Finite volume expectation values in the sine-Gordon model
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Springer
Received: October 14, 2019 Accepted: December 20, 2019 Published: January 21, 2020
´ ad Heged˝ Arp´ us Wigner Research Centre for Physics, Budapest 114, P.O. Box 49, H-1525 Hungary
E-mail: [email protected] 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
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)122
JHEP01(2020)122
Finite volume expectation values in the sine-Gordon model
Contents 1
2 Sine-Gordon model as a perturbed conformal field theory 2.1 Perturbed Liouville CFT formulation 2.2 Compactified free boson description
2 3 3
3 Integral equations for the spectrum
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4 The function ω for excited states
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5 Formulas for the expectation values
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6 Large volume checks 6.1 Compatibility check of the formulas (5.9) and (5.12) 6.2 Connected diagonal form-factors of Φ4 1−ν (0) ν 6.3 Classical limit of connected diagonal form factors
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7 Small volume checks 7.1 The case of primaries Φ2 1−ν (0) and Φ4 1−ν (0) ν ν 7.2 Expectation values of descendant fields 7.2.1 The case of hl−2 Φ2 1−ν i ν 7.2.2 Expectation values of the descendants of the unity
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8 Summary and conclusions
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1
Introduction
The knowledge of finite volume form-factors of integrable quantum field theories became important in string theory and in condensed matter applications, as well. In string-theory, they arise in the AdS/CFT correspondence when heavy-heavy-light 3-point functions are considered [1–5]. In condensed matter physics the finite volume form-factors are necessary to represent correlation functions for describing various quasi 1-dimensional condensed matter systems [6]. So far two basic approaches have been developed to compute finite volume matrix elements of local operators in an integrable quantum field theory. In the first approach [7, 8], the finite-volume form-factors are represented as a large volume series in terms of the infinite volume form-factors of the theory. In this approach the polynomial in volume corrections are given by the Bethe-Yang quantizations of the rapidities, while the exponentially
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JHEP01(2020)122
1 Introduction
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: Z 1 2µ2 i dz ∧ d¯ z (2.1) ASG = ∂z ϕ(z,
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