SRG演进的红外极限与Levinson定理：无限动量网格上的哈密顿对角化

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本文主要探讨了SRG（相似性重正化群）在量子力学中的应用，特别是在红外极限下的行为以及与Levinson定理的关系。SRG是一种强大的技术，它通过在有限动量网格（具有N个积分点pn和权重wn）上进行相似性变换，用于处理复杂的量子系统，如核物理中的核力相互作用。在这个过程中，SRG利用给定生成器G来演化一个初始相互作用，通过能量差的截断参数λ（λ→0表示红外极限），将系统的哈密顿量Hn，m0从动量空间的非对角形式Hn，m0 = pn^2 δn，m + Vn，m逐步趋向于对角化，即Hn，mG，λ→0 = Eπ(n) δn，m，其中En是对应的特征值，π(n)是特征值的置换函数。 Levinson定理是一个关键的概念，它给出了相位变化δ(pn)与束缚态数目nB之间的定量关系：δ(p1) - δ(pN) = nB π。这一定理对于理解动量空间中能量级的分布和系统的基本性质至关重要。作者指出，在构建哈密顿算子时，等效的矩阵操作应当与Levinson定理保持一致，尽管在SRG演化过程中，这些操作是等价的。文章强调了在实际应用中如何通过能量转移的方式来定义相移，但需要确保动量网格上的状态排列与Levinson定理相符。这涉及到对束缚态存在的SRG演化路径的研究，不同生成器G（如威尔逊生成器和韦格纳生成器）会产生不同的等光谱流，进而影响红外极限中的秩序。威尔逊生成器导致的升序排列与Levinson定理不匹配，而韦格纳生成器则提供了一种更合适的排序方式，尽管可能不是最优解。作者以S01和S13通道中的核子-核子（NN）相互作用为例，展示了这些理论概念的实际应用，证明了SRG在处理复杂核力问题时的有效性和局限性。本文不仅深入解析了SRG的数学原理，还突出了它在实际物理问题中的重要应用价值，尤其是在红外区域的物理性质探索中。

Physics Letters B 735 (2014) 149–156

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

The infrared limit of the SRG evolution and Levinson’s theorem

E. Ruiz Arriola

a,∗

, S. Szpigel

, V.S. Timóteo

Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de Fisica Teórica y Computacional, Universidad de Granada, E-18071 Granada, Spain

Centro de Rádio-Astronomia e Astrofísica Mackenzie, Escola de Engenharia, Universidade Presbiteriana Mackenzie, Brazil

Grupo de Óptica e Modelagem Numérica – GOMNI, Faculdade de Tecnologia – FT, Universidade Estadual de Campinas – UNICAMP, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:

Received

23 April 2014

Received

in revised form 10 June 2014

Accepted

12 June 2014

Available

online 17 June 2014

Editor:

W. Haxton

On a ﬁnite momentum grid with N integration points p

and weights w

(n = 1,...,N) the Similarity

Renormalization Group (SRG) with a given generator G unitarily evolves an initial interaction with a

cutoff λ on energy differences, steadily driving the starting Hamiltonian in momentum space H

n,m

+ V

n,m

to a diagonal form in the infrared limit (λ → 0), H

G,λ→0

= E

π(n)

n,m

, where π (n) is

a permutation of the eigenvalues E

which depends on G. Levinson’s theorem establishes a relation

between phase-shifts δ(p

) and the number of bound-states, n

, and reads δ(p

) − δ(p

) = n

π. We

show that unitarily equivalent Hamiltonians on the grid generate reaction matrices which are compatible

with Levinson’s theorem but are phase-inequivalent along the SRG trajectory. An isospectral deﬁnition

of the phase-shift in terms of an energy-shift is possible but requires in addition a proper ordering of

states on a momentum grid such as to fulﬁll Levinson’s theorem. We show how the SRG with different

generators G induces different isospectral ﬂows in the presence of bound-states, leading to distinct

orderings in the infrared limit. While the Wilson generator induces an ascending ordering incompatible

with Levinson’s theorem, the Wegner generator provides a much better ordering, although not the

optimal one. We illustrate the discussion with the nucleon–nucleon (NN) interaction in the

and

channels.

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

1. Introduction

During the last decade the renormalization group equations

have advantageously been used as a technique to simplify micro-

scopic

large scale calculations in Nuclear Structure and Reactions.

More speciﬁcally, the Similarity Renormalization Group (SRG) has

been intensively applied to handle multinucleon forces in order to

soften the short-distance core [1,2] with a rather universal pat-

tern

for nuclear symmetries [3,4] and interactions [5]. The ba-

sic

strategy underlying the SRG method is to evolve a starting

(bare) interaction H

which has been ﬁtted to nucleon–nucleon

(NN) scattering data via a continuous unitary transformation that

runs a cutoff λ on energy differences. Such a transformation gen-

erates

a family of unitarily equivalent smooth interactions H

†

with a band-diagonal structure of a prescribed width

roughly given by the SRG cutoff λ. For most cases of interest a

ﬁnite momentum grid with N integration points p

and resolu-

tion

weights p

= w

(n = 1,...,N) is needed to solve the SRG

ﬂow equations numerically, and for such a ﬁnite basis the SRG

Corresponding author.

E-mail

addresses: earriola@ugr.es (E.R. Arriola), szpigel@mackenzie.br

(S. Szpigel),

varese@ft.unicamp.br (V.S. Timóteo).

transformation corresponds to a continuum generalization of the

well-known Gauss reduction method of a matrix to the diagonal

form. Typical calculations have taken w

∼ 0.01 fm

−1

, λ ∼ 2fm

−1

and N ∼ 200 [1,2]. For reasons to be elaborated below we will ana-

lyze

here a quite different regime, namely smaller SRG cut-offs and

grids, so our discussion will have no implications for the many ex-

isting

calculations.

Unfortunately

the NN force is not yet known from ﬁrst prin-

ciples

and most information on the NN interaction is strongly

constrained by the abundant np and pp scattering data (see e.g.

Ref. [6] for a recent upgrade and references therein). Roughly

speaking this is equivalent to knowing the phase-shifts with their

uncertainties at some center-of-mass (CM) momenta and in a given

range, 0 <p ≡ p

< ··· < p

≡ Λ, and in fact a common prac-

tice

has been to tabulate the phase-shifts at given discrete set of

energy values. The implicit assumption underlying this practice is

that one expects this discrete information to encode and summa-

rize

suﬃcient details on the interaction, in full harmony with the

need of solving SRG ﬂow equations on a ﬁnite grid. The computa-

tional

advantages of using properly chosen few discrete variables

for ﬁnite volume systems such as nuclei [7] have been emphasized

as the number of states gets drastically reduced.

http://dx.doi.org/10.1016/j.physletb.2014.06.032

0370-2693/

SCOAP

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