大规模Thirring模型中ΨΨ期望值的光锥晶格计算

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"来自光锥晶格相关器的大规模Thirring模型中ΨΨ$$\overline{\varPsi}\varPsi$$的精确有限体积期望值" 本文深入探讨了大规模Thirring模型中的一个关键物理现象，即在有限体积条件下的复合算子ΨΨ$$\overline{\varPsi}\varPsi$$的期望值。Thirring模型是量子场论中的一个基本模型，它描述了费米子之间的相互作用，这种模型在凝聚态物理学、弦理论以及对称性破缺等领域有广泛应用。文章指出，采用光锥晶格正则化方法来处理这个问题，这是一种在高维物理系统中处理无穷大空间的方法，它将连续空间替换为离散的晶格结构，使得数学上的处理更为可行。在光锥晶格中，Ψ$$\overline{\varPsi}\varPsi$$的期望值与晶格上相邻位置的自旋算子的2点函数紧密关联。这个算子与应力能量张量的迹成比例，后者是量子场论中描述系统能量和动量的关键对象。作者通过非线性积分方程（NLIE）来计算这个理论在有限体积下的谱，这种方法允许他们准确地评估连续空间中的期望值。NLIE是一组复杂但强大的工具，能够描述系统的动态和统计特性，尤其适用于分析具有周期边界条件的系统。文中提到的NLIE计算与基于晶格相关器的方法所得结果一致，这验证了这两种方法的有效性和一致性。此外，论文还考察了LeClair-Mussardo类型的系列表示法，这是一种处理多体问题的数学技术，用于表示多粒子系统的物理量，如两体或更高体的关联函数。作者对先前关于这种表示的猜想进行了检验，这有助于深化我们对大规模Thirring模型的理解。这篇研究工作展示了光锥晶格正则化和NLIE在处理复杂量子场论问题时的潜力，特别是对于理解在有限体积条件下相互作用费米子系统的性质。同时，它还促进了对LeClair-Mussardo公式在不同上下文中的应用的研究，这对于推动理论物理的发展具有重要意义。

JHEP03(2018)047

(−∞)/2π, Z

(∞)/2π] . The excitations above this sea of real roots are characterized

by complex Bethe-roots and holes. In this paper we will consider only hole excitations,

since they correspond to fermion or soliton excitations of the continuum quantum ﬁeld

theory [26–31]. The holes are such special real solutions of (2.25), which are not Bethe-

roots.

Holes can be interpreted as missing Bethe-roots and the quantum numbers of the

missing Bethe-roots can be assigned to them:

) = 2π I

, I

∈ Z +

1 + δ

, k = 1, . . . , m

, (2.26)

where h

denotes the positions of the holes and their number is denoted by m

2.3 NLIE for the ﬁnite volume spectrum

When one has to deal with a large number of Bethe-roots, it is worth to rephrase the Bethe-

equations (2.21) or equivalently (2.25) in a form of a set of nonlinear-integral equations

(NLIE) [22]–[30].

Here we present the equations only for the pure hole sector of the theory [26] and here

we will use the rapidity convention for the equations. This means a simple rescaling of the

spectral parameter: θ =

λ.

In the pure hole sector, the counting-function in rapidity variable Z

(θ) = Z

(

θ)

satisfy the nonlinear-integral equations as follows:

(θ) =

{arctan [sinh(θ − Θ)] + arctan [sinh(θ + Θ)]} +

k=1

χ(θ − H

)

∞

−∞

dθ

2πi

G(θ − θ

− iη) L

(+)

(θ

+ iη) −

∞

−∞

dθ

2πi

G(θ − θ

+ iη) L

(−)

(θ

− iη),

(2.27)

where χ(θ) is the soliton-soliton scattering phase and G(θ) denotes its derivative;

χ(θ) = 2

∞

dω

sin(ω θ)

sinh(

(p−1) πω

)

2 cosh(

πω

) sinh(

p π ω

)

, (2.28)

G(θ) =

dθ

χ(θ) =

∞

−∞

dω e

−i ωθ

sinh(

(p−1) πω

)

2 cosh(

πω

) sinh(

p π ω

)

, (2.29)

0 < η < min(pπ, π) is an arbitrary positive contour-integral parameter, Θ = ln

2 N

M L

is the

inhomogeneity parameter of the vertex-model and H

denote the positions of the

holes in the rapidity convention. They are subjected to the quantization equations:

) = 2π I

, I

∈ Z +

1 + δ

, k = 1, . . . , m

. (2.30)

The nonlinearity of the equations is encoded into the form of L

(±)

(θ), which takes the form:

(±)

(θ) = ln



1 + (−1)

±i Z

(θ)



. (2.31)

Namely, they do not enter in the deﬁnition of Z

(λ) in (2.23).

– 8 –

JHEP03(2018)047

consequence in (2.36) and (2.37) the integral terms can be dropped in the inﬁnite volume

limit and one ends up with the energy and momentum formulas of m

pieces of fermions

(solitons) with rapidities {H

}

j=1...m

. This implies that the holes in the sea of real roots

describe the fermions (solitons) of the MT (SG) model. This is why in the sequel we will

refer to holes as fermions or solitons.

Finally, we note that the actual value

of the quantum number δ is important from

the point of view of the continuum theory. Its value can make diﬀerence between fermions

(δ = 1) of the MT model and the solitons (δ = 0) of the SG model in the odd U(1) charge

sector of the theory [28]–[30]. In the even charge sector only the δ = 0 value is physical

and there is no diﬀerence between MT fermions and SG solitons [28]–[30].

From the discussion above it follows that only the even charge sector of the MT and

SG models can be regularized by the twistless 6-vertex model. The description of the odd

charge sector requires a twisted vertex-model with a twist angle ω =

[63]. However, in

this paper we restrict ourselves to the twistless case.

2.4 Expectation values of the trace of the stress-energy tensor

In this subsection using the NLIE description of the ﬁnite volume spectrum given by (2.34)

and (2.36), we compute the fermionic (solitonic) expectation values of the trace of the

stress-energy tensor Θ

. The ﬁnite temperature 1-point functions, which correspond to

the ﬁnite volume vacuum expectation value, has been previously computed and discussed

in [70] and [64].

It has been shown in [18] that the diagonal matrix elements of Θ

can be computed

from the volume dependence of the energy of the sandwiching state by the following for-

mula:

hΘ

i = hΘ

∞

i + 2π M



E(`)

dE(`)



. (2.39)

In the sequel we compute hΘ

i when the sandwiching state is an m

-fermion state de-

scribed by the equations (2.34) and (2.36).

As a starting point, it is worth to compute the inﬁnite volume or in other words the

bulk expectation value: hΘ

∞

i. Using Zamolodchikov’s argument [18], it can be expressed

in terms of the eigenstate independent bulk energy of the model by the formula:

hΘ

∞

i = 2π M



bulk

(`)

bulk

(`)



. (2.40)

In the MT (SG) model the bulk energy term is of the form [25]:

bulk

(`) = −

tan



pπ



. (2.41)

Inserting (2.41) into (2.40) one obtains:

hΘ

∞

i = −πM

tan



pπ



. (2.42)

On the lattice the actual value of δ can be inﬂuenced by the parity of

– 10 –

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