Abstract
The Bukhvostov–Lipatov model is an exactly soluble model of two interacting Dirac fermions in 1+1 dimensions. The model describes weakly interacting instantons and anti-instantons in the O(3) non-linear sigma model. In our previous work [arXiv:1607.04839] we have proposed an exact formula for the vacuum energy of the Bukhvostov–Lipatov model in terms of special solutions of the classical sinh-Gordon equation, which can be viewed as an example of a remarkable duality between integrable quantum field theories and integrable classical field theories in two dimensions. Here we present a complete derivation of this duality based on the classical inverse scattering transform method, traditional Bethe ansatz techniques and analytic theory of ordinary differential equations. In particular, we show that the Bethe ansatz equations defining the vacuum state of the quantum theory also define connection coefficients of an auxiliary linear problem for the classical sinh-Gordon equation. Moreover, we also present details of the derivation of the non-linear integral equations determining the vacuum energy and other spectral characteristics of the model in the case when the vacuum state is filled by 2-string solutions of the Bethe ansatz equations.
Highlights
The pair of real numbers (k+, k−) labels different sectors of the theory and, one can address the problem of computing of vacuum energy Ek in each sector
Which is related to the so-called effective central charge, F = −ceff /6
Concerning different regimes of the Fateev model we presented an exact formula for the scaling function
Summary
Consider (1+1)-dimensional quantum field theory of two interacting Dirac fermions Ψa (a = ±), defined by the bare Lagrangian. Both these operators N± are separately conserved quantities This means that the stationary Schrodinger equation H |Φ = E |Φ has solutions with a fixed total number N = N+ + N− of quasiparticles,. Note that if no coordinates {xi} pairwise coincide the interaction potential vanishes and the wave function becomes a linear combination of products of the free quasiparticle solutions (2.9). To which the system (2.20) imposes no restrictions.1 At this point it is worth remembering that the numbers of quasiparticles of each flavor are separately conserved. This means that the vector (2.21) must be an eigenvector of the operator. Where, to (2.8), the Pauli matrix τ3(i) acts only on the flavor of the i-th quasiparticle
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