Bethe-Salpeter equation - Revision history

Tal at 11:41, 21 February 2025

2025-02-21T11:41:51Z

← Older revision		Revision as of 11:41, 21 February 2025
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	Here, the common notation is used <math>1\to{\mathbf{r}_1,t_1}</math>. The interaction kernel contains the full self-energy <math>\Sigma</math> including the Hartree term and reads		Here, the common notation is used <math>1\to{\mathbf{r}_1,t_1}</math>. The interaction kernel contains the full self-energy <math>\Sigma</math> including the Hartree term and reads
	::<math>		::<math>
	\Xi'(2,1,3,4)=\frac{\delta[\Sigma_{\rm H}(12)+\Sigma^{\rm xc}(12)]}{\delta G(34)}=\delta(1,2)\delta(3^+,4)v(1,3)+\Xi(2,1,3,4)		\Xi'(2,1,3,4)=\frac{\delta[\Sigma_{\rm H}(12)+\Sigma^{\rm xc}(12)]}{\delta G(34)}=\delta(1,2)\delta(3^+,4)V(1,3)+\Xi(2,1,3,4)
	</math>.		</math>.
	In practical calculations the <math>\Sigma^{\rm xc}</math> is approximated via the GW self-energy		In practical calculations the <math>\Sigma^{\rm xc}</math> is approximated via the GW self-energy
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	The variation of the screened potential w.r.t. the Green's function is of order <math>W^2</math> and is usually neglected. Thus, the Bethe-Salpeter equation takes a simplified from		The variation of the screened potential w.r.t. the Green's function is of order <math>W^2</math> and is usually neglected. Thus, the Bethe-Salpeter equation takes a simplified from
	::<math>		::<math>
	L(1,3,2,4)=L_0(1,3,2,4)+\int \mathrm{d}5\mathrm{d}6\mathrm{d}7\mathrm{d}8L_0(1,3,6,5)[\delta(5,6)\delta(7,8)v(5,8)-\delta(5,7)\delta(6,8)W(5^+6)]L(7,8,2,4)		L(1,3,2,4)=L_0(1,3,2,4)+\int \mathrm{d}5\mathrm{d}6\mathrm{d}7\mathrm{d}8L_0(1,3,6,5)[\delta(5,6)\delta(7,8)V(5,8)-\delta(5,7)\delta(6,8)W(5^+6)]L(7,8,2,4)
	</math>		</math>
	This equation can be represented by an infinite sum of Feynman diagrams		This equation can be represented by an infinite sum of Feynman diagrams
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	Finally, the macroscopic dielectric function can be found by making the connection to the two-point polarizability <math>\chi(1,2)=L(1,1^+,2,2^+)</math>		Finally, the macroscopic dielectric function can be found by making the connection to the two-point polarizability <math>\chi(1,2)=L(1,1^+,2,2^+)</math>
	::<math>		::<math>
	\varepsilon^{-1}(1,2)=\delta(1,2)+\int \mathrm{d}3 v(1,3)\chi(3,2)		\varepsilon^{-1}(1,2)=\delta(1,2)+\int \mathrm{d}3 V(1,3)\chi(3,2)
	</math>		</math>

Tal at 11:40, 21 February 2025

2025-02-21T11:40:02Z

← Older revision		Revision as of 11:40, 21 February 2025
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	The Bethe-Salpeter equation (BSE) was first derived and applied in the context of particle physics and QED in 1951 {{cite\|salpeter:pr:1951}}. The first application of BSE in solids was done by Hanke and Sham {{cite\|hanke:prl:1979}}, who calculated the absorption spectrum of bulk silicon in qualitative agreement with experiment. Since then numerous works have shown successful applications of BSE for describing optical properties of materials and BSE has become the state of the art for ''ab initio'' simulation of absorption spectra.		The Bethe-Salpeter equation (BSE) was first derived and applied in the context of particle physics and QED in 1951 {{cite\|salpeter:pr:1951}}. The first application of BSE in solids was done by Hanke and Sham {{cite\|hanke:prl:1979}}, who calculated the absorption spectrum of bulk silicon in qualitative agreement with experiment. Since then numerous works have shown successful applications of BSE for describing optical properties of materials and BSE has become the state of the art for ''ab initio'' simulation of absorption spectra.
			__TOC__
	== Theory ==		== Theory ==
	The Bethe-Salpeter equation is a Dyson like equation for the four-point polarization function <math>L(1,3,2,4)</math>, which can be found by summing up an infinite series of non-interacting polarization functions <math>L_0(1,3,2,4)</math> connected via the interaction kernel <math>\Xi'(6,5,7,8)</math>		The Bethe-Salpeter equation is a Dyson like equation for the four-point polarization function <math>L(1,3,2,4)</math>, which can be found by summing up an infinite series of non-interacting polarization functions <math>L_0(1,3,2,4)</math> connected via the interaction kernel <math>\Xi'(6,5,7,8)</math>

Tal at 11:39, 21 February 2025

2025-02-21T11:39:43Z

← Older revision		Revision as of 11:39, 21 February 2025
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	== References ==		== References ==

	~~----~~
	[[Category:Many-body perturbation theory]][[Category:Theory]][[Category:Bethe-Salpeter equations]]		[[Category:Many-body perturbation theory]][[Category:Theory]][[Category:Bethe-Salpeter equations]]

Tal: /* Implementation */

2025-02-21T11:39:04Z

Implementation

← Older revision		Revision as of 11:39, 21 February 2025
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	== Implementation ==		== Implementation ==
			The poles of the response function <math>L</math> correspond to the excitation energies including the excitonic effects.
	For practical reasons, the Bethe-Salpeter equation ~~is usually reformulated~~ in the the transition space and ~~solved~~ as a non-hermitian eigenvalue problem, where the excitation energies correspond to the eigenvalues <math>\omega_\lambda</math>{{cite\|sander:prb:15}}		For practical reasons, it is more efficient to reformulate the Bethe-Salpeter equation in the the transition space and solve it as a non-hermitian eigenvalue problem, where the excitation energies correspond to the eigenvalues <math>\omega_\lambda</math>{{cite\|sander:prb:15}}

	::<math>		::<math>

Tal: /* Theory */

2025-02-21T11:35:49Z

Theory

← Older revision		Revision as of 11:35, 21 February 2025
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	== Theory ==		== Theory ==
	The Bethe-Salpeter equation is a Dyson like equation for the four-point ~~polarizability~~ <math>L(1,3,2,4)</math>		The Bethe-Salpeter equation is a Dyson like equation for the four-point polarization function <math>L(1,3,2,4)</math>, which can be found by summing up an infinite series of non-interacting polarization functions <math>L_0(1,3,2,4)</math> connected via the interaction kernel <math>\Xi'(6,5,7,8)</math>
	::<math>		::<math>
	L(1,3,2,4)=L_0(1,3,2,4)+\int \mathrm{d}5\mathrm{d}6\mathrm{d}7\mathrm{d}8 L_0(1,3,6,5)\Xi'(6,5,7,8)L(7,8,2,4).		L(1,3,2,4)=L_0(1,3,2,4)+\int \mathrm{d}5\mathrm{d}6\mathrm{d}7\mathrm{d}8 L_0(1,3,6,5)\Xi'(6,5,7,8)L(7,8,2,4).

Tal: /* Theory */

2025-02-21T11:30:34Z

Theory

← Older revision		Revision as of 11:30, 21 February 2025
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	This equation can be represented by an infinite sum of Feynman diagrams		This equation can be represented by an infinite sum of Feynman diagrams
	[[File:Bse_graph.png\|700px\|thumb\|center\|Diagrammatic representation of the Bethe-Salpeter equation for the polarizability function <math>L</math>.]]		[[File:Bse_graph.png\|700px\|thumb\|center\|Diagrammatic representation of the Bethe-Salpeter equation for the polarizability function <math>L</math>.]]
	The interaction in BSE is described by two terms, the bare Coulomb interaction <math>v</math> and the attractive screened potential <math>W</math>. ~~The screening in~~ <math>W</math> is ~~calculated in~~ the random phase approximation (RPA) ~~the same way it is included in GW~~. However, the important difference is that in BSE the static approximation is usually used, i.e., <math>W(\omega=0)</math>. The dynamical effects in the screened potential are shown to cancel out to a large extent with the dynamical effects in the polarizability <math>L_0</math>.		The interaction in BSE is described by two terms, the bare Coulomb interaction <math>v</math> and the attractive screened potential <math>W</math>.
			Since the <math>W</math> term originated from the GW self-energy, the same level of approximation is used, i.e., the random phase approximation (RPA). However, the important difference is that in BSE the static approximation is usually used, i.e., <math>W(\omega=0)</math>. The dynamical effects in the screened potential are shown to cancel out to a large extent with the dynamical effects in the polarizability <math>L_0</math>.

			Finally, the macroscopic dielectric function can be found by making the connection to the two-point polarizability <math>\chi(1,2)=L(1,1^+,2,2^+)</math>
			::<math>
			\varepsilon^{-1}(1,2)=\delta(1,2)+\int \mathrm{d}3 v(1,3)\chi(3,2)
			</math>

	== Implementation ==		== Implementation ==

Tal at 11:20, 21 February 2025

2025-02-21T11:20:55Z

← Older revision		Revision as of 11:20, 21 February 2025
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	The Bethe-Salpeter equation (BSE) was first derived and applied in the context of particle physics and QED in 1951 {{cite\|salpeter:pr:1951}}. The first application of BSE in solids was done by Hanke and Sham {{cite\|hanke:prl:1979}}, who calculated the absorption spectrum of bulk silicon in qualitative agreement with experiment. Since then numerous works have shown successful applications of BSE for describing optical properties of materials and BSE has become the state of the art for ''ab initio'' simulation of absorption spectra.		The Bethe-Salpeter equation (BSE) was first derived and applied in the context of particle physics and QED in 1951 {{cite\|salpeter:pr:1951}}. The first application of BSE in solids was done by Hanke and Sham {{cite\|hanke:prl:1979}}, who calculated the absorption spectrum of bulk silicon in qualitative agreement with experiment. Since then numerous works have shown successful applications of BSE for describing optical properties of materials and BSE has become the state of the art for ''ab initio'' simulation of absorption spectra.

	== ~~Bethe-Salpeter equation~~ ==		== Theory ==
	The Bethe-Salpeter equation is a Dyson like equation for the four-point polarizability <math>L(1,3,2,4)</math>		The Bethe-Salpeter equation is a Dyson like equation for the four-point polarizability <math>L(1,3,2,4)</math>
	::<math>		::<math>
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	[[File:Bse_graph.png\|700px\|thumb\|center\|Diagrammatic representation of the Bethe-Salpeter equation for the polarizability function <math>L</math>.]]		[[File:Bse_graph.png\|700px\|thumb\|center\|Diagrammatic representation of the Bethe-Salpeter equation for the polarizability function <math>L</math>.]]
	The interaction in BSE is described by two terms, the bare Coulomb interaction <math>v</math> and the attractive screened potential <math>W</math>. The screening in <math>W</math> is calculated in the random phase approximation (RPA) the same way it is included in GW. However, the important difference is that in BSE the static approximation is usually used, i.e., <math>W(\omega=0)</math>. The dynamical effects in the screened potential are shown to cancel out to a large extent with the dynamical effects in the polarizability <math>L_0</math>.		The interaction in BSE is described by two terms, the bare Coulomb interaction <math>v</math> and the attractive screened potential <math>W</math>. The screening in <math>W</math> is calculated in the random phase approximation (RPA) the same way it is included in GW. However, the important difference is that in BSE the static approximation is usually used, i.e., <math>W(\omega=0)</math>. The dynamical effects in the screened potential are shown to cancel out to a large extent with the dynamical effects in the polarizability <math>L_0</math>.

			== Implementation ==

	For practical reasons, the Bethe-Salpeter equation is usually reformulated in the the transition space and solved as a non-hermitian eigenvalue problem, where the excitation energies correspond to the eigenvalues <math>\omega_\lambda</math>{{cite\|sander:prb:15}}		For practical reasons, the Bethe-Salpeter equation is usually reformulated in the the transition space and solved as a non-hermitian eigenvalue problem, where the excitation energies correspond to the eigenvalues <math>\omega_\lambda</math>{{cite\|sander:prb:15}}

Tal at 11:06, 21 February 2025

2025-02-21T11:06:31Z

← Older revision		Revision as of 11:06, 21 February 2025
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	This equation can be represented by an infinite sum of Feynman diagrams		This equation can be represented by an infinite sum of Feynman diagrams
	[[File:Bse_graph.png\|700px\|thumb\|center\|Diagrammatic representation of the Bethe-Salpeter equation for the polarizability function <math>L</math>.]]		[[File:Bse_graph.png\|700px\|thumb\|center\|Diagrammatic representation of the Bethe-Salpeter equation for the polarizability function <math>L</math>.]]
			The interaction in BSE is described by two terms, the bare Coulomb interaction <math>v</math> and the attractive screened potential <math>W</math>. The screening in <math>W</math> is calculated in the random phase approximation (RPA) the same way it is included in GW. However, the important difference is that in BSE the static approximation is usually used, i.e., <math>W(\omega=0)</math>. The dynamical effects in the screened potential are shown to cancel out to a large extent with the dynamical effects in the polarizability <math>L_0</math>.
	~~Thus, the~~ interaction in BSE is described by two terms, the bare Coulomb interaction <math>v</math> and the attractive screened potential <math>W</math>. The screening in <math>W</math> is calculated in the random phase approximation (RPA) the same way it is included in GW. However, the important difference is that in BSE the static approximation is usually used, i.e., <math>W(\omega=0)</math>. The dynamical effects in the screened potential are shown to cancel out to a large extent with the dynamical effects in the polarizability <math>L_0</math>.

	For practical reasons, the Bethe-Salpeter equation is usually reformulated in the the transition space and solved as a non-hermitian eigenvalue problem, where the excitation energies correspond to the eigenvalues <math>\omega_\lambda</math>{{cite\|sander:prb:15}}		For practical reasons, the Bethe-Salpeter equation is usually reformulated in the the transition space and solved as a non-hermitian eigenvalue problem, where the excitation energies correspond to the eigenvalues <math>\omega_\lambda</math>{{cite\|sander:prb:15}}

Tal: /* Bethe-Salpeter equation */

2025-02-21T10:58:17Z

Bethe-Salpeter equation

← Older revision		Revision as of 10:58, 21 February 2025
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	== Bethe-Salpeter equation ==		== Bethe-Salpeter equation ==
	The Bethe-Salpeter equation for four-point polarizability <math>L(~~1234~~)</math>		The Bethe-Salpeter equation is a Dyson like equation for the four-point polarizability <math>L(1,3,2,4)</math>
	::<math>		::<math>
	L(~~1234~~)=L_0(~~1234~~)+\int \mathrm{d}5\mathrm{d}6\mathrm{d}7\mathrm{d}8 L_0(~~1256~~)\Xi(~~5678~~)L(~~7834~~).		L(1,3,2,4)=L_0(1,3,2,4)+\int \mathrm{d}5\mathrm{d}6\mathrm{d}7\mathrm{d}8 L_0(1,3,6,5)\Xi'(6,5,7,8)L(7,8,2,4).
	</math>		</math>
	Here, the common notation is used <math>1\to{\mathbf{r}_1,t_1}</math>. The interaction kernel ~~is defined as <math>\Xi=\delta\Sigma/\delta G</math>. In practical calculations~~ the self-energy <math>\Sigma</math> ~~is approximated via~~ the ~~GW approximation~~ and ~~after neglecting the higher order terms, the interaction kernel reduces to~~		Here, the common notation is used <math>1\to{\mathbf{r}_1,t_1}</math>. The interaction kernel contains the full self-energy <math>\Sigma</math> including the Hartree term and reads
	::<math>		::<math>
	\Xi(~~1234~~)=\delta(12)\delta(34)v(13)-\delta(14)\delta(23)W(12).		\Xi'(2,1,3,4)=\frac{\delta[\Sigma_{\rm H}(12)+\Sigma^{\rm xc}(12)]}{\delta G(34)}=\delta(1,2)\delta(3^+,4)v(1,3)+\Xi(2,1,3,4)
			</math>.
			In practical calculations the <math>\Sigma^{\rm xc}</math> is approximated via the GW self-energy
			::<math>
			\Xi(2,1,3,4)=-\frac{\delta[G(12)W(1^+2)]}{\delta G(34)}=-\delta(1,3)\delta(2,4)W(1^+2)-G(12)\frac{\delta W(1^+2)}{\delta G(34)}.
	</math>		</math>
			The variation of the screened potential w.r.t. the Green's function is of order <math>W^2</math> and is usually neglected. Thus, the Bethe-Salpeter equation takes a simplified from
			::<math>
			L(1,3,2,4)=L_0(1,3,2,4)+\int \mathrm{d}5\mathrm{d}6\mathrm{d}7\mathrm{d}8L_0(1,3,6,5)[\delta(5,6)\delta(7,8)v(5,8)-\delta(5,7)\delta(6,8)W(5^+6)]L(7,8,2,4)
			</math>
			This equation can be represented by an infinite sum of Feynman diagrams
			[[File:Bse_graph.png\|700px\|thumb\|center\|Diagrammatic representation of the Bethe-Salpeter equation for the polarizability function <math>L</math>.]]

	Thus, the interaction in BSE is described by two terms, the bare Coulomb interaction <math>v</math> and the attractive screened potential <math>W</math>. The screening in <math>W</math> is calculated in the random phase approximation (RPA) the same way it is included in GW. However, the important difference is that in BSE the static approximation is usually used, i.e., <math>W(\omega=0)</math>. The dynamical effects in the screened potential are shown to cancel out to a large extent with the dynamical effects in the polarizability <math>L_0</math>.		Thus, the interaction in BSE is described by two terms, the bare Coulomb interaction <math>v</math> and the attractive screened potential <math>W</math>. The screening in <math>W</math> is calculated in the random phase approximation (RPA) the same way it is included in GW. However, the important difference is that in BSE the static approximation is usually used, i.e., <math>W(\omega=0)</math>. The dynamical effects in the screened potential are shown to cancel out to a large extent with the dynamical effects in the polarizability <math>L_0</math>.

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	Although the dielectric function is frequency-dependent, the static approximation <math>W_{\mathbf{G}, \mathbf{G}^{\prime}}(\mathbf{q}, \omega=0)</math> is considered a standard for practical BSE calculations.		Although the dielectric function is frequency-dependent, the static approximation <math>W_{\mathbf{G}, \mathbf{G}^{\prime}}(\mathbf{q}, \omega=0)</math> is considered a standard for practical BSE calculations.

	== References ==		== References ==

	----		----
	[[Category:Many-body perturbation theory]][[Category:Theory]][[Category:Bethe-Salpeter equations]]		[[Category:Many-body perturbation theory]][[Category:Theory]][[Category:Bethe-Salpeter equations]]

Tal at 08:07, 21 February 2025

2025-02-21T08:07:22Z

← Older revision		Revision as of 08:07, 21 February 2025
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	\Xi(1234)=\delta(12)\delta(34)v(13)-\delta(14)\delta(23)W(12).		\Xi(1234)=\delta(12)\delta(34)v(13)-\delta(14)\delta(23)W(12).
	</math>		</math>
	Thus, the interaction in BSE is described by two terms, the bare Coulomb interaction <math>v</math> and the screened ~~attractive~~ potential <math>W</math>. The ~~screened potential~~ is the same ~~potential used~~ in ~~the~~ GW ~~approximation~~. However, the important difference is that in BSE the static approximation is usually used, i.e., <math>W(\omega=0)</math>. The dynamical effects in the screened potential are shown to cancel out to a large extent with the dynamical effects in the polarizability <math>L_0</math>.		Thus, the interaction in BSE is described by two terms, the bare Coulomb interaction <math>v</math> and the attractive screened potential <math>W</math>. The screening in <math>W</math> is calculated in the random phase approximation (RPA) the same way it is included in GW. However, the important difference is that in BSE the static approximation is usually used, i.e., <math>W(\omega=0)</math>. The dynamical effects in the screened potential are shown to cancel out to a large extent with the dynamical effects in the polarizability <math>L_0</math>.

	For practical reasons, the Bethe-Salpeter equation is usually reformulated in the the transition space and solved as a non-hermitian eigenvalue problem, where the excitation energies correspond to the eigenvalues <math>\omega_\lambda</math>{{cite\|sander:prb:15}}		For practical reasons, the Bethe-Salpeter equation is usually reformulated in the the transition space and solved as a non-hermitian eigenvalue problem, where the excitation energies correspond to the eigenvalues <math>\omega_\lambda</math>{{cite\|sander:prb:15}}