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Moved page from wikipedia:en:Dynamical mean-field theory (history)
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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 |author2=G. Kotliar |author3=W. Krauth |author4=M. Rozenberg |journal=[[Reviews of Modern Physics]] |pages=13 |volume=68 |issue=1 |year=1996 |doi=10.1103/RevModPhys.68.13 |bibcode = 1996RvMP...68...13G }}
</ref>
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 gas limit and the atomic limit of condensed-matter physics.
动力学平均场理论(DMFT)是一种确定强关联材料电子结构的方法。在这种材料中，用于密度泛函理论和通常的能带结构计算的独立电子近似失效了。动力学平均场理论是对电子之间局部相互作用的非微扰处理，它在近自由电子气极限和凝聚态物理学的原子极限之间架起了桥梁。
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=6479–6483 |year=1992 |doi=10.1103/PhysRevB.45.6479|bibcode = 1992PhRvB..45.6479G |pmid=10000408 }}</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 |author1=W. Metzner |author2=D. Vollhardt |journal=[[Physical Review Letters]] |pages=324–327 |volume=62 |issue=3 |year=1989 |doi=10.1103/PhysRevLett.62.324|bibcode = 1989PhRvL..62..324M |pmid=10040203 }}</ref>
DMFT consists in mapping a many-body 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 to be a momentum-independent (local) quantity. This approximation becomes exact in the limit of lattices with an infinite coordination.
DMFT 是将一个多体晶格问题映射到一个多体局部问题，即杂质模型。杂质模型通常可以通过各种方案求解，而晶格问题通常是难以解决的。映射本身并不构成近似值。在一般的 DMFT 格式中，唯一的近似是假设晶格自能是一个与动量无关的(局部)量。这种近似在具有无限协调的格的极限下变得精确。
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 |author2=S. Y. Savrasov |author3=K. Haule |author4=V. S. Oudovenko |author5=O. Parcollet |author6=C. A. Marianetti |journal=[[Reviews of Modern Physics]] |pages=865 |volume=78 |issue=3 |year=2006 |doi=10.1103/RevModPhys.78.865 |arxiv = cond-mat/0511085 |bibcode = 2006RvMP...78..865K }}</ref><ref name=Vollhardt>{{cite journal | title=Dynamical mean-field theory for correlated electrons | author=D. Vollhardt | journal = [[Annalen der Physik]] | volume=524 | issue=1 | pages=1–19 | year=2012 | doi=10.1002/andp.201100250 | bibcode=2012AnP...524....1V| doi-access=free }}</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 correlations is increased. It has been successfully applied to real materials, in combination with the local density approximation of density functional theory.
DMFT 的主要成果之一是描述了当电子关联强度增加时金属和莫特绝缘体之间的相变。它与密度泛函理论的局域密度近似相结合，已成功地应用于实际材料。
==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 conference |book-title=AIP Conference Proceedings |author=Antoine Georges |year=2004 |doi=10.1063/1.1800733 |title=Strongly Correlated Electron Materials: Dynamical Mean-Field Theory and Electronic Structure |volume=715 |pages=3–74 |issue=1 |conference=Lectures on the Physics of Highly Correlated Electron Systems VIII |publisher=American Institute of Physics |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.
The DMFT treatment of lattice quantum models is similar to the mean-field theory (MFT) treatment of classical models such as the Ising model. 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.
= = 与平均场理论的关系 = = 晶格量子模型的 DMFT 处理类似于经典模型如伊辛模型的平均场理论(MFT)处理。在伊辛模型中，晶格问题被映射到一个有效单位问题，其磁化是通过一个有效的“平均场”再现晶格的磁化。这种情况称为自我一致性条件。它规定，单点观测应再现格子“局部”观测通过一个有效的领域。虽然 n 位伊辛哈密顿量很难解析求解(迄今为止，解析解只存在于一维和二维情况下) ，但单位点问题很容易解决。
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.
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. 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.
类似地，DMFT 映射一个格子问题(例如:。哈伯德模型)到一个单站点的问题。在 DMFT 中，局部可观测量是局部格林函数。因此，DMFT 的自洽条件是杂质格林函数通过一个有效的平均场复现格林函数，在 DMFT 中，这个平均场是杂质模型的杂化函数 Delta (tau)。DMFT 之所以得名，是因为它的平均场增量(tau)具有时间依赖性或动力学特性。这也指出了 Ising MFT 和 DMFT 之间的主要区别: Ising MFT 将 n 自旋问题映射为一个单点单自旋问题。DMFT 将晶格问题映射到一个单位点问题上，但后者从根本上仍然是一个 n 体问题，它捕获了由于电子-电子关联而引起的时间波动。
==Description of DMFT for the Hubbard model==
==Description of DMFT for the Hubbard model==
= = 哈伯德模型的 DMFT 描述 = =
=== The DMFT mapping===
=== The DMFT mapping===
= = = DMFT 映射 = = =
====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=238–257 |volume=276 |year=1963 |doi=10.1098/rspa.1963.0204 |issue=1365|bibcode = 1963RSPSA.276..238H }}</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, on suppressing the spin 1/2 indices <math>\sigma</math>, <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 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.
= = = = 单轨道哈伯德模型 = = = = 哈伯德模型通过单参数描述了反自旋电子之间的现场相互作用。哈密顿函数可以采用下列形式: 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 }其中，在抑制自旋1/2指数时，c i ^ dagger { dagger } ，c i 表示位于 i 上的局部电子的创生及消灭算符，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 [[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 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
提出了以下假设:
* 只有一个轨道对电子性质有贡献(如超导铜氧化物中铜原子的情况，其 d 带是非简并的)
* 轨道局域化，只考虑了最近邻的跳跃 t
====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\sigma}</math> and <math>a_{p\sigma}^{\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, on suppressing some spin 1/2 indices <math>\sigma</math>, is:
:<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 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}
= = = = 辅助问题: 安德森模型模型在通常的扰动展开技术下通常是难以处理的。DMFT 将这个格子模型映射到所谓的安德森模型(AIM)上。这个模型通过一个杂化函数描述了一个位置(杂质)与电子能级的“浴”(用湮灭和创造算符 a _ { p sigma }和 a _ { p sigma } ^ { dagger }描述)的相互作用。与我们的单点模型对应的 Anderson 模型是一个单轨道安德森模型，其哈密顿公式是: h _ { text { AIM } = underbrace { sum { p } epsilon _ p a _ p ^ { dagka _ p }{ h _ { text { bath }} + underbrace { sum _ { p sigma } left (v _ { p } ^ { sigma } ^ { dagger } a _ { p sigma } + h.C.右)} _ { h _ { text { mix } + underbrace { u n _ { uparrow } n _ { downarrow }-mu left (n _ { uparrow } + n _ { downarrow } right)} _ { h _ { text { text { loc }}}}}其中
* h _ { text { text { bath }}描述了浴
* h _ { text { loc }的非关联电子能级 epsilon _ p 描述了杂质,当两个电子与能量成本 u
* h { text { mix }相互作用时，用杂化项 v _ p ^ { sigma }描述了杂化(或耦合)现象{ sigma }
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>.
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).
这个模型的 Matsubara Green 函数，定义为 g { text { imp }(tau) =-langle tc (tau) c ^ { dagger }(0) rangle，完全由参数 u，mu 和所谓的杂交函数 Delta _ sigma (i omega _ n) = sum { p } frac { | v _ p ^ | ^ 2}{ i omega _ n-epsilon _ p }确定，它是 Delta { sigma }(tau)的虚时间傅里叶变换。
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>(\mathcal{G}_0)^{-1}(i\omega_n)=i\omega_n+\mu-\Delta(i\omega_n)</math> (1)
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)
这个杂化函数描述了电子跳进跳出浴室的动力学。它必须再现晶格动力学，使杂质格林函数与局域晶格格林函数相同。它与非相互作用格林函数有关: (数学{ 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 <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
* [[Numerical renormalization group]]
* [[Exact diagonalization]]
* [[Iterative perturbation theory]]
* [[Non-crossing approximation]]
* [[Continuous-time quantum Monte Carlo]] algorithms
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
解决安德森模型的关键在于计算可观测量，比如给定杂交函数 Delta (i omega n)和 u，mu 的相互作用格林函数 g (i omega n)。这是一个困难但并非难以解决的问题。有许多方法可以解决这个问题，例如
* 数值重整化群
* 精确对角化
* 迭代摄动理论
* 非交叉近似
* 连续时间量子蒙特卡罗法算法
===Self-consistency equations===
The self-consistency condition requires the impurity Green's function <math>G_\mathrm{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_\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)}</math>
where <math>\Sigma(k,i\omega_n)</math> denotes the lattice self-energy.
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)}
where \Sigma(k,i\omega_n) denotes the lattice self-energy.
= = = = 自洽性方程 = = = 自洽性条件要求杂质格林函数 gmathrm { imp }(tau)与局域格林函数 g { ii }(tau) =-langle tci (tau) c i ^ { dagger }(0) rangle:(i omega _ n) = g { ii }(i omega _ n) = sum _ k frac {1}{ i omega _ n + mu-epsilon (k)-Sigma (k，i omega _ n)}其中 Sigma (k，i omega _ n)表示晶格自能。
===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>
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)
= = DMFT 近似: 晶格自能的局域性 = = = 唯一的 DMFT 近似(除了可以用来解决 Anderson 模型的近似之外)在于忽略晶格自能的空间涨落，将其等同于杂质自能: : Sigma (k，i _ n) approx _ _ { imp }(i _ 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.
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.
这种近似在具有无限协调的格子的极限下变得精确，也就是当每个格子的邻域数目是无限的时候。事实上，我们可以证明，在晶格自能的图解展开中，当进入无限协调极限时，只有局域图存在。
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.
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.
因此，就像经典的平均场理论一样，随着维数(以及邻域的数量)的增加，DMFT 应该得到更精确的结果。换句话说，对于低维度，空间波动将使 DMFT 近似不那么可靠。
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.
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.
空间波动也变得相关的附近的相变。在这里，DMFT 和经典的平均场理论导致了平均场临界指数，相变前的明显变化并没有反映在 DMFT 的自能中。
===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 <math> U</math>, <math>\mu</math> and temperature <math>T</math>:
# Start with a guess for <math>\Sigma(k,i\omega_n)</math> (typically, <math>\Sigma(k,i\omega_n)=0</math>)
# Make the DMFT approximation: <math> \Sigma(k,i\omega_n) \approx \Sigma_\mathrm{imp}(i\omega_n) </math>
# Compute the local Green's function <math>G_\mathrm{loc}(i\omega_n)</math>
# Compute the dynamical mean field <math>\Delta(i\omega) = i\omega_n + \mu - G^{-1}_\mathrm{loc}(i\omega_n) - \Sigma_\mathrm{imp}(i\omega_n) </math>
# Solve the AIM for a new impurity Green's function <math>G_\mathrm{imp}(i\omega_n)</math>, extract its self-energy: <math>\Sigma_\mathrm{imp}(i\omega_n) = (\mathcal{G}_0)^{-1}(i\omega_n) - (G_\mathrm{imp})^{-1}(i\omega_n) </math>
# Go back to step 2 until convergence, namely when <math>G_\mathrm{imp}^n = G_\mathrm{imp}^{n+1}</math>.
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}.
= = = DMFT 循环 = = = 为了找到局域晶格格林函数，必须确定杂化函数，使相应的杂质格林函数与备受追捧的局域晶格格林函数一致。解决这个问题最普遍的方法是使用一种正向递归方法，即对于给定的 u,mu 和温度 t: # 从 Sigma (k，i omega _ n)的猜测开始(通常，Sigma (k，i omega _ n) = 0) # 使 DMFT 近似: Sigma (k，i omega _ n)接近 Sigma _ mathrm { imp }(i omega _ n) # 计算本地 Green 函数 g _ mathrm { loc }(i _ n) # 计算动力平均场Δ (i omega) = i omega _ n + mu-g ^ {-1} _ mathrm { loc }(i omega _ n)-Sigma _ mathrm { imp }(i omega _ n) # 求一个新杂质格林函数 g _ mathrm { imp }(i omega _ n)的目标,提取它的自身能量: Sigma _ mathrm { imp }(i omega _ n) = (cal { g } _ 0) ^ {-1}(i omega _ n)-(g _ mathrm {}) ^ {-1}(i omega _ n) # 回到第二步，直到收敛,即当 g mathrm { imp } ^ n = g mathrm { imp } ^ { n + 1}。
==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)
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 (which gives the band structure)
* the kinetic energy
* the double occupancy of a site
* response functions (compressibility, optical conductivity, specific heat)
= = = 应用 = = 局域格林函数和其他杂质可观测量可用来计算一些物理量，作为关联 u，带宽，填充(化学势 μ)和温度 t 的函数:
* 谱函数(给出能带结构)
* 动能
* 一个位点的双占有率响应函数(压缩率，光导率，比热)
In particular, the drop of the double-occupancy as <math>U</math> increases is a signature of the Mott transition.
In particular, the drop of the double-occupancy as U increases is a signature of the Mott transition.
特别是，随着 u 的增加，双人占有率下降，这是莫特过渡的一个标志。
==Extensions of DMFT==
DMFT has several extensions, extending the above formalism to multi-orbital, multi-site problems, long-range correlations and non-equilibrium.
DMFT has several extensions, extending the above formalism to multi-orbital, multi-site problems, long-range correlations and non-equilibrium.
= = DMFT = DMFT 的扩展有几个扩展，将上述公式扩展到多轨道、多站点问题、长程相关性和非平衡性。
=== 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. The combination with [[density functional theory]] (DFT+DMFT)<ref name="LDA_DMFT" /><ref name=Held>{{cite journal |title=Electronic Structure Calculations using Dynamical Mean Field Theory |author=K. Held |journal=[[Adv. Phys.]] |volume=56 |issue=6 |pages=829–926 |year=2007 |doi=10.1080/00018730701619647|arxiv = cond-mat/0511293| bibcode = 2007AdPhy..56..829H }}</ref> then allows for a realistic calculation of correlated materials.<ref>{{Cite web|url=http://hauleweb.rutgers.edu/tutorials/|title=Embedded Dynamical Mean Field Theory, an electronic structure package implementing DFT+DMFT}}</ref>
DMFT can be extended to Hubbard models with multiple orbitals, namely with electron-electron interactions of the form U_{\alpha \beta} n_{\alpha}n_{\beta} where \alpha and \beta denote different orbitals. The combination with density functional theory (DFT+DMFT) then allows for a realistic calculation of correlated materials.
= = = 多轨道延伸 = = DMFT 可以推广到具有多个轨道的 Hubbard 模型，即具有 u _ { alpha beta } n _ { alpha } n _ { beta }形式的电子-电子相互作用，其中 α 和 β 表示不同的轨道。结合密度泛函理论(dft + dmft) ，然后允许一个现实的计算相关材料。
=== 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]].
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.
= = = 扩展 DMFT = = = 扩展 DMFT 产生局部杂质自能的非局部相互作用，因此允许我们应用 DMFT 更一般的模型，如 t-J 模型。
=== 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.
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.
= = = 团簇 DMFT = = = = = 为了改进 DMFT 近似，Hubbard 模型可以映射到一个多位点杂质(团簇)问题上，这使得人们可以对杂质自能增加一些空间依赖性。团簇包含4至8个低温位点和多达100个高温位点。
=== Diagrammatic extensions ===
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<ref name=Rohringer>{{cite journal |title=Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory |author1=G. Rohringer |author2= H. Hafermann| author3=A. Toschi |author4=A. Katanin | author5= A. E. Antipov| author6= M. I. Katsnelson|author7= A. I. Lichtenstein|author8= A. N. Rubtsov|author9=K. Held |journal=[[Reviews of Modern Physics]] |volume=90 |issue=4 |pages= 025003 |year=2018|doi=10.1103/RevModPhys.90.025003|arxiv = 1705.00024 }}</ref> using a combination of analytical and numerical techniques. The starting point of the dynamical vertex approximation<ref name=Toschi>{{cite journal |title=Dynamical vertex approximation: A step beyond dynamical mean-field theory |author1=A. Toschi |author2=A. Katanin |author3=K. Held |journal=[[Physical Review B]] |volume=75 |issue=4 |pages=045118 |year=2007 |doi=10.1103/PhysRevB.75.045118|arxiv = cond-mat/0603100| bibcode = 2007PhRvB..75d5118T }}</ref> and of the dual fermion approach is the local [[Vertex function|two-particle vertex]].
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 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.
= = = 图解扩展 = = = DMFT 之外自能的空间依赖性，包括相变附近的长程关联，也可以通过使用解析和数值技术相结合的 DMFT 图解扩展来获得。动态顶点近似和对偶费米子近似的出发点是局部两粒子顶点。
=== Non-equilibrium ===
DMFT has been employed to study non-equilibrium transport and optical excitations. Here, the reliable calculation of the AIM's [[Keldysh formalism|Green function out of equilibrium]] remains a big challenge.
DMFT has been employed to study non-equilibrium transport and optical excitations. Here, the reliable calculation of the AIM's Green function out of equilibrium remains a big challenge.
= = = 非平衡态 = = = DMFT 被用来研究非平衡态输运和光激发。在这里，如何可靠地计算 AIM 平衡外的格林函数仍然是一个很大的挑战。
==References and notes==
<references/>
==See also==
*[[Strongly correlated material]]
*Strongly correlated material
= = = = = = =
* 强相关材料
==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.)
* [http://www.cond-mat.de/events/correl14/manuscripts/ Lecture notes DMFT at 25: Infinite Dimensions] Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.)
* [http://www.cond-mat.de/events/correl18/manuscripts/ Lecture notes DMFT – From Infinite Dimensions to Real Materials] Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.)
* 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.)
= = = 外部链接 = =
* 强相关材料: 来自动态平均场理论的见解 g. Kotliar 和 d. Vollhardt
* 对强相关材料 lda + dmft 方法的讲义 Eva Pavarini，Erik Koch，Dieter Vollhardt 和 Alexander Lichtenstein (合编)
* 讲稿 DMFT 在25: 无限维度伊娃帕瓦里尼，埃里克科赫，迪特沃尔哈特，和亚历山大利希滕斯坦(编。)
* 讲稿 DMFT-从无限维度到真正的材料伊娃帕瓦里尼，埃里克科赫，迪特沃尔哈特，和亚历山大利希滕斯坦(编。)
[[Category:Correlated electrons]]
[[Category:Materials science]]
[[Category:Condensed matter physics]]
[[Category:Quantum mechanics]]
Category:Correlated electrons
Category:Materials science
Category:Condensed matter physics
Category:Quantum mechanics
类别: 相关电子类别: 材料科学类别: 凝聚态物理学科学类别: 量子力学
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<small>This page was moved from [[wikipedia:en:Dynamical mean-field theory]]. Its edit history can be viewed at [[动力学平均场理论/edithistory]]</small></noinclude>
[[Category:待整理页面]]
'''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 |author2=G. Kotliar |author3=W. Krauth |author4=M. Rozenberg |journal=[[Reviews of Modern Physics]] |pages=13 |volume=68 |issue=1 |year=1996 |doi=10.1103/RevModPhys.68.13 |bibcode = 1996RvMP...68...13G }}
</ref>
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 gas limit and the atomic limit of condensed-matter physics.
动力学平均场理论(DMFT)是一种确定强关联材料电子结构的方法。在这种材料中，用于密度泛函理论和通常的能带结构计算的独立电子近似失效了。动力学平均场理论是对电子之间局部相互作用的非微扰处理，它在近自由电子气极限和凝聚态物理学的原子极限之间架起了桥梁。
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=6479–6483 |year=1992 |doi=10.1103/PhysRevB.45.6479|bibcode = 1992PhRvB..45.6479G |pmid=10000408 }}</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 |author1=W. Metzner |author2=D. Vollhardt |journal=[[Physical Review Letters]] |pages=324–327 |volume=62 |issue=3 |year=1989 |doi=10.1103/PhysRevLett.62.324|bibcode = 1989PhRvL..62..324M |pmid=10040203 }}</ref>
DMFT consists in mapping a many-body 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 to be a momentum-independent (local) quantity. This approximation becomes exact in the limit of lattices with an infinite coordination.
DMFT 是将一个多体晶格问题映射到一个多体局部问题，即杂质模型。杂质模型通常可以通过各种方案求解，而晶格问题通常是难以解决的。映射本身并不构成近似值。在一般的 DMFT 格式中，唯一的近似是假设晶格自能是一个与动量无关的(局部)量。这种近似在具有无限协调的格的极限下变得精确。
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 |author2=S. Y. Savrasov |author3=K. Haule |author4=V. S. Oudovenko |author5=O. Parcollet |author6=C. A. Marianetti |journal=[[Reviews of Modern Physics]] |pages=865 |volume=78 |issue=3 |year=2006 |doi=10.1103/RevModPhys.78.865 |arxiv = cond-mat/0511085 |bibcode = 2006RvMP...78..865K }}</ref><ref name=Vollhardt>{{cite journal | title=Dynamical mean-field theory for correlated electrons | author=D. Vollhardt | journal = [[Annalen der Physik]] | volume=524 | issue=1 | pages=1–19 | year=2012 | doi=10.1002/andp.201100250 | bibcode=2012AnP...524....1V| doi-access=free }}</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 correlations is increased. It has been successfully applied to real materials, in combination with the local density approximation of density functional theory.
DMFT 的主要成果之一是描述了当电子关联强度增加时金属和莫特绝缘体之间的相变。它与密度泛函理论的局域密度近似相结合，已成功地应用于实际材料。
==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 conference |book-title=AIP Conference Proceedings |author=Antoine Georges |year=2004 |doi=10.1063/1.1800733 |title=Strongly Correlated Electron Materials: Dynamical Mean-Field Theory and Electronic Structure |volume=715 |pages=3–74 |issue=1 |conference=Lectures on the Physics of Highly Correlated Electron Systems VIII |publisher=American Institute of Physics |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.
The DMFT treatment of lattice quantum models is similar to the mean-field theory (MFT) treatment of classical models such as the Ising model. 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.
= = 与平均场理论的关系 = = 晶格量子模型的 DMFT 处理类似于经典模型如伊辛模型的平均场理论(MFT)处理。在伊辛模型中，晶格问题被映射到一个有效单位问题，其磁化是通过一个有效的“平均场”再现晶格的磁化。这种情况称为自我一致性条件。它规定，单点观测应再现格子“局部”观测通过一个有效的领域。虽然 n 位伊辛哈密顿量很难解析求解(迄今为止，解析解只存在于一维和二维情况下) ，但单位点问题很容易解决。
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.
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. 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.
类似地，DMFT 映射一个格子问题(例如:。哈伯德模型)到一个单站点的问题。在 DMFT 中，局部可观测量是局部格林函数。因此，DMFT 的自洽条件是杂质格林函数通过一个有效的平均场复现格林函数，在 DMFT 中，这个平均场是杂质模型的杂化函数 Delta (tau)。DMFT 之所以得名，是因为它的平均场增量(tau)具有时间依赖性或动力学特性。这也指出了 Ising MFT 和 DMFT 之间的主要区别: Ising MFT 将 n 自旋问题映射为一个单点单自旋问题。DMFT 将晶格问题映射到一个单位点问题上，但后者从根本上仍然是一个 n 体问题，它捕获了由于电子-电子关联而引起的时间波动。
==Description of DMFT for the Hubbard model==
==Description of DMFT for the Hubbard model==
= = 哈伯德模型的 DMFT 描述 = =
=== The DMFT mapping===
=== The DMFT mapping===
= = = DMFT 映射 = = =
====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=238–257 |volume=276 |year=1963 |doi=10.1098/rspa.1963.0204 |issue=1365|bibcode = 1963RSPSA.276..238H }}</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, on suppressing the spin 1/2 indices <math>\sigma</math>, <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 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.
= = = = 单轨道哈伯德模型 = = = = 哈伯德模型通过单参数描述了反自旋电子之间的现场相互作用。哈密顿函数可以采用下列形式: 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 }其中，在抑制自旋1/2指数时，c i ^ dagger { dagger } ，c i 表示位于 i 上的局部电子的创生及消灭算符，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 [[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 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
提出了以下假设:
* 只有一个轨道对电子性质有贡献(如超导铜氧化物中铜原子的情况，其 d 带是非简并的)
* 轨道局域化，只考虑了最近邻的跳跃 t
====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\sigma}</math> and <math>a_{p\sigma}^{\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, on suppressing some spin 1/2 indices <math>\sigma</math>, is:
:<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 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}
= = = = 辅助问题: 安德森模型模型在通常的扰动展开技术下通常是难以处理的。DMFT 将这个格子模型映射到所谓的安德森模型(AIM)上。这个模型通过一个杂化函数描述了一个位置(杂质)与电子能级的“浴”(用湮灭和创造算符 a _ { p sigma }和 a _ { p sigma } ^ { dagger }描述)的相互作用。与我们的单点模型对应的 Anderson 模型是一个单轨道安德森模型，其哈密顿公式是: h _ { text { AIM } = underbrace { sum { p } epsilon _ p a _ p ^ { dagka _ p }{ h _ { text { bath }} + underbrace { sum _ { p sigma } left (v _ { p } ^ { sigma } ^ { dagger } a _ { p sigma } + h.C.右)} _ { h _ { text { mix } + underbrace { u n _ { uparrow } n _ { downarrow }-mu left (n _ { uparrow } + n _ { downarrow } right)} _ { h _ { text { text { loc }}}}}其中
* h _ { text { text { bath }}描述了浴
* h _ { text { loc }的非关联电子能级 epsilon _ p 描述了杂质,当两个电子与能量成本 u
* h { text { mix }相互作用时，用杂化项 v _ p ^ { sigma }描述了杂化(或耦合)现象{ sigma }
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>.
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).
这个模型的 Matsubara Green 函数，定义为 g { text { imp }(tau) =-langle tc (tau) c ^ { dagger }(0) rangle，完全由参数 u，mu 和所谓的杂交函数 Delta _ sigma (i omega _ n) = sum { p } frac { | v _ p ^ | ^ 2}{ i omega _ n-epsilon _ p }确定，它是 Delta { sigma }(tau)的虚时间傅里叶变换。
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>(\mathcal{G}_0)^{-1}(i\omega_n)=i\omega_n+\mu-\Delta(i\omega_n)</math> (1)
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)
这个杂化函数描述了电子跳进跳出浴室的动力学。它必须再现晶格动力学，使杂质格林函数与局域晶格格林函数相同。它与非相互作用格林函数有关: (数学{ 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 <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
* [[Numerical renormalization group]]
* [[Exact diagonalization]]
* [[Iterative perturbation theory]]
* [[Non-crossing approximation]]
* [[Continuous-time quantum Monte Carlo]] algorithms
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
解决安德森模型的关键在于计算可观测量，比如给定杂交函数 Delta (i omega n)和 u，mu 的相互作用格林函数 g (i omega n)。这是一个困难但并非难以解决的问题。有许多方法可以解决这个问题，例如
* 数值重整化群
* 精确对角化
* 迭代摄动理论
* 非交叉近似
* 连续时间量子蒙特卡罗法算法
===Self-consistency equations===
The self-consistency condition requires the impurity Green's function <math>G_\mathrm{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_\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)}</math>
where <math>\Sigma(k,i\omega_n)</math> denotes the lattice self-energy.
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)}
where \Sigma(k,i\omega_n) denotes the lattice self-energy.
= = = = 自洽性方程 = = = 自洽性条件要求杂质格林函数 gmathrm { imp }(tau)与局域格林函数 g { ii }(tau) =-langle tci (tau) c i ^ { dagger }(0) rangle:(i omega _ n) = g { ii }(i omega _ n) = sum _ k frac {1}{ i omega _ n + mu-epsilon (k)-Sigma (k，i omega _ n)}其中 Sigma (k，i omega _ n)表示晶格自能。
===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>
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)
= = DMFT 近似: 晶格自能的局域性 = = = 唯一的 DMFT 近似(除了可以用来解决 Anderson 模型的近似之外)在于忽略晶格自能的空间涨落，将其等同于杂质自能: : Sigma (k，i _ n) approx _ _ { imp }(i _ 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.
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.
这种近似在具有无限协调的格子的极限下变得精确，也就是当每个格子的邻域数目是无限的时候。事实上，我们可以证明，在晶格自能的图解展开中，当进入无限协调极限时，只有局域图存在。
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.
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.
因此，就像经典的平均场理论一样，随着维数(以及邻域的数量)的增加，DMFT 应该得到更精确的结果。换句话说，对于低维度，空间波动将使 DMFT 近似不那么可靠。
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.
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.
空间波动也变得相关的附近的相变。在这里，DMFT 和经典的平均场理论导致了平均场临界指数，相变前的明显变化并没有反映在 DMFT 的自能中。
===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 <math> U</math>, <math>\mu</math> and temperature <math>T</math>:
# Start with a guess for <math>\Sigma(k,i\omega_n)</math> (typically, <math>\Sigma(k,i\omega_n)=0</math>)
# Make the DMFT approximation: <math> \Sigma(k,i\omega_n) \approx \Sigma_\mathrm{imp}(i\omega_n) </math>
# Compute the local Green's function <math>G_\mathrm{loc}(i\omega_n)</math>
# Compute the dynamical mean field <math>\Delta(i\omega) = i\omega_n + \mu - G^{-1}_\mathrm{loc}(i\omega_n) - \Sigma_\mathrm{imp}(i\omega_n) </math>
# Solve the AIM for a new impurity Green's function <math>G_\mathrm{imp}(i\omega_n)</math>, extract its self-energy: <math>\Sigma_\mathrm{imp}(i\omega_n) = (\mathcal{G}_0)^{-1}(i\omega_n) - (G_\mathrm{imp})^{-1}(i\omega_n) </math>
# Go back to step 2 until convergence, namely when <math>G_\mathrm{imp}^n = G_\mathrm{imp}^{n+1}</math>.
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}.
= = = DMFT 循环 = = = 为了找到局域晶格格林函数，必须确定杂化函数，使相应的杂质格林函数与备受追捧的局域晶格格林函数一致。解决这个问题最普遍的方法是使用一种正向递归方法，即对于给定的 u,mu 和温度 t: # 从 Sigma (k，i omega _ n)的猜测开始(通常，Sigma (k，i omega _ n) = 0) # 使 DMFT 近似: Sigma (k，i omega _ n)接近 Sigma _ mathrm { imp }(i omega _ n) # 计算本地 Green 函数 g _ mathrm { loc }(i _ n) # 计算动力平均场Δ (i omega) = i omega _ n + mu-g ^ {-1} _ mathrm { loc }(i omega _ n)-Sigma _ mathrm { imp }(i omega _ n) # 求一个新杂质格林函数 g _ mathrm { imp }(i omega _ n)的目标,提取它的自身能量: Sigma _ mathrm { imp }(i omega _ n) = (cal { g } _ 0) ^ {-1}(i omega _ n)-(g _ mathrm {}) ^ {-1}(i omega _ n) # 回到第二步，直到收敛,即当 g mathrm { imp } ^ n = g mathrm { imp } ^ { n + 1}。
==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)
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 (which gives the band structure)
* the kinetic energy
* the double occupancy of a site
* response functions (compressibility, optical conductivity, specific heat)
= = = 应用 = = 局域格林函数和其他杂质可观测量可用来计算一些物理量，作为关联 u，带宽，填充(化学势 μ)和温度 t 的函数:
* 谱函数(给出能带结构)
* 动能
* 一个位点的双占有率响应函数(压缩率，光导率，比热)
In particular, the drop of the double-occupancy as <math>U</math> increases is a signature of the Mott transition.
In particular, the drop of the double-occupancy as U increases is a signature of the Mott transition.
特别是，随着 u 的增加，双人占有率下降，这是莫特过渡的一个标志。
==Extensions of DMFT==
DMFT has several extensions, extending the above formalism to multi-orbital, multi-site problems, long-range correlations and non-equilibrium.
DMFT has several extensions, extending the above formalism to multi-orbital, multi-site problems, long-range correlations and non-equilibrium.
= = DMFT = DMFT 的扩展有几个扩展，将上述公式扩展到多轨道、多站点问题、长程相关性和非平衡性。
=== 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. The combination with [[density functional theory]] (DFT+DMFT)<ref name="LDA_DMFT" /><ref name=Held>{{cite journal |title=Electronic Structure Calculations using Dynamical Mean Field Theory |author=K. Held |journal=[[Adv. Phys.]] |volume=56 |issue=6 |pages=829–926 |year=2007 |doi=10.1080/00018730701619647|arxiv = cond-mat/0511293| bibcode = 2007AdPhy..56..829H }}</ref> then allows for a realistic calculation of correlated materials.<ref>{{Cite web|url=http://hauleweb.rutgers.edu/tutorials/|title=Embedded Dynamical Mean Field Theory, an electronic structure package implementing DFT+DMFT}}</ref>
DMFT can be extended to Hubbard models with multiple orbitals, namely with electron-electron interactions of the form U_{\alpha \beta} n_{\alpha}n_{\beta} where \alpha and \beta denote different orbitals. The combination with density functional theory (DFT+DMFT) then allows for a realistic calculation of correlated materials.
= = = 多轨道延伸 = = DMFT 可以推广到具有多个轨道的 Hubbard 模型，即具有 u _ { alpha beta } n _ { alpha } n _ { beta }形式的电子-电子相互作用，其中 α 和 β 表示不同的轨道。结合密度泛函理论(dft + dmft) ，然后允许一个现实的计算相关材料。
=== 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]].
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.
= = = 扩展 DMFT = = = 扩展 DMFT 产生局部杂质自能的非局部相互作用，因此允许我们应用 DMFT 更一般的模型，如 t-J 模型。
=== 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.
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.
= = = 团簇 DMFT = = = = = 为了改进 DMFT 近似，Hubbard 模型可以映射到一个多位点杂质(团簇)问题上，这使得人们可以对杂质自能增加一些空间依赖性。团簇包含4至8个低温位点和多达100个高温位点。
=== Diagrammatic extensions ===
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<ref name=Rohringer>{{cite journal |title=Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory |author1=G. Rohringer |author2= H. Hafermann| author3=A. Toschi |author4=A. Katanin | author5= A. E. Antipov| author6= M. I. Katsnelson|author7= A. I. Lichtenstein|author8= A. N. Rubtsov|author9=K. Held |journal=[[Reviews of Modern Physics]] |volume=90 |issue=4 |pages= 025003 |year=2018|doi=10.1103/RevModPhys.90.025003|arxiv = 1705.00024 }}</ref> using a combination of analytical and numerical techniques. The starting point of the dynamical vertex approximation<ref name=Toschi>{{cite journal |title=Dynamical vertex approximation: A step beyond dynamical mean-field theory |author1=A. Toschi |author2=A. Katanin |author3=K. Held |journal=[[Physical Review B]] |volume=75 |issue=4 |pages=045118 |year=2007 |doi=10.1103/PhysRevB.75.045118|arxiv = cond-mat/0603100| bibcode = 2007PhRvB..75d5118T }}</ref> and of the dual fermion approach is the local [[Vertex function|two-particle vertex]].
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 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.
= = = 图解扩展 = = = DMFT 之外自能的空间依赖性，包括相变附近的长程关联，也可以通过使用解析和数值技术相结合的 DMFT 图解扩展来获得。动态顶点近似和对偶费米子近似的出发点是局部两粒子顶点。
=== Non-equilibrium ===
DMFT has been employed to study non-equilibrium transport and optical excitations. Here, the reliable calculation of the AIM's [[Keldysh formalism|Green function out of equilibrium]] remains a big challenge.
DMFT has been employed to study non-equilibrium transport and optical excitations. Here, the reliable calculation of the AIM's Green function out of equilibrium remains a big challenge.
= = = 非平衡态 = = = DMFT 被用来研究非平衡态输运和光激发。在这里，如何可靠地计算 AIM 平衡外的格林函数仍然是一个很大的挑战。
==References and notes==
<references/>
==See also==
*[[Strongly correlated material]]
*Strongly correlated material
= = = = = = =
* 强相关材料
==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.)
* [http://www.cond-mat.de/events/correl14/manuscripts/ Lecture notes DMFT at 25: Infinite Dimensions] Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.)
* [http://www.cond-mat.de/events/correl18/manuscripts/ Lecture notes DMFT – From Infinite Dimensions to Real Materials] Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.)
* 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.)
= = = 外部链接 = =
* 强相关材料: 来自动态平均场理论的见解 g. Kotliar 和 d. Vollhardt
* 对强相关材料 lda + dmft 方法的讲义 Eva Pavarini，Erik Koch，Dieter Vollhardt 和 Alexander Lichtenstein (合编)
* 讲稿 DMFT 在25: 无限维度伊娃帕瓦里尼，埃里克科赫，迪特沃尔哈特，和亚历山大利希滕斯坦(编。)
* 讲稿 DMFT-从无限维度到真正的材料伊娃帕瓦里尼，埃里克科赫，迪特沃尔哈特，和亚历山大利希滕斯坦(编。)
[[Category:Correlated electrons]]
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[[Category:Condensed matter physics]]
[[Category:Quantum mechanics]]
Category:Correlated electrons
Category:Materials science
Category:Condensed matter physics
Category:Quantum mechanics
类别: 相关电子类别: 材料科学类别: 凝聚态物理学科学类别: 量子力学
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