Impurity Lattice Monte Carlo for Hypernuclei
EEPJ manuscript No. (will be inserted by the editor)
Impurity Lattice Monte Carlo for Hypernuclei
Dillon Frame , Timo A. L¨ahde , Dean Lee , and Ulf-G. Meißner , , Institut f¨ur Kernphysik, Institute for Advanced Simulation and J¨ulich Center for Hadron Physics,Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany Facility for Rare Isotope Beams and Department of Physics and Astronomy, Michigan State University,MI 48824, USA Helmholtz-Institut f¨ur Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universit¨at Bonn,D-53115 Bonn, Germany Tbilisi State University, 0186 Tbilisi, GeorgiaReceived: date / Revised version: date
Abstract.
We consider the problem of including Λ hyperons into the ab initio framework of nuclear latticeeffective field theory. In order to avoid large sign oscillations in Monte Carlo simulations, we make useof the fact that the number of hyperons is typically small compared to the number of nucleons in thehypernuclei of interest. This allows us to use the impurity lattice Monte Carlo method, where the minorityspecies of fermions in the full nuclear Hamiltonian is integrated out and treated as a worldline in Euclideanprojection time. The majority fermions (nucleons) are treated as explicit degrees of freedom, with theirmutual interactions described by auxiliary fields. This is the first application of the impurity lattice MonteCarlo method to systems where the majority particles are interacting. Here, we show how the impurityMonte Carlo method can be applied to compute the binding energy of the light hypernuclei. In thisexploratory work we use spin-independent nucleon-nucleon and hyperon-nucleon interactions to test thecomputational power of the method. We find that the computational effort scales approximately linearlyin the number of nucleons. The results are very promising for future studies of larger hypernuclear systemsusing chiral effective field theory and realistic hyperon-nucleon interactions, as well as applications to otherquantum many-body systems. PACS.
Hypernuclei are bound states of one or two hyperons to-gether with a core composed of nucleons. They extend thenuclear chart into a third dimension, augmenting the usualtwo dimensions of proton number and neutron number.We will use the notation Y for a Λ or Σ hyperon and N fora nucleon. Due to the scarcity of direct hyperon-nucleon( Y N ) and hyperon-hyperon (
Y Y ) scattering data, theseunusual forms of baryonic matter play an important rolein pinning down the fundamental baryon-baryon forces.This requires on the one hand an effective field theory(EFT) description of the underlying forces, as pioneeredin Ref. [1,2], and on the other hand a numerically pre-cise and consistent method to solve the nuclear A -bodyproblem, such as nuclear lattice EFT (NLEFT) [3,4]. Forcalculations combining these chiral EFT forces at LO andNLO [5,6] with other many-body methods, see e.g . Ref. [7,8,9,10,11,12].In view of the success of NLEFT in the descriptionof nuclear spectra and reactions, it seems natural to ex-tend this method to hypernuclei. However, this is notquite straightforward. While one can extend the four spin- isospin degrees of freedom comprising the nucleons to in-clude the Λ and Σ states [13], this has not been donebecause there is no longer an approximate symmetry suchas Wigner’s SU(4) symmetry [23] that protects the MonteCarlo (MC) simulations against strong sign oscillationswhen using auxiliary fields. The physics of hypernucleitherefore requires a different approach, and in this paperwe show how the computational problems are solved usingthe impurity lattice Monte Carlo (ILMC) method.The ILMC method was introduced in Ref. [15] in thecontext of a Hamiltonian theory of spin-up and spin-downfermions, and applied to the intrinsically non-perturbativephysics of Fermi polarons in two dimensions in Ref. [16].The ILMC method is particularly useful for the case whereonly one fermion (of either species) is immersed in a “sea”of the other species. Within the standard auxiliary fieldMonte Carlo method, such an extreme imbalance wouldlead to unacceptable sign oscillations in the Monte Carloprobability weight. In the ILMC method, the minority In the SU(3) limit of equal up, down and strange quarkmasses, such a spin-flavor symmetry might be restored [14],but this limit is far from the physical world. a r X i v : . [ nu c l - t h ] J u l Dillon Frame et al.: Impurity Lattice Monte Carlo for Hypernuclei particle is “integrated out”, resulting in a formalism whereonly the majority species fermions appear as explicit de-grees of freedom, while the minority fermion is representedby a “worldline” in Euclidean projection time. The spatialposition of this worldline is updated using Monte Carlo up-dates, while the interactions between the majority fermionsare described by the auxiliary field formalism [4].Here, we apply the ILMC method to the inclusion ofhyperons into NLEFT simulations. We identify the Λ hy-peron as the minority species, which we represent by aworldline in Euclidean time. This Λ worldline is treated asimmersed in an environment consisting of some number ofnucleons. We focus on the Monte Carlo calculation of thebinding energy of light hypernuclei, by means of a simpli-fied Y N interaction, consisting of a single contact interac-tion, tuned to a best description of the the empirical bind-ing energies of the s -shell hypernuclei with A = 3 , , For the
N N interaction, we use a simple leading order in-teraction similar to that described in Ref. [17]. We bench-mark our ILMC results against Lanczos calculations oftransfer matrix and exact Euclidean projection calcula-tions with initial states and number of time steps thatmatch the ILMC calculations. We note that our MonteCarlo method is free from any approximation about thenodal structure of the many-body wave function. This isthe first application of such unconstrained Monte Carlosimulations to hypernuclei.This paper is organized as follows. In Sec. 2, we presentthe
Y N and
N N interactions used for this study. In Sec. 3,we derive the impurity worldline formalism for the chosen
Y N interaction, and introduce the concept of the “reducedtransfer matrix”, which refers to the nucleon degrees offreedom only. In Sec. 4, we discuss the Monte Carlo up-dating of the hyperon worldline and the auxiliary fields,which encode the interactions between nucleons. In Sec. 5,we present results for the ground state energies of the s -shell nuclei and hypernuclei. In Sec. 6, we conclude with adiscussion of future improvements and applications of theimpurity lattice Monte Carlo method to hypernuclei andother quantum many-body systems. We develop the ILMC formalism following Ref. [15], whoconsidered a system of spin-up and spin-down fermions,with a contact interaction which operates between fermionsof opposite spin. The situation here is completely analo-gous, we have one majority species, the nucleons, and one impurity, the Λ . As usual in NLEFT, we consider positionson a spatial lattice denoted by n and lattice spacing a . Wealso assume that Euclidean time has been discretized, suchthat slices of the Euclidean time are denoted by n t withtemporal lattice spacing a t . The partition function can be We are well aware of the importance of the ΛN - Σ N tran-sition. However, we choose a simple starting point for this ex-ploratory study and will consider more realistic interactions ina later publication. expressed in terms of the Grassmann path integral Z = (cid:90) (cid:34) (cid:89) n ,n t s = N,Y dζ s ( n , n t ) dζ ∗ s ( n , n t ) (cid:35) exp( − S [ ζ, ζ ∗ ]) , (1)where the subscripts N and Y refer to all nucleon andhyperon degrees of freedom, respectively. In this studywe consider only Λ hyperons. In future work we will alsoconsider Σ hyperons or account for their influence viathree-baryon interactions involving a Λ and two nucle-ons. We also make the simplifying assumption that thehyperon-nucleon and nucleon-nucleon interaction are spin-independent and neglect Coulomb interactions.Assuming that the exponent of the Euclidean actionin Eq. (1) is treated by a Trotter decomposition, we find S [ ζ, ζ ∗ ] ≡ (cid:88) n t (cid:26) S t [ ζ, ζ ∗ , n t ] + S Y [ ζ , ζ ∗ , n t ]+ S N [ ζ , ζ ∗ , n t ] + S Y N [ ζ, ζ ∗ , n t ] + S NN [ ζ, ζ ∗ , n t ] (cid:27) , (2)where the component due to the time derivative is S t [ ζ, ζ ∗ , n t ] ≡ (cid:88) n ,s = N,Y ζ ∗ s ( n , n t ) × (cid:20) ζ s ( n , n t + 1) − ζ s ( n , n t ) (cid:21) , (3)while S Y and S N describe the kinetic energies of the hy-perons and nucleons, respectively. Further, S Y N providesthe
Y N interaction, and S NN the N N interaction.Derivations of Feynman rules are usually easier to per-form in the Grassmann field formalism. However, actualNLEFT calculations are performed using the transfer ma-trix Monte Carlo method. As noted in Ref. [15], the Grass-mann and transfer matrix formulations are related byTr (cid:8) : f N t − [ a s ( n ) , a † s (cid:48) ( n (cid:48) )] : · · · : f [ a s ( n ) , a † s (cid:48) ( n (cid:48) )] : (cid:9) = (cid:90) (cid:34) (cid:89) n ,n t s = N,Y dζ s ( n , n t ) dζ ∗ s ( n , n t ) (cid:35) exp (cid:32) − (cid:88) n t S t [ ζ, ζ ∗ , n t ] (cid:33) × N t − (cid:89) n t =0 f n t (cid:2) ζ s ( n , n t ) , ζ ∗ s (cid:48) ( n (cid:48) , n t ) (cid:3) , (4)where f is an arbitrary function, a † s and a s denote cre-ation and annihilation operators for the fermion degreesof freedom, and colons signify normal ordering. We shallnow consider the explicit forms of the Y N and
N N in-teractions, and use Eq. (4) to relate expressions in theGrassmann and transfer matrix formulations. illon Frame et al.: Impurity Lattice Monte Carlo for Hypernuclei 3
For the hyperons, we take for simplicity the lowest-order(unimproved) kinetic energy S Y [ ζ, ζ ∗ , n t ] ≡ h (cid:88) n ζ ∗ Y ( n , n t ) ζ Y ( n , n t ) − h (cid:88) n (cid:88) l =1 ζ ∗ Y ( n , n t ) (cid:20) ζ Y ( n + ˆ e l , n t ) + ζ Y ( n − ˆ e l , n t ) (cid:21) , (5)with h ≡ α t m Y , (6)where m Y is the hyperon mass, and we have defined α t ≡ a t /a as the ratio of temporal and spatial lattice spacings.The Y N interaction is given by S Y N [ ζ, ζ ∗ , n t ] ≡ α t C Y N (cid:88) n ρ N ( n , n t ) ρ Y ( n , n t ) , (7)in terms of the nucleon and hyperon densities, respectively.The tuning of the coupling constant C Y N is discussed inSection 5.Using Eq. (4), the hyperon contributions are describedby the transfer matrix operatorˆ M = : exp (cid:32) − α t ˆ H − α t C Y N (cid:88) n ˆ ρ N ( n )ˆ ρ Y ( n ) (cid:33) : , (8)where ˆ ρ N ( n ) ≡ (cid:88) i,j ˆ ρ i,j ( n ) ≡ (cid:88) i,j a † i,j ( n ) a i,j ( n ) , (9)and ˆ ρ Y ( n ) ≡ a † Y ( n ) a Y ( n ) , (10)are density operators for nucleons and hyperons, respec-tively. The a i,j ( n ) and a † i,j ( n ) are lattice annihilation andcreation operators for nucleons on site n with spin i = 0 , j = 0 , H ≡ ˆ H N + ˆ H Y , (11)whereˆ H Y ≡ m Y (cid:88) n (cid:88) l =1 (cid:20) a † Y ( n ) a Y ( n ) − a † Y ( n ) a Y ( n + ˆ e l ) − a † Y ( n ) a Y ( n − ˆ e l ) (cid:21) , (12)denotes the (lowest order) kinetic energy for the hyperonsin the operator formalism [3]. Here, the ˆ e l are unit vectorsin lattice direction l . These lattice operators correspondto the continuum expressionsˆ H Y = 12 m Y (cid:90) d r ∇ a † Y ( r ) · ∇ a Y ( r ) , (13) for the kinetic energy, andˆ V Y N = C Y N (cid:90) d r ˆ ρ Y ( r )ˆ ρ N ( r ) , (14)for the Y N interaction (with m Y and C Y N in physicalrather than lattice units). Note that this is a simplifiedversion of the pionless EFT calculation of Ref. [18], whichalso included a three-body interaction at LO. Such an in-teraction is sub-leading in chiral EFT approaches (such asNLEFT). See also the recent work in Ref. [19].
For the free Hamiltonian of the nucleon degrees of free-dom, we likewise use the lowest-order expressionˆ H N ≡ m N (cid:88) n ˆ ρ N ( n , n ) − m N (cid:88) n (cid:88) l =1 [ˆ ρ N ( n , n + ˆ e l ) + ˆ ρ N ( n , n − ˆ e l )] , (15)where ˆ ρ N ( n , n (cid:48) ) ≡ (cid:88) i,j a † i,j ( n ) a i,j ( n (cid:48) ) , (16)and m N is the nucleon mass.The Wigner SU(4)-symmetric part of the leading-order(LO) N N interaction of Refs. [20,21,22] is used for thepresent work. This is an approximate symmetry [23] of thelow-energy nucleon-nucleon interactions, where the nucle-onic spin and isospin degrees of freedom can be rotated asfour components of an SU(4) multiplet. Hence, we haveˆ V NN ≡ C NN (cid:88) n , n (cid:48) , n (cid:48)(cid:48) ˆ ρ sN ( n (cid:48) ) f s L ( n (cid:48) − n ) × f s L ( n − n (cid:48)(cid:48) )ˆ ρ sN ( n (cid:48)(cid:48) ) : , (17)where ˆ ρ sN ( n ) ≡ (cid:88) i,j a s NL † i,j ( n ) a s NL i,j ( n ) , (18)is the smeared nucleon density operator, and the (local)smearing function f s L is defined as f s L ( n ) ≡ | n | = 0 , ≡ s L for | n | = 1 , ≡ , (19)and the operators a s NL † i,j ( n ) and a s NL i,j ( n ) are defined interms of (non-local) smearing a s NL i,j ( n ) ≡ a i,j ( n ) + s NL (cid:88) | n (cid:48) | =1 a i,j ( n + n (cid:48) ) , (20) Dillon Frame et al.: Impurity Lattice Monte Carlo for Hypernuclei and a s NL † i,j ( n ) ≡ a † i,j ( n ) + s NL (cid:88) | n (cid:48) | =1 a † i,j ( n + n (cid:48) ) , (21)where the values of the parameters C NN , s L and s NL for the present work are discussed in Section 5 (see alsoRef. [17] for a full treatment).For ILMC calculations with the N N interaction in-cluded, we reduce the expressions quadratic in the densityoperators using the relation: exp (cid:18) − C NN α t ˜ ρ ( n ) (cid:19) : = 1 √ π (cid:90) ∞−∞ dφ exp( − φ / × : exp (cid:16)(cid:112) − C NN α t φ ˜ ρ ( n ) (cid:17) : , (22)where ˜ ρ ( n ) ≡ (cid:88) n (cid:48) f s L ( n − n (cid:48) )ˆ ρ sN ( n (cid:48) ) , (23)such that φ ( n , n t ) is treated as a scalar auxiliary (Hubbard-Stratonovich) field. The N N term in the transfer matrixcan then be written as: exp( − α t ˆ V NN ) : = (cid:90) (cid:89) n (cid:20) dφ ( n , n t ) √ π (cid:21) exp (cid:32) − (cid:88) n φ ( n , n t ) (cid:33) : exp( − ˆ V φN ) : , (24)for Euclidean time slice n t , where: exp( − ˆ V φN ) : =: exp (cid:112) − C NN α t (cid:88) n , n (cid:48) φ ( n , n t ) f s L ( n − n (cid:48) )ˆ ρ sN ( n (cid:48) ) : . (25)In the ILMC calculations, the path integral over theauxiliary field φ is evaluated using either local Metropo-lis algorithm updates or global lattice updates using thehybrid Monte Carlo algorithm. See Ref. [17] for details onefficient updating of the products of auxiliary-field trans-fer matrices. We shall now integrate out the hyperon degrees of free-dom and derive a “reduced” transfer matrix, which refersto the nucleon degrees of freedom only. For simplicity (andwithout loss of generality), we shall neglect the
N N inter-action term for the purpose of the derivation, and considerthe case of a single hyperon Y and nucleon N (which canbe thought of as representing any one of the spin-isospincombinations i, j of the full theory). Let us write down the transfer matrix element betweentime slices n t and n t + 1 in terms of | χ Nn t , χ Yn t (cid:105) ≡ (cid:89) n (cid:26)(cid:104) a † N ( n ) (cid:105) χ Nnt ( n ) (cid:104) a † Y ( n ) (cid:105) χ Ynt ( n ) (cid:27) | (cid:105) , (26)where the χ sn t ( n ) count the occupation numbers for nu-cleons and hyperons on time slice n t and spatial latticesite n . Following the relations established in Ref. [15], weexpress the transfer matrix element as (cid:104) χ Nn t +1 , χ Yn t +1 | ˆ M | χ Nn t , χ Yn t (cid:105) = (cid:89) n (cid:34) −→ ∂∂ζ ∗ N ( n , n t ) (cid:35) a (cid:34) −→ ∂∂ζ ∗ Y ( n , n t ) (cid:35) b X ( n t ) M ( n t ) × (cid:89) n (cid:48) (cid:34) ←− ∂∂ζ ∗ N ( n (cid:48) , n t ) (cid:35) c (cid:34) ←− ∂∂ζ ∗ Y ( n (cid:48) , n t ) (cid:35) d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ ∗ N = ζ N =0 ζ ∗ Y = ζ Y =0 , (27)where a = χ Nn t +1 ( n ) , b = χ Yn t +1 ( n ) , (28)and c = χ Nn t ( n (cid:48) ) , d = χ Yn t ( n (cid:48) ) , (29)are integers which assume values of either 0 or 1. Further, X ( n t ) ≡ (cid:89) n exp( ζ ∗ N ( n , n t ) ζ N ( n , n t )) × exp( ζ ∗ Y ( n , n t ) ζ Y ( n , n t )) , (30)and M ( n t ) ≡ exp( − S kin [ ζ, ζ ∗ , n t ]) exp( − S int [ ζ, ζ ∗ , n t ]) , (31)are Grassmann functions (to be defined below).The impurity worldline is considered static for the pur-poses of this derivation, although it will be updated bythe Metropolis algorithm in the actual Monte Carlo simu-lations. From one time slice to the next, the impurity mayeither remain on the same lattice site, or hop to a nearest-neighbor site. For the case where the impurity remains ona given lattice site n (cid:48)(cid:48) , we have (cid:104) χ Nn t +1 , χ Yn t +1 | ˆ M | χ Nn t , χ Yn t (cid:105) = (cid:89) n (cid:34) −→ ∂∂ζ ∗ N ( n , n t ) (cid:35) χ Nnt +1 ( n ) /X ( n t ) /M n (cid:48)(cid:48) , n (cid:48)(cid:48) ( n t ) × (cid:89) n (cid:48) (cid:34) ←− ∂∂ζ ∗ N ( n (cid:48) , n t ) (cid:35) χ Nnt ( n (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ ∗ N = ζ N =0 , (32)with χ Yn t ( n (cid:48)(cid:48) ) = 1 , χ Yn t +1 ( n (cid:48)(cid:48) ) = 1 , (33) illon Frame et al.: Impurity Lattice Monte Carlo for Hypernuclei 5 and hence /X ( n t ) /M n (cid:48)(cid:48) , n (cid:48)(cid:48) ( n t ) = −→ ∂∂ζ ∗ Y ( n (cid:48)(cid:48) , n t ) X ( n t ) exp( − S kin [ ζ, ζ ∗ , n t ]) × exp( − S int [ ζ, ζ ∗ , n t ]) ←− ∂∂ζ Y ( n (cid:48)(cid:48) , n t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ ∗ Y = ζ Y =0 , (34)where /X ( n t ) ≡ (cid:89) n exp( ζ ∗ N ( n , n t ) ζ N ( n , n t )) . (35)and we define /M ( n t ) as the “reduced” transfer matrix.Specifically, we take S kin [ ζ, ζ ∗ , n t ] ≡ S N [ ζ, ζ ∗ , n t ] + S Y [ ζ , ζ ∗ , n t ] , (36)with the nearest-neighbor expression Eq. (5) for the hy-peron kinetic term. We also take S int [ ζ, ζ ∗ , n t ] ≡ S Y N [ ζ, ζ ∗ , n t ] , (37)as the Y N interaction. By evaluating the derivatives inEq. (34), we find /M n (cid:48)(cid:48) , n (cid:48)(cid:48) ( n t ) = exp( − S N [ ζ, ζ ∗ , n t ]) × (1 − h − α t C Y N ρ N ( n (cid:48)(cid:48) , n t )) , (38)which we write as /M n (cid:48)(cid:48) , n (cid:48)(cid:48) ( n t ) = (1 − h ) exp( − S N [ ζ, ζ ∗ , n t ]) × (cid:18) − α t C Y N − h ρ N ( n (cid:48)(cid:48) , n t ) (cid:19) , (39)or /M n (cid:48)(cid:48) , n (cid:48)(cid:48) ( n t ) (cid:39) (1 − h ) × exp (cid:18) − S N [ ζ, ζ ∗ , n t ] − α t C Y N − h ρ N ( n (cid:48)(cid:48) , n t ) (cid:19) , (40)where the last factor, which encodes the interaction be-tween the nucleons and the single hyperon impurity, hasbeen exponentiated. Thus, Eq. (40) is the reduced Grass-mann transfer matrix for the case where the impurityworldline remains stationary. For the case of a long-range Y N interaction, Eq. (40) should be replaced by an expres-sion of the form /M n (cid:48)(cid:48) , n (cid:48)(cid:48) ( n t ) (cid:39) (1 − h ) exp (cid:18) − S N [ ζ, ζ ∗ , n t ] − α t − h (cid:88) n (cid:48) G ( n (cid:48) − n (cid:48)(cid:48) ) ρ N ( n (cid:48) , n t ) (cid:19) , (41)whereby the hyperon impurity now also interacts with nu-cleons not on the same spatial lattice site. If we take G ( n (cid:48) − n ) = C Y N δ ( n (cid:48) − n ) , (42) then Eq. (40) for a contact interaction is recovered.Another possibility permitted by the nearest-neighbor Y N kinetic term is χ Yn t ( n (cid:48)(cid:48) ) = 1 , χ Yn t +1 ( n (cid:48)(cid:48) ± ˆ e l ) = 1 , (43)such that /X ( n t ) /M n (cid:48)(cid:48) ± ˆ e l , n (cid:48)(cid:48) ( n t ) = −→ ∂∂ζ ∗ Y ( n (cid:48)(cid:48) ± ˆ e l , n t ) X ( n t ) exp( − S kin [ ζ, ζ ∗ , n t ]) × exp( − S int [ ζ, ζ ∗ , n t ]) ←− ∂∂ζ Y ( n (cid:48)(cid:48) , n t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ ∗ Y = ζ Y =0 , (44)which gives /M n (cid:48)(cid:48) ± ˆ e l , n (cid:48)(cid:48) ( n t ) = h exp ( − S N [ ζ, ζ ∗ , n t ]) , (45)for the reduced Grassmann transfer matrix, when the im-purity hops to a neighboring lattice site.Having determined the form of the reduced Grassmanntransfer matrices, we may translate these to the transfermatrix formulation. The corresponding operators areˆ /M n (cid:48)(cid:48) , n (cid:48)(cid:48) = (1 − h ) : exp (cid:18) − α t ˆ H N − α t C Y N − h ˆ ρ N ( n (cid:48)(cid:48) ) (cid:19) : , (46)from Eq. (40), andˆ /M n (cid:48)(cid:48) ± ˆ e l , n (cid:48)(cid:48) = h : exp( − α t ˆ H N ) : , (47)from Eq. (45). The nucleon kinetic energy ˆ H N in Eqs. (46)and (47) is given by Eq. (15), and the nucleon densityoperator ˆ ρ N in Eq. (46) by Eq. (9).A few comments are in order about our implementa-tion of the ILMC formalism. In our MC codes, Eq. (46) isevaluated asˆ /M n (cid:48)(cid:48) , n (cid:48)(cid:48) ∼ (cid:18) − α t ˆ H N − α t C Y N − h ˆ ρ N ( n (cid:48)(cid:48) ) − ˆ V φN (cid:19) , (48)where the prefactor (1 − h ) and the Gaussian term fromthe Hubbard-Stratonovich transformation has not beenwritten out. Note that Eq. (48) also includes the N N in-teraction through Eq. (25). For ILMC, the nucleons aretreated as distinguishable particles, and the hyperon asa classical worldline during the Euclidean time evolution.This induces a three-body interaction when two nucleonsand the hyperon occupy the same site, which is absent inEq. (8). As we shall benchmark our ILMC codes againstexact Euclidean time projection calculations of Eq. (8),we include the induced interactionˆ H Y NN = − α t C Y N − h ) (cid:88) n ˆ ρ N ( n )ˆ ρ N ( n )ˆ ρ Y ( n ) , (49)to the original transfer matrix (8). This induced three-body interaction is a lattice artifact which disappears when α t → Dillon Frame et al.: Impurity Lattice Monte Carlo for Hypernuclei
We now describe how ILMC calculations are performedusing the Projection Monte Carlo (PMC) method. Let usfirst assume that the impurity has been fixed at a givenspatial lattice site, and that no “hopping” of the impurityoccurs during the Euclidean time evolution. We shall thenrelax this constraint, and discuss a practical algorithm forupdating the configuration of the hyperon worldline.
For a stationary hyperon impurity, the reduced transfermatrix is given by Eq. (46), and for the purposes of thePMC calculation, we define the Euclidean projection am-plitude Z jk ( N t ) ≡ (cid:104) ψ j | ˆ /M N t | ψ k (cid:105) , (50)for a product of N t Euclidean time slices, where j and k denote different initial cluster states. As usual, this isexpressed as a determinant of single-particle amplitudes,which gives Z jk ( N t ) = det M jkp × p , (51)where M jkp × p = (cid:104) φ ,j | ˆ /M N t | φ ,k (cid:105) (cid:104) φ ,j | ˆ /M N t | φ ,k (cid:105) · · ·(cid:104) φ ,j | ˆ /M N t | φ ,k (cid:105) (cid:104) φ ,j | ˆ /M N t | φ ,k (cid:105) · · · ... ... . . . , (52)for p nucleons. By means of the projection amplitudes (51),we construct[ ˆ M a ( N t )] qq (cid:48) ≡ (cid:88) q (cid:48)(cid:48) Z − qq (cid:48)(cid:48) ( N t ) Z q (cid:48)(cid:48) q (cid:48) ( N t + 1) , (53)which is known as the “adiabatic transfer matrix”. If wedenote the eigenvalues of (53) by λ i ( N t ), we find λ i ( N t ) = exp( − α t E i ( N t + 1 / , (54)such that the low-energy spectrum is given by the “tran-sient” energies E i ( N t + 1 /
2) = − log( λ i ( N t )) α t , (55)at finite temporal lattice spacing a t . For the case of a singletrial cluster state with p nucleons, Eq. (51) reduces to Z ( N t ) = det M p × p , (56)for the case of a single trial state. The ground-state energyis obtained from E ( N t + 1 /
2) = − log( Z ( N t + 1) /Z ( N t )) α t , (57)in the limit N t → ∞ , where the exact low-energy spec-trum of the transfer matrix will be recovered. Note that the argument N t + 1 / N t + 1 and N t .As an example, for the hypertriton we have p = 2nucleons after the impurity hyperon has been integratedout. We start the Euclidean time projection with a sin-gle initial trial cluster state ( j = k = 0) consisting of aspin-up proton, and a spin-up neutron. As there are noterms that mix spin or isospin, the other components ofeach single-particle state are set to zero, and remain soduring the PMC calculation. For the spatial parts of thenucleon wave functions, we may choose, for example, thezero-momentum state | φ , (cid:105) = | φ , (cid:105) = (cid:104) , , (cid:105) , (58)in the notation of Ref. [15], which denotes plane-wave or-bitals in a cubic box. In principle, we may also chooseany other plane-wave state with non-zero momentum (seeTable 1 of Ref. [15]), or any other more complicated trialstate. For the heavier nuclei, it is indeed better to choosean initial state where the nucleons are clustered together.In this case we sum over all possible translations of thecluster in order construct an initial state with zero totalmomentum. If the hyperon impurity is allowed to hop between nearest-neighbor sites (from one Euclidean time slice to the next),the Euclidean projection amplitude becomes a sum overhyperon worldline configurations. This gives Z jk ( N t ) ≡ (cid:88) n ,..., n Nt (cid:104) ψ j | ˆ /M N t { n j } | ψ k (cid:105) , (59)where the productˆ /M N t { n j } ≡ ˆ /M n Nt , n Nt − ˆ /M n Nt − , n Nt − . . . ˆ /M n , n ˆ /M n , n , (60)is expressed in terms of the reduced transfer matrices (46)and (47). Here, n j denotes the spatial position of the hy-peron impurity (which has been integrated out) on timeslice j . The expressions for the projection amplitude anddeterminant are generalized to Z jk ( N t ) = (cid:88) n ,..., n Nt det M jkp × p , (61)where M jkp × p = (cid:104) φ ,j | ˆ /M N t { n j } | φ ,k (cid:105) (cid:104) φ ,j | ˆ /M N t { n j } | φ ,k (cid:105) · · ·(cid:104) φ ,j | ˆ /M N t { n j } | φ ,k (cid:105) (cid:104) φ ,j | ˆ /M N t { n j } | φ ,k (cid:105) · · · ... ... . . . , (62) illon Frame et al.: Impurity Lattice Monte Carlo for Hypernuclei 7 such that the determinant is now to be computed over allpossible hyperon wordline configurations.We note that the worldline configuration is to be up-dated stochastically using a Metropolis algorithm. Thus,proposed changes in the impurity worldline are acceptedor rejected by importance sampling with | Z jj ( N t ) | as theprobability weight function. Here, j denotes one of theinitial trial nucleon cluster states. The updating of the impurity worldline is handled in twosteps: The generation of a new proposed worldline, anda Metropolis accept/reject step to determine whether touse the generated worldline. For this work, the worldline W ( n , n t ) is a function of only the lattice site n and theEuclidean time step n t , and is equal to 1 where the impu-rity is present, and 0 at all other lattice points. From theexpressions of the reduced transfer matrices, the worldlineat two adjacent time steps, W ( n (cid:48) , n t ) and W (cid:48) ( n (cid:48) , n t + 1)must obey the relation | n − n (cid:48) | ≤
1. For an illustration ofthe impurity (hyperon) worldline, see Fig. 1. space E u c li dean t i m e Λ Fig. 1.
Illustration of the hyperon worldline. In the reducedtransfer matrix formalism, the hyperon has been “integratedout”, and the interaction between the hyperon and the nucle-ons is mediated by an effective “background field” generatedby the hyperon worldline.
For the non-interacting worldline, we can generate newconfigurations from the free probabilities, as determinedfrom the reduced transfer matrices. In this case, P h = h isthe hopping probability, and P s = (1 − h ) is the probabil-ity to remain stationary. When initializing the worldlineat the beginning of the MC simulation, we may start from a configuration where the worldline is completely station-ary (“cold start”) or one where the worldline either hopsor remains stationary at each time step according to theprobabilities P h and P s (“warm start”).At the beginning of every sweep through the lattice,we propose a new worldline to use for that sweep. This isdone by taking the previous worldline and choosing a ran-dom time at which we cut the worldline and regeneratingit either in the forwards and backwards time direction.The new worldline is then accepted or rejected using aMetropolis accept or reject condition to preserve detailedbalance associated with the absolute value of the ampli-tude. For the results presented in what follows, we use a spa-tial lattice spacing a = 1 / (100 MeV) and temporal lat-tice spacing of a t = 1 / (300 MeV). The non-local smear-ing parameter is chosen to be s NL = 0 .
2, and the localsmearing parameter is set to s L = 0 .
0. Since we onlyconsider s -shell nuclei and hypernuclei in this study, thelocal attraction provided by s L for heavier nuclei is notneeded [20]. The coupling constant C NN is set to − . × − MeV − , and this combination of parameters yieldsa nucleon-nucleon scattering length a NN = 6 .
86 fm andeffective range r NN = 1 .
77 fm. As stated before, in thisstudy the spin-dependent terms of the nucleon-nucleon in-teraction are not accounted for.For the
Y N interaction, we set C Y N according to thebest overall fit to the light hypernuclei. Fitting to the Λ separation energies for Λ H, Λ H/He, and Λ He, we find C Y N = − . × − MeV − . This gives a Y N = − .
45 fmfor the scattering length and r Y N = − .
45 fm for theeffective range. In Table 1, we present benchmark calcula-tions of the ILMC results for Λ H in comparison with exacttransfer matrix calculations. We show the results for theenergy as a function of Euclidean projection time.
Table 1.
ILMC results for the energy of Λ H versus Euclideantime in comparison with exact transfer matrix results for peri-odic box length 15.8 fm. N t t (MeV − ) ILMC (MeV) Exact (MeV)50 0.1667 − − − − − − − − − − − − We see that the agreement is quite good. The initialnucleon trial states for these calculations are taken to bespatially constant functions, which correspond to single-particle states of zero momentum in a periodic cubic box.
Dillon Frame et al.: Impurity Lattice Monte Carlo for Hypernuclei
The hyperon initial wave function is also taken be a con-stant function. These exact transfer matrix calculationsinclude the induced three-baryon interaction described inEq. (49).In Table 2, we present exact Lanczos transfer matrixcalculations of the ground state of H, Λ H, and separationenergy B Λ , as a function of periodic box length. In thiswork, we also present the exact Lanczos transfer matrixcalculation wherever it is computationally possible andusing Monte Carlo for cases where it is not. Given the ex-tremely small Λ separation energy, it is necessary to go tovery large volumes in order to remove finite volume arti-facts. Interestingly, B Λ is found to be relatively constantwith the periodic box size L . This suppression of the finitevolume dependence is an indication that the asymptoticnormalization coefficient of the hypertriton wave functionis small [24,25]. Table 2.
Exact transfer matrix results for H, Λ H, and theseparation energy B Λ versus periodic box length. L (fm) H (MeV) Λ H (MeV) B Λ (MeV)15.8 − − − − − − − − − − − − − − − − In Fig. 2, we present ILMC results for the Λ H/He en-ergy versus Euclidean time. These calculations use a pe-riodic box size of L = 15 . N t = 300 Eu-clidean time steps. In order to extract the ground stateenergy, we use the extrapolation ansatz E ( t ) = E + c exp( − ∆Et ) , (63)which takes into account the residual dependence of thefirst excited state that couples to our initial state. For thiscalculation, we use an initial state where the nucleon stateshave a spatially decaying exponential form with respect tothe nucleus center of mass, while the initial hyperon wavefunction is a constant function.In Fig. 3, we show lattice Monte Carlo (LMC) resultsfor the He energy versus Euclidean time. As there are nohyperons in this system, these are auxiliary field MonteCarlo calculations without impurity worldlines. These cal-culations use a periodic box size of L = 9 . N t = 150 Euclidean time steps. In order to extract theground state energy, we again use the exponential ansatz in Eq. (63). For this calculation, we again use an initialstate where the nucleons have a spatially-decaying expo-nential form with respect to the nucleus center of mass.In Fig. 4, ILMC results are shown for the Λ He energyversus Euclidean time. These calculations use a periodic -9.5-9-8.5-8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 E ( M e V ) t (MeV -1 )impurity lattice MCextrapolation fitextrapolated value Fig. 2.
ILMC results for the Λ H/He energy versus Euclideanprojection time in a periodic box size of L = 15 . ansatz for the asymptotic time dependence. -27-26.5-26-25.5-25-24.5-24-23.5-23 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 E ( M e V ) t (MeV -1 ) lattice MCextrapolation fitextrapolated value Fig. 3.
LMC results for the He energy versus Euclidean pro-jection time in a periodic box size of L = 9 . ansatz for theasymptotic time dependence. box size of L = 9 . N t = 250 Euclidean timesteps. We again use the exponential ansatz from Eq. 63to extract the ground state energy. Similar to the Λ H/Hecalculation, here we use an initial state where the nucleonshave a spatially decaying exponential form with respect tothe nucleus center of mass, while the initial hyperon wavefunction is a constant function.In Table 3, we present the lattice results for all of the s -shell nuclei and hypernuclei. The exact transfer matrixresults are shown without error bars, while the ILMC andLMC results are shown with error bars that take into ac-count stochastic errors and extrapolation errors. There isalso a residual systematic error due to finite volume ef- illon Frame et al.: Impurity Lattice Monte Carlo for Hypernuclei 9 -32-31-30-29-28-27-26-25-24 0 0.2 0.4 0.6 0.8 1 1.2 1.4 E ( M e V ) t (MeV -1 )impurity lattice MCextrapolation fitextrapolated value Fig. 4.
ILMC results for the Λ He energy versus Euclidean timein a periodic box size of L = 9 . ansatz for the asymptotictime dependence. fects. For a box size of L = 29 . H is 0 .
04 MeV, and the estimated finite volumeerror for Λ H is also (cid:39) .
04 MeV. As both corrections are inthe same direction (with more binding at finite volume),the resulting finite volume error on the separation energyis < .
002 MeV.For a box size of L = 15 . H/He is (cid:39) .
10 MeV, and the estimated finite volumeerrors for Λ H/He are also (cid:39) .
10 MeV. For a box size of L = 9 . He is (cid:39) . Λ H/He are (cid:39) . Table 3.
Summary of lattice results (exact transfer matrix,ILMC and LMC) for the energies of light nuclei and hyper-nuclei, and for separation energies. Comparisons with exper-imental separation energies are given where such data exists.These comparisons are averaged over Wigner SU(4) and Λ spincomponents. For the case of Λ H/He, we average over the 0 + and 1 + separation energies for Λ H and Λ He weighted by num-ber of spin components. More data can be found in the reviewRef. [32].
Nucleus L (fm) E (MeV) B Λ (MeV) B exp Λ (MeV) H 29.6 − Λ H 29.6 − H/He 15.8 − Λ H/He 15.8 − He 9.9 − Λ He 9.9 − For the comparison with the experimental results, weaverage over Wigner SU(4) and Λ spin components wherethe data exists. We see that while the B exp Λ is larger thanthe experimental values for Λ H and Λ He, the separation is smaller than experimental value for Λ H/He. This is anindication that there are deficiencies in our very simpletreatment of the
Y N and
N N interactions. However, thisserves as a good starting point for determining the essen-tial features of the
Y N interactions needed to describe thestructure and properties of hypernuclei.
We have shown, as a proof of principle, how state-of-the-art NLEFT calculations can be extended to include hy-perons. As the number of hyperons in realistic hypernucleiis small (typically one or two) relative to the number ofnucleons, we have applied the ILMC method whereby thehyperon “impurity” is integrated out and represented by ahyperon “worldline”, the position of which is updated dur-ing the MC calculation. Effectively, the standard NLEFTcalculations for nucleons are augmented by a “backgroundfield” induced by the hyperon worldline. We have bench-marked the ILMC method by presenting preliminary MCresults for the s -shell hypernuclei, using a simplified inter-action similar to pionless EFT.One of the most promising aspects of this work is thefact that the ILMC simulations scale very favorably withthe number of nucleons. We have found that nearly all ofthe computational effort is consumed in calculating single-nucleon amplitudes as a function of the auxiliary field. Asthis part of the code scales linearly with the number ofnucleons, it should be possible to perform calculations ofhypernuclei with up to one hundred or more nucleons.We note also that the particular set of interactions thatwe have used here can also be directly applied to study-ing the properties of a bosonic impurity immersed in asuperfluid Fermi gas. By modifying the included P -waveinteractions of the impurity, we would also be able to de-scribe the properties of an alpha particle immersed in agas of superfluid neutrons. The possible applications ofthis method clearly go well beyond hypernuclear struc-ture calculations and have general utility for numerousquantum many-body systems.Returning to hypernuclear systems, the obvious nextextension of this work is to include spin-dependent Y N interactions. The importance of the spin-dependence ofthe
Y N interaction can be seen clearly in the splittingsbetween the 0 + and 1 + states in Λ H and Λ He Ref. [26].One should also include explicit ΛN - Σ N transitions, see e.g . [33], as well as one-meson exchange interactions thatwould put the Y N interaction in the same EFT formal-ism [5,6] as currently used for the
N N interaction inNLEFT [22].The number of adjustable parameters in the
Y N in-teraction will then increase. The most natural approach,in line with the treatment of the
N N interaction, wouldbe to fit such parameters to ΛN scattering phase shifts.However, due to the paucity of such data (especially at lowenergies), we expect to need at least the hypertriton bind-ing energy as an additional constraint, as it is also donein continuum chiral EFT, see e.g. Ref. [6]. As the effectsof ΛN - Σ N transitions are included, it may be necessary to use further empirical data on other light hypernucleito constrain the relevant LECs. A further extension con-cerns the extension to S = − Y Y interactions [34,35,36]and on the other hand a modified ILMC algorithm fortwo interacting worldlines. Work along these lines is un-derway.
Acknowledgments
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