| * only one orbital contributes to the electronic properties (as might be the case of copper atoms in superconducting [[High-temperature_superconductivity#Cuprates|cuprates]], whose <math>d</math>-bands are non-degenerate),
| |
| * the orbitals are so localized that only nearest-neighbor hopping <math>t</math> 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 <math>a_p</math> and <math>a_p^{\dagger}</math>) through a hybridization function. The Anderson model corresponding to our single-site model is a single-orbital Anderson impurity model, whose hamiltonian formulation is the following:
| |
| :<math>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}}}</math>
| |
| where
| |
| * <math>H_{\text{bath}} </math> describes the non-correlated electronic levels <math>\epsilon_p</math> of the bath
| |
| * <math>H_{\text{loc}}</math> describes the impurity, where two electrons interact with the energetical cost <math>U</math>
| |
| *<math> H_{\text{mix}}</math> describes the hybridization (or coupling) between the impurity and the bath through hybridization terms <math>V_p^{\sigma}</math>
| |
| | |
| The Matsubara Green's function of this model, defined by <math> G_{\text{imp}}(\tau) = - \langle T c(\tau) c^{\dagger}(0)\rangle </math>, is entirely determined by the parameters <math>U,\mu</math> and the so-called hybridization function <math> \Delta_\sigma(i\omega_n) = \sum_{p}\frac{|V_p^\sigma|^2}{i\omega_n-\epsilon_p}</math>, which is the imaginary-time Fourier-transform of <math>\Delta_{\sigma}(\tau)</math>.
| |
| | |
| 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:
| |
| :<math>(G_0)^{-1}(i\omega_n)=i\omega_n+\mu-\Delta(i\omega_n)</math> (1)
| |
| | |
| Solving the Anderson impurity model consists in computing observables such as the interacting Green's function <math>G(i\omega_n)</math> for a given hybridization function <math>\Delta(i\omega_n)</math> and <math> U,\mu</math>. It is a difficult but not intractable problem. There exists a number of ways to solve the AIM, such as
| |
| * The [[Numerical_renormalization_group|Numerical Renormalization Group]]
| |
| * [[Exact diagonalization]]
| |
| * [[Iterative Perturbation Theory]]
| |
| * [[Non-Crossing Approximation]]
| |
| * Continuous-Time [[Quantum Monte Carlo]] algorithms<ref name=Rubtsov>{{cite journal |title=Continuous-time quantum Monte Carlo method for fermions |author=A. N. Rubtsov, V. V. Savkin, and A. I. Lichtenstein | journal=[[Physical Review B]] | pages=035122 |volume=72 | year=2005 |doi=10.1103/PhysRevB.72.035122 |issue=3}}</ref><ref name=Werner>{{cite journal |title=Continuous-Time Solver for Quantum Impurity Models |author=Philipp Werner, Armin Comanac, Luca de’ Medici, Matthias Troyer, and Andrew J. Millis |journal=[[Physical Review Letters]] |pages=076405 |volume=97 |year=2006 |doi=10.1103/PhysRevLett.97.076405 |issue=7}}</ref><ref name=Werner2>{{cite journal |title=Hybridization expansion impurity solver: General formulation and application to Kondo lattice and two-orbital models |author=Werner, Philipp and Millis, Andrew J. |journal=[[Physical Review B]] |pages=155107 |volume=74 |year=2006 |doi=10.1103/PhysRevB.74.155107 |issue=15}}</ref><ref name=Haule>{{cite journal| title=Quantum Monte Carlo Impurity Solver for Cluster DMFT and Electronic Structure Calculations| author=K. Haule| journal=[[Physical Review B]]|pages=155113|volume=75|year=2007|doi=10.1103/PhysRevB.75.155113}}
| |
| </ref><ref name=Gull>{{cite journal| title=Continuous-time Monte~Carlo methods for quantum impurity models| author=Gull, Emanuel and Millis, Andrew J. and Lichtenstein, Alexander I. and Rubtsov, Alexey N. and Troyer, Matthias and Werner, Philipp| journal=[[Reviews of Modern Physics]]|pages=349|volume=83|year=2011|doi=10.1103/RevModPhys.83.349}} | |
| </ref>
| |
| | |
| ===Self-consistency equations===
| |
| The self-consistency condition requires the impurity Green's function <math>G_{imp}(\tau)</math> to coincide with the local lattice Green's function <math>G_{ii}(\tau) = -\langle T c_i(\tau)c_i^{\dagger}(0)\rangle </math>:
| |
| :<math> G(i\omega_n) = G_{ii}(i\omega_n) = \sum_k \frac {1}{i\omega_n +\mu - \epsilon(k) - \Sigma(k,i\omega_n)}</math>
| |
| where <math>\Sigma(k,i\omega_n)</math> 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:
| |
| :<math> \Sigma(k,i\omega_n) \approx \Sigma_{imp}(i\omega_n) </math>
| |
| | |
| 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.
| |
| | |
| ===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. An intuitive method would be the following: for a given <math> U</math> , <math>\mu</math> and temperature <math>T</math>
| |
| # First, compute the non-interacting lattice Green's function <math>G_0(k,i\omega_n)</math>, and extract its local part <math>G_{0,loc}</math>
| |
| # The self-consistency condition requires it to be equal to the impurity Green's function, <math> \mathcal{G}^0(\tau) = G_{0,loc}</math>
| |
| # Compute the corresponding hybridization function through (1)
| |
| # Solve the AIM for a new impurity Green's function <math>G_{imp}^0(\tau)</math>, extract its self-energy: <math>\Sigma_{imp}(i\omega_n) = (G_{imp}^0)^{-1}(i\omega_n) - (\mathcal{G}^0)^{-1}(i\omega_n)</math>
| |
| # Make the DMFT approximation: <math> \Sigma(k,i\omega_n) \approx \Sigma_{imp}(i\omega_n) </math>
| |
| # Compute the new lattice Green's function, extract its local part and go back to step 2 with a new <math> \mathcal{G}^1</math>
| |
| | |
| Self-consistency is reached when <math>G_{imp}^n = G_{imp}^{n+1}</math>.
| |
| | |
| ==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 <math>U</math>, bandwidth, filling (chemical potential <math>\mu</math>), and temperature <math>T</math>:
| |
| * the [[spectral function]] (which gives the band structure)
| |
| * the [[kinetic energy]]
| |
| * the double occupancy of a site
| |
| * [[Linear response|response functions]] (compressibility, optical conductivity, specific heat)
| |
| | |
| In particular, the drop of the double-occupancy as <math>U</math> 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.
| |
| | |
| === Multi-orbital extension===
| |
| DMFT can be extended to Hubbard models with multiple orbitals, namely with electron-electron interactions of the form <math>U_{\alpha \beta} n_{\alpha}n_{\beta}</math> where <math>\alpha</math> and <math>\beta</math> denote different orbitals. This is especially relevant for compounds whose <math>d</math>-orbitals are degenerate, such as iron in the newly discovered high-temperature [[iron-based superconductor]]s (pnictides).
| |
| | |
| === 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 T and up to 100 sites at high T.
| |
| | |
| === Extended DMFT===
| |
| DMFT can be applied to more general models such as the [[t-J model]].
| |
| | |
| ==References and notes==
| |
| <references/>
| |
| | |
| ==See also==
| |
| *[[Strongly correlated materials]]
| |
| | |
| ==External links==
| |
| * [http://www.physics.rutgers.edu/~kotliar/papers/PT-Kotliar_57_53.pdf Strongly Correlated Materials: Insights From Dynamical Mean-Field Theory] G. Kotliar and D. Vollhardt
| |
| * [http://www.cond-mat.de/events/correl11/manuscript Lecture notes on the LDA+DMFT approach to strongly correlated materials] Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.)
| |
| | |
| [[Category:Materials science]]
| |
| [[Category:Condensed matter physics]]
| |
| [[Category:Quantum mechanics]]
| |
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