automated angular-momentum coupling in many-body theory · wick’s theorem, symmetry reduction and...
TRANSCRIPT
Alexander Tichai
Max-Planck-Institut für Kernphysik Heidelberg
Automated Angular-Momentum Coupling in
Many-Body Theory
Tichai, Wirth, Ripoche, Duguet, arXiv:2002.05011
Progress in Ab Initio Techniques in Nuclear PhysicsTRIUMF workshop
March 2020
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Your daily source of depression…
Many-body theory is (to a large extent):Wick’s theorem, symmetry reduction and tensor contractions
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Hors d’œuvre
Development process in many-body theory
Formalism
Truncation scheme
Schematicimplementation
time
Formal stage ProductionTesting
Optimisation
Symmetry reduction
up to several years ~ 6 months ~ 6 months ~ 6 months
Testing of (potentially great) ideas can easily take 2 years of work!
Reduction
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Hors d’œuvre
Development process in many-body theory
Formalism time
Formal stage ProductionTesting
Optimisation
Symmetry reduction
Gorkov ADC(2)Gorkov ADC(3)
CCSD (T)CCSDT (Q)
(MR-)IMSRG(2)MR-IMSRG(3) IMSRG(3)
(B)MBPT(3)BMBPT(5)BMBPT(6)
Reduction
here SU(2)
Schematicimplementation
Truncation scheme
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Hors d’œuvre
Development process in many-body theory
Formalism
Truncation scheme
time
Formal stage ProductionTesting
Optimisation
Symmetry reduction
Schematicimplementation
ADG‘Automated Diagram Generator’
Arthuis et al., CPC 240:202-227missing link(s)
Reduction
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
RADNUT
RApid Development toolkit for NUclear Theory
Formalism
Truncation scheme
time
Formal stage ProductionTesting
Optimisation
Symmetry reduction
Schematicimplementation
ADG‘Automatic Diagram Generator’
Arthuis et al., CPC 240:202-227
AMC‘Angular-Momentum Coupling’
Tichai, Wirth, Ripoche, Duguet arXiv:2002.05011
ADG + AMCAll J-coupled expressions at 5-th order
BMBPT derived in a few seconds (takes even experts months by hand!)
Reduction
https://github.com/radnut
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Outline
Many-body theory and symmetry reduction
Angular-momentum coupling
• States, operators and tensor networks
• Graph-theory-based reformulation
The AMC program
Future perspectives
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Interplay of symmetries
• Mathematically encoded via linear representations of symmetry groups ( SU(2), U(1), Z2, … )
[H,U(g)] = 0 (8g 2 G)<latexit sha1_base64="5tKRi+yuxQmEtg6yU3I51XZhdic=">AAACEXicbVDLSgNBEJz1GeMr6tFLYxASkLAbRb0IAQ/mGME8ILuE2dlJMmR2dp2ZFcKSX/Dir3jxoIhXb978GyePgyYWNBRV3XR3+TFnStv2t7W0vLK6tp7ZyG5ube/s5vb2GypKJKF1EvFItnysKGeC1jXTnLZiSXHoc9r0B9djv/lApWKRuNPDmHoh7gnWZQRrI3VyhTZU4QTqhV4RPLiywb1PcAAFtxtJzDn0wGUCborZTi5vl+wJYJE4M5JHM9Q6uS83iEgSUqEJx0q1HTvWXoqlZoTTUdZNFI0xGeAebRsqcEiVl04+GsGxUQIwN5gSGibq74kUh0oNQ990hlj31bw3Fv/z2onuXnopE3GiqSDTRd2Eg45gHA8ETFKi+dAQTCQztwLpY4mJNiGOQ3DmX14kjXLJOS2Vb8/ylfNZHBl0iI5QATnoAlVQFdVQHRH0iJ7RK3qznqwX6936mLYuWbOZA/QH1ucPzwKZOA==</latexit>
• Symmetries encode fundamental invariances of a (quantum-mechanical) system
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Interplay of symmetries
• Mathematically encoded via linear representations of symmetry groups ( SU(2), U(1), Z2, … )
[H,U(g)] = 0 (8g 2 G)<latexit sha1_base64="5tKRi+yuxQmEtg6yU3I51XZhdic=">AAACEXicbVDLSgNBEJz1GeMr6tFLYxASkLAbRb0IAQ/mGME8ILuE2dlJMmR2dp2ZFcKSX/Dir3jxoIhXb978GyePgyYWNBRV3XR3+TFnStv2t7W0vLK6tp7ZyG5ube/s5vb2GypKJKF1EvFItnysKGeC1jXTnLZiSXHoc9r0B9djv/lApWKRuNPDmHoh7gnWZQRrI3VyhTZU4QTqhV4RPLiywb1PcAAFtxtJzDn0wGUCborZTi5vl+wJYJE4M5JHM9Q6uS83iEgSUqEJx0q1HTvWXoqlZoTTUdZNFI0xGeAebRsqcEiVl04+GsGxUQIwN5gSGibq74kUh0oNQ990hlj31bw3Fv/z2onuXnopE3GiqSDTRd2Eg45gHA8ETFKi+dAQTCQztwLpY4mJNiGOQ3DmX14kjXLJOS2Vb8/ylfNZHBl0iI5QATnoAlVQFdVQHRH0iJ7RK3qznqwX6936mLYuWbOZA/QH1ucPzwKZOA==</latexit>
• Symmetries encode fundamental invariances of a (quantum-mechanical) system
• Symmetry groups affect many-body treatment at various stages of the formalism
Computationalbasis
Reference state
Hamiltonian
Gbas<latexit sha1_base64="I+kx0c3erLqj22n6fJlCdHTgYl8=">AAAB83icbVBNSwMxEJ31s9avqkcvwSJ4Kru1oMeCBz1WsB/QXUo2TdvQbHZJZsWy9G948aCIV/+MN/+NabsHbX0w8HhvJpl5YSKFQdf9dtbWNza3tgs7xd29/YPD0tFxy8SpZrzJYhnrTkgNl0LxJgqUvJNoTqNQ8nY4vpn57UeujYjVA04SHkR0qMRAMIpW8m97PvInzOwT016p7FbcOcgq8XJShhyNXunL78csjbhCJqkxXc9NMMioRsEknxb91PCEsjEd8q6likbcBNl85yk5t0qfDGJtSyGZq78nMhoZM4lC2xlRHJllbyb+53VTHFwHmVBJilyxxUeDVBKMySwA0heaM5QTSyjTwu5K2IhqytDGVLQheMsnr5JWteJdVqr3tXK9lsdRgFM4gwvw4ArqcAcNaAKDBJ7hFd6c1Hlx3p2PReuak8+cwB84nz9hQ5Hd</latexit>
Gref<latexit sha1_base64="tQo6uGq7kxxckNlnnZvW0eBJdW8=">AAAB83icbVBNS8NAEN34WetX1aOXxSJ4Kkkt6LHgQY8V7Ac0pWy2k3bpZhN2J2IJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEhh0HW/nbX1jc2t7cJOcXdv/+CwdHTcMnGqOTR5LGPdCZgBKRQ0UaCETqKBRYGEdjC+mfntR9BGxOoBJwn0IjZUIhScoZX8276P8ISZhnDaL5XdijsHXSVeTsokR6Nf+vIHMU8jUMglM6bruQn2MqZRcAnTop8aSBgfsyF0LVUsAtPL5jdP6blVBjSMtS2FdK7+nshYZMwkCmxnxHBklr2Z+J/XTTG87mVCJSmC4otFYSopxnQWAB0IDRzlxBLGtbC3Uj5imnG0MRVtCN7yy6ukVa14l5Xqfa1cr+VxFMgpOSMXxCNXpE7uSIM0CScJeSav5M1JnRfn3flYtK45+cwJ+QPn8wdsCpHk</latexit>
GHam<latexit sha1_base64="hMcsqPLDQHDepYx+tQRmtae28S4=">AAAB83icbVBNSwMxEM3Wr1q/qh69BIvgqezWgh4LHuyxgv2A7lKyabYNTbJLMiuWpX/DiwdFvPpnvPlvTNs9aOuDgcd7M8zMCxPBDbjut1PY2Nza3inulvb2Dw6PyscnHROnmrI2jUWseyExTHDF2sBBsF6iGZGhYN1wcjv3u49MGx6rB5gmLJBkpHjEKQEr+XcDH9gTZE0iZ4Nyxa26C+B14uWkgnK0BuUvfxjTVDIFVBBj+p6bQJARDZwKNiv5qWEJoRMyYn1LFZHMBNni5hm+sMoQR7G2pQAv1N8TGZHGTGVoOyWBsVn15uJ/Xj+F6CbIuEpSYIouF0WpwBDjeQB4yDWjIKaWEKq5vRXTMdGEgo2pZEPwVl9eJ51a1buq1u7rlUY9j6OIztA5ukQeukYN1EQt1EYUJegZvaI3J3VenHfnY9lacPKZU/QHzucPMG+RvQ==</latexit>
‘unperturbed Hamiltonian’(e.g. spherical HO/HF)
unperturbed A-body vacuum(Slater, NCSM, Bogoliubov, ….)
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Interplay of symmetries
• Mathematically encoded via linear representations of symmetry groups ( SU(2), U(1), Z2, … )
[H,U(g)] = 0 (8g 2 G)<latexit sha1_base64="5tKRi+yuxQmEtg6yU3I51XZhdic=">AAACEXicbVDLSgNBEJz1GeMr6tFLYxASkLAbRb0IAQ/mGME8ILuE2dlJMmR2dp2ZFcKSX/Dir3jxoIhXb978GyePgyYWNBRV3XR3+TFnStv2t7W0vLK6tp7ZyG5ube/s5vb2GypKJKF1EvFItnysKGeC1jXTnLZiSXHoc9r0B9djv/lApWKRuNPDmHoh7gnWZQRrI3VyhTZU4QTqhV4RPLiywb1PcAAFtxtJzDn0wGUCborZTi5vl+wJYJE4M5JHM9Q6uS83iEgSUqEJx0q1HTvWXoqlZoTTUdZNFI0xGeAebRsqcEiVl04+GsGxUQIwN5gSGibq74kUh0oNQ990hlj31bw3Fv/z2onuXnopE3GiqSDTRd2Eg45gHA8ETFKi+dAQTCQztwLpY4mJNiGOQ3DmX14kjXLJOS2Vb8/ylfNZHBl0iI5QATnoAlVQFdVQHRH0iJ7RK3qznqwX6936mLYuWbOZA/QH1ucPzwKZOA==</latexit>
• Symmetries encode fundamental invariances of a (quantum-mechanical) system
• Simplifications can be done in the case where a common symmetry group exists
Gsym = GHam = Gbas = Gref<latexit sha1_base64="4G0aiIzx2jLILyDcYY6O8bQfjqI=">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</latexit>
• Symmetry groups affect many-body treatment at various stages of the formalism
Computationalbasis
Reference state
Hamiltonian
Gbas<latexit sha1_base64="I+kx0c3erLqj22n6fJlCdHTgYl8=">AAAB83icbVBNSwMxEJ31s9avqkcvwSJ4Kru1oMeCBz1WsB/QXUo2TdvQbHZJZsWy9G948aCIV/+MN/+NabsHbX0w8HhvJpl5YSKFQdf9dtbWNza3tgs7xd29/YPD0tFxy8SpZrzJYhnrTkgNl0LxJgqUvJNoTqNQ8nY4vpn57UeujYjVA04SHkR0qMRAMIpW8m97PvInzOwT016p7FbcOcgq8XJShhyNXunL78csjbhCJqkxXc9NMMioRsEknxb91PCEsjEd8q6likbcBNl85yk5t0qfDGJtSyGZq78nMhoZM4lC2xlRHJllbyb+53VTHFwHmVBJilyxxUeDVBKMySwA0heaM5QTSyjTwu5K2IhqytDGVLQheMsnr5JWteJdVqr3tXK9lsdRgFM4gwvw4ArqcAcNaAKDBJ7hFd6c1Hlx3p2PReuak8+cwB84nz9hQ5Hd</latexit>
Gref<latexit sha1_base64="tQo6uGq7kxxckNlnnZvW0eBJdW8=">AAAB83icbVBNS8NAEN34WetX1aOXxSJ4Kkkt6LHgQY8V7Ac0pWy2k3bpZhN2J2IJ/RtePCji1T/jzX/jts1BWx8MPN6bYWZekEhh0HW/nbX1jc2t7cJOcXdv/+CwdHTcMnGqOTR5LGPdCZgBKRQ0UaCETqKBRYGEdjC+mfntR9BGxOoBJwn0IjZUIhScoZX8276P8ISZhnDaL5XdijsHXSVeTsokR6Nf+vIHMU8jUMglM6bruQn2MqZRcAnTop8aSBgfsyF0LVUsAtPL5jdP6blVBjSMtS2FdK7+nshYZMwkCmxnxHBklr2Z+J/XTTG87mVCJSmC4otFYSopxnQWAB0IDRzlxBLGtbC3Uj5imnG0MRVtCN7yy6ukVa14l5Xqfa1cr+VxFMgpOSMXxCNXpE7uSIM0CScJeSav5M1JnRfn3flYtK45+cwJ+QPn8wdsCpHk</latexit>
GHam<latexit sha1_base64="hMcsqPLDQHDepYx+tQRmtae28S4=">AAAB83icbVBNSwMxEM3Wr1q/qh69BIvgqezWgh4LHuyxgv2A7lKyabYNTbJLMiuWpX/DiwdFvPpnvPlvTNs9aOuDgcd7M8zMCxPBDbjut1PY2Nza3inulvb2Dw6PyscnHROnmrI2jUWseyExTHDF2sBBsF6iGZGhYN1wcjv3u49MGx6rB5gmLJBkpHjEKQEr+XcDH9gTZE0iZ4Nyxa26C+B14uWkgnK0BuUvfxjTVDIFVBBj+p6bQJARDZwKNiv5qWEJoRMyYn1LFZHMBNni5hm+sMoQR7G2pQAv1N8TGZHGTGVoOyWBsVn15uJ/Xj+F6CbIuEpSYIouF0WpwBDjeQB4yDWjIKaWEKq5vRXTMdGEgo2pZEPwVl9eJ51a1buq1u7rlUY9j6OIztA5ukQeukYN1EQt1EYUJegZvaI3J3VenHfnY9lacPKZU/QHzucPMG+RvQ==</latexit>
‘unperturbed Hamiltonian’(e.g. spherical HO/HF)
unperturbed A-body vacuum(Slater, NCSM, Bogoliubov, ….)
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Symmetry reduction
• Starting point for many-body theories is a symmetry-unrestricted tensor network (SU-TN)
Rabij = ...+X
kl
X
cd
Hklcdtdjtaktcbil + ...
<latexit sha1_base64="CLDJXVb0thBCpZDfNaxaAIlI6Ns=">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</latexit>
internal summations(contractions)
external indices
symmetry-unrestricted tensors (SU-T’s)
(CCSD residual)
holes: i,j,k,l,… particles: a,b,c,d,…
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Symmetry reduction
• Starting point for many-body theories is a symmetry-unrestricted tensor network (SU-TN)
Rabij = ...+X
kl
X
cd
Hklcdtdjtaktcbil + ...
<latexit sha1_base64="CLDJXVb0thBCpZDfNaxaAIlI6Ns=">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</latexit>
internal summations(contractions)
external indices
symmetry-unrestricted tensors (SU-T’s)
• Symmetry reduction mediated by transformation generating symmetry-restricted tensors (SR-T)
Tk1...kn
fGsym������! T�k1...kn
<latexit sha1_base64="dJnDazv0bmIa+X60ALLzsU4lZhU=">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</latexit>
X
k
A...k...B...k...C...k...fGsym�����!
X
�k
A�...k...
B�...k...
C�...k...
<latexit sha1_base64="eogyQuta/dZttJ9WQrQ9ZbnUjcM=">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</latexit>
(CCSD residual)
holes: i,j,k,l,… particles: a,b,c,d,…
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Symmetry reduction
• Starting point for many-body theories is a symmetry-unrestricted tensor network (SU-TN)
Rabij = ...+X
kl
X
cd
Hklcdtdjtaktcbil + ...
<latexit sha1_base64="CLDJXVb0thBCpZDfNaxaAIlI6Ns=">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</latexit>
internal summations(contractions)
external indices
symmetry-unrestricted tensors (SU-T’s)
• Symmetry reduction mediated by transformation generating symmetry-restricted tensors (SR-T)
Tk1...kn
fGsym������! T�k1...kn
<latexit sha1_base64="dJnDazv0bmIa+X60ALLzsU4lZhU=">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</latexit>
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<latexit sha1_base64="eogyQuta/dZttJ9WQrQ9ZbnUjcM=">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</latexit>
(CCSD residual)
• Storage and runtime requirements are reduced by (up to) several orders of magnitudes
holes: i,j,k,l,… particles: a,b,c,d,…
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Symmetry reduction
• Starting point for many-body theories is a symmetry-unrestricted tensor network (SU-TN)
Rabij = ...+X
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<latexit sha1_base64="CLDJXVb0thBCpZDfNaxaAIlI6Ns=">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</latexit>
internal summations(contractions)
external indices
symmetry-unrestricted tensors (SU-T’s)
• Symmetry reduction mediated by transformation generating symmetry-restricted tensors (SR-T)
Tk1...kn
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<latexit sha1_base64="dJnDazv0bmIa+X60ALLzsU4lZhU=">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</latexit>
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<latexit sha1_base64="eogyQuta/dZttJ9WQrQ9ZbnUjcM=">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</latexit>
(CCSD residual)
• Storage and runtime requirements are reduced by (up to) several orders of magnitudes
holes: i,j,k,l,… particles: a,b,c,d,…
• Many-body objects are manifestly invariant with respect to symmetry properties
Symmetry conservation throughout numerical solution
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
SU(2) symmetry
• SU(2) symmetry encodes rotational invariance of quantum objects
|ki ⌘ |nk, lk, jk, tk,mki = |k,mki<latexit sha1_base64="4KrVf+KJ3MXBR53Hmb3dQ+VEzsU=">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</latexit>
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
SU(2) symmetry
• SU(2) symmetry encodes rotational invariance of quantum objects
|ki ⌘ |nk, lk, jk, tk,mki = |k,mki<latexit sha1_base64="4KrVf+KJ3MXBR53Hmb3dQ+VEzsU=">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</latexit>
• Definition of angular-momentum-coupled states from symmetry transformation
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<latexit sha1_base64="q7vv4oJHWvLHXssBjC786FOENnc=">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</latexit>
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
SU(2) symmetry
• SU(2) symmetry encodes rotational invariance of quantum objects
|ki ⌘ |nk, lk, jk, tk,mki = |k,mki<latexit sha1_base64="4KrVf+KJ3MXBR53Hmb3dQ+VEzsU=">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</latexit>
• Definition of angular-momentum-coupled states from symmetry transformation
|k1i ⌦ |k2ifSU(2)�����! |k1k2(J)i ⌘
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<latexit sha1_base64="q7vv4oJHWvLHXssBjC786FOENnc=">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</latexit>
OJMJ 0M 0
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ok1k2k3k4
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<latexit sha1_base64="QzUBs9Q+i3TiLuuN/8pOJDr/v4g=">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</latexit>
• Symmetry-restricted tensors: angular-momentum-coupled matrix elements
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
SU(2) symmetry
• SU(2) symmetry encodes rotational invariance of quantum objects
|ki ⌘ |nk, lk, jk, tk,mki = |k,mki<latexit sha1_base64="4KrVf+KJ3MXBR53Hmb3dQ+VEzsU=">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</latexit>
• Definition of angular-momentum-coupled states from symmetry transformation
|k1i ⌦ |k2ifSU(2)�����! |k1k2(J)i ⌘
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<latexit sha1_base64="q7vv4oJHWvLHXssBjC786FOENnc=">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</latexit>
OJMJ 0M 0
k1k2k3k4=
X
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ok1k2k3k4
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<latexit sha1_base64="QzUBs9Q+i3TiLuuN/8pOJDr/v4g=">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</latexit>
• Symmetry-restricted tensors: angular-momentum-coupled matrix elements
• SU(2)-irreducible tensor operators can be processed via Wigner-Eckart theorem
h⇠1j1m1|T JM |⇠2j2m2i = (�1)2J
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✓j2 J j1m2 M m1
◆(⇠1j1|TJ |⇠2j2)
<latexit sha1_base64="By7xh+qoxO1KMQZGIlPtSDeHUrI=">AAAEE3iclZJLbxMxEIA3DY+yvFq4wcViBUoPjeJtKVyQKjgAkZCK1JcUp5HX8W7c7npXthcauf4Z/BNu3BBXfgD/hnE2FenjgiVb43l4Ps9MUuVCm17vT2upfePmrdvLd8K79+4/eLiy+mhfl7VifI+VeakOE6p5LiTfM8Lk/LBSnBZJzg+Sk3fefvCFKy1KuWumFR8WNJMiFYwaUI1WvpOcyizniJyKEUbHsAvYZ2j3yPbdyH5yIHtbDLYYbDEiqokI36DOOl47snHfIZIqyix2lhxPqIGzoGbiRhgsLOeJZhML8c4W/ug7eBfu2N+xCzsL2SGdJT44Se2u8xAXCdAaCkcrUa/bmy10VcBzIQrma2e0uvSejEtWF1wallOtB7hXmaGlygjAcyGpNa8oO6EZH4AoacH10M7K69Bz0IxRWirY0qCZdjHC0kLraZGApwfXl22zUlxjG9QmfT20Qla14ZI1idI6R6ZEvldoLBRnJp+CQJkSwIrYhEKhDXQ0DInkX1lZFFSOm7K7AR5awqWuFffZLPG9iKAJzl1ynzfFDbYgIuGZkJkq64poDiMkMx9LlaJT+CzwOtvrYl44kvPUdMhEn4gKrS96NG/YmcZZxs7gOxFGL1C04Y+XiBAUxV7c9McWAszx3P3a95TIJmbNe83A3EV8LU6P/4M99uwN4VsojBKn53Rxg+jpNhvQf3TnrgsQfvTw5UG7KuzHXbzRjT9vRttb8yFcDp4Gz4JOgINXwXbwIdgJ9gLWetLabn1s9dvf2j/aP9u/Gtel1jzmcXBhtX//BQrwT6k=</latexit>
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
SU(2) symmetry
• SU(2) symmetry encodes rotational invariance of quantum objects
|ki ⌘ |nk, lk, jk, tk,mki = |k,mki<latexit sha1_base64="4KrVf+KJ3MXBR53Hmb3dQ+VEzsU=">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</latexit>
• Definition of angular-momentum-coupled states from symmetry transformation
|k1i ⌦ |k2ifSU(2)�����! |k1k2(J)i ⌘
X
mk1mk2
✓jk1 jk2 Jmk1 mk2 M
◆|k1k2i
<latexit sha1_base64="q7vv4oJHWvLHXssBjC786FOENnc=">AAADi3icbVLdbtMwGHVWfkZhrINLbiwqUCdBlaSjTIiLSQiBJiENQbdJdVU5rpOaxk6wnZbK9UvwdlzyJjhJK7ofK3GOznf8+TjHUZ4ypX3/j7fTuHP33v3dB82Hj/Ye77cOnpyrrJCEDkiWZvIywoqmTNCBZjqll7mkmEcpvYhmH8r6xZxKxTLxXS9zOuI4ESxmBGtHjVt/V3A2DiCSWCQphSjTjFMFSzbcsE30S7JkqrGU2cJAxFWOCTXHvLDx2HwbdMJDe4WFtrmCSLN0QuvuGxjCzunh/83oz4LNIVIFHxv3OKmF1Te0riFJaaTI1PyoK3YjsTUTbpgSnFrzxS15BevjbJsft9p+168GvAmCNWiD9TgbH3i/0SQjBadCkxQrNQz8XI8Mlpo5T7aJCkXdWWc4oUMHBXZ/bGSqLCx84ZgJjDPpXqFhxW6vMJgrteSRU3Ksp+p6rSRvqw0LHR+PDBN5oakg9UZxkUKdwTJYOGGSEp0uHcBEMucVkimWmGgXf7OJBF2QjHMsJmbza+2wPzIoogkTicyKHCnqLpBI9NQgFzZeOvfOgDV+N6DcopTGuoOmasZy+HpbUfcwFWMNISvnrx3Al7DdK6c3ECHYDkt4VE59F72zUctv7Vfdt8NSVRmzZYrB9cxugvOwG/S64dej9kl/necueAaegw4IwFtwAj6DMzAAxPvkcW/uLRp7jV7jXeN9Ld3x1muegiuj8fEfj7QiFg==</latexit>
OJMJ 0M 0
k1k2k3k4=
X
mk1 ...mk4
ok1k2k3k4
✓jk1 jk2 J
mk1 mk2 M
◆✓jk3 jk4 J
0
mk3 mk4 M0
◆
<latexit sha1_base64="QzUBs9Q+i3TiLuuN/8pOJDr/v4g=">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</latexit>
• Symmetry-restricted tensors: angular-momentum-coupled matrix elements
• SU(2)-irreducible tensor operators can be processed via Wigner-Eckart theorem
h⇠1j1m1|T JM |⇠2j2m2i = (�1)2J
1
|1
✓j2 J j1m2 M m1
◆(⇠1j1|TJ |⇠2j2)
<latexit sha1_base64="By7xh+qoxO1KMQZGIlPtSDeHUrI=">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</latexit>
• Analogous treatment can be performed for more complicated Lie groups and algebras
“The many-nucleon theory of collective structure and its macroscopic limits: an algebraic perspective’’ Rowe, McCoy, Caprio, Phys. Scr. 04961 (2016)
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Intermezzo - IMSRG(2)
• m-scheme expression of IMSRG(2) flow arising from two two-body operators S and T
Parzuchowski, Morris, Bogner, PRC 95, 044304 (2017)
• Plugging in all Clebsch-Gordan coefficients in the case of non-scalar operators yields
(pqJ1|C�|rsJ2) =1
2
X
µ1µ2µ
X
{mi}
X
J1,...,J6M1,...,M6
1
J1J3J5
✓�1 �2 �µ1 µ2 µ
◆
⇥✓jp jq J1mp mq M1
◆✓jr js J2mr ms M2
◆✓J2 � J1M2 µM1
◆
⇥✓jp jq J3mp mq M3
◆✓jt ju J4mt mu M4
◆✓J4 �1 J3M4 µ1 M3
◆
⇥✓jt ju J5mt mu M5
◆✓jr js J6mr ms M6
◆✓J6 �2 J5M6 µ2 M5
◆
⇥ ntnu(pqJ3|S�1 |tuJ4)(tuJ5|T�2 |rsJ6)<latexit sha1_base64="r/Cg6x6R48fRffwB60QGg16y9i8=">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</latexit>
fromWigner-Eckart
theorem
reducedmatrix elements
Cpqrs =1
2
X
tu
ntnuSpqtuTturs
<latexit sha1_base64="R7U5WjT1MWCGgw5SSkVbLlI0IqM=">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</latexit>
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Intermezzo - IMSRG(2)
• m-scheme expression of IMSRG(2) flow arising from two two-body operators S and T
Parzuchowski, Morris, Bogner, PRC 95, 044304 (2017)
• Plugging in all Clebsch-Gordan coefficients in the case of non-scalar operators yields
• IMSRG(3) involves ~100 additional terms with more even more couplings coefficients
More systematic solution required !
(pqJ1|C�|rsJ2) =1
2
X
µ1µ2µ
X
{mi}
X
J1,...,J6M1,...,M6
1
J1J3J5
✓�1 �2 �µ1 µ2 µ
◆
⇥✓jp jq J1mp mq M1
◆✓jr js J2mr ms M2
◆✓J2 � J1M2 µM1
◆
⇥✓jp jq J3mp mq M3
◆✓jt ju J4mt mu M4
◆✓J4 �1 J3M4 µ1 M3
◆
⇥✓jt ju J5mt mu M5
◆✓jr js J6mr ms M6
◆✓J6 �2 J5M6 µ2 M5
◆
⇥ ntnu(pqJ3|S�1 |tuJ4)(tuJ5|T�2 |rsJ6)<latexit sha1_base64="r/Cg6x6R48fRffwB60QGg16y9i8=">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</latexit>
fromWigner-Eckart
theorem
reducedmatrix elements
Cpqrs =1
2
X
tu
ntnuSpqtuTturs
<latexit sha1_base64="R7U5WjT1MWCGgw5SSkVbLlI0IqM=">AAAC7XicbVFNb9QwEPWGr7J8dAtHLhYRqBxYJdt24YJUaS8ci+i2ldaryPFOEmtjJ9hO0cr4TyBxQ1wRv4Yr/Tc42UUqbUey9fTmzfh5Jq1Lrk0UXfSCW7fv3L23db//4OGjx9uDnScnumoUgymrykqdpVRDySVMDTclnNUKqEhLOE2XkzZ/eg5K80oem1UNc0FzyTPOqPFUMphMElt/Utrhd5hkijIbOztyRDcisaZxmKRUYZmYf6DBH9uKNnXcKnxpMgijYdQFvg7iDQjRJo6Snd5XsqhYI0AaVlKtZ3FUm7mlynBWguuTRkNN2ZLmMPNQUgF6brvfOvzCMwucVcofaXDHXq6wVGi9EqlXCmoKfTXXkjflZo3J3s4tl3VjQLL1Q1lTYlPhdnR4wRUwU648oExx7xWzgvqZGT/gfp9I+MwqIahcWOK/kWpWuNl4bkkKOZe5qpqaaPArkrkpLKFK0ZV37w04Gw1jEI6UkJldUuglr/Hry4p1D9sxzjL2xfsLY/wSh3vtdYAJweGohfvtNcYEvI21/MZ+iueFedWqOmOu77cYX93ZdXAyGsZ7w9GH/fBwvNnnFnqGnqNdFKM36BC9R0doihj6hX6jP+giqIJvwffgx1oa9DY1T9F/Efz8C+5y7KM=</latexit>
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Diagrammatic notation
• Introduction of diagrammatic notion of angular-momentum-coupling objects
�
j1 j2 j3m1 m2 m3
!=
j3m3
j1m1
j2m2
. Vertex of degree threeWigner
3jm-symbol
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Diagrammatic notation
• Introduction of diagrammatic notion of angular-momentum-coupling objects
�
j1 j2 j3m1 m2 m3
!=
j3m3
j1m1
j2m2
. Vertex of degree threeWigner
3jm-symbol
• Contractions among vertices correspond to summation over projection quantum numbers
� �j3
j2m2
j1m1
j10m10
j20m20
� j3m3
j2m2
j1m1
�j3m3
j10m10
j20m20
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Diagrammatic notation
• Introduction of diagrammatic notion of angular-momentum-coupling objects
�
j1 j2 j3m1 m2 m3
!=
j3m3
j1m1
j2m2
. Vertex of degree threeWigner
3jm-symbol
• Wigner nj-symbols yield irreducible topologies with 2,4,6,… vertices and 3,6,9,… summations
+
+ +
+
j2 j3
j1
j4
j6 j5
+
� �
+ +
�
j1
j7
j8
j3
j6
j5
j4
j9
j2
+ �j3
j1
j2
3j-symbol(triangular inequality) 6j-symbol 9j-symbol
• Contractions among vertices correspond to summation over projection quantum numbers
� �j3
j2m2
j1m1
j10m10
j20m20
� j3m3
j2m2
j1m1
�j3m3
j10m10
j20m20
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
IMSRG(2) - revisited
� +
+ �
�
�
�
+
� +
jp
jqJ3
jr
js
J6
J1
J2
�
�1
�2
jtju
J4
J5
Single-particle angular momenta
Two-bodytotal angular momenta
Spherical tensor ranks
Angular-momentum network (Yutsis graph)
Subdiagram(apply 2-cycle rule)
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Reduction rules
• Reduction rules simplify the Yutsis graph while inducing irreducible 3nj-Wigner symbols
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Reduction rules
• Reduction rules simplify the Yutsis graph while inducing irreducible 3nj-Wigner symbols
� + =
nj1 j2 j3
o
|23
j1
j2
j3m3 j30m30 j3m3
• 2-cycle rule: Simplest reduction corresponds to orthogonality relation
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Reduction rules
• Reduction rules simplify the Yutsis graph while inducing irreducible 3nj-Wigner symbols
� + =
nj1 j2 j3
o
|23
j1
j2
j3m3 j30m30 j3m3
• 2-cycle rule: Simplest reduction corresponds to orthogonality relation
�
� �j3m3 j2m2
j1m1
j4
j6j5=
(j1 j2 j3j4 j5 j6
)+
j3m3
j2m2
j1m1
• 3-cycle rule: generation of 6j-symbol from removing three summations
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Reduction rules
• Reduction rules simplify the Yutsis graph while inducing irreducible 3nj-Wigner symbols
� + =
nj1 j2 j3
o
|23
j1
j2
j3m3 j30m30 j3m3
• 2-cycle rule: Simplest reduction corresponds to orthogonality relation
�
� �j3m3 j2m2
j1m1
j4
j6j5=
(j1 j2 j3j4 j5 j6
)+
j3m3
j2m2
j1m1
• 3-cycle rule: generation of 6j-symbol from removing three summations
� �
� �j2m2 j1m1
j3m3 j4m4
j5
j8j6
j7
=X
jx
(�1) j7� j1� j4� j5 |2x
(j1 jx j4j7 j8 j5
)(j2 jx j3j7 j6 j5
)⇥ + �jx
j2m2
j3m3
j1m1
j4m4
• 4-cycle: generation of two 6j-symbols and an additional dummy summation
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Reduction rules
• Reduction rules simplify the Yutsis graph while inducing irreducible 3nj-Wigner symbols
� + =
nj1 j2 j3
o
|23
j1
j2
j3m3 j30m30 j3m3
• 2-cycle rule: Simplest reduction corresponds to orthogonality relation
�
� �j3m3 j2m2
j1m1
j4
j6j5=
(j1 j2 j3j4 j5 j6
)+
j3m3
j2m2
j1m1
• 3-cycle rule: generation of 6j-symbol from removing three summations
� �
� �j2m2 j1m1
j3m3 j4m4
j5
j8j6
j7
=X
jx
(�1) j7� j1� j4� j5 |2x
(j1 jx j4j7 j8 j5
)(j2 jx j3j7 j6 j5
)⇥ + �jx
j2m2
j3m3
j1m1
j4m4
• 4-cycle: generation of two 6j-symbols and an additional dummy summation
• Higher-order rules do exist but where never encountered in any application
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
IMSRG(2) - revisited
� +
+ �
�
�
�
+
� +
jp
jqJ3
jr
js
J6
J1
J2
�
�1
�2
jtju
J4
J5
Single-particle angular momenta
Two-bodytotal angular momenta
Spherical tensor ranks
Angular-momentum network (Yutsis graph)
Subdiagram(apply 2-cycle rule)
(pqJ1|C�|rsJ2) =1
2�(�1)J1+J2+�
X
J3
⇢�1 �2 �J2 J1 J3
�X
tu
ntnu(pqJ1|S�1 |tuJ3)(tuJ3|T�2 |rsJ2)<latexit sha1_base64="gTKuFdDetBqAF3COisyytPB5wG4=">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</latexit>
Final result
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Intrinsic workflow
• Symmetry transformation: replace m-scheme expression via J-scheme matrix elements
• Canonicalization: transform all Clebsch-Gordan coefficients to Wigner 3jm-symbols
✓j1 j2 j3m1 m2 m3
◆⌘ 1
|1(�1)j2�j3�m1
✓j1 j2 j3
�m1 m2 m3
◆
<latexit sha1_base64="QDPu1gB+MGXPKzYInhv553YSB2Q=">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</latexit>
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Intrinsic workflow
• Symmetry transformation: replace m-scheme expression via J-scheme matrix elements
• Canonicalization: transform all Clebsch-Gordan coefficients to Wigner 3jm-symbols
• Formation of Jutsis graph: build contractions from internal summation
Output: cubic angular-momentum network
✓j1 j2 j3m1 m2 m3
◆⌘ 1
|1(�1)j2�j3�m1
✓j1 j2 j3
�m1 m2 m3
◆
<latexit sha1_base64="QDPu1gB+MGXPKzYInhv553YSB2Q=">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</latexit>
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Intrinsic workflow
• Symmetry transformation: replace m-scheme expression via J-scheme matrix elements
• Canonicalization: transform all Clebsch-Gordan coefficients to Wigner 3jm-symbols
• Formation of Jutsis graph: build contractions from internal summation
Output: cubic angular-momentum network
✓j1 j2 j3m1 m2 m3
◆⌘ 1
|1(�1)j2�j3�m1
✓j1 j2 j3
�m1 m2 m3
◆
<latexit sha1_base64="QDPu1gB+MGXPKzYInhv553YSB2Q=">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</latexit>
• Symmetry reduction: search for closed subgraphs in the angular-momentum network
Application of 2-, 3- and 4-cycle rules
1
|global· �global
<latexit sha1_base64="jzxRxrP167Sbk2wQqon7utRFRnA=">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</latexit>
keep track of global prefactor
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Intrinsic workflow
• Symmetry transformation: replace m-scheme expression via J-scheme matrix elements
• Canonicalization: transform all Clebsch-Gordan coefficients to Wigner 3jm-symbols
• Output: generation of abstract syntax tree containing the symmetry-reduced tensor network
Many-body framework written in terms of SU(2)-invariant objects
• Formation of Jutsis graph: build contractions from internal summation
Output: cubic angular-momentum network
✓j1 j2 j3m1 m2 m3
◆⌘ 1
|1(�1)j2�j3�m1
✓j1 j2 j3
�m1 m2 m3
◆
<latexit sha1_base64="QDPu1gB+MGXPKzYInhv553YSB2Q=">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</latexit>
• Symmetry reduction: search for closed subgraphs in the angular-momentum network
Application of 2-, 3- and 4-cycle rules
1
|global· �global
<latexit sha1_base64="jzxRxrP167Sbk2wQqon7utRFRnA=">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</latexit>
keep track of global prefactor
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
How to work with AMC
• Text file: specify tensor mode, tensorial properties and LaTeX output
declare C { mode=(2,2), latex="C", scalar=False }declare S { mode=(2,2), latex="S", scalar=False }declare T { mode=(2,2), latex="T", scalar=False }declare nbar { mode=2, diagonal=true, latex="\bar{n}" }
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
How to work with AMC
• Text file: specify tensor mode, tensorial properties and LaTeX output
declare C { mode=(2,2), latex="C", scalar=False }declare S { mode=(2,2), latex="S", scalar=False }declare T { mode=(2,2), latex="T", scalar=False }declare nbar { mode=2, diagonal=true, latex="\bar{n}" }
C_pqrs = 1/2 * sum_tu(nbar_t * nbar_u * S_pqtu * T_turs);
• Text file: specify symmetry-unrestricted working equations of many-body formalism
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
How to work with AMC
• Text file: specify tensor mode, tensorial properties and LaTeX output
• Execution: call the AMC program generating an output LaTeX file
amc imsrgtens.txt imsrgtens.tex -option
declare C { mode=(2,2), latex="C", scalar=False }declare S { mode=(2,2), latex="S", scalar=False }declare T { mode=(2,2), latex="T", scalar=False }declare nbar { mode=2, diagonal=true, latex="\bar{n}" }
C_pqrs = 1/2 * sum_tu(nbar_t * nbar_u * S_pqtu * T_turs);
• Text file: specify symmetry-unrestricted working equations of many-body formalism
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
How to work with AMC
• Text file: specify tensor mode, tensorial properties and LaTeX output
• Execution: call the AMC program generating an output LaTeX file
amc imsrgtens.txt imsrgtens.tex -option
• Possible options enable for fine-tuning of output for various end-users
- Switches for using reduced/unreduced matrix elements
- Various phase conventions for Wigner-Eckart theorem
- Using 9j-symbols instead of products of 6j-symbols
declare C { mode=(2,2), latex="C", scalar=False }declare S { mode=(2,2), latex="S", scalar=False }declare T { mode=(2,2), latex="T", scalar=False }declare nbar { mode=2, diagonal=true, latex="\bar{n}" }
C_pqrs = 1/2 * sum_tu(nbar_t * nbar_u * S_pqtu * T_turs);
• Text file: specify symmetry-unrestricted working equations of many-body formalism
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Possible output
plus 14 more pages …(derived in less than a second)
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Conclusions
Development process• Improving accuracy of many-body solution requires relaxing many-body truncation • Formal complexity will soon exceed boundaries of human capacities • Systematic approach needed to accelerate (nuclear) many-body developments
The AMC program
• Graph-theory-based framework for angular-momentum simplifications • Easy interface with state-of-the-art many-body frameworks in nuclear physics • Structure of code already anticipates future generation of source code
Future extensions• Application to partial-wave-decomposed form of PT expressions in nuclear matter • Formulation of other symmetry reductions by extending to other symmetry groups • Generation of source code of symmetry-restricted tensor networks
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
One thing left to do …
pip3 install amc
March 2020A. Tichai — Automated Angular-Momentum Coupling in Many-Body Theory
Epilogue
• Collaborators - T. Duguet, M. Frosini, F. Raimondi, V. Somà
CEA Saclay, France
- P. Arthuis, C. Barbieri University of Surrey, UK
- S. König NC State University, USA
- H. Hergert, D. Lee, R. Wirth Michigan State University, USA
- G. Hagen, T. Papenbrock Oak Ridge National Lab/University of Tennessee, USA
- P. Demol KU Leuven, Belgium
- R. Roth Technische Universität Darmstadt, Germany
- A. Ekström Chalmers University of Technology, Sweden
• ‘STRONGINT’ group - F. Alp, C. Brase, H. Göttling, S. Greif, K. Hebeler,
M. Heinz, J. Hoppe, J. Keller, Y. Lim, M. Plößer, S. Schäfer, A. Schwenk, R. Seutin, C. Wellenhofer, L. Zurek
Technische Universität Darmstadt, Germany