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Hubbard Uparameters from constrained random-phase ... · Hubbard Ufrom first principles...
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Mitglied der Helmholtz-Gemeinschaft Hubbard U parameters from constrained random-phase approximation C. Friedrich Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany
Transcript of Hubbard Uparameters from constrained random-phase ... · Hubbard Ufrom first principles...
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Hubbard U parameters fromconstrained random-phaseapproximationC. Friedrich
Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany
GW
Hybrids (PBE0, HSE) OEP (EXX)
RPA total energy
Bethe-Salpeter eq.(magnons)
GT self-energy
COHSEX
GSOCWSOC
QSGW
Hubbard U(LDA+U, DMFT)
Wannier functions (interpolation) Hartree-Fock
Dielectric function
Correlation strength0.1 1.0 10
Alkali metals Transition metals (TMs) Semiconductors TM-Oxides Rare Earths Li, Na, K … Fe, Co, Ni, FePd, … GaAs, GaN, … MnO, FeO, … Gd, Eu, EuO, …
DFT-LDA Hubbard ModelGW, ………. LDA+U, Hubbard I, LDA+DMFT
U/t<latexit sha1_base64="i4eQe9EiZuHiCdT6lnx8xJ4y0Zg=">AAAB6nicdVDLSgNBEJyNrxhfUY9eBoPgaZ1dozG3oBePEd0kkCxhdjJJhsw+mOkVwpJP8OJBEa9+kTf/xslDUNGChqKqm+6uIJFCAyEfVm5peWV1Lb9e2Njc2t4p7u41dJwqxj0Wy1i1Aqq5FBH3QIDkrURxGgaSN4PR1dRv3nOlRRzdwTjhfkgHkegLRsFIt94JdIslYlfK59UzgontuA6plg0hLnGrFezYZIYSWqDeLb53ejFLQx4Bk1TrtkMS8DOqQDDJJ4VOqnlC2YgOeNvQiIZc+9ns1Ak+MkoP92NlKgI8U79PZDTUehwGpjOkMNS/van4l9dOoX/hZyJKUuARmy/qpxJDjKd/455QnIEcG0KZEuZWzIZUUQYmnYIJ4etT/D9puLZzars35VLtchFHHh2gQ3SMHFRBNXSN6shDDA3QA3pCz5a0Hq0X63XemrMWM/voB6y3T1b1jdg=</latexit>
Approximation or Downfolding
vext(r)
ApproximateHamiltonian
HF, DFT (LSDA, GGA), GW LDA+U, LDA+DMFT, LDA+Gutzwiller
DownfoldHamiltonian(correlated subspaceand rest)
DownfoldingFirst quantization
Second quantization
Hubbard U parameter: electron-electron interaction andscreening by the other electrons
2
664�X
RR0
ab,�
tab�RR0 c†Ra� cR0b� +1
2
X
RR0R00R000
abcd,��0
V abcdRR0R00R000 c†Ra� c
†R0b�0 cR000d�0 cR00c�
3
775 n = En n
<latexit sha1_base64="cpjjoqWkfHYsIBJVibI2rwx92l0=">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</latexit>
Screening
U
Downfolding → One-band Hubbard model2
4�tX
hR,R0i
c†RcR0 + ✏0X
R
c†RcR + UX
R
nR"nR#
3
5 n = En n
<latexit sha1_base64="SHh1odX0l/slNbqqA8qNJ5VtdZ0=">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</latexit>
Combining DFT and many-body methods
Correlated Hilbert space
DFT (LDA) Model Hamiltonian
l space
r space
r space
r space: O2p + V3d (eg) + ...
l space: V3d (t2g)
DFT
DFT
Devide Hilbert space into two parts:• Localized Hilbert space (l space)• Rest Hilbert space (r space)
SrVO3SrVO3
Hubbard U from first principles
Constrained local-density approximation (cLDA) [Anisimov and Gunnarsson, PRB 43, 7570 (1991); Cococchioni and de Gironcoli, PRB 71, 035105 (2005)]
Constrained random-phase approximation (cRPA) [Springer and Aryasetiawan et al., PRB 57, 4364 (1998); Kotani, J. Phys. Condens. Matter 12, 2413 (2000)]
• Formulation in many-body perturbation theory.• Frequency dependence U(w) accessible.• Individual matrix elements (U, J, off-site U).• BUT: more expensive.
U =@2E
@n2d
� @2EKS
@n2d
<latexit sha1_base64="7dSCYVZxp8afa8yvVcFDwir5v1g=">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</latexit>
• Easy to implement.• Cheap computation.• BUT: not general.
Hubbard U parameters fromconstrained random-phase approximation (cRPA)
l space
r space
r spaceSrVO3
P (r, r0;!) =occX
m
unoccX
m0
�m(r)�⇤m0(r)�⇤
m(r0)�m0(r0)
1
! � ✏m0 + ✏m + i⌘� 1
! + ✏m0 � ✏m � i⌘
�
<latexit sha1_base64="okpF86VZtyuK7/pjJtR+a+ptZG8=">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</latexit>
P = Pl + Pr
occX
l
unoccX
l0<latexit sha1_base64="QT+20EIjP2yhosYwgJtOiQrZq/E=">AAACFnicbZDLSsNAFIYnXmu8RV26GSyiG0tSBV0W3bisYC/QxDCZTtqhM5MwMxFK6FO48VXcuFDErbjzbUzSLGzrDwM/3zmHOecPYkaVtu0fY2l5ZXVtvbJhbm5t7+xae/ttFSUSkxaOWCS7AVKEUUFammpGurEkiAeMdILRTV7vPBKpaCTu9TgmHkcDQUOKkc6Qb525KuEPLkd6KHkaYTzxmTnDElHQlJ1MTN+q2jW7EFw0TmmqoFTTt77dfoQTToTGDCnVc+xYeymSmmJGJqabKBIjPEID0susQJwoLy3OmsDjjPRhGMnsCQ0L+nciRVypMQ+yznxXNV/L4X+1XqLDKy+lIk40EXj6UZgwqCOYZwT7VBKs2TgzCEua7QrxEEmEdZZkHoIzf/KiaddrznmtfndRbVyXcVTAITgCp8ABl6ABbkETtAAGT+AFvIF349l4NT6Mz2nrklHOHIAZGV+/zOugXw==</latexit>
occX
r
unoccX
r0
+occX
r
unoccX
l0
+occX
l
unoccX
r0<latexit sha1_base64="MxnL8zgvhozJ1ynWskIqDpfv1ak=">AAACdHicjVFNS8MwGE7r16xfVQ8e9BAdQ0Eo7RT0OPTicYL7gHWONMu2sCQtSSqM0l/gv/Pmz/Di2XbrwW0OfCHw8DzPm7x53iBiVGnX/TTMtfWNza3StrWzu7d/YB8eNVUYS0waOGShbAdIEUYFaWiqGWlHkiAeMNIKxo+53nojUtFQvOhJRLocDQUdUIx0RvXsd1/F/NXnSI8kT0KM05605rhYTNlEXqbWtfVvO1thZ6tv79ll13GnBZeBV4AyKKresz/8fohjToTGDCnV8dxIdxMkNcWMpJYfKxIhPEZD0smgQJyobjINLYWVjOnDQSizIzScsr87EsSVmvAgc+azqkUtJ//SOrEe3HcTKqJYE4FnDw1iBnUI8w3APpUEazbJAMKSZrNCPEISYZ3tKQ/BW/zyMmhWHe/GqT7flmsPRRwlcAouwBXwwB2ogSdQBw2AwZdxYkDj3Pg2z8yyWZlZTaPoOQZzZTo/CILBcA==</latexit>
l space
r space
r spaceSrVO3
P = Pl + Pr<latexit sha1_base64="pd0njhbwQOC8aCSvyUeIuEeVJww=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRZBEMpuFfQiFL14XMF+SLss2TRtQ5PskmSFsvRXePGgiFd/jjf/jWm7B219MPB4b4aZeVHCmTau++0UVlbX1jeKm6Wt7Z3dvfL+QVPHqSK0QWIeq3aENeVM0oZhhtN2oigWEaetaHQ79VtPVGkWywczTmgg8ECyPiPYWOnRv/ZDfuaHKixX3Ko7A1omXk4qkMMPy1/dXkxSQaUhHGvd8dzEBBlWhhFOJ6VuqmmCyQgPaMdSiQXVQTY7eIJOrNJD/VjZkgbN1N8TGRZaj0VkOwU2Q73oTcX/vE5q+ldBxmSSGirJfFE/5cjEaPo96jFFieFjSzBRzN6KyBArTIzNqGRD8BZfXibNWtU7r9buLyr1mzyOIhzBMZyCB5dQhzvwoQEEBDzDK7w5ynlx3p2PeWvByWcO4Q+czx/eMI/M</latexit>
U(!) =v
1� vPr(!)<latexit sha1_base64="tuZoSyd19de/g0GRHPdXv+rfpl4=">AAACC3icbVBNS8NAEN3Ur1q/oh69hBahHixJFfQiFL14rGDaQhPKZrtpl+5mw+6mUELuXvwrXjwo4tU/4M1/47aNoK0PBh7vzTAzL4gpkcq2v4zCyura+kZxs7S1vbO7Z+4ftCRPBMIu4pSLTgAlpiTCriKK4k4sMGQBxe1gdDP122MsJOHRvZrE2GdwEJGQIKi01DPLbtXjDA/gyZUXCojScZY6p+NmT/zoWc+s2DV7BmuZODmpgBzNnvnp9TlKGI4UolDKrmPHyk+hUARRnJW8ROIYohEc4K6mEWRY+unsl8w61krfCrnQFSlrpv6eSCGTcsIC3cmgGspFbyr+53UTFV76KYniROEIzReFCbUUt6bBWH0iMFJ0oglEguhbLTSEOhKl4yvpEJzFl5dJq15zzmr1u/NK4zqPowiOQBlUgQMuQAPcgiZwAQIP4Am8gFfj0Xg23oz3eWvByGcOwR8YH9/l85pU</latexit>
→ projected ontoWannier functions:
wRa(r) =1
N
X
km
TkmRa �km(r)
<latexit sha1_base64="B7mHOw6OIxI7dUoI6J6m/3IYTeA=">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</latexit>
transformation matrix Tdefined such that Wannierfunctions are (maximally) localized
Constrained RPA (cRPA)• U is basis independent• U, J, and off-site U easy to calculate• U(w) accessible• subspace screening easy to eliminate if
bands are disentangled (!)
Hubbard U parameters fromconstrained random-phase approximation (cRPA)
Entangled bands
P (r, r0;!) =occX
m
unoccX
m0
�m(r)�⇤m0(r)�⇤
m(r0)�m0(r0)
1
! � ✏m0 + ✏m + i⌘� 1
! + ✏m0 � ✏m � i⌘
�
<latexit sha1_base64="okpF86VZtyuK7/pjJtR+a+ptZG8=">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</latexit>
P (r, r0;!) =occX
m
unoccX
m0
pmpm0�m(r)�⇤m0(r)�⇤
m(r0)�m0(r0)
1
! � ✏m0 + ✏m + i⌘� 1
! + ✏m0 � ✏m � i⌘
�
<latexit sha1_base64="7lu7twMXDbbYeDcR7gm/kHQ8QDk=">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</latexit>
Projection method[Sasioglu et al., PRB 83, 121101 (2011)]
Disentanglement method[Miyake et al., PRB 80, 155134 (2009)]
Hybridization between subspace and rest switched off.→ Bands are disentangled.→ Equation (*) is applicable.
(*)
Effective parameters
“Hubbard-Hund“ parameters (full d shell)
Kanamori parameters (t2g or eg Hamiltonian)
Example: Parameters for d states
U��0
m1,m2,m3,m4(!)
<latexit sha1_base64="7CH6cSZCvyiFkNwg3i4tr67PbSE=">AAACFHicbVDLSsNAFJ3UV62vqEs3wSJWlJK0BV0W3bisYNpCE8NkOolDZ5IwMxFKyEe48VfcuFDErQt3/o3TNgttPXCHwzn3cuceP6FESNP81kpLyyura+X1ysbm1vaOvrvXFXHKEbZRTGPe96HAlETYlkRS3E84hsynuOePriZ+7wFzQeLoVo4T7DIYRiQgCEolefqpfZc5goQMzt7j3MuYZ50xr6GqqaqV15yY4RCeeHrVrJtTGIvEKkgVFOh4+pczjFHKcCQRhUIMLDORbga5JIjivOKkAicQjWCIB4pGkGHhZtOjcuNIKUMjiLmqSBpT9fdEBpkQY+arTgblvZj3JuJ/3iCVwYWbkShJJY7QbFGQUkPGxiQhY0g4RpKOFYGIE/VXA91DDpFUOVZUCNb8yYuk26hbzXrjplVtXxZxlMEBOAQ1YIFz0AbXoANsgMAjeAav4E170l60d+1j1lrSipl98Afa5w8ggp2U</latexit>
Output file ”spex.cou”
• 3 different matrix elements• 2 are independent
end of standard output
• 15 different matrix elements (or 8)• 3 are independent
• 625 different matrix elements
3d transition metals
I ·N(✏F) > 1<latexit sha1_base64="coq2ccslVRZ1wctyIidc3mt74ho=">AAACCHicbVDLSsNAFJ3UV62vqEsXDhahbkpSBV1JURDdSAX7gCaUyWTSDp1kwsxEKKFLN/6KGxeKuPUT3Pk3TtostPXAhcM593LvPV7MqFSW9W0UFhaXlleKq6W19Y3NLXN7pyV5IjBpYs646HhIEkYj0lRUMdKJBUGhx0jbG15mfvuBCEl5dK9GMXFD1I9oQDFSWuqZ+zcO9rmCtxWHxJIyrTkhUgMRplfjo3O7Z5atqjUBnCd2TsogR6Nnfjk+x0lIIoUZkrJrW7FyUyQUxYyMS04iSYzwEPVJV9MIhUS66eSRMTzUig8DLnRFCk7U3xMpCqUchZ7uzI6Us14m/ud1ExWcuSmN4kSRCE8XBQmDisMsFehTQbBiI00QFlTfCvEACYSVzq6kQ7BnX54nrVrVPq7W7k7K9Ys8jiLYAwegAmxwCurgGjRAE2DwCJ7BK3gznowX4934mLYWjHxmF/yB8fkD2iSZOg==</latexit>
Stoner criterion
(40% reduction due to correlation)I = (U + 6J)/5
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Stollhoff et al., PRB 41, 7028:
J
WJJW
E. Sasioglu, Spring School 2014
Hubbard U at surfaces
Sasioglu et al., PRL 109, 146401 (2012)
• FLEUR: Self-consistent field calculationè Density, Exchange-correlation potential
• SPEX: Generate special equidistant k-point setk, k', k+k’, and 0 must be elements
• FLEUR: Diagonalize Hamiltonian on new k points (non iterative)è Kohn-Sham energies and wavefunctions
• SPEX: GW calculationè Quasiparticle energies
Computational procedureOne-Shot GW
k
k'k+k'
spex.inp: JOB GW FULL X:(1-4)
• FLEUR: Self-consistent field calculationè Density, Exchange-correlation potential
• SPEX: Generate special equidistant k-point setk, k', k+k’, and 0 must be elements
• FLEUR: Diagonalize Hamiltonian on new k points (non iterative)è Kohn-Sham energies and wavefunctions
• SPEX: cRPA calculation� Construction of Wannier orbitals (Wannier90 library used for MLWFs)� Calculation of Pr� Calculation of U and projection onto Wannier basisè Hubbard U parameters
Computational procedureHubbard U (cRPA)
k
k'k+k'
spex.inp: JOB SCREENW {0}
Summary
● For strongly correlated systems, methods like LDA+U or LDA+DMFT,which are based on the Hubbard Hamiltonian of a correlated subspace, might be more appropriate than DFT (LDA, GGA) or GW.
● These methods require an effective interaction parameter, the HubbardU parameter, which incorporates the screening processes of the electrons that are not included in the correlated subspace.
● The constrained random-phase-approximation (cRPA) is a first-principlesmethod to determine the Hubbard U parameter.
● Spex has an implementation of cRPA. Correlated subspace spanned inWannier basis. Possibility of treating entangled bands.
