Introduction The Framework Finite Volume Effects Discretization Effects Conclusion Appendix 57th International Winter Meeting on Nuclear Physics - Bormio, Italy Breaking and restoration of rotational symmetry in the spectrum of α − conjugate nuclei on the lattice P RESENTATION S ESSION 23rd January 2019 G. S TELLIN , S. E LHATISARI , U.-G. M EISSNER Rheinische Friedrich-Wilhelms- Universität Bonn H ELMHOLTZ I NSTITUT FÜR S TRAHLEN - UND K ERNPHYSIK U.-G. Meißner’s Workgroup 1 / 24 Breaking and restoration of rotational symmetry on the lattice
Introduction The Framework Finite Volume Effects Discretization Effects Conclusion Appendix Motivation We investigate rotational symmetry breaking in the low-energy spectra of ✄ � light α -conjugate nuclei: 8 Be, 12 C, 16 O, ... ✂ ✁ on a cubic lattice G.S. et al. EPJ A 54, 232 (2018) . In particular, we aim at ♣ identifying lattice eigenstates in terms of SO(3) irreps ⇒ Phys. Lett. B 114, 147-151 (1982) , PRL 103, 261001 (2009) = ♣ exploring the dependence of physical observables on spacing and size ⇒ PRD 90, 034507 (2014) , PRD 92, 014506 (2015) = ♣ developing memory-saving and fast algorithms for the diagonalization of the lattice Hamiltonian = ⇒ Phys. Lett. B 768, 337 (2017) ♣ testing techniques for the suppression of discretization artifacts = ⇒ Lect. Notes in Phys. 788 (2010) Applications Nuclear Lattice EFT: ab initio nuclear structure PRL 104, 142501 (2010) , PRL 112, 102501 (2014) , PRL 117, 132501 (2016) and scattering Nature 528, 111-114 (2015) 2 / 24 Breaking and restoration of rotational symmetry on the lattice
Introduction The Framework Finite Volume Effects Discretization Effects Conclusion Appendix The Hamiltonian of the system The macroscopic α -cluster model 1 of B. Lu et al. PR D 90, 034507 (2014) is adopted ⇒ nuclei are decomposed into M structureless α -particles = M M M H = − � 2 ∇ 2 � � � i + [ V C ( r ij ) + V AB ( r ij )] + V T ( r ij , r ik , r jk ) 2 m α i = 1 i > j = 1 i > j > k = 1 with r ij = | r i − r j | . The potentials are of the type Gaussian Ali-Bodmer 2 Coulomb 2 � √ V 0 e − λ ( r 2 ij + r 2 ik + r 2 jk ) V a f e − η 2 a r 2 ij + V r e − η 2 r r 2 4 e 2 3 r ij � 1 ij erf 4 πǫ 0 2 R α r ij with λ = 0 . 00506 fm − 2 , with η − 1 = 1 . 89036 fm, with R α = 1 . 44 fm r V 0 = − 4 . 41 MeV for 12 C 3 V r = 353 . 508 MeV rms radius of the 4 He s.t. E g . s . = − ∆ E Hoyle and η − 1 = 2 . 29358 fm, NB: Erf adsorbs the a and V 0 = − 11 . 91 MeV for 16 O 4 V a = − 216 . 346MeV, singularity at r = 0 auxiliary param. f = 1 s.t. E g . s . = − ∆ E 4 α 1 G.S. et al. JP G 43, 8 (2016) , 2 NP 80, 99-112 (1966) , 3 Z. Physik A 290, 93-105 (1979) , 4 G.S. (2017) 3 / 24 Breaking and restoration of rotational symmetry on the lattice
Introduction The Framework Finite Volume Effects Discretization Effects Conclusion Appendix The lattice environment The configuration space in relative d.o.f. of an M − body physical system into a cubic lattice reduces to R 3 M − 3 − → N 3 M − 3 where: ⇒ number of points per dimension ( ≡ lattice size ) N = and a = ⇒ lattice spacing L ≡ Na 5 continuum Consequences: discretization effects N=1 4 N=2 2 m α T ( p x ) / � 2 N=3 N=4 1. the action of differential operators is 3 represented via finite differences: = ⇒ Lect. Notes in Phys. 788 (2010) 2 2. breaking of Galiean invariance 1 3. breaking of continuous translational invariance (free-particle case) 0 − 3 − 2 − 1 0 1 2 3 p x 4 / 24 Breaking and restoration of rotational symmetry on the lattice
Introduction The Framework Finite Volume Effects Discretization Effects Conclusion Appendix The lattice environment The configuration space in relative d.o.f. of an M − body physical system into a cubic lattice reduces to R 3 M − 3 − → N 3 M − 3 where: ⇒ number of points per dimension ( ≡ lattice size ) N = a = ⇒ lattice spacing and finite-volume effects on physical observables With periodic boundary conditions: 1. configuration space becomes isomorphic to a torus in 3 M − 3-dimensions 2. lattice momenta become p = � 2 π n Na where n is a vector of integers 4 / 24 Breaking and restoration of rotational symmetry on the lattice
Introduction The Framework Finite Volume Effects Discretization Effects Conclusion Appendix Symmetries On the lattice SO(3) symmetry reduces to the invariance under the cubic group O . Accordingly «Only eight [five: A 1 , A 2 , E , T 1 , T 2 ] different possibilities exist for rotational classification of states on a cubic lattice. So, the question arises: how do these correspond to the angular momentum states in the continuum? [...] To be sure of higher spin assigments and mass predictions it seems necessary to follow all the relevant irreps simultaneously to the continuum limit. » R.C. Johnson, Phys. Lett. B 114, 147-151, (1982). Integer spin irreps D ℓ of SO(3) decompose into irreps of O as follows: D 0 = A 1 D 1 = T 1 D 2 = E ⊕ T 2 D 3 = A 2 ⊕ T 1 ⊕ T 2 D 4 = A 1 ⊕ E ⊕ T 1 ⊕ T 2 D 5 = E ⊕ T 1 ⊕ T 1 ⊕ T 2 D 6 = A 1 ⊕ A 2 ⊕ E ⊕ T 1 ⊕ T 2 ⊕ T 2 5 / 24 Breaking and restoration of rotational symmetry on the lattice
Introduction The Framework Finite Volume Effects Discretization Effects Conclusion Appendix Symmetries Degenerate states belonging to the same O irrep can be labeled with the irreps I z of the cyclic group C 4 , generated by an order-three element of O (e.g. R π/ 2 ): z SO ( 3 ) SO ( 2 ) ⊃ O ⊃ C 4 ↓ ↓ ↓ ↓ = ⇒ l m , Γ I z , Conversely, the discrete symmetries of the Hamiltonian are preserved: time reversal, parity, exchange symmetry Applications Within an iterative approach for the diagonalization of H , the states belonging to an irrep Γ of a point group G can be extracted applying the projector � P Γ = χ Γ ( g ) D ( g ) g ∈G where D ( g ) is a representation of dimension 3 M − 3 for the operation g ∈ G 5 / 24 Breaking and restoration of rotational symmetry on the lattice
Introduction The Framework Finite Volume Effects Discretization Effects Conclusion Appendix Finite volume energy corrections LO finite volume energy corrections for relative two-body bosonic states with reduced mass µ , angular momentum ℓ and belonging to the Γ irrep of O are given by PRL 107, 112011 (2011) � 1 | γ | 2 e − κ 0 L √ � � 2 κ 0 N � ∆ E ( ℓ, Γ) ≡ E ( ℓ, Γ) ( ∞ ) − E ( ℓ, Γ) e − ( L ) = β + O B B B κ 0 L µ L with γ ⇒ asymptotic normalization constant κ 0 ⇒ binding momentum and β ( x ) ⇒ a polynomial ℓ Γ β ( x ) A + 0 − 3 1 1 T − + 3 1 30 x + 135 x 2 + 315 x 3 + 315 x 4 T + 2 2 ( 15 + 90 x + 405 x 2 + 945 x 3 + 945 x 4 ) − 1 2 E + 315 x 2 + 2835 x 3 + 122285 x 4 + 28350 x 5 + 28350 x 6 A − 2 2 ( 105 x + 945 x 2 + 5355 x 3 + 19530 x 4 + 42525 x 5 + 42525 x 6 ) − 1 3 T − 2 2 ( 14 + 105 x + 735 x 2 + 3465 x 3 + 11340 x 4 + 23625 x 5 + 23625 x 6 ) − 1 T − 1 Although no analythic LO FVEC formula for the three-body case exists, results for zero- range potentials PRL 114, 091602 (2015) and the asymptotic ( ≡ large N ) behaviour are available Phys. Lett. B 779, 9-15 (2018) . 6 / 24 Breaking and restoration of rotational symmetry on the lattice
Introduction The Framework Finite Volume Effects Discretization Effects Conclusion Appendix Finite volume energy corrections LO finite volume energy corrections for relative two-body bosonic states with reduced mass µ , angular momentum ℓ and belonging to the Γ irrep of O are given by PRL 107, 112011 (2011) � 1 | γ | 2 e − κ 0 L √ � � 2 κ 0 N � ∆ E ( ℓ, Γ) ≡ E ( ℓ, Γ) ( ∞ ) − E ( ℓ, Γ) e − ( L ) = β + O B B B κ 0 L µ L with γ ⇒ asymptotic normalization constant κ 0 ⇒ binding momentum and β ( x ) ⇒ a polynomial Multiplet averaging of the energies. ⇒ the finite volume energy corrections assume an universal form, independent in magnitude on the SO(3) irreps LO � = ( − 1 ) ℓ + 1 3 | γ | 2 e − κ 0 L χ Γ( ✶ ) 2 ℓ + 1 E ( ℓ P , Γ) E ∞ ( ℓ P A ) − E L ( ℓ P with A ) E ( ℓ P � A ) ≡ � ( L ) µ L Γ ∈O B � at LO, i.e. order exp ( − κ 0 L ) . where: Γ ⇒ irrep of the cubic group χ Γ ( ✶ ) ⇒ character of Γ w.r.t. the identity conjugacy class ( ≡ dim Γ ) P ⇒ eigenvalue of the inversion operator P 6 / 24 Breaking and restoration of rotational symmetry on the lattice
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