Volume Dependence of N-Body Bound States

Transcription

1 Volume Dependence of N-Body Bound States Sebastian König in collaboration with Dean Lee INT Workshop 18-70W, University of Washington Seattle, WA February 9, 2018 arxiv: [hep-lat], to appear in PLB Volume Dependence of N-Body Bound States p. 1

2 Finite periodic boxes physical system enclosed in finite volume (box) typically used: periodic boundary conditions volume-dependent energies Lüscher formalism Physical properties encoded in the L-dependent energy levels! infinite-volume S-matrix governs discrete finite-volume spectrum PBC natural for lattice calculations but can also be implemented with other methods Volume Dependence of N-Body Bound States p. 2

3 Overview of recent results 1 Two-body sector nonzero angular momentum SK, Lee, Hammer, PRL (2011); Annals Phys (2011) moving frames (twisted boundary conditions) Davoudi, Savage, PRD (2011) coupled channels, spin, resonances,... e.g., Döring et al., Eur. Phys. J. A (2012); Briceño et al., Phys. Rev. D (2014)... Volume Dependence of N-Body Bound States p. 3

4 Overview of recent results 1 Two-body sector nonzero angular momentum SK, Lee, Hammer, PRL (2011); Annals Phys (2011) moving frames (twisted boundary conditions) Davoudi, Savage, PRD (2011) coupled channels, spin, resonances,... e.g., Döring et al., Eur. Phys. J. A (2012); Briceño et al., Phys. Rev. D (2014)... 2 Three-body sector Efimov physics (bosons, triton) in finite box Kreuzer+Hammer, PLB 673, 260 (2008); 694, 424(2011); Kreuzer+Grießhammer, EPJA (2012) topological correction factors Bour, SK, Lee, Hammer, Meißner, PRD (R) (2011); Rokash et al., JPG (2014) explicit result for three bosons at unitarity Meißner, Ríos, Rusetsky, PRL (2015) twisted boundary conditions Körber+Luu, PRC (2016) many formal results (quantization condition) Polejaeva+Rusetsky, EPJA (2012) Hansen+Sharpe, PRD (2014),..., Briceño, Hansen, Sharpe, PRD (2017) Hammer, Pang, Rusetsky, JHEP ; JHEP (2017) Mai+Döring, EPJA (2017) Volume Dependence of N-Body Bound States p. 3

5 Two-body review Ĥ ψ B = κ2 2µ ψ B binding momentum κ intrinsic length scale Asymptotic wavefunction overlap B(L) = d 3 r ψb(r) V (r) ψ B (r + nl) + O ( e 2κL ) n =1 M. Lüscher, Commun. Math. Phys (1986) for S-wave states, one finds B(L) = 3π γ 2 e κl µl + O( e in general, the prefactor is a polynomial in 1/κL 2κL ) SK, Lee, Hammer, PRL (2011); Annals Phys. 327, 1450 (2012) Volume Dependence of N-Body Bound States p. 4

6 Two-body review B(L) = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) L Volume Dependence of N-Body Bound States p. 5

7 Two-body review B(L) = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) L Volume Dependence of N-Body Bound States p. 5

8 Two-body review B(L) = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) L Volume Dependence of N-Body Bound States p. 5

9 Two-body review B(L) = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) L It s all determined by the tail! Volume Dependence of N-Body Bound States p. 5

10 Two-body review B(L) = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) ψ B(r)V (r) = [ r κ 2] ψ B(r) Y m l (θ, φ) ĥ+ l (iκr) = ( i) l R m l r ( ) 1 e κr κ r r [ r κ 2] e κr r = 4πδ (3) (r) L It s all determined by the tail! Volume Dependence of N-Body Bound States p. 5

11 Two-body review Ĥ ψ B = κ2 2µ ψ B binding momentum κ intrinsic length scale Asymptotic wavefunction overlap B(L) = d 3 r ψb(r) V (r) ψ B (r + nl) + O ( e 2κL ) n =1 M. Lüscher, Commun. Math. Phys (1986) for S-wave states, one finds B(L) = 3π γ 2 e κl µl + O( e in general, the prefactor is a polynomial in 1/κL 2κL ) SK, Lee, Hammer, PRL (2011); Annals Phys. 327, 1450 (2012) Volume Dependence of N-Body Bound States p. 6

12 N-body setup 2- up to N-body interactions: V 1 N (r 1, r N ; r 1, r N) = W i,j (r i, r j ; r i, r j)1 /i,/j + W i,j,k (r i, r j, r k ; r i, r j, r k)1 /i,/j,/k + i<j i<j<k can be local or nonlocal (as written above) all with finite range, set R = max{r i,j, }, assume L R Volume Dependence of N-Body Bound States p. 7

13 Cluster separation 1 consider one particle (WLOG the first) separated from all others S = {(r 1, r N ) : r 1 r i > R i = 2, N} Volume Dependence of N-Body Bound States p. 8

14 Cluster separation 1 consider one particle (WLOG the first) separated from all others S = {(r 1, r N ) : r 1 r i > R i = 2, N} 2 look at Hamiltonian restricted to S: Ĥ N [ CM S = ˆKi ˆK 2 N + ˆV ] rel 2 N + ˆK 1 N 1 no interaction ˆV 1! i=2 Volume Dependence of N-Body Bound States p. 8

15 Cluster separation 1 consider one particle (WLOG the first) separated from all others S = {(r 1, r N ) : r 1 r i > R i = 2, N} 2 look at Hamiltonian restricted to S: Ĥ N [ CM S = ˆKi ˆK 2 N + ˆV ] rel 2 N + ˆK 1 N 1 no interaction ˆV 1! i=2 3 separation ansatz: Ψ(r 1, r N ) = α f α (r 2, r N )g α (r 1 N 1 ) overall Schrödinger equation: ĤΨ S = B N Ψ S lowest f 0 is eigenstate of sub-hamiltonian with energy B N 1 g 0 is Bessel function with scale set by B N B N 1 Volume Dependence of N-Body Bound States p. 8

16 General case 4 now separate A particles from the rest and follow the same steps Relevant variables r A N A = m1r1+ +mara m 1+ +m A ma+1ra+1+ +mn rn m A+1+ +m N µ A N A = m 1+ +m A + m A+1+ +m N κ A N A = 2µ A N A (B N B A B N A ) ψ B N(r 1, r N ) ψ B A(r 1, r A )ψ B N A(r A+1, r N ) (κ A N A r A N A ) 1 d/2 K d/2 1 (κ A N A r A N A ) note: this assumes both clusters to be bound Volume Dependence of N-Body Bound States p. 9

17 General bound-state volume dependence volume dependence overlap of asymptotic wave functions κ A N A = 2µ A N A (B N B A B N A ) Volume dependence of N-body bound state B N (L) (κ A N A L) 1 d/2 K d/2 1 (κ A N A L) ( ) exp κ A N A L /L (d 1)/2 as L (L = box size, d no. of spatial dimensions, K n = Bessel function) SK and D. Lee, arxiv: [hep-lat] channel with smallest κ A N A determines asymptotic behavior Volume Dependence of N-Body Bound States p. 10

18 Analytical examples Three bosons at unitarity two-body interaction with zero range and infinite scattering length B 3 (L) (κ 1 2 L) 1/2 K 1/2 (κ 1 2 L)P(κ 1 2 L) ( ) ( exp 4mB 3 3 L 4mB 3 3 L) 1 P(κ 1 2 L) same exp. dependence as exact result Meißner et al., PRL (2015) by comparison, power-law factor P(x) = x 1/2 Volume Dependence of N-Body Bound States p. 11

19 Analytical examples Three bosons at unitarity two-body interaction with zero range and infinite scattering length B 3 (L) (κ 1 2 L) 1/2 K 1/2 (κ 1 2 L)P(κ 1 2 L) ( ) ( exp 4mB 3 3 L 4mB 3 3 L) 1 P(κ 1 2 L) same exp. dependence as exact result Meißner et al., PRL (2015) by comparison, power-law factor P(x) = x 1/2 N particles with N-body interaction only spinless N-particle bound state with only an N-particle interaction no bound cluster substructures! ψ(r 1, ) (κ 1 N 1 r 1 N 1 ) 1 d(n 1)/2 K d(n 1)/2 1 (κ 1 N 1 r 1 N 1 ) again agrees with prediction, read off P(x) = x d(n 2)/2 Volume Dependence of N-Body Bound States p. 11

20 Generator code onlinewebfonts.com numerical code to check derived volume dependence published with paper, using scientific copyleft terms fully general: arbitrary dimensions, number of particles automatic code generation for each specific system setup Haskell MATLAB/Octave output Volume Dependence of N-Body Bound States p. 12

21 Generator code onlinewebfonts.com numerical code to check derived volume dependence published with paper, using scientific copyleft terms fully general: arbitrary dimensions, number of particles automatic code generation for each specific system setup Haskell MATLAB/Octave output A case for functional programming tell computer what you want, not how to calculate it no loops, only recursion ideal for certain problems no mutable variables, only functions Haskell compiles to maschine code, can be linked with C/C++ (NB: full-thruster Mathematica uses functional techniques as well) Volume Dependence of N-Body Bound States p. 12

22 Numerical results log(δb) D = 1, alatt = 1/3, k = 2 N = 2 N = 3 N = 4 N = L log(l ΔB) D = 3, alatt = 1/2, k = 2 N = 2 N = L straight lines excellent agreement with prediction N B N L min... L max κ fit κ 1 N 1 d = 1, V 0 = 1.0, R = (3) (14) (3) (21) d = 3, V 0 = 5.0, R = (2) (3) Volume Dependence of N-Body Bound States p. 13

23 More complicated example Typically, one exponential dominates, but not necessarily: Setup 1 attractive two-body force B 2 < 0 2 add repulsive three-body force 3 add attractive four-body force B 4 < 0 D = 1, a latt = 1/3, k = 4 log(δb) N = 2 N = L κ fit = κ 2 2 = three-body system unbound asymptotic slope from 2 2 separation Volume Dependence of N-Body Bound States p. 14

24 Finite-volume shift and ANC recall: ψn(r B 1, r N ) ψa( B )ψn A( B ) ψ asympt (r A N A ) 2κ A N A ψ asympt (r A N A ) = γ (r π A N A ) 1 d/2 K d/2 1 (κ A N A r A N A ) γ = asymptotic normalization coefficient (ANC) Finite-volume energy shift 2 B N (L) = ( 1)l+1 π f(d) γ 2 κ 2 d/2 µ A N A L1 d/2 K d/2 1 (κ A N A L) A N A extract ANC from volume dependence Volume Dependence of N-Body Bound States p. 15

25 Finite-volume shift and ANC recall: ψn(r B 1, r N ) ψa( B )ψn A( B ) ψ asympt (r A N A ) 2κ A N A ψ asympt (r A N A ) = γ (r π A N A ) 1 d/2 K d/2 1 (κ A N A r A N A ) γ = asymptotic normalization coefficient (ANC) Finite-volume energy shift 2 B N (L) = ( 1)l+1 π f(d) γ 2 κ 2 d/2 µ A N A L1 d/2 K d/2 1 (κ A N A L) A N A extract ANC from volume dependence ψ N r for comparison, extract ANC from finite-volume wavefunction (mind the PBC) Volume Dependence of N-Body Bound States p. 15

26 ANC comparison Compare ANC result to direct extrapolation from wavefunction: N B N L max γ FV γ WF d = 1, V 0 = 1.0, R = (4) (4) (27) 1.638(16) (6) 2.56(8) (62) 3.63(18) d = 2, V 0 = 1.5, R = (2) 1.921(9) (4) 5.24(2) (4) 10.99(4) d = 3, V 0 = 5.0, R = (3) 1.89(1) (97) 7.83(11) good agreement log(δb) D = 1, alatt = 1/3, k = 2 N = 2 N = 3 N = 4 N = L ψ N r Volume Dependence of N-Body Bound States p. 16

27 Single-volume extrapolation Strategy 1 extract N-body and (N A)-body wavefunctions ψn r ψn look along a given fixed direction, account for periodic boundary divide ψ N (r) by ψ N 1 (0) to adjust normalization 2 get ANC from ratio of tail to known asymptotic form r 3 use B N (L) = ± 2 π f(d) γ2 µ A N A κ 2 d/2 A N A L1 d/2 K d/2 1 (κ A N A L) sign determined by angular momentum (or leading parity) κ A N A extracted as part of ANC fit, initial value from energies at L Volume Dependence of N-Body Bound States p. 17

28 Single-volume extrapolation N B N L B N (L) estimate B N (L) actual d = 1, V 0 = 1.0, R = (2) (4) (7) (2) d = 2, V 0 = 1.5, R = (6) (6) (6) d = 3, V 0 = 5.0, R = (3) (6) overall good agreement with known actual energies! uncertainty included fit error and variation of tail fit range in practice, noisy data will give larger uncertainties Volume Dependence of N-Body Bound States p. 18

29 Summary and outlook leading volume dependence known for arbitrary bound states reproduces known results, checked numerically calculate ANCs, single-volume extrapolations possible! applications to lattice QCD, EFT, cold-atomic systems Volume Dependence of N-Body Bound States p. 19

30 Summary and outlook leading volume dependence known for arbitrary bound states reproduces known results, checked numerically calculate ANCs, single-volume extrapolations possible! applications to lattice QCD, EFT, cold-atomic systems Things to do general derivation of power-law correction factors connection with / inspiration for general quant. conditions include Coulomb interaction expect asymptotic behavior given by Whittaker function Volume Dependence of N-Body Bound States p. 19

31 The end Thank you! Volume Dependence of N-Body Bound States p. 20