Symmetry-restored coupled cluster theory
One strategy to tackle near-degenerate and open-shell systems:
I. Break symmetries
II. Restore symmetries
Thomas Duguet
CEA/IRFU/SPhN, Saclay, France
IKS, KU Leuven, Belgium
NSCL, Michigan State University, USA
FUSTIPEN Workshop on New Directions for Nuclear Structure and Reaction Theories
March 16th – 20th 2015, GANIL, Caen
1/18
Outline
I. Introduction
II. Angular-momentum-restored coupled cluster formalism
[T. Duguet, J. Phys. G42, 025107 (2015)]
III. Preliminary results in the doubly-open shell 24Mg nucleus
[S. Binder, T. Duguet, G. Hagen, T. Papenbrock, unpublished]
2/18
Part I
Introduction
3/18
Towards ab-initio methods for open-shell nuclei
1. Design methods to study complete isotopic/isotonic chains, i.e. singly open-shell nuclei
2. Further extend methods to tackle doubly open-shell nuclei
 Many 100s of nuclei eventually
Nuclear structure at/far from b stability
 Magic numbers and their evolution?
 Limits of stability beyond Z=8?
 Mechanisms for nuclear superfluidity?
 Evolution of quadrupole collectivity?
 Role and validation of AN forces?
ASn
ANi
ACa
Option 1: single-reference methods
 Gorkov-SCGF
AO
[V. Somà, T. Duguet, C. Barbieri, PRC 84, 064317 (2011)]
 Bogoliubov CC
[A. Signoracci, T. Duguet, G. Hagen, G. R. Jansen, arXiv:1412.2696]
Option 2: multi-reference methods
 MR-IMSRG
[H. Hergert et al., PRL 110, 242501 (2013)]
 IMSRG-based valence shell model
[S. K. Bogner et al., PRL 113, 142501 (2014)]
 CC-based valence shell model
[G. R. Jansen et al., PRL 113 142502 (2014)]
 NCSM-based valence shell model
[E. Dikmen et al., arXiv:1502.00700]
4/18
Breaking and restoring symmetries
Expansion around a single reference state
Target
state
Wave
operator
Reference
state
Expand
A-body ground state
Size extensive
Ground-state energy
Closed shell
Singly/doubly open shell
Singly/doubly open shell
RHF reference
UHF(B) reference
UHF(B) manyfold
conserves A, J2, M
breaks A / J2 and M
restores A / J2 and M
Dy-SCGF
CC
Go-SCGF
BCC
First implementation
Breaks down for
open-shell nuclei
PNR-BCC/AMR-CC
New formalism
SCGF: to be done
Symmetry contaminants
Multi-reference character
ph degeneracy
<->
Zero mode
Finite inertia
<->
Resolve zero mode
5/18
Part II
Symmetry-restored coupled-cluster theory
Angular-momentum-restored coupled-cluster formalism
[T. Duguet, J. Phys. G42, 025107 (2015)]
Particle-number-restored Bogoliubov coupled-cluster formalism
[T. Duguet, A. Signoracci, in preparation (2015)]
6/18
Master equations (1)
Symmetry group of H includes SU(2) = non abelian compact Lie group – Lie algebra
labeled by J and spanned by M
and IRREPs are
Wigner D functions
Eigen-states of H
1) UHF reference state
leads to
2) Rotated reference states
Imaginary-time dependent scheme
Evolution operator
Time-evolved state
Thouless transformation
Non-pert. sum of p-h excitations
R(W)
Standard many-body methods
Off-diagonal kernels
Diagonal kernels (W = 0)
Dynamical equation
Reduced kernels
Intermediate normalization at W = 0
7/18
Master equations (2)
Expansion of off-diagonal kernels over IRREPs of SU(2)
Time propagation selects the proper IRREP
Straight ratio
Schrödinger equation for G.S.
Truncating kernels expanded around a symmetry-breaking reference
IRREPs still mixed as t -> ∞
<=>
The good symmetry is lost
Standard many-body methods
Symmetry-restored energy
Superfluous in exact limit but not after truncation
No fingerprint of mixing left to be used
Benefit of inserting rotation operator in kernels 8/18
Angular-momentum-restored CC theory
Objective: extend symmetry restoration techniques beyond PHF to any order in CC such that
1. It keeps the simplicity of a single-reference-like CC theory
2. It is valid for any symmetry (spontaneously) broken by the reference state
3. It is valid for any system, i.e. closed shell, near degenerate and open shell
4. It accesses not only the ground state but also the lowest state of each IRREP
1) Static correlations from integral over SU(2)
2) Dynamic correlations from CC expansions of kernels
+ consistent interference
Technical points of importance
•
Wick Theorem for off-diagonal matrix element
of strings of operators
[R. Balian, E. Brezin, NC 64, 37 (1969)]
•
Care must be taken of both the rotated energy
and norm
kernels
Expansion and truncation must be consistent
Problematic to find a naturally terminating expansion
Does not stay normalized when W varies
9/18
Expansion of the operator kernels
All kernels are reduced ones at t = ∞
only one Euler angle b
Expanding the kernel of an operator
The norm kernel factorizes exactly
This is proved via the MBPT expansions of
and
is the connected/linked kernel to operator
Generalized diagrammatic
b-dependent unperturbed line
Possess generalized expansions
1. MBPT expansion
2. Naturally terminating CC expansion
b-dependent part connects holes and particles
Signals the symmetry breaking
Diagonal density matrix of UHF reference state
Standard methods recovered at b = 0
In the exact limit
After truncation
10/18
Coupled cluster energy kernel
Algebraic expressions
1) Same formal structure as in standard CC
2) Natural termination of an exponential
3) No use of Baker-Campbell-Hausdorff
4) Expand with off diagonal Wick theorem
Cumbersome expressions
Very rich content in b
Transformed operators
Bi-orthogonal system
-Exact same expressions as in standard CC
-Similar CPU cost at a given truncation level
Same extension for CC amplitudes equations CC expansion of the off-diagonal energy kernel
11/18
Norm kernel
Energy of the yrast states
Two key questions about
1. No naturally terminating expansion
2. Consistent expansion with operator kernels
?
Solution: apply the symmetry-restored scheme to
and require exact symmetry restoration
This leads to the ODE satisfied by
Displays a naturally
terminating CC expansion
Initial conditions
This ensures that the symmetry is exactly/consistently restored at any truncation order of
Alternatively one can solve a first-order ODE invoking
Note: Extends to any CC order a known result of projected HF
e.g. [K. Enami et al., PRC59 (1999) 135]
12/18
Limits of interest
1. Standard SR-CC theory recovered at b = 0 or if
does not break the symmetry
2. First order = Projected Hartree-Fock approximation
ODE
where
3. Second-order MBPT
Recover standard MBPT2 at b = 0
Perturbative cluster amplitudes
ODE
Cannot be factorized in terms of a projected state
13/18
Part III
Preliminary results for 24Mg
Set up
 NNLOopt 2NF (L = 500 MeV/c)
[A. Ekstrom et al., PRL110, 192502 (2013)]
 No 3NF yet
 HO basis
 emax = 3,4,5
 hw = 22 MeV
 m-scheme code
S. Binder, T. Duguet, G. Hagen, T. Papenbrock, unpublished
14/18
Off-diagonal kernels
ODE
Hartree Fock (n=1)
MBPT2 (n=2)
emax=5 (not converged yet)
combined with
Scalar/vector operator’s kernel is symmetric/antisymmetric with respect to p/2
Deformed
HF
Deformed
MBPT2
Integrate over b
Integrate over b
First (1) ab initio (2) beyond PHF such calculations 15/18
Angular momentum restoration
MBPT2
Hartree Fock
Hartree Fock
Sym. Unr.
Sym. Rest.
DE (Sym. Rest.)
HF
-66.4
-71.0
4.6
MBPT2
-162.9
-167.8
4.9
GS E (MeV)
MBPT2
Dynamic & non-dynamic correlations weakly interfere (DE ↗ 6%)
N2LOopt is not perturbative: MBPT & Emax = 5 not enough
Perfect restoration of
J2
16/18
Ground-state rotational band
State-of-the-art MR-EDF: PNR & AMR + GCM - SLy4 + DDDI
[M. Bender, P.-H. Heenen, PRC78 (2008) 024309]
Quite converged with respect to emax
Moment of inertia significantly impacted by dynamic correlations (have to look at PES)
Certainly look very reasonable and encouraging
No 3N interaction yet + non-perturbative CC needed for realistic Hamiltonians (in progress)
Present low-order MBPT could be the basis of a “better controlled” nuclear EDF method
[T. Duguet, M. Bender, J.-P. Ebran, T. Lesinski, V. Somà, arXiv:1502.03672]
Future
Angular-momentum-restored potential energy surfaces,
EOM-like extension for non-yrast states,
17/18
Use of triaxial/cranked reference states…
Conclusions and perspectives
 First consistent symmetry-restoration MBPT/CC theory beyond PHF
[T. Duguet, J. Phys. G42, 025107 (2015)]
Conclusions
 Main features
 Applies to any symmetry and any system
 Includes standard single-reference CC theory as a particular case
 Reduces to Projected Hartree-Fock theory at lowest order
 Accesses yrast spectroscopy
 Denotes a multi-reference scheme amenable to parallelization
 Ab initio calculations of doubly open-shell nuclei in progress (encouraging)
[S. Binder, T. Duguet, T. Papenbrock, G. Hagen, unpublished]
 Particle-number Bogoliubov CC formalism
[T. D., A. Signoracci, in preparation]
Future
 Develop a similar formalism to extend Go-SCGF
[T. D. V. Somà, C. Barbieri, work in progress]
 Ab-initio-driven nuclear energy density functional method
[T. Duguet, M. Bender, J.-P. Ebran, T. Lesinski, V. Somà, arXiv:1502.03672]
18/18
Appendix
Complementary slides
19/18
AMR-CC scheme in one slide
1. Off-diagonal operator kernels
Recovers single-reference CC at W = 0
Recovers Projected HF at lowest order
with
2. Off-diagonal connected kernels
6. Symmetry-restored energy
3. Transformed matrix elements
where
Bi-orthogonal system
5. Norm kernel
4. Amplitude CC equations
Set of SR-CC calculations for Nsym~(10)angles values of W
20/18
Non-perturbative ab-initio many-body theories
Comp
data
Input
Ab-initio many-body theories
 Effective structure-less nucleons
 2N + 3N + … inter-nucleon interactions
 Solve A-body Schrödinger equation
 Thorough assessment of errors needed
High predictive power
Limited applicability domain
100Sn
56Ni
40Ca
48Ca
2003-2014
CC, Dy-SCGF, IMSRG
Near doubly-magic nuclei
A<132
Inter-nucleon interactions
 Link to QCD – cEFT
[E. Epelbaum, PPNP67, 343 (2012)]
 Soften through RG
[S.K. Bogner et al., PPNP65, 94 (2010)]
132Sn
16O
22,24O
[Dean, Hjorth-Jensen, Piecuch, Hagen, Papenbrock, Roth]
[Barbieri, Dickhoff]
[Tsukiyama, Bogner, Schwenk, Hergert]
1980-2014
FY, GFMC, NCSM
All nuclei A<12
[Carlson, Pieper, Wiringa]
[Barrett, Vary, Navratil, Ormand]
Based on expansion scheme
 Systematic error
 Cross-benchmarks needed
21/18
Symmetry and symmetry breaking
Some symmetries of H
Symmetry breaking and associated physics
Group | YX >
Invariance
DE
Correlations
Excitation
All nuclei…
Gap
but doubly magic
Spatial trans.
T(1)
𝑷
Pairing
<2MeV
Gauge rot.
U(1)
N,Z
Angular loc.
<20MeV Rot. band
Spatial rot.
SO(3)
J,M
Dealt with by decoupling
internal and CM motions
Restoration of symmetries
| YX >
DE
One way is to enforce the symmetry throughout the description
Another way is to let symmetry break in low order description
Symmetry restricted
description
particle-hole
(quasi)degeneracy
Spectro
N,Z
~1MeV Pairing rot.
J,M
~2MeV Rot. band
Crucial for specific observables
but singly magic
Good symmetry 
Hard to get DE 
Symmetry restored
description
Finite inertia
<->
Resolve Goldstone mode
Symmetry breaking
description
ph degeneracy
<->
Zero mode
Account of DE 
Order parameter
Symmetry lost 
Fictitious in nuclei!
Extra DE 
Good symmetry 22/18
Many-body perturbation theory (1)
Symmetry-breaking unperturbed system
Such that
where
and
with
Rotated state
with
and
where
Off diagonal unperturbed one-body density matrix
W-dependent part couples p and h spaces
Density matrix of sym. unrest. reference state
23/18
Many-body perturbation theory (2)
Off-diagonal unperturbed propagator = basic contraction for Wick Theorem
Couples p and h
Off-diagonal operator kernel (e.g. O=H)
Off-diagonal Wick theorem
Evolution operator
[R. Balian, E. Brezin, NC 64, 37 (1969)]
Factorization of the norm kernel
= connected vacuum-to-vacuum diagrams linked to
at time 0
Expansion in Feynman diagrams
Usual rules but off-diagonal propagators
24/18
Many-body perturbation theory (3)
Potential energy diagrams – example at zero order
= standard symmetry unrestricted MBPT-0 contribution
Genuinely W -dependent part
Signals the symmetry breaking
Larget limit
In the exact limit
After truncation
25/18
Coupled cluster energy kernel
t- and W-dependent cluster operators (or rather their hermitian conjuguate…)
Kinetic energy kernel = connected diagrams linked to T
CC
MBPT-0
Time integration
MBPT-1,2…∞
…
Similar for the potential energy kernel
All connected
diagrams with
one ph pair
leaving down
26/18
Account of single-reference CC method (1)
Nuclear Hamiltonian
Wave-function ansatz
Anti-symmetrized matrix elements
Product state of reference
Particle states a,b,c…
Hole states i,j,k…
Cluster operator
N-tuple connected component
Norm kernel in intermediate normalization
Only non-zero ph matrix elements
Exponential generates
connected + disconnected
n-tuple excitations
Size extensive
27/18
Account of single-reference CC method (2)
Time-independent Schrödinger equation
xe-T
Similarity-transformed Hamiltonian
Baker-Campbell-Hausdorff + Wick theorem
Exponential naturally terminates
Energy equation
Variant without xe-T first
Amplitude equation to determine Tn
Define energy
and norm kernels
Obtain algebraic expressions via Wick theorem
Approx <-> truncate T beyond a certain Tn
Exemple: retaining T1 and T2 defines CCSD
28/18
Issues with near degenerate systems
Problematic reference states
MBPT fails because of ea - ei -> 0
High-order CC based on RHF or ROHF
-CCSDT or CR-CC(2,3) for single bond breaking
[J. Noga, R.J. Bartlett, JCP 86, 7041 (1987)]
[P. Piecuch, M. Wloch, JCP 123, 224105 (2005)]
-CCSDTQ or CR-CC(2,4) for double bond breaking
[J. Olsen et al., JCP 104, 8007 (1996)]
-EOM-CC for states near closed shell reference
[J.F. Stanton, R. J. Bartlett, JCP 98, 7029 (1993)]
[G. Jansen et al., PRC 83, 054306 (2011)]
-Spin-adapted CC theory for high spin states
[M. Heckert et al., JCP 124, 124105 (2006)]
Near degenerate
Multi reference
Open shell
MBPT and CC based on UHF
[R. J. Bartlett, ARPC 32, 359 (1981)]
CC and SCGF based on HFB
[V. Somà, T. Duguet, C. Barbieri, PRC 84, 064317 (2011)]
[A. Signoracci, T. Duguet, G. Hagen, G. R. Jansen, arXiv:1412.2696]
Error from CI for H2O symmetric stretch (bond angle fixed at 110.6°)
MR-MBPT
MR-CC
MR-IMSRG
Large spin contamination near 2Re
->very slow convergence
->need non-perturbative step
[A.G. Taube, MP 108, 2951 (2010)]
Good at equilibrium/dissociation29/18
Symmetry breaking reference state
Purpose of symmetry breaking reference state
•
•
•
•
Breaking
Opens the gap at eF
Non degenerate reference
Diagrammatic methods well behaved
Incorporates non-perturbative physics
symmetry
Open shell
Near degenerate
Non degenerate
Lowdin operator in low-order MBPT based on UHF
[P.-O. Lowdin, PR 97, 1509 (1955)]
[H.B. Schlegel, JCP 92, 3075 (1988)]
[P.J. Knowles, N.C. Hardy, JCP 88, 6991 (1988)]
No generic and consistent symmetry broken & restored CC theory…
30/18