The magneto-optical Kerr effect - Revision history

Davide: Warning moved into a note

2025-02-11T10:35:28Z

Warning moved into a note

← Older revision		Revision as of 10:35, 11 February 2025
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	The MOKE consist in a different absorpion of the right (+) and left (-) circularly polarized light expirienced in magnetic materials.		The MOKE consist in a different absorpion of the right (+) and left (-) circularly polarized light expirienced in magnetic materials.
	The dielectric function ε±(ω) can be constructed from the dielectric tensor<ref name="Sangalli2012"/>: <br>		The dielectric function ε±(ω) can be constructed from the dielectric tensor<ref name="Sangalli2012"/> (<span style="color:red">Please also check this warning footnote<ref name="Warning-Note" /></span>): <br>
	[[File:Epsilon left right.png\|x40px\|frameless\|alt=Epsilon left right]]		[[File:Epsilon left right.png\|x40px\|frameless\|alt=Epsilon left right]]

	<span style="color:red">Warning, the above equation, which is also reported in many text-books, holds for the case in which <math>\epsilon_{xy}(\omega)=-\epsilon_{yx}(\omega)</math> and <math>\epsilon_{xx}(\omega)=\epsilon_{yy}(\omega)</math>. Otherwise it needs to be generalized. </span>

	Here "z" is the direction of the magnetization and of the propagation of the light, while the polarization lays in the "xy"-plane. The absorption is often expressed in terms of the optical conductivity which is related to the dielectric tensor trough the equation <br>		Here "z" is the direction of the magnetization and of the propagation of the light, while the polarization lays in the "xy"-plane. The absorption is often expressed in terms of the optical conductivity which is related to the dielectric tensor trough the equation <br>
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	<ref name="Delin1999">A. Delin, O. Eriksson, B. Johanssonand, S. Auluck, and J. Wills, PRB 60, 14105 (1999)</ref>		<ref name="Delin1999">A. Delin, O. Eriksson, B. Johanssonand, S. Auluck, and J. Wills, PRB 60, 14105 (1999)</ref>
	<ref name="Sangalli2012">D. Sangalli, A. Marini and A. Debernardi, ''Pseudopotential-based first-principles approach to the magneto-optical Kerr effect: From metals to the inclusion of local fields and excitonic effects'' [https://doi.org/10.1103/PhysRevB.86.125139 PRB '''86''', 125139 (2012)]</ref>		<ref name="Sangalli2012">D. Sangalli, A. Marini and A. Debernardi, ''Pseudopotential-based first-principles approach to the magneto-optical Kerr effect: From metals to the inclusion of local fields and excitonic effects'' [https://doi.org/10.1103/PhysRevB.86.125139 PRB '''86''', 125139 (2012)]</ref>
			<ref name="Warning-Note">The relation between Kerr angle and components of the dielectric tensor, which is also reported in many text-books, holds for the case in which <math>\epsilon_{xy}(\omega)=-\epsilon_{yx}(\omega)</math> and <math>\epsilon_{xx}(\omega)=\epsilon_{yy}(\omega)</math>. Otherwise it needs to be generalized. These conditions are fulfilled in bulk Iron</ref>
	<ref name="MolinaSanchez2020">A. Molina-Sánchez, G. Catarina, D. Sangalli, J. Fernandez-Rossier, ''Magneto-optical response of chromium trihalide monolayers: chemical trends'', [https://doi.org/10.1039/D0TC01322F J. of Mat. Chem. C '''8''', 8856 (2020)]</ref>		<ref name="MolinaSanchez2020">A. Molina-Sánchez, G. Catarina, D. Sangalli, J. Fernandez-Rossier, ''Magneto-optical response of chromium trihalide monolayers: chemical trends'', [https://doi.org/10.1039/D0TC01322F J. of Mat. Chem. C '''8''', 8856 (2020)]</ref>
	</references>		</references>

Davide: /* Few equations */ Added warning on kerr effect

2025-02-11T10:29:11Z

Few equations: Added warning on kerr effect

← Older revision		Revision as of 10:29, 11 February 2025
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	The dielectric function ε±(ω) can be constructed from the dielectric tensor<ref name="Sangalli2012"/>: <br>		The dielectric function ε±(ω) can be constructed from the dielectric tensor<ref name="Sangalli2012"/>: <br>
	[[File:Epsilon left right.png\|x40px\|frameless\|alt=Epsilon left right]]		[[File:Epsilon left right.png\|x40px\|frameless\|alt=Epsilon left right]]

			<span style="color:red">Warning, the above equation, which is also reported in many text-books, holds for the case in which <math>\epsilon_{xy}(\omega)=-\epsilon_{yx}(\omega)</math> and <math>\epsilon_{xx}(\omega)=\epsilon_{yy}(\omega)</math>. Otherwise it needs to be generalized. </span>

	Here "z" is the direction of the magnetization and of the propagation of the light, while the polarization lays in the "xy"-plane. The absorption is often expressed in terms of the optical conductivity which is related to the dielectric tensor trough the equation <br>		Here "z" is the direction of the magnetization and of the propagation of the light, while the polarization lays in the "xy"-plane. The absorption is often expressed in terms of the optical conductivity which is related to the dielectric tensor trough the equation <br>

Davide: /* Optical Properties of Bulk Iron */ fixed link

2023-11-20T09:44:30Z

Optical Properties of Bulk Iron: fixed link

← Older revision		Revision as of 09:44, 20 November 2023
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	[[File:Absorption_Iron_tutorial.png\|600px\|frameless\|center\|alt=Absorption Iron tutorial]]		[[File:Absorption_Iron_tutorial.png\|600px\|frameless\|center\|alt=Absorption Iron tutorial]]

	Compare the results of the three runs. The absence of the [x,Vnl] gives a significant change in the intensity of the absorption. The length and the velocity gauge formulation instead should be perfectly equivalent. They are if you are using a recent version of the code. The plot was, instead, generated with an earlier implementation, and the two results differ slightly. See		Compare the results of the three runs. The absence of the [x,Vnl] gives a significant change in the intensity of the absorption. The length and the velocity gauge formulation instead should be perfectly equivalent. They are if you are using a recent version of the code. The plot was, instead, generated with an earlier implementation, and the two results differ slightly. See [[The_magneto-optical_Kerr_effect#Length_and_velocity_gauge]]

	== The full dielectric tensor and the Kerr effect ==		== The full dielectric tensor and the Kerr effect ==

Davide: /* Length and velocity gauge */

2023-11-20T09:43:31Z

Length and velocity gauge

← Older revision		Revision as of 09:43, 20 November 2023
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	The length and velocity gauge dielectric function are defined via		The length and velocity gauge dielectric function are defined via
	<math>\epsilon(\omega)=1-4\pi/q^2 \chi_{\rho\rho}(\omega)</math>.
			<math>\epsilon(\omega)=1-4\pi/q^2 \chi_{\rho\rho}(\omega)</math>

	and		and
	<math>\epsilon(\omega)=1-4\pi/\omega^2 \chi_{jj}(\omega)</math>.
			<math>\epsilon(\omega)=1-4\pi/\omega^2 \chi_{jj}(\omega)</math>

	The two should in principle give identical results. However, in the earlier implementations, the same complex <math>\omega+i\eta</math> expression was used both in the response function and in the pre-factor appearing in the velocity gauge, leading to small differences in the final results.		The two should in principle give identical results. However, in the earlier implementations, the same complex <math>\omega+i\eta</math> expression was used both in the response function and in the pre-factor appearing in the velocity gauge, leading to small differences in the final results.
	This has been fixed in more recent versions of the code		This has been fixed in more recent versions of the code

Davide: Added comment on length vs velocity gauge

2023-11-20T09:42:55Z

Added comment on length vs velocity gauge

← Older revision		Revision as of 09:42, 20 November 2023
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	By default yambo computes the absorption of linearly polarized light. With specific keywords in the input file, however, yambo is also able to compute the absorption of circularly polarized light, through the computation of the off-diagonal components of the dielectric tensor. In this tutorial we will learn how to take advantage of it for the description of the magneto-optical Kerr effect (MOKE) in the polar geometry<ref name="wikipedia"/>.		By default yambo computes the absorption of linearly polarized light. With specific keywords in the input file, however, yambo is also able to compute the absorption of circularly polarized light, through the computation of the off-diagonal components of the dielectric tensor. In this tutorial we will learn how to take advantage of it for the description of the magneto-optical Kerr effect (MOKE) in the polar geometry<ref name="wikipedia"/>.

	~~''' WARNING: This tutorial was created with an older version of the code, where length and velocity guage where giving slightly different results. In more recent versions this has been fixed '''~~

	== Prerequisites ==		== Prerequisites ==
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	We also suggest to have a look to these references for a detailed discussion of the different components of the dielectric tensor, as well as for a discussion about the length and the velocity gauges <ref name="DelSole1984"/>, <ref name="Strinati1986"/>, <ref name="Sipe1993"/>, <ref name="Delin1999"/>. In this tutorial you will compute the kerr effect only at the independent particle version. However in the released version of the code it is also possible to compute the Kerr effect at the BSE level <ref name="MolinaSanchez2020"/>.		We also suggest to have a look to these references for a detailed discussion of the different components of the dielectric tensor, as well as for a discussion about the length and the velocity gauges <ref name="DelSole1984"/>, <ref name="Strinati1986"/>, <ref name="Sipe1993"/>, <ref name="Delin1999"/>. In this tutorial you will compute the kerr effect only at the independent particle version. However in the released version of the code it is also possible to compute the Kerr effect at the BSE level <ref name="MolinaSanchez2020"/>.

			===Length and velocity gauge===

			The length and velocity gauge dielectric function are defined via
			<math>\epsilon(\omega)=1-4\pi/q^2 \chi_{\rho\rho}(\omega)</math>.
			and
			<math>\epsilon(\omega)=1-4\pi/\omega^2 \chi_{jj}(\omega)</math>.
			The two should in principle give identical results. However, in the earlier implementations, the same complex <math>\omega+i\eta</math> expression was used both in the response function and in the pre-factor appearing in the velocity gauge, leading to small differences in the final results.
			This has been fixed in more recent versions of the code

	== Optical Properties of Bulk Iron ==		== Optical Properties of Bulk Iron ==
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	[[File:Absorption_Iron_tutorial.png\|600px\|frameless\|center\|alt=Absorption Iron tutorial]]		[[File:Absorption_Iron_tutorial.png\|600px\|frameless\|center\|alt=Absorption Iron tutorial]]

	Compare the results of the three runs. The absence of the [x,Vnl] gives a significant change in the intensity of the absorption. The length and the velocity gauge formulation instead should be perfectly equivalent. They are if you are using a recent version of the code. ~~In the~~ earlier implementation ~~instead they~~ differ slightly. ~~The velocity gauge dielectric function is defined via~~		Compare the results of the three runs. The absence of the [x,Vnl] gives a significant change in the intensity of the absorption. The length and the velocity gauge formulation instead should be perfectly equivalent. They are if you are using a recent version of the code. The plot was, instead, generated with an earlier implementation, and the two results differ slightly. See
	~~<math>\epsilon(\omega)=1-4\pi/\omega^2 \chi_{jj}(\omega)</math>.~~
	~~In the earlier implementations the same complex <math>\omega+i\eta</math> was used both in the response function and in the pre-factor, leading to these small differences.~~

	== The full dielectric tensor and the Kerr effect ==		== The full dielectric tensor and the Kerr effect ==

Davide at 09:38, 20 November 2023

2023-11-20T09:38:52Z

← Older revision		Revision as of 09:38, 20 November 2023
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	[[File:MO KERR effect yambo.jpg\|x300px\|frameless\|right\|The Magneto Optical Kerr effect in bulk Iron]]		[[File:MO KERR effect yambo.jpg\|x300px\|frameless\|right\|The Magneto Optical Kerr effect in bulk Iron]]
	By default yambo computes the absorption of linearly polarized light. With specific keywords in the input file, however, yambo is also able to compute the absorption of circularly polarized light, through the computation of the off-diagonal components of the dielectric tensor. In this tutorial we will learn how to take advantage of it for the description of the magneto-optical Kerr effect (MOKE) in the polar geometry<ref name="wikipedia"/>.		By default yambo computes the absorption of linearly polarized light. With specific keywords in the input file, however, yambo is also able to compute the absorption of circularly polarized light, through the computation of the off-diagonal components of the dielectric tensor. In this tutorial we will learn how to take advantage of it for the description of the magneto-optical Kerr effect (MOKE) in the polar geometry<ref name="wikipedia"/>.

			''' WARNING: This tutorial was created with an older version of the code, where length and velocity guage where giving slightly different results. In more recent versions this has been fixed '''

	== Prerequisites ==		== Prerequisites ==

Davide: /* Databases */

2023-11-19T22:01:17Z

Databases

← Older revision		Revision as of 22:01, 19 November 2023
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	For this tutorials we provide two <code>yambo</code> databases (DBs) for bulk iron. The first DB is the results of a Spin-DFT calculation while the second of a spinorial DFT calculation with the SO effect included both in the Hamiltonian and in the pseudo-potential. Both DBs are generated with the HGH pseudo-potentials.		For this tutorials we provide two <code>yambo</code> databases (DBs) for bulk iron. The first DB is the results of a Spin-DFT calculation while the second of a spinorial DFT calculation with the SO effect included both in the Hamiltonian and in the pseudo-potential. Both DBs are generated with the HGH pseudo-potentials.

	* ~~GROUND STATE~~ files (optional): [http://media.yambo-code.eu/educational/tutorials/files/Iron_abinit_gs_files.tar.gz Iron_abinit_gs_files.tar.gz]		* Ground State files (optional): [http://media.yambo-code.eu/educational/tutorials/files/Iron_abinit_gs_files.tar.gz Iron_abinit_gs_files.tar.gz]
	* ~~REFERENCE~~ files : [http://media.yambo-code.eu/educational/tutorials/files/Iron_Inputs.tar.gz Iron_Inputs.tar.gz]		* Input and Reference files: [http://media.yambo-code.eu/educational/tutorials/files/Iron_Inputs.tar.gz Iron_Inputs.tar.gz]
	* ~~YAMBO CORE DATABASES and REFERENCE files (suggested)~~ (48 Mb): [http://media.yambo-code.eu/educational/tutorials/files/Iron_DBs.tar.gz Iron_DBs.tar.gz]		* Yambo core databases (48 Mb): [http://media.yambo-code.eu/educational/tutorials/files/Iron_DBs.tar.gz Iron_DBs.tar.gz]

	Alternatively you can re-generate the DBs by your-self (also because the DB provided have been generated with small convergence parameters). To this end you just need to download and compile abinit, and generate the DBs using the input files and the pseudo-potentials provided. If you prefer, you can also try to generate the same DBs with the PWscf package (however you need to find a proper pseudo-potential).		Alternatively you can re-generate the DBs by your-self (also because the DB provided have been generated with small convergence parameters). To this end you just need to download and compile abinit, and generate the DBs using the input files and the pseudo-potentials provided. If you prefer, you can also try to generate the same DBs with the PWscf package (however you need to find a proper pseudo-potential).

Davide: Few small fixes

2023-11-19T22:00:25Z

Few small fixes

Davide: /* Optical Properties of Bulk Iron */

2023-11-19T21:50:59Z

Optical Properties of Bulk Iron

← Older revision		Revision as of 21:50, 19 November 2023
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	[[File:Absorption_Iron_tutorial.png\|600px\|frameless\|center\|alt=Absorption Iron tutorial]]		[[File:Absorption_Iron_tutorial.png\|600px\|frameless\|center\|alt=Absorption Iron tutorial]]

	Compare the results of the three runs. The absence of the [x,Vnl] gives a significant change in the intensity of the absorption. The length and the velocity gauge formulation instead~~, they~~ should be perfectly equivalent. ~~However we see small differences between~~ the ~~two plots~~. ~~Try to guess why~~.		Compare the results of the three runs. The absence of the [x,Vnl] gives a significant change in the intensity of the absorption. The length and the velocity gauge formulation instead should be perfectly equivalent. They are if you are using a recent version of the code. In the earlier implementation instead they differ slightly. The velocity gauge dielectric function is defined via
	~~[Hint: the two gauges are perfectly equivalent if we use the relation 1 /~~ (~~ω+iη~~) = 1/~~ω - 2πi δ~~(ω) ]		<math>\epsilon(\omega)=1-4\pi/\omega^2 \chi_{jj}(\omega)</math>.
			In the earlier implementations the same complex <math>\omega+i\eta</math> was used both in the response function and in the pre-factor, leading to these small differences.

	== The full dielectric tensor and the Kerr effect ==		== The full dielectric tensor and the Kerr effect ==

Davide: /* Databases */

2023-11-19T21:42:16Z

Databases

← Older revision		Revision as of 21:42, 19 November 2023
Line 9:		Line 9:
	For this tutorials we provide two <code>yambo</code> databases (DBs) for bulk iron. The first DB is the results of a Spin-DFT calculation while the second of a spinorial DFT calculation with the SO effect included both in the Hamiltonian and in the pseudo-potential. Both DBs are generated with the HGH pseudo-potentials.		For this tutorials we provide two <code>yambo</code> databases (DBs) for bulk iron. The first DB is the results of a Spin-DFT calculation while the second of a spinorial DFT calculation with the SO effect included both in the Hamiltonian and in the pseudo-potential. Both DBs are generated with the HGH pseudo-potentials.

	~~'''WARNING: Links are broken, to be fixed'''~~
	* GROUND STATE files (optional): [http://media.yambo-code.eu/educational/tutorials/files/Iron_abinit_gs_files.tar.gz Iron_abinit_gs_files.tar.gz]		* GROUND STATE files (optional): [http://media.yambo-code.eu/educational/tutorials/files/Iron_abinit_gs_files.tar.gz Iron_abinit_gs_files.tar.gz]
	* REFERENCE files : [http://media.yambo-code.eu/educational/tutorials/files/~~Solid_Fe_reference_files~~.~~zip Solid_Fe_reference_files~~.~~zip~~]		* REFERENCE files : [http://media.yambo-code.eu/educational/tutorials/files/Iron_Inputs.tar.gz Iron_Inputs.tar.gz]
	* YAMBO CORE DATABASES and REFERENCE files (suggested) (48 Mb): [http://media.yambo-code.eu/educational/tutorials/files/Iron_DBs.tar.gz Iron_DBs.tar.gz]		* YAMBO CORE DATABASES and REFERENCE files (suggested) (48 Mb): [http://media.yambo-code.eu/educational/tutorials/files/Iron_DBs.tar.gz Iron_DBs.tar.gz]



	Alternatively you can re-generate the DBs by your-self (also because the DB provided have been generated with small convergence parameters). To this end you just need to download and compile abinit, and generate the DBs using the input files and the pseudo-potentials provided. If you prefer, you can also try to generate the same DBs with the PWscf package (however you need to find a proper pseudo-potential).		Alternatively you can re-generate the DBs by your-self (also because the DB provided have been generated with small convergence parameters). To this end you just need to download and compile abinit, and generate the DBs using the input files and the pseudo-potentials provided. If you prefer, you can also try to generate the same DBs with the PWscf package (however you need to find a proper pseudo-potential).

← Older revision		Revision as of 22:00, 19 November 2023
Line 1:		Line 1:
	~~'''WARNIGN: This page has been recently recoverd from the old version of the yambo website. If you try the tutorial and find any issue, please contact us in the yambo forum'''~~


	[[File:MO KERR effect yambo.jpg\|x300px\|frameless\|right\|The Magneto Optical Kerr effect in bulk Iron]]		[[File:MO KERR effect yambo.jpg\|x300px\|frameless\|right\|The Magneto Optical Kerr effect in bulk Iron]]
	By default yambo computes the absorption of linearly polarized light. With specific keywords in the input file, however, yambo is also able to compute the absorption of circularly polarized light, through the computation of the off-diagonal components of the dielectric tensor. In this tutorial we will learn how to take advantage of it for the description of the magneto-optical Kerr effect (MOKE) in the polar geometry<ref name="wikipedia"/>.		By default yambo computes the absorption of linearly polarized light. With specific keywords in the input file, however, yambo is also able to compute the absorption of circularly polarized light, through the computation of the off-diagonal components of the dielectric tensor. In this tutorial we will learn how to take advantage of it for the description of the magneto-optical Kerr effect (MOKE) in the polar geometry<ref name="wikipedia"/>.
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	yambo -i -F Input/01_init		yambo -i -F Input/01_init
	for the initialization and		for the initialization and
	yambo -opsics bse -V resp -F Input/"0n_filename".		yambo -opsics bse -V resp -F Input/"0n_filename"
	Among the varaibles activated with the -V resp option we will use these options which are specific for metals:		Among the varaibles activated with the -V resp option we will use these options which are specific for metals:
	Gauge= "length" # [BSE] Gauge (length\|velocity). In metals length misses An-Hall		Gauge= "length" # [BSE] Gauge (length\|velocity). In metals length misses An-Hall
	DrudeWBS= (4.80, 5.65) eV # [BSE] Drude plasmon ;		DrudeWBS= (4.80, 5.65) eV # [BSE] Drude plasmon
	For the computation of the kerr parameters yambo_kerr will just need two special input varaibles		For the computation of the kerr parameters yambo_kerr will just need two special input varaibles
	BSEprop="abs kerr" # [BSE] Can be any among		BSEprop="abs kerr" # [BSE] Can be any among
	AnHall # [BSE] Compute the anomalous Hall effect and if length add it to eps ,		AnHall # [BSE] Compute the anomalous Hall effect and if length add it to eps
	which are activated by default using the <code>yambo</code> executable. Other variables not used in the present tutorial have been erased to simplify the input files.		which are activated by default using the <code>yambo</code> executable. Other variables not used in the present tutorial have been erased to simplify the input files.

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	yambo -F Inputs/02_IP-RPA_len -J 02_IP-RPA_len ;		yambo -F Inputs/02_IP-RPA_len -J 02_IP-RPA_len ;
	<code>yambo</code> will compute the absorption in the length gauge (this is the standard gauge used by yambo) including the term [x,Vnl] by default. To check the importance of this term first hide the <code>ns.kb_pp</code> DB,		<code>yambo</code> will compute the absorption in the length gauge (this is the standard gauge used by yambo) including the term [x,Vnl] by default. To check the importance of this term first hide the <code>ns.kb_pp</code> DB,
	mv SAVE/ns.kb_pp SAVE/hide_ns.kb_pp ,		mv SAVE/ns.kb_pp SAVE/hide_ns.kb_pp

	and then run the code again		and then run the code again
	yambo -F Inputs/02_IP-RPA_len -J 02_IP-RPA_len_noVnl .		yambo -F Inputs/02_IP-RPA_len -J 02_IP-RPA_len_noVnl .

	[Hint: the most time consuming part of this calulation is the computation of the dipoles. You can avoid yambo_kerr to re-compute each time the dipoles preparing the database in the correct directory. In this case just do, before running yambo, :		[Hint: the most time consuming part of this calulation is the computation of the dipoles. You can avoid yambo_kerr to re-compute each time the dipoles preparing the database in the correct directory. In this case just do, before running yambo:
	mkdir 02_IP-RPA_len_noVnl \| cp 02_IP-RPA_len/ndb.dipoles 02_IP-RPA_len_noVnl].		mkdir 02_IP-RPA_len_noVnl \| cp 02_IP-RPA_len/ndb.dipoles 02_IP-RPA_len_noVnl].

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	1.000000 \| 0.000000 \| 0.000000 \| # [BSS] [cc] Electric Field		1.000000 \| 0.000000 \| 0.000000 \| # [BSS] [cc] Electric Field
	%		%
	and thus we are computing ~~εxx~~(ω).		and thus we are computing <math>\epsilon_{xx}(\omega)</math>.

	To derive the absorption of circularly polarized light we need the off-diagonal component of the dielectric tensor εxy(ω). To this end, we need to compute the current-current response function, that is to move to the velocity gauge. Let's check how it works. Restore the [x,Vnl] DB,		To derive the absorption of circularly polarized light we need the off-diagonal component of the dielectric tensor εxy(ω). To this end, we need to compute the current-current response function, that is to move to the velocity gauge. Let's check how it works. Restore the [x,Vnl] DB,
	mv SAVE/hide_ns.kb_pp SAVE/ns.kb_pp ,		mv SAVE/hide_ns.kb_pp SAVE/ns.kb_pp

	and then run yambo_kerr in the velocity gauge:		and then run yambo_kerr in the velocity gauge:
	yambo -F Inputs/02_IP-RPA_vel -J 02_IP-RPA_vel .		yambo -F Inputs/02_IP-RPA_vel -J 02_IP-RPA_vel

	The new input file differ from the last only for the variable		The new input file differ from the last only for the variable
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	Now compute both the diagonal and the off-diagonal elements of the dielectric tensor, and the Kerr parameters. We just need to run		Now compute both the diagonal and the off-diagonal elements of the dielectric tensor, and the Kerr parameters. We just need to run

	yambo -F Inputs/02_KERR_IP-RPA_vel -J 02_KERR_IP-RPA_vel,		yambo -F Inputs/02_KERR_IP-RPA_vel -J 02_KERR_IP-RPA_vel
	and then plot the results which we can compare with the case without SO. The input file is identical to the previous one but now we are computing the epsilon starting from a DB with SO.		and then plot the results which we can compare with the case without SO. The input file is identical to the previous one but now we are computing the epsilon starting from a DB with SO.

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	The MOKE and more in general the absorption of circular polarized light can be described in the velocity gauge. However the equations for the velocity gauge suffer of numerical instabilities. While these is not a severe problem at the IP-RPA level, the instabilities could be more problematic with the inclusion of local fields or of the electron-hole interaction. Moreover the description of the full dielectric tensor in the velocity gauge requires the computation of the full current-current response function which cannot be obtained within a pure density based approach (indeed we are using <code>yambo -o b</code> and not <code>yambo -o c</code>). It is thus tempting to try to compute the MOKE in the more stable length gauge. To this end we use a generalized version of the equation for the lenght gauge, which however is formally not correct. Run		The MOKE and more in general the absorption of circular polarized light can be described in the velocity gauge. However the equations for the velocity gauge suffer of numerical instabilities. While these is not a severe problem at the IP-RPA level, the instabilities could be more problematic with the inclusion of local fields or of the electron-hole interaction. Moreover the description of the full dielectric tensor in the velocity gauge requires the computation of the full current-current response function which cannot be obtained within a pure density based approach (indeed we are using <code>yambo -o b</code> and not <code>yambo -o c</code>). It is thus tempting to try to compute the MOKE in the more stable length gauge. To this end we use a generalized version of the equation for the lenght gauge, which however is formally not correct. Run

	yambo -F Inputs/03_KERR_IP-RPA_len -J 03_KERR_IP-RPA_len,		yambo -F Inputs/03_KERR_IP-RPA_len -J 03_KERR_IP-RPA_len
	and compare the results with the felocity gauge. The length expression is well defined for the diagonal part of the dielectric tensor which describes the optical absorption. The off-diagonal term however is not properly described with this approach.		and compare the results with the felocity gauge. The length expression is well defined for the diagonal part of the dielectric tensor which describes the optical absorption. The off-diagonal term however is not properly described with this approach.

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	and thus we run		and thus we run
	yambo -F Inputs/03_KERR_IP-RPA_len+AnHall -J 03_KERR_IP-RPA_len+AnHall,		yambo -F Inputs/03_KERR_IP-RPA_len+AnHall -J 03_KERR_IP-RPA_len+AnHall

	we finally obtain the correct result. The Anomalous Hall term is different from zero only in metals and in the so called Chern insulators, e.g. materials with non zero Chern number. In all other cases the length based approach is exact, at least at the IP-RPA level.		we finally obtain the correct result. The Anomalous Hall term is different from zero only in metals and in the so called Chern insulators, e.g. materials with non zero Chern number. In all other cases the length based approach is exact, at least at the IP-RPA level.
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	== Final Considerations ==		== Final Considerations ==

	Remember that the spectra obtained in the present tutorial are from DBs with poor convergence parameters. Below you can find the results obtained with full convergence. Data published in... However you can notice that, already with the parameters used in the tutorial the main features of the spectra		Remember that the spectra obtained in the present tutorial are from DBs with poor convergence parameters. Below you can find the results obtained with full convergence. Data published in Ref.<ref name="Sangalli2012"/>. However you can notice that, already with the parameters used in the tutorial, the main features of the spectra

	[[File:Sigma Iron converged.jpg\|600px\|frameless\|center\|alt=Sigma Iron converged]]		[[File:Sigma Iron converged.jpg\|600px\|frameless\|center\|alt=Sigma Iron converged]]