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JHEP01(2020)122

Published for SISSA by Springer

Received: October 14, 2019

Accepted: December 20, 2019

Published: January 21, 2020

Finite volume expectation values in the sine-Gordon

model

Arp´ad Heged˝us

Wigner Research Centre for Physics,

Budapest 114, P.O. Box 49, H-1525 Hungary

E-mail: hegedus.arpad@wigner.mta.hu

Abstract: Using the fermionic basis discovered in the 6-vertex model, we derive exact

formulas for the expectation values of local operators of the sine-Gordon theory in any

eigenstate of the Hamiltonian. We tested our formulas in the pure multi-soliton sector

of the theory. In the ultraviolet limit, we checked our results against Liouville 3-point

functions, while in the infrared limit, we evaluated our formulas in the semi-classical limit

and compared them up to 2-particle contributions against the semi-classical limit of the

previously conjectured LeClair-Mussardo type formula. Complete agreement was found in

both cases.

Keywords: Bethe Ansatz, Integrable Field Theories

ArXiv ePrint: 1909.08467

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP01(2020)122

JHEP01(2020)122

small in volume L¨uscher corrections come from contributions of virtual particles propagat-

ing around the compact dimension [9–11]. In a diagonally scattering theory, the diagonal

matrix elements of local operators can be computed by means of the LeClair-Mussardo

series [19–21], provided the connected diagonal form-factors of the operator under consid-

eration are known. In a non-diagonally scattering theory, so far the availability of such a

series representation is restricted to the sine-Gordon model [24, 25].

The second approach, which works well for non-diagonally scattering theories, as well,

is based on some integrable lattice regularization of the model under consideration. The re-

sults of [13] for the sine-Gordon model made it possible to conjecture exact formulas for the

ﬁnite volume diagonal matrix elements of the sinh-Gordon model [15, 16], as well. The main

advantage of the lattice approach is that, it provides exact formulas for the (speciﬁc ratios

of) ﬁnite-volume form-factors. From this approach, the corresponding LeClair-Mussardo

series can be obtained by the large volume series expansion of the formulas. In [13] the ﬁ-

nite volume ground state expectation values of the exponential ﬁelds and their descendants

have been determined in the sine-Gordon model. In this paper we extend these results to

get formulas for the expectation values of local operators in any excited state of the theory.

The outline of the paper is as follows: in section 2, we formulate the sine-Gordon model

as perturbed conformal ﬁeld theory. In section 3, we review the equations governing the

ﬁnite volume spectrum of the theory. In section 4, we derive equations for the function ω,

which is the fundamental building block of the expectation value formulas of local operators.

In section 5, we recall from [13], how the expectation values of the exponential ﬁelds and

their descendants are built up from the function ω. In section 6, we perform the large

volume checks of our formulas for the expectation values. The most important of them is a

comparison to the classical limit of the previously conjectured [24] LeClair-Mussardo type

large volume series representation. Section 7, contains the ultraviolet tests of our formulas.

Finally, the paper is closed by our conclusions.

2 Sine-Gordon model as a perturbed conformal ﬁeld theory

In this section, we recall the perturbed conformal ﬁeld theory (PCFT) descriptions of the

sine-Gordon model deﬁned by the Euclidean action:



4π

∂

ϕ(z, ¯z) ∂

¯z

ϕ(z, ¯z) +

2µ

sin πβ

cos(β ϕ(z, ¯z))



i dz ∧ d¯z

(2.1)

where z = x + i y and ¯z = x − i y, with x, y being the coordinates of the Euclidean

space-time.

In the paper, we will use two perturbed conformal ﬁeld theory formulations of this

model. The ﬁrst one is, when it is considered as a perturbed complex Liouville CFT, and

the second one is when it is described as a perturbed c = 1 compactiﬁed boson.

In the original paper [13], the formulation with complex Liouville CFT is used. Never-

theless, it turned out from the detailed [37–39] UV analysis of the ﬁnite volume spectrum,

that all eigenstates of the Hamiltonian are in one to one correspondence to the operators of

the c = 1 modular invariant compactiﬁed boson CFT. Thus, we ﬁnd eligible to describe the

– 2 –

JHEP01(2020)122

The primary states of this CFT are created by the vertex operators V

n,m

(z, ¯z) which are

labeled by two quantum numbers n ∈ R and m ∈ Z. Their left and right conformal

dimensions are given by:

∆

n,m





(2.9)

Here n is the momentum quantum number, and m is the winding number or topological

charge. The requirement of the locality of the operator product algebra of the CFT imposes

further severe restrictions on the possible values of the pair of quantum numbers (n, m).

It turns out [54], that only a bosonic and a fermionic maximal subalgebras of the vertex

operators V

n,m

(z, ¯z) are allowed. The bosonic subalgebra is characterized by the quantum

numbers {n ∈ Z, m ∈ Z}. It corresponds to the modular invariant partition function and

this CFT describes the UV limit of the sine-Gordon model [54].

In the fermionic subalgebra the allowed set of quantum numbers is given by {n ∈

Z, m ∈ 2Z}∪



n ∈ Z +

, m ∈ 2Z + 1



. It corresponds to a Γ

invariant partition function

and this CFT describes the UV limit of the massive-Thirring model [54].

The perturbing term in the action (2.7) is given in terms of these vertex operators as

follows:

pert

sin πβ

i dz ∧ d¯z

1,0

(z, ¯z) + V

−1,0

(z, ¯z)) .

(2.10)

In this paper we will mostly focus on computing the diagonal matrix elements of the

primaries and their descendants belonging to the bosonic subalgebra of the c = 1 CFT.

Diagonal matrix elements are non-zero only in the zero winding number sector (m = 0),

thus our primary goal is to derive formulas for the diagonal matrix elements of the vertex

operators V

n,0

(z, ¯z), and of their descendants with n ∈ Z. In the Liouville formulation they

correspond to the primaries Φ

2 n

1−ν

(z, ¯z) and their descendants.

3 Integral equations for the spectrum

In this section, we summarize the nonlinear integral equations (NLIE) governing the ﬁnite

volume spectrum of the sine-Gordon theory.

Originally, the equations were derived from an inhomogeneous 6-vertex model with,

which serves as an integrable light-cone lattice regularization [30] of the sine-Gordon model.

Later, the same type of equations were obtained from an approriate lattice regularization

for two coupled quantum KdV equations [41]–[43].

In the series of papers [32, 33]–[39], the starting point of the derivation of the equa-

tions is the Bethe-Ansatz solution of the inhomogeneous 6-vertex model, since it has been

shown, at the level of the spectrum in refs. [26]–[29], and later at ﬁeld theory level in [30],

that the continuum limit of the 6-vertex model with appropriately chosen alternating in-

homogeneities is the massive Thirring model. The equivalence between the sine-Gordon

and massive Thirring models [53, 54] allows one to consider the 6-vertex model as an

appropriate lattice regularization for the sine-Gordon theory, as well.

– 4 –

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