Ecient few-body calculations in nite volume S K onig

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Eﬃcient few-body calculations in ﬁnite volume

S K¨onig

Department of Physics, North Carolina State University, Raleigh, NC 27695, USA

E-mail: skoenig@ncsu.edu

Abstract. Simulating quantum systems in a ﬁnite volume is a powerful theoretical tool to

extract information about them. Real-world properties of the system are encoded in how

its discrete energy levels change with the size of the volume. This approach is relevant not

only for nuclear physics, where lattice methods for few- and many-nucleon states complement

phenomenological shell-model descriptions and ab initio calculations of atomic nuclei based

on harmonic oscillator expansions, but also for other ﬁelds such as simulations of cold atomic

systems. This contribution presents recent progress concerning ﬁnite-volume simulations of few-

body systems. In particular, it discusses details regarding the eﬃcient numerical implementation

of separable interactions and it presents eigenvector continuation as a method for performing

robust and eﬃcient volume extrapolations.

1. Introduction

Simulating quantum systems in ﬁnite volume (FV), such as a cubic box with periodic boundary

conditions, is a well established theoretical approach to extract information about them, going

back to the early work of L¨uscher [1, 2, 3] who showed that the real-world (i.e., inﬁnite-volume)

properties a the system are encoded in how its (discrete) energy levels change as the size of

the volume is varied. The method has become a standard approach for example in Lattice

Quantum Chromodynamics (LQCD) to extract scattering information for hadronic systems,

and extending it in diﬀerent directions, in particular to three-body systems, is an area of active

research [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Moreover, few-body approaches

formulated in FV can be used to match and extrapolate LQCD results to an eﬀective ﬁeld theory

(EFT) description [21, 22, 23, 24].

Bound-state energy levels have an exponential dependence on the size Lof the periodic box

that encodes asymptotic properties of the state’s wavefunction in inﬁnite volume [1, 25, 26]. For

a general bound state of N≥2 particles with lowest breakup channel into two clusters with A

and N−Aparticles, respectively, the volume dependence of the binding energy has been found

to be [27, 28]

∆BN(L)≡BN(L)−BN(∞) = (−1)`+1q2

πf(d)|γ|2

µA|N−A

κ2−d/2

A|N−AL1−d/2Kd/2−1(κA|N−AL),(1)

where ddenotes the number spatial dimensions, f(d) is a normalization factor, Kd/2−1is a

modiﬁed Bessel function, and

κA|N−A=q2µA|N−A(BN−BA−BN−A) (2)

arXiv:2211.00395v1 [nucl-th] 1 Nov 2022

with the reduced mass µA|N−Aof the two-cluster system. Moreover, γis the asymptotic

normalization coeﬃcient of the cluster wavefunction, a quantity that plays an important role for

the description of low-energy capture processes. Equation (1) implies that both γand κA|N−A

can be extracted by ﬁtting the volume dependence of numerical simulations. In that regard it

should be noted that most systems in nuclear physics of practical interest feature more than

one proton, and therefore Eq. (1) does not directly apply because it assumes pure short-range

interactions between all particles. Work that derives the volume dependence for charged-particle

bound states, i.e., including the long-range Coulomb interaction, is nearly concluded at the time

this contribution is being written [29]. The following sections discuss recent progress regarding

the eﬃcient numerical implementation of few-body systems in ﬁnite volume. The material is

based primarily on Refs. [30, 31], but includes additional details in particular in Sec. 2.1.

2. Discrete Variable Representation

Few-body calculations in periodic ﬁnite boxes can be implemented elegantly with a “discrete

variable representation (DVR)” based on plane-wave states. This method has been used and

described in Refs. [32, 28, 30, 31]. The following discussion elaborates on the use of separable

interactions within this framework, ﬁrst introduced in Ref. [30], with focus here on an eﬃcient

numerical implementation. To that end, we keep the general introduction of the method brief

(referring the reader to the papers cited above), but provide previously unpublished details

regarding the computation.

The DVR construction starts from plane-wave states deﬁned on an interval L,

φ(L)

j(x) = 1

√Lexp i2πj

Lx,(3)

with j=−N/2,···N/2−1 for even number of modes N > 2 and where xdenotes the relative

coordinate describing a two-body (n= 2) system in one dimension (d= 1). Given a set of

equidistant points xk∈[−L/2, L/2) and weights wk=L/n (independent of k), DVR states are

constructed as [33]

ψk(x) =

N/2−1

j=−N/2U∗

kj φj(x),(4)

with Uki =√wkφi(xk) deﬁning a unitary matrix. The index kin Eq. (4) covers the same range of

integers as the jlabeling the original plane-wave modes, and ψk(x) is a wavefunctions peaked at

xk. Eﬀectively, the plane-wave DVR can be thought of as a lattice discretization that maintains

the exact continuum energy-momentum dispersion relation.

Let now B(n,d)

N={|si} denote a basis of DVR states for Aparticles in dspatial dimensions,

with truncation parameter N. In the following, the superscript (n, d) is dropped to simplify the

notation. A basis state |sican then be written as

|si=|(k1,1,··k1,d),··,(kn−1,1,··kn−1,d); (σ1,··σn)i,(5)

where the σilabel optional spin degrees of freedom. The formalism can be extended in a

straightforward way to include additional discrete degrees of freedom such as isospin. In

coordinate-space representation, where we use xto collectively denote all relative coordinates,

the spatial part of a DVR state |siis a tensor product of one-dimensional DVR wavefunctions:

ψs(x) = hx|si=Y

i=1,··n−1

c=1,··d

ψki,c (xi,c).(6)

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摘要：

Ecientfew-bodycalculationsinnitevolumeSKonigDepartmentofPhysics,NorthCarolinaStateUniversity,Raleigh,NC27695,USAE-mail:skoenig@ncsu.eduAbstract.Simulatingquantumsystemsinanitevolumeisapowerfultheoreticaltooltoextractinformationaboutthem.Real-worldpropertiesofthesystemareencodedinhowitsdiscreteen...

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