Quantum Monte Carlo calculations of two neutrons in finite volume

Quantum Monte Carlo calculations of two neutrons in finite volume Philipp Klos with J. E. Lynn, I. Tews, S. Gandolfi, A. Gezerlis, H.-W. Hammer, M. Hoferichter, and A. Schwenk Nuclear Physics from Lattice QCD INT, Seattle, April 6, 2016 April 6, 2016 Institut für Kernphysik Philipp Klos 1

Motivation Lüscher formula Quantum Monte Carlo Ground and excited states in AFDMC Chiral effective field theory Extraction of scattering parameters from finite volume Summary and outlook April 6, 2016 Institut für Kernphysik Philipp Klos 2

Two neutrons in a box April 6, 2016 Institut für Kernphysik Philipp Klos 3

Motivation Lattice QCD to describe nuclei from QCD www.jicfus.jp April 6, 2016 Institut für Kernphysik Philipp Klos 4

Motivation I Lattice QCD to describe nuclei from QCD www.jicfus.jp I marekich/wikimedia However, constrained to finite volume and very few particles April 6, 2016 Institut für Kernphysik Philipp Klos 4

Motivation I Lattice QCD to describe nuclei from QCD I Strategy: I Lattice QCD calculations for few nucleon systems I Matching of nuclear forces (EFT) in finite volume I EFT calculations of nuclear properties using advanced many-body methods April 6, 2016 Institut für Kernphysik Philipp Klos 4

Motivation Why two neutrons? Comparison to Lüscher formula possible "proof of principle" Low-energy constants (LEC) in chiral EFT are fitted to experimental data Neutron-neutron scattering length cannot be measured directly Why QMC? Exact method to solve Schrödinger equation Very successful for light nuclei Naturally extendable to more particles (3n, 4n,...) April 6, 2016 Institut für Kernphysik Philipp Klos 5

Two neutrons in finite volume Lüscher formula Consider interaction of nucleons N through contact interactions ( ) L = Ñ i t + 2 Ñ + 2M S-wave scattering amplitude A Now use A = 4π M 1 p cot δ ip. ( ) 4 D µ [C 0 (Ñ Ñ) 2 + C (1) 2 2 (Ñ Ñ) 2 + C (2) 2 [i (Ñ Ñ)] 2 +...] and periodic boundary conditions of the box to obtain energies. April 6, 2016 Institut für Kernphysik Philipp Klos 7

Two neutrons in finite volume Lüscher formula Energy E = p2 of two nucleons in a box predicted by infinite volume phase shift δ(p) m M. Lüscher, Commun. Math. Phys. 105, 153 (1986). p cot δ(p) = 1 πl S S(η) = lim Λ (( ) 2 ) Lp 2π ( Λ Beane et al., PLB 585, 106 (2006). j 1 j 2 η 4πΛ ) pionless EFT Possible to infer scattering length, effective range,..., from finite volume calculations (p cot δ = 1 a + 1 2 r ep 2 +...) Finite volume energies scattering length a, effective range r e,... April 6, 2016 Institut für Kernphysik Philipp Klos 8

Quantum Monte Carlo QMC in three lines: Ground state: H Ψ 0 = E 0 Ψ 0 Trial state: Ψ T = α i Ψ i i Propagate: lim τ e Hτ Ψ T Ψ 0 QMC in more than three lines: J. Carlson et al., RMP 87, 1067 (2015). April 6, 2016 Institut für Kernphysik Philipp Klos 10

Quantum Monte Carlo Trial wave function The trial wave function written in terms of eigenstates of H Ψ T = α i Ψ i Propagate in imaginary time to project out the ground state Ψ 0 Ψ(τ) = e (H ET )τ Ψ ( T ) = e (E0 ET )τ α 0 Ψ 0 + α i e (Ei E0)τ Ψ i Ψ(τ) τ Ψ 0 i i 0 April 6, 2016 Institut für Kernphysik Philipp Klos 11

Quantum Monte Carlo Example Particle in a{ box 0, 0 < x < L V =, otherwise Trial wave function Ψ T (x) = Mixed estimator c n Ψ n (x) n=1 Ψ n (x) = 2 sin(nπx) E(τ) = Ψ T He (H ET )τ Ψ T Ψ T e (H ET )τ Ψ T c n ψ n (x) E(τ) 0.06 0.04 0.02 0-0.02 n=3-0.04 n=5 n=7-0.06 0 0.2 0.4 0.6 0.8 1 x 5 4.99 4.98 4.97 4.96 4.95 4.94 4.93 0 0.2 0.4 0.6 0.8 1 1.2 1.4 τ [1/E sep. ] E 1 April 6, 2016 Institut für Kernphysik Philipp Klos 12

Quantum Monte Carlo Trial wave function Trial wave function for N-body system: Ψ T (R, S) = f J (r ij )Ψ SD (R, S) i<j R = (r 1,..., r N ) and S = (s 1,..., s N ) Jastrow function f J (r ij ) Slater determinant Ψ SD (R, S) = det({φ α (r i, s i )}) φ α (r i, s i ) = e ikα ri χ s,ms (s i ) April 6, 2016 Institut für Kernphysik Philipp Klos 13

Two neutrons in finite volume Contact interaction Comparison: AFDMC vs. Lüscher prediction with contact interaction [ ( r ) ] 4 V (r) = C 0 exp R0-0.05-0.06 ground state pionless EFT -0.07-0.08 E = q 2 4π2 ML 2 q 2-0.09-0.10-0.11 AFDMC, C 0 physical a AFDMC, C 0 a= 101.7 fm Lüscher, physical a Lüscher, a= 101.7 fm -0.12 0 10 20 30 40 50 60 L [fm] April 6, 2016 Institut für Kernphysik Philipp Klos 15

Two neutrons in finite volume Excited state Challenging problem: Excited state has nodal surface ψ(r node ) = 0 QMC requires nodal surface of wave function as input (fixed node approximation, plus small release possible) spherical nodal surface April 6, 2016 Institut für Kernphysik Philipp Klos 16

Two neutrons in finite volume Excited state Introduce node in Jastrow function f J (r 12 ) in Ψ T (R, S) = f J (r 12 )Ψ SD (R, S) f J (r 12 ) 8 7 6 5 4 3 2 1 Ground state jastrow GS 0 0 1 2 3 4 5 6 7 8 9 10 r 12 [fm] f J (r 12 ) 2 0-2 -4-6 -8-10 -12-14 -16 Excited state jastrow EX 0 1 2 3 4 5 6 7 8 9 10 r 12 [fm] April 6, 2016 Institut für Kernphysik Philipp Klos 17

Two neutrons in finite volume Excited state How to determine radius of the nodal surface? For local potential Schrödinger equation Hψ(R) = Eψ(R) yields the same energy E independent of coordinates R 0 r node r max Perform separate simulations in the two pockets until E left = E right April 6, 2016 Institut für Kernphysik Philipp Klos 18

Two neutrons in finite volume Excited state Adjust rnode such that Eleft = Eright 0 rnode rmax 3500 1.8 1.6 3000 1.4 1.2 E [MeV] counts 2500 2000 1500 1 0.8 left pocket right pocket 0.6 1000 0.4 500 0.2 5 10 r [fm] April 6, 2016 Institut für Kernphysik Philipp Klos 19 15 0 0 1 2 3-1 t [MeV ] 4

Two neutrons in finite volume Excited state Comparison: AFDMC vs. Lüscher prediction with contact interaction [ ( r ) ] 4 V (r) = C 0 exp R0 0.70 0.68 1st excited state 0.66 q 2 0.64 0.62 0.60 0.58 AFDMC, C 0, physical a Lüscher, physical a 0 10 20 30 40 50 60 L [fm] April 6, 2016 Institut für Kernphysik Philipp Klos 20

Nodal surface Determine exact nodal surface through diagonalization of H ψ i = E i ψ i. 0.70 1st excited state 0.68 0.66 q 2 0.64 0.62 0.60 0.58 AFDMC, C 0 physical a Diagonalization Lüscher, physical a Lüscher fit, a=-18.9(2) fm 0 10 20 30 40 50 60 L [fm] April 6, 2016 Institut für Kernphysik Philipp Klos 21

Nodal surface Extract nodal surface r node (θ, ϕ) from first excited state ψ ex (r node, θ, ϕ) = 0 Nodal surface not spherical! April 6, 2016 Institut für Kernphysik Philipp Klos 22

Nodal surface April 6, 2016 Institut für Kernphysik Philipp Klos 23

Decomposition in cubic harmonics Nodal surface can be decomposed in spherical harmonics r node (θ, φ) = c lm Y lm (θ, φ) Rotation symmetry group is broken down to the cubic symmetry group O h J. Muggli, Z. Angew. Math. Mech. 23, 311 (1972). cubic harmonics Y c l = l m=0,4,8,... c m Y lm Express nodal surface in terms of cubic harmonics r node (θ, φ) = c l Y c l (θ, φ). l April 6, 2016 Institut für Kernphysik Philipp Klos 24

Decomposition in cubic harmonics Nodal surface decomposed in cubic harmonics r node (θ, φ) = c l Y c l (θ, φ). l cubic harmonics Y c l = c m Y lm m=0,4,8,... We find c l ( r node ) l L Two-body scattering in infinite volume R(r) j l (pr) pr 1 (pr) l c l 1 0.1 0.01 L=10 fm L=20 fm L=30 fm 0.001 0 2 4 6 8 10 12 14 l April 6, 2016 Institut für Kernphysik Philipp Klos 25

Decomposition in cubic harmonics Nodal surface decomposed in cubic harmonics r node (θ, φ) = c l Y c l (θ, φ). l cubic harmonics Y c l = c m Y lm m=0,4,8,... Estimate correction due to l = 4 contribution E = ψ l=0 H ψ l=0 + c 2 4 ψ l=4 H ψ l=4 +... 1 0.1 L=10 fm L=20 fm L=30 fm c l E E (c 4) 2 1% 0.01 in agreement with results 0.001 0 2 4 6 8 10 12 14 l April 6, 2016 Institut für Kernphysik Philipp Klos 25

Chiral effective field theory Separation of scales: Momentum Q breakdown scale Λ b Most general Lagrangian consistent with QCD symmetries Expansion in powers of Q Λ b Local forces up to N 2 LO by using linearly indep. operators Gezerlis et al., PRL 111, 032501 (2013) [Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Hammer, Kaiser, Meißner,...] April 6, 2016 Institut für Kernphysik Philipp Klos 27

Limitations of Lüscher for pionful theory Chiral leading-order (LO) potential q 2-0.05-0.06-0.07-0.08-0.09-0.10-0.11 ground state AFDMC, LO, R 0 =1.0 fm AFDMC, LO, R 0 =1.2 fm Lüscher, a=-18.9 fm, r=2.01 fm Lüscher, a=-18.9 fm, r=2.15 fm -0.12 0 10 20 30 40 50 60 L [fm] q 2 0.78 0.76 0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60 1st excited state AFDMC, LO R 0 =1.0 fm Lüscher, a= 18.9 fm, r=2.01 fm 0.58 0 10 20 30 40 50 60 L [fm] Lüscher formula assumes zero-range interaction (pionless EFT, p < m π /2) Not applicable for nuclear interactions at small box sizes L Direct matching of lattice QCD and chiral EFT necessary for small L April 6, 2016 Institut für Kernphysik Philipp Klos 28

Limitations of Lüscher for pionful theory Chiral leading-order (LO) potential q 2 0.78 0.76 0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60 1st excited state AFDMC, LO R 0 =1.0 fm Lüscher, a= 18.9 fm, r=2.01 fm 0.58 0 10 20 30 40 50 60 L [fm] δ 0 (p) 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 AFDMC, LO R 0 =1.0 fm, excited state δ 0 (p) chiral LO ERE with a= 18.9 fm, r e =2.01 fm 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 p [fm -1 ] Compare directly to phase shift Shows challenges for p > m π /2 p cot δ(p) = 1 (( ) 2 ) Lp πl S 2π April 6, 2016 Institut für Kernphysik Philipp Klos 29

Extraction of scattering parameters from finite volume results Energy E = p2 of two nucleons in a box predicted by infinite volume phase shift δ(p) m Beane et al., PLB 585, 106 (2006). p cot δ(p) = 1 πl S (( ) 2 ) Lp 2π Effective range expansion: 1 a + 1 2 r ep 2 +... = 1 (( ) 2 ) Lp πl S 2π pionless EFT Finite volume energies scattering length a, effective range r e,... April 6, 2016 Institut für Kernphysik Philipp Klos 31

Fit to AFDMC data Chiral NLO and N 2 LO potentials Fit to ground + excited state results 0.70 0.68 1st excited state 0.78 0.76 0.74 1st excited state 0.66 0.72 q 2 0.64 q 2 0.70 0.68 0.62 0.66 0.60 0.58 AFDMC, C 0 physical a Diagonalization Lüscher, physical a Lüscher fit, a=-18.9(2) fm 0 10 20 30 40 50 60 L [fm] Contact interaction C 0 0.64 0.62 AFDMC, LO R 0 =1.0 fm 0.60 Lüscher, a= 18.9 fm, r=2.01 fm Lüscher fit, a=-18.5(4) fm 0.58 0 10 20 30 40 50 60 L [fm] Chiral LO interaction Fit: a = 18.9(2) fm, r e = 1.10(1) fm Exact: a = 18.9 fm, r e = 1.096 fm Fit: a = 18.5(4) fm, r e = 2.00(7) fm Exact: a = 18.9 fm, r e = 2.01 fm April 6, 2016 Institut für Kernphysik Philipp Klos 32

Summary First results for two-neutron finite-volume ground and excited states in AFDMC Approximate construction of excited state vs. exact diagonalization Extraction of scattering parameters from AFDMC simulations yields accurate results QMC techniques can serve to match chiral EFT and lattice results beyond limitations of the Lüscher formula Outlook Generalizable to more particles (3n, 4n,...) where extensions of Lüscher s formula are only partially available Extraction of resonance properties through calculations of excited states arxiv:1604.01387 Thank you! April 6, 2016 Institut für Kernphysik Philipp Klos 34