Automated Angular-Momentum Coupling in Many-Body Theory · Wick’s theorem, symmetry reduction and tensor contractions. ... Hamiltonian G bas ... • Symmetry groups affect many-body

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


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


Hors d’œuvre


Formalism time


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)


Truncation scheme


Hors d’œuvre


Formalism

Truncation scheme

time


Optimisation

Symmetry reduction


ADG‘Automated Diagram Generator’

Arthuis et al., CPC 240:202-227missing link(s)

Reduction


RADNUT

RApid Development toolkit for NUclear Theory

Formalism

Truncation scheme

time


Optimisation

Symmetry reduction


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


Outline

Many-body theory and symmetry reduction

Angular-momentum coupling

• States, operators and tensor networks

• Graph-theory-based reformulation

The AMC program

Future perspectives


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, ….)






• 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, ….)


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,…


Symmetry reduction


Rabij = ...+X

kl

X

cd




external indices


• 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)



Symmetry reduction


Rabij = ...+X

kl

X

cd




external indices



Tk1...kn



X

k

A...k...B...k...C...k...fGsym��!

X

�k

A�...k...

B�...k...

C�...k...


(CCSD residual)

• Storage and runtime requirements are reduced by (up to) several orders of magnitudes



Symmetry reduction


Rabij = ...+X

kl

X

cd




external indices



Tk1...kn



X

k

A...k...B...k...C...k...fGsym��!

X

�k

A�...k...

B�...k...

C�...k...


(CCSD residual)

• Storage and runtime requirements are reduced by (up to) several orders of magnitudes


• Many-body objects are manifestly invariant with respect to symmetry properties

Symmetry conservation throughout numerical solution


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>


SU(2) symmetry



• 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=">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</latexit>


SU(2) symmetry




|k1i ⌦ |k2ifSU(2)��! |k1k2(J)i ⌘

X

mk1mk2


◆|k1k2i


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=">AAADuHicbVLbbtNAELVrLsXcUnjkZYQFKg9YcRICQkKqxAuKFFEk0hZl07DebOwlvml3DYqWFa/wMXwQ38EPsL6EtmlX8ujonDmzMzsOi4QJ2e3+sXeca9dv3Ny95d6+c/fe/c7egyORl5zQCcmTnJ+EWNCEZXQimUzoScEpTsOEHoert5V+/JVywfLso1wXdJbiKGNLRrA01LzzFwBJliwovD9Vo/HoFBUcxlXUc9Uqq3kA/2HvDPbP4EDDG0CiTOcKlAnGosH3faixUbU2F4WYQ16LdaGqQi0aiSQ0FCRWXxqv3hTRDdPbMBUYaTXWW5b+JqG/sQw2TAXquYytGazjdf1ufeAyCFrgWe05nO/Zv9EiJ2VKM0kSLMQ06BZypjCXzHShXVQKWmCywhGdGpjhlIqZqrej4YlhFrDMufkyCTV73qFwKsQ6DU1mimUstrWKvEqblnL5aqZYVpSSZqS5aFkmIHOoVg0LximRydoATDgzvQKJMcdEmh/CdV2U0W8kT1OcLVT9PCaw1IxzUWnfWU+HM4VCGrEs4nlZIEHN35ZFMlYIc47XZjDTm1ZdP6CpRgldyn0UixUr4Pn5jKaGqhmtCPluWvcCeApevwovACHwehUcVGEIiJo2mvQr63EWxfJZlVU3pl2z4GB7nZfBUc8P+n7vw8A7GLar3rUeWY+tfSuwXloH1jvr0JpYxP5k/7B/2r+c185nJ3JYk7pjt56H1oXj8H9KrjTl</latexit>

• Symmetry-restricted tensors: angular-momentum-coupled matrix elements


SU(2) symmetry




|k1i ⌦ |k2ifSU(2)��! |k1k2(J)i ⌘

X

mk1mk2


◆|k1k2i


OJMJ 0M 0

k1k2k3k4=

X

mk1 ...mk4

ok1k2k3k4

✓jk1 jk2 J

mk1 mk2 M

◆✓jk3 jk4 J

0

mk3 mk4 M0

◆



• 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>


SU(2) symmetry




|k1i ⌦ |k2ifSU(2)��! |k1k2(J)i ⌘

X

mk1mk2


◆|k1k2i


OJMJ 0M 0

k1k2k3k4=

X

mk1 ...mk4

ok1k2k3k4

✓jk1 jk2 J

mk1 mk2 M

◆✓jk3 jk4 J

0

mk3 mk4 M0

◆



• 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)


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

◆


◆✓jt ju J4mt mu M4

◆✓J4 �1 J3M4 µ1 M3

◆

⇥✓jt ju J5mt mu M5


◆✓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>


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

◆



◆✓J2 � J1M2 µM1

◆


◆✓jt ju J4mt mu M4

◆✓J4 �1 J3M4 µ1 M3

◆

⇥✓jt ju J5mt mu M5


◆✓J6 �2 J5M6 µ2 M5

◆

⇥ ntnu(pqJ3|S�1 |tuJ4)(tuJ5|T�2 |rsJ6)<latexit sha1_base64="r/Cg6x6R48fRffwB60QGg16y9i8=">AAAGTHicjVTLbhMxFJ0+UtrwamHJxiKiaqUSZSYP2CBV6gZVqlQEhUpxOvI4TjLtvGp7QJXrf+ML2MN3sENI3JtM0klJBSPFOb4+9/jYvnaQRaHSjcb3peWV1cravfWN6v0HDx893tx68lGlueTihKdRKk8DpkQUJuJEhzoSp5kULA4i8Sm4OMDxT5+FVGGafNBXmejFbJiEg5AzDSF/a0XtUB1GfUEyUoBLcui75JrQmOlRMDAH9oxGoNhnGJxw5BQoIHtkl7ypbtOBZNy41niWqjz2DY1zEMLWw9aSImxiP6R21lV5oDTjFwam3avX63uHfodQSo6K7hF0CdKn+vR8xLQ5tCg+hc0b2Ca2SnkkAsVHpnDuYxrasbOIN4ng314Ro3vjTpwDQAfbsMhYqBu1cz+z4B6ac/8SETSHKApeLSnzJI5K5ClECnke8rwybxabuhp7Kmv+v4smZjRtmaZxUCMtRzQWbiGtZcseitCCvZqp3uFj0QRtzGjbf21GB2mdOR9FaMEJzVTnfARMmsRCBU1KUdvbkdxWF1Z3s1Td7+1ZaeE3Fa6nIIeEFtmdKc0NtEtKH0pKnr3jrnTgrlT9zVqj3hh/5G/gFqDmFN+xv7X0lfZTnsci0TxiSnXdRqZ7hkkdwv7BLuZKZHCH2FB0ASYMNqhnxs+DJS8g0ieDVMIv0WQcLWcYFit1FQfAxKWo22MYXDTWzfXgdc+ESZZrkfDJRIM8Ijol+NaQfigF19EVAMZlCF4JHzG4xRpepGqVJuILT+OYJf3JnbZdt2eoSFQuBc5mKF7pGhyLtfPsomhstwMJgRiGyVCmeUaVgBcwGWIqk5JdwVrBrjWNuitiSyMx0Dt0pC7CjLwsMyYaZhyxhvNrWE3NJduk1sSmjXVX8xC2sIH3SYCNCX2hngyHI72LrLExi2fu3j7hv8FHr+426967Vm2/U5z+uvPMee7sOK7zytl33jrHzonDV36sOqsbq9XKt8rPyq/K7wl1eanIeerMfWtrfwDWSCgB</latexit>

fromWigner-Eckart

theorem

reducedmatrix elements

Cpqrs =1

2

X

tu

ntnuSpqtuTturs

<latexit sha1_base64="R7U5WjT1MWCGgw5SSkVbLlI0IqM=">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</latexit>


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




�

j1 j2 j3m1 m2 m3

!=

j3m3

j1m1

j2m2


3jm-symbol

• Contractions among vertices correspond to summation over projection quantum numbers

� �j3

j2m2

j1m1

j10m10

j20m20

� j3m3

j2m2

j1m1

�j3m3

j10m10

j20m20




�

j1 j2 j3m1 m2 m3

!=

j3m3

j1m1

j2m2


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


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)


Reduction rules

• Reduction rules simplify the Yutsis graph while inducing irreducible 3nj-Wigner symbols


Reduction rules


� + =

nj1 j2 j3

o

|23

j1

j2

j3m3 j30m30 j3m3

• 2-cycle rule: Simplest reduction corresponds to orthogonality relation


Reduction rules


� + =

nj1 j2 j3

o

|23

j1

j2

j3m3 j30m30 j3m3


�

� �j3m3 j2m2

j1m1

j4

j6j5=

(j1 j2 j3j4 j5 j6

)+

j3m3

j2m2

j1m1

• 3-cycle rule: generation of 6j-symbol from removing three summations


Reduction rules


� + =

nj1 j2 j3

o

|23

j1

j2

j3m3 j30m30 j3m3


�

� �j3m3 j2m2

j1m1

j4

j6j5=

(j1 j2 j3j4 j5 j6

)+

j3m3

j2m2

j1m1


� �

� �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


Reduction rules


� + =

nj1 j2 j3

o

|23

j1

j2

j3m3 j30m30 j3m3


�

� �j3m3 j2m2

j1m1

j4

j6j5=

(j1 j2 j3j4 j5 j6

)+

j3m3

j2m2

j1m1


� �

� �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


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


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>


Intrinsic workflow



• 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

◆



Intrinsic workflow





✓j1 j2 j3m1 m2 m3

◆⌘ 1

|1(�1)j2�j3�m1

✓j1 j2 j3

�m1 m2 m3

◆


• Symmetry reduction: search for closed subgraphs in the angular-momentum network

Application of 2-, 3- and 4-cycle rules

1

|global· �global

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keep track of global prefactor


Intrinsic workflow



• Output: generation of abstract syntax tree containing the symmetry-reduced tensor network

Many-body framework written in terms of SU(2)-invariant objects



✓j1 j2 j3m1 m2 m3

◆⌘ 1

|1(�1)j2�j3�m1

✓j1 j2 j3

�m1 m2 m3

◆


• Symmetry reduction: search for closed subgraphs in the angular-momentum network

Application of 2-, 3- and 4-cycle rules

1

|global· �global

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keep track of global prefactor


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




• Execution: call the AMC program generating an output LaTeX file

amc imsrgtens.txt imsrgtens.tex -option







• 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





Possible output

plus 14 more pages …(derived in less than a second)


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


One thing left to do …

pip3 install amc


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

https://theorie.ikp.physik.tu-darmstadt.de/strongint/people_ploesser.html

https://theorie.ikp.physik.tu-darmstadt.de/strongint/people_schaefer.html

Documents

Automated Angular-Momentum Coupling in Many-Body Theory · Wick’s theorem, symmetry reduction and tensor contractions. ... Hamiltonian G bas ... • Symmetry groups affect many-body