Category:Strongly correlated electrons - Revision history

Huebsch: Huebsch moved page Construction:Category:Strongly correlated electrons to Category:Strongly correlated electrons without leaving a redirect

2026-03-27T12:41:13Z

Huebsch moved page Construction:Category:Strongly correlated electrons to Category:Strongly correlated electrons without leaving a redirect

← Older revision	Revision as of 12:41, 27 March 2026
(No difference)

Hampel at 09:38, 27 March 2026

2026-03-27T09:38:45Z

← Older revision		Revision as of 09:38, 27 March 2026
Line 39:		Line 39:

	== Dynamical Mean-Field Theory (DMFT) ==		== Dynamical Mean-Field Theory (DMFT) ==
	DMFT is an dynamical extension of DFT+U~~, where U is considered to be frequency dependent ([[DFT+DMFT calculations\|DFT+DMFT]]). DMFT~~ provides an accurate treatment of strongly correlated materials~~, which are fully neglected in~~ DFT+U {{cite\|kotliar:rmp:2006}}. ~~Within~~ DMFT a lattice problem ~~is mapped~~ onto a ~~self-consistent~~ quantum impurity model ~~by embedding~~ a single correlated site in an effective bath ~~that represents~~ the rest of the ~~system, and the key~~ approximation is that the self-energy is local (frequency-dependent but momentum-independent). ~~DMFT constitutes a state-of-~~the~~-art approach for the accurate description of strongly correlated systems,~~ capturing ~~essential~~ many-body effects such as quasiparticle renormalization, Hubbard bands, and Mott metal–insulator ~~transitions~~. DMFT can be ~~used in combination with~~ cRPA~~, where~~ cRPA ~~is used as a preliminary step to determine the effective screening $U(\omega~~)~~$ without the contribution from the target space to avoid double-counting in the subsequent calculation of the self-energy within DMFT~~.		DMFT is a dynamical extension of DFT+U that provides a more accurate treatment of strongly correlated materials where DFT+U is insufficient {{cite\|kotliar:rmp:2006}}. Like DFT+U, DMFT augments the DFT calculation with an additional local correlated subproblem — typically a specific $d$- or $f$-shell. The key idea is to map the full lattice problem onto a quantum impurity model: a single correlated site embedded in a self-consistently determined effective bath representing the rest of the lattice. The central approximation is that the self-energy is purely local — frequency-dependent but momentum-independent. This makes the problem tractable while still capturing many-body effects beyond the reach of DFT+U, such as quasiparticle mass renormalization, Hubbard bands, and the Mott metal–insulator transition. The interaction parameters entering DMFT can be determined from first principles using cRPA (see [[Constrained–random-phase–approximation formalism\|cRPA section above]]).

	== Other methods ==		== Other methods ==

Kaltakm: /* Other methods */

2026-03-26T08:08:47Z

Other methods

← Older revision		Revision as of 08:08, 26 March 2026
Line 42:		Line 42:

	== Other methods ==		== Other methods ==
	There are other methods that ~~are~~ not ~~specialized~~ for strongly correlated ~~materials but nevertheless~~ have been ~~shown~~ to improve ~~the~~ description ~~of the~~ electronic structure ~~of the strongly correlated systems~~.		There are other methods that, while not specifically designed for strongly correlated systems, have nonetheless been demonstrated to improve their description and electronic structure.

	=== Hybrid functionals ===		=== Hybrid functionals ===
	By ~~including~~ a fraction of exact exchange, ~~the~~ hybrid ~~functional approach can reduce~~ the self-interaction error in DFT~~, which is required~~ to ~~improve the~~ description of ~~the physics of strong correlations~~ {{cite\|Silva2007}}{{cite\|liu2019assessing}}.		By incorporating a fraction of exact exchange, hybrid functionals partially mitigate the self-interaction error inherent to standard DFT. This reduction of the self-interaction error has been shown to yield an improved description of strongly correlated systems {{cite\|Silva2007}}{{cite\|liu2019assessing}}.

	=== Hybrid functionals + U ===		=== Hybrid functionals + U ===
	A ~~shortcoming~~ of hybrid functionals is their uniform ~~description~~ of all states, which		A well-known limitation of hybrid functionals is their uniform treatment of all electronic states, which can result in markedly different levels of accuracy for states with varying degrees of localization. The inclusion of the Hubbard on-site interaction term within the hybrid functional framework has been shown to address inaccuracies arising from the overscreening of localized states {{cite\|Ivady2014}}.
	can ~~show very~~ different accuracy for states with ~~different~~ degrees of
	localization. The ~~introduction~~ of the Hubbard on-site interaction within the
	hybrid functional ~~approach was~~ shown to ~~resolve issues caused by~~
	overscreening of localized states {{cite\|Ivady2014}}.

	=== QPGW ===		=== QPGW ===
	The [[:Category:GW\|GW]] approximation in its simplest form (one-shot approach) ~~is strongly dependent~~ on the		The [[:Category:GW\|GW]] approximation in its simplest, non-self-consistent form (i.e., the one-shot approach) exhibits a strong dependence on the choice of starting point, and thus inherits the limitations of the underlying DFT description of localized states. In contrast, self-consistent GW schemes such as QPGW, which are independent of the starting electronic structure, have been shown to provide an accurate description of correlated electrons {{cite\|Cunningham2023}}.
	starting point and thus ~~suffers from~~ the ~~shortcomings~~ of the DFT ~~for describing~~
	localized states. ~~However~~, a self-consistent GW ~~approach~~ such as QPGW which ~~does~~
	~~not depend on~~ the starting electronic structure ~~can yield~~ an accurate
	description of ~~the~~ correlated electrons {{cite\|Cunningham2023}}.

	== Tutorials ==		== Tutorials ==

Tal: /* Other methods */

2026-03-26T08:05:15Z

Other methods

← Older revision		Revision as of 08:05, 26 March 2026
Line 42:		Line 42:

	== Other methods ==		== Other methods ==
	There are other methods that are not specialized for strongly correlated ~~system~~ but nevertheless have been shown to improve the description of the electronic structure of the strongly correlated systems.		There are other methods that are not specialized for strongly correlated materials but nevertheless have been shown to improve the description of the electronic structure of the strongly correlated systems.

	=== Hybrid functionals ===		=== Hybrid functionals ===

Kaltakm: /* Dynamical Mean-Field Theory (DMFT) */

2026-03-26T07:57:18Z

Dynamical Mean-Field Theory (DMFT)

← Older revision		Revision as of 07:57, 26 March 2026
Line 39:		Line 39:

	== Dynamical Mean-Field Theory (DMFT) ==		== Dynamical Mean-Field Theory (DMFT) ==
	DMFT is an ~~advanced~~ extension of DFT ([[DFT+DMFT calculations\|DFT+DMFT]]) ~~that~~ provides an accurate treatment of strongly correlated materials ~~including dynamical effects~~, which are fully neglected in DFT+U {{cite\|kotliar:rmp:2006}}. Within DMFT a lattice problem is mapped onto a self-consistent quantum impurity model by embedding a single correlated site in an effective bath that represents the rest of the system, and the key approximation is that the self-energy is local (frequency-dependent but momentum-independent). DMFT constitutes a state-of-the-art approach for the accurate description of strongly correlated systems, capturing essential many-body effects such as quasiparticle renormalization, Hubbard bands, and Mott metal–insulator transitions. DMFT can be used in combination with cRPA, where cRPA is used as a preliminary step to determine the effective screening $U(\omega)$ without the contribution from the target space to avoid double-counting in the subsequent calculation of the self-energy within DMFT.		DMFT is an dynamical extension of DFT+U, where U is considered to be frequency dependent ([[DFT+DMFT calculations\|DFT+DMFT]]). DMFT provides an accurate treatment of strongly correlated materials, which are fully neglected in DFT+U {{cite\|kotliar:rmp:2006}}. Within DMFT a lattice problem is mapped onto a self-consistent quantum impurity model by embedding a single correlated site in an effective bath that represents the rest of the system, and the key approximation is that the self-energy is local (frequency-dependent but momentum-independent). DMFT constitutes a state-of-the-art approach for the accurate description of strongly correlated systems, capturing essential many-body effects such as quasiparticle renormalization, Hubbard bands, and Mott metal–insulator transitions. DMFT can be used in combination with cRPA, where cRPA is used as a preliminary step to determine the effective screening $U(\omega)$ without the contribution from the target space to avoid double-counting in the subsequent calculation of the self-energy within DMFT.

	== Other methods ==		== Other methods ==

Kaltakm at 07:50, 26 March 2026

2026-03-26T07:50:43Z

← Older revision		Revision as of 07:50, 26 March 2026
Line 1:		Line 1:
	[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]		[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]
	Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT. These systems typically include elements with partially filled $d$ and $f$ electron orbitals which are localized~~, and correlation~~ effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.		Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT. These systems typically include elements with partially filled $d$ and $f$ electron orbitals which are localized. Correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.
	To model such systems, several extensions of DFT have been developed.		To model such systems, several extensions of DFT have been developed.

Kaltakm at 07:50, 26 March 2026

2026-03-26T07:50:16Z

← Older revision		Revision as of 07:50, 26 March 2026
Line 1:		Line 1:
	[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]		[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]
	Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT. These systems typically include elements with partially filled $d$ ~~or/~~and $f$ electron orbitals which are localized, and correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.		Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT. These systems typically include elements with partially filled $d$ and $f$ electron orbitals which are localized, and correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.
	To model such systems, several extensions of DFT have been developed.		To model such systems, several extensions of DFT have been developed.

Kaltakm at 07:49, 26 March 2026

2026-03-26T07:49:54Z

← Older revision		Revision as of 07:49, 26 March 2026
Line 1:		Line 1:
	[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]		[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]
	Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT. These systems typically include elements with partially filled $d$ and $f$ electron orbitals which are localized, and correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.		Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT. These systems typically include elements with partially filled $d$ or/and $f$ electron orbitals which are localized, and correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.
	To model such systems, several extensions of DFT have been developed.		To model such systems, several extensions of DFT have been developed.

Kaltakm at 07:49, 26 March 2026

2026-03-26T07:49:40Z

← Older revision		Revision as of 07:49, 26 March 2026
Line 6:		Line 6:
	[[:Category:DFT+U\|DFT+U]] is the simplest and the most computationally efficient approach to treat strong correlations within electronic structure calculations. Within this approach, standard DFT is augmented with an on-site Hubbard interaction term $U$ that explicitly penalizes fractional occupation of localized orbitals. The Hubbard interaction is typically applied to $d$ or $f$ states.		[[:Category:DFT+U\|DFT+U]] is the simplest and the most computationally efficient approach to treat strong correlations within electronic structure calculations. Within this approach, standard DFT is augmented with an on-site Hubbard interaction term $U$ that explicitly penalizes fractional occupation of localized orbitals. The Hubbard interaction is typically applied to $d$ or $f$ states.
	{{NB\|mind\|It is not currently possible to apply $U$ to both $d$ and $f$ states of the same atom}}		{{NB\|mind\|It is not currently possible to apply $U$ to both $d$ and $f$ states of the same atom}}
	The total energy within DFT+U can be written		The total energy within DFT+U can be written as
	\begin{equation}		\begin{equation}
	E=E_{DFT}+\sum_I\left[\frac{U^I}{2} \sum_{m, \sigma \neq m^{\prime}, \sigma^{\prime}} n_m^{I \sigma} n_{m^{\prime}}^{I \sigma^{\prime}}-\frac{U^I}{2} n^I\left(n^I-1\right)\right],		E=E_{DFT}+\sum_I\left[\frac{U^I}{2} \sum_{m, \sigma \neq m^{\prime}, \sigma^{\prime}} n_m^{I \sigma} n_{m^{\prime}}^{I \sigma^{\prime}}-\frac{U^I}{2} n^I\left(n^I-1\right)\right],

Hampel at 14:00, 25 March 2026

2026-03-25T14:00:18Z

← Older revision		Revision as of 14:00, 25 March 2026
Line 1:		Line 1:
	[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]		[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]
	Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT. These systems typically include elements with $d$ and $f$ ~~electrons~~ which are localized, and correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.		Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT. These systems typically include elements with partially filled $d$ and $f$ electron orbitals which are localized, and correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.
	To model such systems, several extensions of DFT have been developed.		To model such systems, several extensions of DFT have been developed.