Auxiliary-field quantum Monte Carlo methods in heavy nuclei

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1 Mika Mustonen and Yoram Alhassid (Yale University) Introduction Auxiliary-field quantum Monte Carlo methods in heavy nuclei Auxiliary-field Monte Carlo (AFMC) methods at finite temperature Sign problem and good-sign interactions Circumventing the odd particle-number sign problem Heavy nuclei Nuclear deformation in the spherical shell model: quadrupole distributions in the laboratory frame and in the intrinsic frame Possible application of AFMC to neutrinoless double beta decay Recent review: Y. Alhassid, arxiv:17.187, a chapter in a book edited by K.D. Launey (217)

2 Introduction The challenge: microscopic calculations of nuclear properties from underlying effective interactions Ab initio methods have been developed: Green function Monte Carlo No-core shell model Lattice effective field theory Coupled cluster (CC) method However, they are mostly limited to light nuclei or to nuclei near shell closure (CC). Most microscopic treatments of mid-mass and heavy nuclei are based on mean-field methods, e.g., density functional theory. However, important correlations can be missed.

3 The configuration-interaction (CI) shell model takes into account correlations beyond the mean-field but the combinatorial increase of the dimensionality of its model space has hindered its applications in mid-mass and heavy nuclei. Conventional diagonalization methods for the shell model are limited to spaces of dimensionality ~ The auxiliary-field Monte Carlo (AFMC method) enables microscopic calculations in spaces that are many orders of magnitude larger (~ 1 ) than those that can be treated by conventional methods. Also known in nuclear physics as the shell model Monte Carlo (SMMC) method. G.H. Lang, C.W. Johnson, S.E. Koonin, W.E. Ormand, PRC 48, 1518 (199); Y. Alhassid, D.J. Dean, S.E. Koonin, G.H. Lang, W.E. Ormand, PRL 72, 1 (1994).

4 Hubbard-Stratonovich (HS) transformation Assume an effective Hamiltonian in Fock space with a one-body part and a two-body interaction : i are single-particle energies and densities. ˆ ij = a i a j R Ĥ = P i iˆn i ˆ P v ˆ 2 are linear combinations of one-body The HS transformation describes the Gibbs ensemble at inverse temperature β = 1/T as a path integral over time-dependent auxiliary fields e β H = D σ G σ U σ G = e 1 2 v 2 ( )d is a Gaussian weight and U σ is a one-body propagator of non-interacting nucleons in time-dependent auxiliary fields Û = T e R ĥ ( )d e β H ( ) with a one-body Hamiltonian ĥ ( ) = P i iˆn i + P s v ( )ˆ s =1 for v <, and s = i for v >

5 Thermal expectation values of observables hôi = Tr (Ôe Ĥ ) Tr (e Ĥ ) = R D[ ]G h Ôi Tr Û R D[ ]G Tr Û where hôi Tr (ÔÛ )/Tr Û Grand canonical quantities in the integrands can be expressed in terms of the single-particle representation matrix U of the propagator : Tr Û =det(1 + U ) ha i a ji Tr (a i a jû ) Tr Û = h 1 1+U 1 iji Canonical (i.e., fixed N,Z) quantities are calculated by an exact particle-number projection (using a discrete Fourier transform). The integrand reduces to matrix algebra in the single-particle space (of typical dimension ~1).

6 Quantum Monte Carlo methods and sign problem σ The high-dimensional integration is done by Monte Carlo methods, sampling the fields according to a weight W G Tr A Û Tr A U / Tr A U is the Monte Carlo sign function. For a generic interaction, the sign can be negative for some of the field configurations. When the average sign is small, the fluctuations become very large the Monte Carlo sign problem. Good-sign Ĥ = P interactions i iˆn i + 1 P We can rewrite 2 v ( + ) where is the time-reversed density. ) Sign rule: when all v <, Tr U > for any configuration and the interaction is known as a good-sign interaction. v < Proof: when all, we have ĥ = P i iˆn i + P (v + v ) and h = h ) the eigenvalues of U appear in complex conjugate pairs and. { i, i } Tr Û = Q i 1+ i 2 >

7 V α (MeV) A practical method for overcoming the sign problem Alhassid, Dean, Koonin, Lang, Ormand, PRL 72, 1 (1994). The dominant collective components of effective nuclear interactions have a good sign Eigenvalues of effective interaction in fp shell K= α A family of good-sign interactions is constructed by multiplying the bad-sign components by a negative parameter g H = H G + gh B g =1 Observables are calculated for 1 <g< and extrapolated to. In the calculation of statistical and collective properties of nuclei, we have used good-sign interactions.

8 Circumventing the odd-particle sign problem in AFMC Mukherjee and Alhassid, PRL 19, 25 (212) Applications of AFMC to odd-even and odd-odd nuclei has been hampered by a sign problem that originates from the projection on odd number of particles. We introduced a method to calculate the ground-state energy of the odd-particle system that circumvents this sign problem. Consider the imaginary-time single-particle Green s functions for even-even nuclei: G Ta a for orbitals =Σ ν = nl j ν( τ) m νm( τ) νm() The energy difference between the lowest energy of the odd-particle system for a given spin j and the ground-state energy of the evenparticle system can be extracted from the slope of ln G ν ( τ ). Minimize Ej ( A± 1) to find the groundstate energy and its spin j. G ν (τ )~ e [ E j ( A±1) E gs ( A)] τ

9 Statistical errors of ground-state energy of Direct AFMC versus Green s function method Green s function method Direct AFMC Green s function method direct SMMC Direct AFMC Pairing gaps in mid-mass nuclei from odd-even mass differences AFMC in the complete fpg 9/2 shell (in good agreement with experiments)

10 Statistical properties in the AFMC method Nakada and Alhassid, PRL 79, 299 (1997) Partition function Calculate the thermal energy E(β) = < H > versus β and integrate ln Z/ β = E( β) to find the partition function Z( β ). Level density The level density (E) Laplace transform: The average state density is found from approximation: ρ ( E) is related to the partition function by an inverse (E) = 1 2i 1 2 2πT C e R i1 i1 d e E Z( ) ( ) S E Z( β ) in the saddle-point S(E) = canonical entropy SE ( ) = lnz+β E C = canonical heat capacity 2 C β E/ = β

11 Heavy nuclei (lanthanides) CI shell model space: protons: 5-82 shell plus 1f 7/2 ; neutrons: shell plus h 11/2 and 1g 9/2 Single-particle Hamiltonian: from Woods-Saxon potential plus spin-orbit Interaction: pairing (g p,g n ) plus multipole-multipole interaction terms quadrupole, octupole, and hexadecupole. Heavy nuclei exhibit various types of collectivity (vibrational, rotational, ) that are well described by empirical models. However, a microscopic description in a CI shell model has been lacking. Can we describe vibrational and rotational collectivity in heavy nuclei using a spherical CI shell model approach in a truncated space?

12 The various types of collectivity are usually identified by their corresponding spectra, but AFMC does not provide detailed spectroscopy. T The behavior of!" 2 J versus is sensitive to the type of collectivity: 12 Dy 12 Dy J!" 2 = E 2 + T!" 2 e E 2 + /T J = is rotational 148 Sm (1 e E 2 + /T ) 2 is vibrational Alhassid, Fang, Nakada, PRL 11, 8251 (28) Ozen, Alhassid, Nakada, PRL 11, 4252 (21)

13 Crossover from vibrational to rotational collectivity in heavy nuclei Ozen, Alhassid, Nakada, PRL 11, 4252 (21) J!" 2 versus T in samarium isotopes Level densities in samarium isotopes Good agreement of AFMC densities with various experimental data sets (level counting, neutron resonance data). ρ(e x ) (MeV -1 ) Level counting n resonance AFMC Sm 15 Sm 152 Sm 154 Sm E x (MeV)

14 Nuclear deformation in the spherical shell model: quadrupole distributions in the laboratory frame Alhassid, Gilbreth, Bertsch, PRL 11, 225 (214) Modeling of shape dynamics, e.g., fission, requires level density as a function of deformation. Deformation is a key concept in understanding heavy nuclei but it is based on a mean-field approximation that breaks rotational invariance. The challenge is to study nuclear deformation in a framework that preserves rotational invariance (e.g., in the CI shell model) without resorting to mean-field approximations. We calculated the distribution of the axial mass quadrupole in the lab frame using an exact projection on (novel in that [Q 2, H ] ). Q 2 Q 2 P β (q) = δ (Q 2 q) = 1 Tr e βh dϕ e iϕ q Tr (e iϕ Q 2π 2 e βh )

15 Application to heavy nuclei T =.1 MeV T = 1.2 MeV T = 4. MeV 154 Sm Sm Rigid rotor SMMC (deformed) P T (q).1.5 At low temperatures, the distribution is similar to that of a prolate rigid rotor a model-independent signature of deformation q (fm 2 ) q (fm 2 ) q (fm 2 ) T =.1 MeV T =.41 MeV T = 4. MeV 148 Sm Sm (spherical) P T (q) q (fm 2 ) q (fm 2 ) q (fm 2 ) The distribution is close to a Gaussian even at low temperatures.

16 Quadrupole distributions P T (β,γ ) Mustonen, Alhassid, Gilbreth, Bertsch vs. intrinsic deformation Information on intrinsic deformation β,γ can be obtained from the expectation values of rotationally invariant combinations of the quadrupole tensor. q q β 2 (q q) q β cos(γ ) (q q) 2 β 4 invariants to 4 th order: ; ; q 2µ ln P T (β,γ ) at a given temperature T is an invariant and can be expanded in the quadrupole invariants [a Landau-like expansion, used for the free energy to describe shape transitions in Alhassid, Levit, Zingman, PRL 57, 59 (198)] ln P T = aβ 2 + bβ cos γ + cβ The expansion coefficients a,b,c can be determined from the expectation values of the invariants, which in turn can be calculated from the low-order moments of q 2 = q in the lab frame. < q q >= 5 < q 2 2 >; <(q q) q >= < q >; <(q q) 2 >= 5 2 < q 4 > 2

17 Expressing the invariants in terms of in the lab frame and integrating over the µ components, we recover P(q 2 ) in the lab frame. q 2µ T = 4. MeV T = 1.19 MeV T =.1 MeV Sm SMMC expansion P(q) q (fm 2 ) -1 1 q (fm 2 ) -1 1 q (fm 2 ) We find excellent agreement with P(q 2 ) calculated in AFMC! T =4. T =1.2 T =.1 lnp(β,γ ) 154 Sm Mimics a shape transition from a deformed to a spherical shape without using a mean-field approximation!

18 Shape distributions in the intrinsic 148 Sm 15 Sm 152 Sm β,γ variables 154 Sm T =4. T =4. T = T = T = T = T = T Thermal shape transition vs. temperature (or excitation energy) T =.71 T =.71 T =.71 T = N Quantum shape transition (at T=) vs. neutron number

19 β,γ We divide the plane into three regions: spherical, prolate and oblate. Integrate over each deformation region to determine the probability of shapes versus temperature using the appropriate metric µ dq 2µ β 4 sin(γ ) dβ dγ oblate prolate spherical Compare deformed ( 154 Sm), transitional ( 15 Sm) and spherical ( 148 Sm) nuclei

20 Level density versus intrinsic deformation Use the saddle-point approximation to convert P T ( ) to level densities vs. E x, (canonical micro canonical) β,γ β,γ Deformed Transitional Spherical E x (MeV) In strongly deformed nuclei, the contributions from prolate shapes dominate the level density below the shape transition energy. In spherical nuclei, both spherical and prolate shapes make significant contributions.

21 Double Beta Decay in AFMC We can do heavy nuclei in the pn-formalism using AFMC AFMC shell-model estimates for open-shell and heavy nuclei Estimate contribution of the model space truncation to the g A quenching E m E i + i mother intermediate daughter E f + f

22 Matrix elements from response functions in AFMC Ô(t) =e th Ôe th hô (t)ô()i = 1  i e be i  e be i e t(e f E i ) hf Ô ii 2 fi At the limit b!, t b, and large t, only the lowest possible E f = Ef and E i = Ei contribute: hô (t)ô()i = e t(e f E i ) hf Ô i i 2 If Ô is the mass quadrupole ˆQ 2,foraneven-evennucleus hô (t)ô()i = e te 2+ x h2 + 1 ˆQ 2 + g.s.i 2 where is the lowest 2+ state (C.N. Gilbreth et al.)

23 2nbb in AFMC Closure approximation: M 2n = Â m h + f G ~ mi hm G ~ + i i 1 E m 2 (E f + Ei ) h+ f G ~ G ~ + i i 1 Ē 2 (E f + Ei ) where G = Â a ~s a t a. With Ô = ~ G ~ G (a separable two-body operator): hô (t)ô()i = e t(e f E i ) h + f ~ G ~ G + i i 2 To go beyond the closure approximation, a more complicated approach is needed (P.B. Radha et al., PRL 7, 242 (199) for 48 Ca, 7 Ge)

24 nbb in AFMC Can be calculated in the closure approximation The Gamow-Teller part: Ô = 2R Z p Non-separable two-body operator AFMC could then be used to evaluate dq q  ab j (qr ab )h GT (q)~s a ~s b t a t b q + Ē (E i + E hô (t)ô()i = e t(e f Ei ) h + f Ô + i i {z } MGT n The method used for calculating response functions of one-body operators can be generalized to two-body operators f ) 2

25 Conclusion Finite-temperature AFMC is a powerful method for the microscopic calculation of statistical and collective properties of nuclei in very large model spaces; applications in nuclei as heavy as the lanthanides. Microscopic description of collectivity in heavy nuclei Statistical properties as a function of intrinsic deformation in a rotationally invariant framework (CI shell model) without the use of a mean-field approximation Outlook Applications to other mass regions (actinides, unstable nuclei, ). Matrix elements for neutrinoless double beta decay in very large shell model spaces