GL(3)不变R-矩阵下的量子可积模型形因子的 determinant 表达式

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本文主要探讨的是具有GL(3)不变R-矩阵的量子可积模型中的形状因子问题。GL(3)是一种重要的数学群，在量子力学中，尤其是量子统计系统如量子链模型中，R-矩阵的不变性对于模型的完全解是关键，因为它保证了系统的对称性和可积性。这里的GL(3)-不变R-矩阵意味着该模型具有三维格拉姆-李特尔伍德-列文(Lie algebra)结构，这是一种特殊的矩阵对称性，对于理解复杂系统的动力学行为至关重要。作者通过嵌套代数贝塞尔方法(nested algebraic Bethe ansatz, NBA)来研究这些模型，这是一种强大的数值和分析工具，用于求解具有局部相互作用的一维量子系统中的精确波函数。NBA允许将复杂的模型分解成更小的子问题，使得计算变得可行。文章的核心成果是得到了单峰矩阵非对角线元素的形状因子的行列式表示。形状因子在量子统计物理学中扮演着重要角色，它们是粒子之间相互作用的强度和概率的数学描述，对于计算系统的动态性质，如散射矩阵、动态关联函数等具有重要意义。行列式表示简化了形状因子的计算过程，因为行列式的结构通常比直接的无限级数或积分更容易处理。这些结果对于XXX SU(3)不变Heisenberg模型尤其适用，这是量子可积模型的一个经典例子，其SU(3)对称性反映了夸克颜色动力学的某些特性。XXX模型通常指的是XXX Heisenberg链，它是一种具有简单对称性的量子反常磁性模型，具有丰富的物理现象，如磁共振、磁化过程以及多粒子纠缠等。总结来说，这篇文章的贡献在于为GL(3)不变R-矩阵量子可积模型提供了关键的计算工具——形状因子的行列式表示，这对于深入理解和计算这些模型的物理性质，特别是在XXX SU(3)模型中的动态行为，有着重要的实际应用价值。通过这项工作，研究者能够进一步探索量子系统的复杂性，并可能为未来的设计和分析新型量子材料和量子计算体系提供理论基础。

S. Pakuliak et al. / Nuclear Physics B 881 (2014) 343–368 347

where λ

(w) are some scalar functions. In the framework of the generalized model, λ

(w) remain

free functional parameters. Actually, it is always possible to normalize the monodromy matrix

T(w)→ λ

−1

(w)T (w) so as to deal only with the ratios

(w) =

(w)

(w) =

(w)

. (2.10)

If the parameters ¯u and ¯v of a Bethe vector

satisfy a special system of equations (Bethe

equations), then it becomes an eigenvector of the transfer matrix (on-shell Bethe vector). The

system of Bethe equations can be written in the following form:

( ¯u

) =

f(¯u

, ¯u

)

f(¯u

, ¯u

)

f(¯v, ¯u

( ¯v

) =

f(¯v

, ¯v

)

f(¯v

, ¯v

)

f(¯v

, ¯u). (2.11)

These equations should hold for arbitrary partitions of the sets ¯u and ¯v into subsets {¯u

, ¯u

} and

{¯v

, ¯v

} respectively. Obviously, it is enough to demand that the system (2.11) is valid for the

particular case when the sets ¯u

and ¯v

consist of only one element. Then (2.11) coincides with

the standard form of Bethe equations [23].

If ¯u and ¯v satisfy

the system (2.11), then

tr T(w)B

a,b

( ¯u;¯v) = τ(w|¯u, ¯v)B

a,b

( ¯u;¯v),

a,b

( ¯u;¯v)tr T(w)= τ(w|¯u, ¯v)C

a,b

( ¯u;¯v), (2.12)

where

τ(w)≡ τ(w|¯u, ¯v) = r

(w)f ( ¯u, w) + f(w,¯u)f ( ¯v,w) + r

(w)f (w, ¯v). (2.13)

2.4. Scalar products and form factors

The scalar products of Bethe vectors are deﬁned as

a,b

≡ S

a,b



¯u

, ¯v



¯u

, ¯v



= C

a,b



¯u

;¯v



a,b



¯u

;¯v



. (2.14)

We use here superscripts B and C in order to distinguish the sets of parameters entering these two

vectors. In other words, unless explicitly speciﬁed, the variables {¯u

;¯v

} in B

a,b

and {¯u

;¯v

}

in C

a,b

are not supposed to be related.

Before giving an explicit formula for the scalar product we introduce the notion of highest

coef

ﬁcient Z

a,b

(

t;¯x|¯s;¯y). This function depends on four sets of variables with cardinalities #

t =

# ¯x = a,#¯s = # ¯y = b, and a,b = 0, 1,.... There exist several explicit representations for the

highest coefﬁcient in terms of DWPF [34,35]. In this paper we use two of them. The ﬁrst one

reads

a,b

(

t;¯x|¯s;¯y) = (−1)



(¯s − c|¯w

( ¯w

t)K

( ¯y|¯w

)f ( ¯w

, ¯w

). (2.15)

Here ¯w ={¯s, ¯x}. The sum is taken with respect to all partitions of the set ¯w into subsets ¯w

and

¯w

with # ¯w

= b and # ¯w

= a.

For simplicity here and below we do not distinguish between vectors and dual vectors.

348 S. Pakuliak et al. / Nuclear Physics B 881 (2014) 343–368

The second representation has the following form:

a,b

(

t;¯x|¯s;¯y) = (−1)

f(¯y, ¯x)f (¯s,



( ¯α

− c|¯y + c)K

( ¯x|¯α

( ¯α

− c|¯s)

× f(¯α

, ¯α

). (2.16)

Here ¯α ={¯y + c,

t}. The sum is taken with respect to all partitions of the set ¯α into subsets ¯α

and ¯α

with # ¯α

= a and # ¯α

= b.

The scalar product (2.14) is

a bilinear combination of the highest coefﬁcients. It was calculated

in the work [29]

a,b



¯u

, ¯v



¯u

, ¯v







¯u





¯u





¯v





¯v





¯u

, ¯u





¯u

, ¯u





¯v

, ¯v



× f



¯v

, ¯v



f(¯v

, ¯u

)f ( ¯v

, ¯u

)

f(¯v

, ¯u

)f ( ¯v

, ¯u

)

a−k,n



¯u

;¯u



¯v

;¯v



× Z

k,b−n



¯u

;¯u



¯v

;¯v



. (2.17)

Here the sum is taken over the partitions of the sets ¯u

, ¯u

, ¯v

, and ¯v

¯u

⇒



¯u

, ¯u



, ¯v

⇒



¯v

, ¯v



¯u

⇒



¯u

, ¯u



, ¯v

⇒



¯v

, ¯v



. (2.18)

The partitions are independent except that # ¯u

= # ¯u

= k with k = 0,...,a, and # ¯v

= # ¯v

= n

with n = 0,...,b.

In this formula the parameters ¯u

, ¯u

, ¯v

, and ¯v

are arbitrary complex numbers, that

is B

a,b

( ¯u

;¯v

) and C

a,b

( ¯u

;¯v

) are generic Bethe vectors. If one of these vectors, say

a,b

( ¯u

;¯v

), is on-shell, then the parameters ¯u

and ¯v

satisfy the Bethe equations. In this

case one can express the products r

( ¯u

) and r

( ¯v

) in terms of the function f via (2.11).

Form factors of the monodromy matrix entries are deﬁned as

(i,j )

a,b

(z) ≡ F

(i,j )

a,b





¯u

, ¯v

;¯u

, ¯v



= C





¯u

;¯v



(z)B

a,b



¯u

;¯v



, (2.19)

where both C



( ¯u

;¯v

) and B

a,b

( ¯u

;¯v

) are on-shell Bethe vectors, and



= a + δ

− δ



= b + δ

− δ

. (2.20)

The parameter z is an arbitrary complex number. Acting with the operator T

(z) on B

a,b

( ¯u

;¯v

)

via formulas obtained in [28] we reduce the form factor to a linear combination of scalar products,

in which C



( ¯u

;¯v

) is on-shell vector.

2.5. Relations between form factors

Obviously, there exist nine form factors of T

(z) in the models with GL(3)-invariant

R-matrix. However, not all of them are independent. In particular, due to the invariance of the

R-matrix under transposition with respect to both spaces, the mapping

ψ : T

(u) → T

(u) (2.21)

deﬁnes an antimorphism of the algebra (2.2). Acting on the Bethe vectors this antimorphism

maps them into the dual ones and vice versa

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GL(3)不变R-矩阵下的量子可积模型形因子的 determinant 表达式

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