Ab Initio Nuclear Structure Theory

Transcription

1 Ab Initio Nuclear Structure Theory Lecture 1: Hamiltonian Robert Roth

2 Overview Lecture 1: Hamiltonian Prelude Many-Body Quantum Mechanics Nuclear Hamiltonian Matrix Elements Lecture 2: Correlations Two-Body Problem Correlations & Unitary Transformations Similarity Renormalization Group Lecture 3: Light Nuclei Many-Body Problem Configuration Interaction No-Core Shell Model Applications Lecture 4: Beyond Light Nuclei Normal Ordering Coupled-Cluster Theory In-Medium Similarity Renormalization Group Robert Roth - TU Darmstadt - August 217 2

3 Prelude

4 Theoretical Context better resolution / more fundamental Quantum Chromodynamics Nuclear Structure finite nuclei few-nucleon systems nuclear interaction hadron structure quarks & gluons deconfinement Robert Roth - TU Darmstadt - August 217 4

5 New Era of Nuclear Structure Theory QCD at low energies improved understanding through lattice simulations & effective field theories Robert Roth - TU Darmstadt - August 217 5

6 New Era of Nuclear Structure Theory QCD at low energies improved understanding through lattice simulations & effective field theories quantum many-body methods advances in ab initio treatment of the nuclear many-body problem Robert Roth - TU Darmstadt - August 217 6

7 New Era of Nuclear Structure Theory QCD at low energies improved understanding through lattice simulations & effective field theories quantum many-body methods advances in ab initio treatment of the nuclear many-body problem computing and algorithms increase of computational resources and developments of algorithms Robert Roth - TU Darmstadt - August 217 7

8 New Era of Nuclear Structure Theory QCD at low energies improved understanding through lattice simulations & effective field theories quantum many-body methods advances in ab initio treatment of the nuclear many-body problem computing and algorithms increase of computational resources and developments of algorithms experimental facilities amazing perspectives for the exploration of nuclei far-off stability Robert Roth - TU Darmstadt - August 217 8

9 The Problem H ni = E n ni Assumptions use nucleons as effective degrees of freedom use non-relativistic framework, relativistic corrections are absorbed in Hamiltonian use Hamiltonian formulation, i.e., conventional many-body quantum mechanics focus on bound states, though continuum aspects are very interesting Robert Roth - TU Darmstadt - August 217 9

10 The Problem H ni = E n ni What is this many-body Hamiltonian? nuclear forces, chiral effective field theory, three-body interactions, consistency, convergence, What about these many-body states? many-body quantum mechanics, antisymmetry, second quantisation, many-body basis, truncations, How to solve this equation? ab initio methods, correlations, similarity transformations, largescale diagonalization, coupledcluster theory, Robert Roth - TU Darmstadt - August 217 1

11 Many-Body Quantum Mechanics... a very quick reminder

12 Single-Particle Basis effective constituents are nucleons characterized by position, spin and isospin degrees of freedom i = position i spin i isospin i i i typical basis choice for configuration-type bound-state methods position i i = nlm l ii spin i i = s = 1 i 2, m s i spherical harmonic oscillator or other spherical single-particle potential eigenstates of s 2 and sz with s=1/2 isospin i = t = 1 2, m t i eigenstates of t 2 and t3 with t=1/2 i i use spin-orbit coupling at the single-particle level n(l 1 2 )jm; 1 l 1/2 2 m j t i = nlm m l m s m l i 1 2 m s i 1 2 m t i m l,m s c Robert Roth - TU Darmstadt - August

13 Many-Body Basis p p i i i Slater determinants: antisymmetrized A-body product states A i = 1 p A! sgn( ) P ( 1 i 2 i A i) given a complete single-particle basis { i} then the set of Slater determinants formed by all possible combinations of A different single-particle states is a complete basis of the antisymmetric A-body Hilbert space resolution of the identity operator 1 = Aih A = 1< 2 <...< A 1 A! 1, 2,..., A Aih A expansion of general antisymmetric state in Slater determinant basis i = C A Ai = C { A} i 1< 2 <...< A Robert Roth - TU Darmstadt - August

14 Second Quantization: Basics define Fock-space as direct sum of A-particle Hilbert spaces F = H H 1 H 2 H A vacuum state: the only state in the zero-particle Hilbert space i 2H h i = 1 i 6= creation operators: add a particle in single-particle state i Slater determinant yielding an (A+1)-body Slater determinant to an A-body i = i Ai = Ai ; /2 { A} ; otherwise resulting states are automatically normalized and antisymmetrized new single-particle state is added in the first slot, can be moved elsewhere through transpositions Robert Roth - TU Darmstadt - August

15 Second Quantization: Basics annihilation operators: remove a particle with single-particle state i an A-body Slater determinant yielding an (A-1)-body Slater determinant from i = Ai = ( 1) Ai ; = ; otherwise annihilation operator acts on first slot, need transpositions to get correct singleparticle state there based on these definitions one can easily show that creation and annihilations operators satisfy anticommutation relations {, } = {, } = {, } = complication of handling permutations in "first quantization" are translated to the commutation behaviour of strings of operators Robert Roth - TU Darmstadt - August

16 Second Quantization: States Slater determinants can be written as string of creation operators acting on vacuum state Ai = 1 2 A i alternatively one can define an A-body reference Slater determinant i = Ai = 1 2 A i and construct arbitrary Slater determinants through particle-hole excitations on top of the reference state p i = p i pq b i =. p q b i index convention: a,b,c, : hole states, occupied in reference state p,q,r, : particle state, unoccupied in reference states Robert Roth - TU Darmstadt - August

17 Second Quantization: Operators operators can be expressed in terms of creation and annihilation operators as well, e.g., for one-body kinetic energy and two-body interactions: first quantization second quantization T = A t T = h t i =1 V = A <j=1 v j V = h 1 2 v 1 2 i set of one or two-body matrix elements fully defines the one or two-body operator in Fock space second quantization is extremely convenient to compute matrix elements of operators with Slater determinants Robert Roth - TU Darmstadt - August

18 Nuclear Hamiltonian

19 Nuclear Hamiltonian general form of many-body Hamiltonian can be split into a center-of-mass and an intrinsic part H = T + V NN + V 3N + = T cm + T int + V NN + V 3N + = T cm + H int intrinsic Hamiltonian is invariant under translation, rotation, Galilei boost, parity, time evolution, time reversal,... H int = T int + V NN + V 3N + = A <j 1 2mA (~p ~p j) 2 + A v NN, j + <j A <j<k v 3N, jk + these symmetries constrain the possible operator structures that can appear in the interaction terms but how can we really determine the nuclear interaction? Robert Roth - TU Darmstadt - August

20 Nature of the Nuclear Interaction nuclear interaction is not fundamental residual force analogous to van der Waals interaction between neutral atoms based on QCD and induced via polarization of quark and gluon distributions of nucleons encapsulates all the complications of the QCD dynamics and the structure of nucleons acts only if the nucleons overlap, i.e. at short ranges 1.6fm ρ 1/3 = 1.8fm irreducible three-nucleon interactions are important Robert Roth - TU Darmstadt - August 217 2

21 Yesterday... from Phenomenology Wiringa, Machleidt, until 25: high-precision phenomenological NN interactions were stateof-the-art in ab initio nuclear structure theory Argonne V18: long-range one-pion exchange plus phenomenological parametrization of medium- and short-range terms, local operator form CD Bonn 2: more systematic one meson-exchange parametrization including pseudo-scalar, scalar and vector mesons, inherently nonlocal parameters of the NN potential (~4) fit to NN phase shifts up to ~3 MeV and reproduce them with high accuracy no consistency no systematics no connection to QCD supplemented by phenomenological 3N interactions consisting of a Fujita-Miyazawa-type term plus various handpicked contributions fit to ground states and spectra of light nuclei, sometimes up to A 8 Robert Roth - TU Darmstadt - August

22 Argonne V18 Potential Wiringa, et al., PRC 51, 38 (1995) [MeV] (r) (r) L 2 Argonne V18 Potential (r) S 12 (r) L S (r)( L S) 2 c v NN = ST (r) ST + (S, S,TT) S,T t + (1, ) T (r) S 12 1T + (1, T 1) T (, ) s2 + T (r) ( ~L ~S) 2 1T +... (, 1) T 2 ST (r) ~L 2 ST s T (r) ( ~L ~S) 1T [MeV] r [fm] [MeV] Robert Roth - TU Darmstadt - August r [fm] (r) (r) L r [fm] (S, T) (1, ) (1, 1) (, ) (, 1) (r) S 12 (r) L S (r)( L S) 2 22

23 Tomorrow... from Lattice QCD Hatsuda, Aoki, Ishii, Beane, Savage, Bedaque,... NN wave function φ(r) V C (r) [MeV] x[fm] 1 S 3 S S r [fm] 3 S φ(x,y,z=; 1 S ) y[fm] N. Ishii et al., PRL 99, 221 (27) first attempts towards construction of nuclear interactions directly from lattice QCD simulations compute relative two-nucleon wave function on the lattice invert Schrödinger equation to extract effective two-nucleon potential only schematic results so far (unphysical masses and mass dependence, model dependence, ) alternatives: phase-shifts or lowenergy constants from lattice QCD r [fm] Robert Roth - TU Darmstadt - August

24 Today... from Chiral EFT low-energy effective field theory for relevant degrees of freedom (π,n) based on symmetries of QCD explicit long-range pion dynamics Weinberg, van Kolck, Machleidt, Entem, Meißner, Epelbaum, Krebs, Bernard,... LO NN 3N 4N unresolved short-range physics absorbed in contact terms, low-energy constants fit to experiment systematic expansion in a small parameter with power counting enable controlled improvements and error quantification hierarchy of consistent NN, 3N, 4N,... interactions consistent electromagnetic and weak operators can be constructed in the same framework NLO N 2 LO N 3 LO Robert Roth - TU Darmstadt - August

25 Momentum-Space Matrix Elements hq(ls) JM; TM T v NN q (L S) JM; TM T i SRG Evolution in Two-Body Space Argonne V18 chiral NN (N3LO, E&M, 5 MeV) J=1 L= L = S=1 T= J=1 L= L =2 S=1 T= momentum-space matrix elements 3 S 1 3 S 1 3 D 1 ϕ L (r)[arb. units]. α Argonne =.V18 fm 4 Λ = fm 1 J π = 1 +,T = Robert Roth - TU Darmstadt - August S 1 deuteron wave-function L = L = 2 3 S 1 3 D r [fm] 6 8

26 Matrix Elements

27 Partial-Wave Matrix Elements relative partial-wave matrix elements of NN and 3N interaction are universal input for many-body calculations selection of relevant partial-wave bases in two and three-body space with all M quantum numbers suppressed: two-body relative momentum: two-body relative HO: three-body Jacobi momentum: three-body Jacobi HO: antisym. three-body Jacobi HO: q (LS) JTi N (LS) JTi 1 2; [(L 1 S 1 ) J 1, (L ) J 2] J 12 ; (T ) T 12i N 1 N 2 ; [(L 1 S 1 ) J 1, (L ) J 2] J 12 ; (T ) T 12i E 12 J 12 T 12 i lots of transformations between the different bases are needed in practice exception: Quantum Monte-Carlo methods working in coordinate representation need local operator form Robert Roth - TU Darmstadt - August

28 Symmetries and Matrix Elements relative partial-wave matrix elements make maximum use of the symmetries of the nuclear interaction consider, e.g., the relative two-body matrix elements in HO basis hn (LS) JM; TM T v NN N (L S ) J M ; T M T i the matrix elements of the NN interaction do not connect different J do not connect different M and are independent of M do not connect different parities do not connect different S do not connect different T do not connect different MT hn (LS) J; TM T v NN N (L S) J; TM T i relative matrix elements are efficient and simple to compute Robert Roth - TU Darmstadt - August

29 Transformation to Single-Particle Basis most many-body calculations need matrix elements with single-particle quantum numbers (cf. second quantization) h 1 2 v NN 1 2 i = = hn 1 1 j 1 m 1 m t1,n 2 2 j 2 m 2 m t2 v NN n 1 1 j 1 m 1 m t1,n 2 2 j 2 m 2 m t2 i obtained from relative HO matrix elements via Moshinsky-transformation hn 1 1 j 1,n 2 2 j 2 ; JT v NN n 1 1 j 1,n 2 2 j 2 ; JTi = = (2j 1 + 1)(2j 2 + 1)(2j 1 + 1)(2j 2 + 1) 8 < 1 2 L 1 1 S = < : 2 ; : j 1 j 2 J 1 2 j 1 L 2 1 S 2 j J 2 9 = L ; S J j L,L,S N,, L S J j, this analytic transformation from relative to single-particle matrix elements only exists for the harmonic oscillator basis hhn, n 1 1,n 2 2 ; Lii hhn, n 1 1,n 2 2 ; L ii (2j + 1)(2S + 1)(2L + 1)(2L + 1) ( 1) L+L {1 ( 1) +S+T } j h ( S)jT v NN Robert Roth - TU Darmstadt - August 217 ( S)jTi 29

30 Matrix Element Machinery beneath any ab initio many-body method there is a machinery for computing, transforming and storing matrix elements of all operators entering the calculation compute and store relative two-body HO matrix elements of NN interaction compute and store Jacobi three-body HO matrix elements of 3N interaction perform unitary transformations of the two- and three-body relative matrix elements (e.g. Similarity Renormalization Group) same for 4N with four-body matrix elements transform to single-particle JT-coupled two-body HO matrix elements and store transform to single-particle JT-coupled three-body HO matrix elements and store Robert Roth - TU Darmstadt - August 217 3

31 Two-Body Problem

32 Solving the Two-Body Problem simplest ab initio problem: the only two-nucleon bound state, the deuteron start from Hamiltonian in two-body space, change to center of mass and intrinsic coordinates H = H cm + H int = T cm + T int + V NN = 1 P 2M ~ 2 cm ~q2 + V NN separate two-body state into center of mass and intrinsic part i = cm i int i i i i solve eigenvalue problem for intrinsic part (effective one-body problem) H int int i = E int i Robert Roth - TU Darmstadt - August

33 Solving the Two-Body Problem expand eigenstates in a relative partial-wave HO basis inti = C NLSJMTMT N (LS) JM; TM T i NLSJMTM T N (LS) JM; TM T i = M L M S c L S M L M S J M NLM Li SM S i TM T i symmetries simplify the problem dramatically: Hint does not connect/mix different J, M, S, T, MT and parity π angular mom. coupling only allows J=L+1, L, L-1 for S=1 or J=L for S= total antisymmetry requires L+S+T=odd for given J π at most two sets of angular-spin-isospin quantum numbers contribute to the expansion Robert Roth - TU Darmstadt - August

34 Deuteron Problem assume J π = 1 + for the deuteron ground state, then the basis expansion reduces to int, J = 1 + i = N (1) 1M;i + N (21) 1M;i N C () N N C (2) N inserting into Schrödinger equation and multiplying with basis bra leads to matrix eigenvalue problem hn (1)... H int N(1)...i hn (21)... H int N(1)...i hn (1)... H int N(21)...i hn (21)... H int N(21)...i eigenvectors yield expansions coefficients and eigenvalues the energies truncate matrices to N Nmax and choose Nmax large enough so that observables are converged, i.e., do not depend on Nmax anymore 1 C A C () N C (2) N simplest possible Jacobi-NCSM calculation 1 = E C A C () N C (2) N 1 C A Robert Roth - TU Darmstadt - August

35 Argonne V18 Deuteron Solution J π = 1 +,T =.5 deuteron Argonne wave-function V18 chiral NN ϕ L (r)[arb. units] L = L = 2 ϕ L (r)[arb. units] L = L = r [fm] r [fm] deuteron wave function show two characteristics that are signatures of correlations in the two-body system: suppression at small distances due to short-range repulsion L=2 admixture generated by tensor part of the NN interaction 22. Robert Roth - TU Darmstadt - August