Matsubara formalism - Revision history

Huebsch at 06:32, 21 February 2024

2024-02-21T06:32:33Z

← Older revision		Revision as of 06:32, 21 February 2024
Line 22:		Line 22:
	Although formally convenient, the Matsubara series converges poorly with the number of considered terms in practice. VASP, therefore, uses a compressed representation of the Fourier modes by employing the Minimax-Isometry method.{{cite\|Kaltak:PRB:2020}} This approach converges exponentially with the number of considered frequency points.		Although formally convenient, the Matsubara series converges poorly with the number of considered terms in practice. VASP, therefore, uses a compressed representation of the Fourier modes by employing the Minimax-Isometry method.{{cite\|Kaltak:PRB:2020}} This approach converges exponentially with the number of considered frequency points.

	[[Category:Theory~~]][[Category:VASP6~~]][[Category:Low-scaling GW and RPA]][[Category:Many-body perturbation theory]]		[[Category:Theory]][[Category:Low-scaling GW and RPA]][[Category:Many-body perturbation theory]]

Kaltakm at 13:29, 22 January 2024

2024-01-22T13:29:08Z

← Older revision		Revision as of 13:29, 22 January 2024
Line 18:		Line 18:
	</math>		</math>

	The Matsubara formalism has the advantage that all contributions to the Green's function and the polarizability are mathematically well-defined, including contributions from states close to the chemical potential <math>\epsilon_{n{\bf k}}\approx \mu</math>, such that Matsubara series also ~~converges~~ for metallic systems.		The Matsubara formalism has the advantage that all contributions to the Green's function and the polarizability are mathematically well-defined, including contributions from states close to the chemical potential <math>\epsilon_{n{\bf k}}\approx \mu</math>, such that Matsubara series also converge for metallic systems.

	Although formally convenient, the Matsubara series converges poorly with the number of considered terms in practice. VASP, therefore, uses a compressed representation of the Fourier modes by employing the Minimax-Isometry method.{{cite\|Kaltak:PRB:2020}} This approach converges exponentially with the number of considered frequency points.		Although formally convenient, the Matsubara series converges poorly with the number of considered terms in practice. VASP, therefore, uses a compressed representation of the Fourier modes by employing the Minimax-Isometry method.{{cite\|Kaltak:PRB:2020}} This approach converges exponentially with the number of considered frequency points.

	[[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Many-body perturbation theory]]		[[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Many-body perturbation theory]]

Huebsch at 06:45, 29 September 2023

2023-09-29T06:45:26Z

← Older revision		Revision as of 06:45, 29 September 2023
Line 20:		Line 20:
	The Matsubara formalism has the advantage that all contributions to the Green's function and the polarizability are mathematically well-defined, including contributions from states close to the chemical potential <math>\epsilon_{n{\bf k}}\approx \mu</math>, such that Matsubara series also converges for metallic systems.		The Matsubara formalism has the advantage that all contributions to the Green's function and the polarizability are mathematically well-defined, including contributions from states close to the chemical potential <math>\epsilon_{n{\bf k}}\approx \mu</math>, such that Matsubara series also converges for metallic systems.

	Although formally convenient, the Matsubara series converges poorly with the number of considered terms in practice. ~~The~~ VASP ~~code~~, therefore, uses a compressed representation of the Fourier modes by employing the Minimax-Isometry method.{{cite\|Kaltak:PRB:2020}} This approach converges exponentially with the number of considered frequency points.		Although formally convenient, the Matsubara series converges poorly with the number of considered terms in practice. VASP, therefore, uses a compressed representation of the Fourier modes by employing the Minimax-Isometry method.{{cite\|Kaltak:PRB:2020}} This approach converges exponentially with the number of considered frequency points.

	[[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Many-body perturbation theory]]		[[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Many-body perturbation theory]]

Huebsch: Huebsch moved page Matsubara Formalism to Matsubara formalism

2023-09-29T06:44:38Z

Huebsch moved page Matsubara Formalism to Matsubara formalism

← Older revision	Revision as of 06:44, 29 September 2023
(No difference)

Huebsch at 06:44, 29 September 2023

2023-09-29T06:44:11Z

← Older revision		Revision as of 06:44, 29 September 2023
Line 1:		Line 1:
	The zero-temperature formalism of many-body perturbation theory breaks down for metals (systems with zero energy band-gap) as pointed out by Kohn and Luttinger.{{cite\|KohnLuttinger:PR:1960}} This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature <math>T>0</math>, which may be understood by an analytical continuation of the real-time <math>t</math> to the imaginary time axis <math>-i\tau</math>. Matsubara has shown that this Wick-rotation in time <math>t\to-i\tau</math> reveals an intriguing connection to the inverse temperature <math>\beta=1/T</math> of the system.{{cite\|Matsubara:PTP:1955}}		The zero-temperature formalism of many-body perturbation theory breaks down for metals (systems with zero energy band-gap) as pointed out by Kohn and Luttinger.{{cite\|KohnLuttinger:PR:1960}} This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature <math>T>0</math>, which may be understood by an analytical continuation of the real-time <math>t</math> to the imaginary time axis <math>-i\tau</math>. Matsubara has shown that this Wick rotation in time <math>t\to-i\tau</math> reveals an intriguing connection to the inverse temperature <math>\beta=1/T</math> of the system.{{cite\|Matsubara:PTP:1955}}
	More precisely, Matsubara has shown that all terms in perturbation theory at finite temperature can be expressed as integrals of imaginary time quantities (such as the polarizability <math>\chi(-i\tau)</math>) over the fundamental interval <math>-\beta\le\tau\le\beta</math>.		More precisely, Matsubara has shown that all terms in perturbation theory at finite temperature can be expressed as integrals of imaginary time quantities (such as the polarizability <math>\chi(-i\tau)</math>) over the fundamental interval <math>-\beta\le\tau\le\beta</math>.

	As a consequence, one decomposes imaginary time quantities into a Fourier series with period <math>\beta</math>		As a consequence, one decomposes imaginary time quantities into a Fourier series with period <math>\beta</math>
	that determines the spacing of the ~~Fouier~~ modes. For instance the imaginary polarizability can be written as		that determines the spacing of the Fourier modes. For instance the imaginary polarizability can be written as

	<math>		<math>
Line 9:		Line 9:
	</math>		</math>

	and the corresponding random phase approximation of the correlation energy at finite temperature becomes a series over (in this case bosonic) Matsubara frequencies		and the corresponding random-phase approximation of the correlation energy at finite temperature becomes a series over (in this case, bosonic) Matsubara frequencies

	<math>		<math>
Line 18:		Line 18:
	</math>		</math>

	The Matsubara formalism has the advantage that all contributions to the Green's function and the polarizability are mathematically well-defined, including contributions from states close to the chemical potential <math>\epsilon_{n{\bf k}}\approx \mu</math>, such that Matsubara series ~~converge~~ also for metallic systems.		The Matsubara formalism has the advantage that all contributions to the Green's function and the polarizability are mathematically well-defined, including contributions from states close to the chemical potential <math>\epsilon_{n{\bf k}}\approx \mu</math>, such that Matsubara series also converges for metallic systems.

	Although formally convenient, Matsubara series ~~converge~~ poorly with the number of considered terms in practice. The VASP code, therefore, uses a compressed representation of the Fourier modes by employing the Minimax Isometry method.{{cite\|Kaltak:PRB:2020}} This approach converges exponentially with the number of considered frequency points.		Although formally convenient, the Matsubara series converges poorly with the number of considered terms in practice. The VASP code, therefore, uses a compressed representation of the Fourier modes by employing the Minimax-Isometry method.{{cite\|Kaltak:PRB:2020}} This approach converges exponentially with the number of considered frequency points.

	[[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Many-body perturbation theory]]		[[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Many-body perturbation theory]]

Huebsch at 10:12, 19 July 2022

2022-07-19T10:12:54Z

← Older revision		Revision as of 10:12, 19 July 2022
Line 22:		Line 22:
	Although formally convenient, Matsubara series converge poorly with the number of considered terms in practice. The VASP code, therefore, uses a compressed representation of the Fourier modes by employing the Minimax Isometry method.{{cite\|Kaltak:PRB:2020}} This approach converges exponentially with the number of considered frequency points.		Although formally convenient, Matsubara series converge poorly with the number of considered terms in practice. The VASP code, therefore, uses a compressed representation of the Fourier modes by employing the Minimax Isometry method.{{cite\|Kaltak:PRB:2020}} This approach converges exponentially with the number of considered frequency points.

	[[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]] [[Category:Many-~~Body Perturbation Theory~~]]		[[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Many-body perturbation theory]]

Kaltakm at 14:33, 16 June 2020

2020-06-16T14:33:57Z

← Older revision		Revision as of 14:33, 16 June 2020
Line 20:		Line 20:
	The Matsubara formalism has the advantage that all contributions to the Green's function and the polarizability are mathematically well-defined, including contributions from states close to the chemical potential <math>\epsilon_{n{\bf k}}\approx \mu</math>, such that Matsubara series converge also for metallic systems.		The Matsubara formalism has the advantage that all contributions to the Green's function and the polarizability are mathematically well-defined, including contributions from states close to the chemical potential <math>\epsilon_{n{\bf k}}\approx \mu</math>, such that Matsubara series converge also for metallic systems.

	Although formally convenient, Matsubara series converge poorly with the number of considered terms in practice. The code, therefore, uses a compressed representation of the Fourier modes by employing the Minimax Isometry method.{{cite\|Kaltak:PRB:2020}} This approach converges exponentially with the number of considered frequency points.		Although formally convenient, Matsubara series converge poorly with the number of considered terms in practice. The VASP code, therefore, uses a compressed representation of the Fourier modes by employing the Minimax Isometry method.{{cite\|Kaltak:PRB:2020}} This approach converges exponentially with the number of considered frequency points.

	[[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]] [[Category:Many-Body Perturbation Theory]]		[[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]] [[Category:Many-Body Perturbation Theory]]

Kaltakm at 14:32, 16 June 2020

2020-06-16T14:32:37Z

← Older revision		Revision as of 14:32, 16 June 2020
Line 1:		Line 1:
	The zero-temperature formalism of many-body perturbation theory breaks down for metals (systems with zero energy band-gap) as pointed out by Kohn and Luttinger.{{cite\|KohnLuttinger:PR:1960}} This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature <math>T>0</math>, which may be understood by an analytical continuation of the real-time <math>t</math> to the imaginary time axis <math>-i\tau</math>. Matsubara has shown that this Wick-rotation in time <math>t\to-i\tau</math> reveals an intriguing connection to the inverse temperature <math>\beta=1/T</math> of the system ~~in the grand-canonical ensemble~~.{{cite\|Matsubara:PTP:1955}}		The zero-temperature formalism of many-body perturbation theory breaks down for metals (systems with zero energy band-gap) as pointed out by Kohn and Luttinger.{{cite\|KohnLuttinger:PR:1960}} This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature <math>T>0</math>, which may be understood by an analytical continuation of the real-time <math>t</math> to the imaginary time axis <math>-i\tau</math>. Matsubara has shown that this Wick-rotation in time <math>t\to-i\tau</math> reveals an intriguing connection to the inverse temperature <math>\beta=1/T</math> of the system.{{cite\|Matsubara:PTP:1955}}
	More precisely, Matsubara has shown that all terms in perturbation theory at finite temperature can be expressed as integrals of imaginary time quantities (such as the polarizability <math>\chi(-i\tau)</math>) over the fundamental interval <math>-\beta\le\tau\le\beta</math>.		More precisely, Matsubara has shown that all terms in perturbation theory at finite temperature can be expressed as integrals of imaginary time quantities (such as the polarizability <math>\chi(-i\tau)</math>) over the fundamental interval <math>-\beta\le\tau\le\beta</math>.

Kaltakm at 14:31, 16 June 2020

2020-06-16T14:31:51Z

← Older revision		Revision as of 14:31, 16 June 2020
Line 1:		Line 1:
	The zero-temperature formalism of many-body perturbation theory breaks down for metals (systems with zero energy band-gap) as pointed out by Kohn and Luttinger.{{cite\|KohnLuttinger:PR:1960}} This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature <math>T>0</math>, which ~~is best~~ understood by an analytical continuation of the real-time <math>t</math> to the imaginary time axis <math>-i\tau</math>. Matsubara has shown that this Wick-rotation in time <math>t\to-i\tau</math> reveals an intriguing connection to the inverse temperature <math>\beta=1/T</math> of the system in the grand-canonical ensemble.{{cite\|Matsubara:PTP:1955}}		The zero-temperature formalism of many-body perturbation theory breaks down for metals (systems with zero energy band-gap) as pointed out by Kohn and Luttinger.{{cite\|KohnLuttinger:PR:1960}} This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature <math>T>0</math>, which may be understood by an analytical continuation of the real-time <math>t</math> to the imaginary time axis <math>-i\tau</math>. Matsubara has shown that this Wick-rotation in time <math>t\to-i\tau</math> reveals an intriguing connection to the inverse temperature <math>\beta=1/T</math> of the system in the grand-canonical ensemble.{{cite\|Matsubara:PTP:1955}}
	More precisely, Matsubara has shown that all terms in perturbation theory at finite temperature can be expressed as integrals of imaginary time quantities (such as the polarizability <math>\chi(-i\tau)</math>) over the fundamental interval <math>-\beta\le\tau\le\beta</math>.		More precisely, Matsubara has shown that all terms in perturbation theory at finite temperature can be expressed as integrals of imaginary time quantities (such as the polarizability <math>\chi(-i\tau)</math>) over the fundamental interval <math>-\beta\le\tau\le\beta</math>.

Kaltakm at 14:31, 16 June 2020

2020-06-16T14:31:18Z

← Older revision		Revision as of 14:31, 16 June 2020
Line 1:		Line 1:
	The zero-temperature formalism of many-body perturbation theory breaks down for metals (~~system~~ with zero band gap) as pointed out by Kohn and Luttinger.{{cite\|KohnLuttinger:PR:1960}} This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature <math>T>0</math>, which is best understood by an analytical continuation of the real-time <math>t</math> to the imaginary time axis <math>-i\tau</math>. Matsubara has shown that this Wick-rotation in time <math>t\to-i\tau</math> reveals an intriguing connection to the inverse temperature <math>\beta=1/T</math> of the system in the grand-canonical ensemble.{{cite\|Matsubara:PTP:1955}}		The zero-temperature formalism of many-body perturbation theory breaks down for metals (systems with zero energy band-gap) as pointed out by Kohn and Luttinger.{{cite\|KohnLuttinger:PR:1960}} This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature <math>T>0</math>, which is best understood by an analytical continuation of the real-time <math>t</math> to the imaginary time axis <math>-i\tau</math>. Matsubara has shown that this Wick-rotation in time <math>t\to-i\tau</math> reveals an intriguing connection to the inverse temperature <math>\beta=1/T</math> of the system in the grand-canonical ensemble.{{cite\|Matsubara:PTP:1955}}
	More precisely, Matsubara has shown that all terms in perturbation theory at finite temperature can be expressed as integrals of imaginary time quantities (such as the polarizability <math>\chi(-i\tau)</math>) over the fundamental interval <math>-\beta\le\tau\le\beta</math>.		More precisely, Matsubara has shown that all terms in perturbation theory at finite temperature can be expressed as integrals of imaginary time quantities (such as the polarizability <math>\chi(-i\tau)</math>) over the fundamental interval <math>-\beta\le\tau\le\beta</math>.