The scheme dependence of the renormalization group (RG) flow has been investigated in the local potential approximation for two-dimensional periodic, sine-Gordon type field-theoretic models discussing the applicability of various functional RG methods in detail. It was shown that scheme-independent determination of such physical parameters is possible as the critical frequency (temperature) at which Kosterlitz-Thouless-Berezinskii type phase transition takes place in the sine-Gordon and the layered sine-Gordon models, and the critical ratio characterizing the Ising-type phase transition of the massive sine-Gordon model. For the latter case, the Maxwell construction represents a strong constraint on the RG flow, which results in a scheme-independent infrared value for the critical ratio. For the massive sine-Gordon model also the shrinking of the domain of the phase with spontaneously broken periodicity is shown to take place due to the quantum fluctuations.
Functional Renormalization Group (FRG) flow equations of Sine-Gordon model in 1, 2 and 3 dimensions are obtained by using power law regulator in additive and multiplicative approaches in the leading order and next to leading order in the Taylor expansion of the Fourier amplitude with LPA′ and single Fourier mode approximations. The 2-dimensional FRG flow equations are compared to the Perturbative Renormalization Group (PRG) flow equations of Daniel J Amit, Yadin Y Goldschmidt, and S Grinstein. Renormalisation group analysis of the phase transition in the 2d coulomb gas, sine-gordon theory and xy-model. Journal of Physics A: Mathematical and General, 13(2):585, 1980., where a universal behaviour is claimed. This thesis explored whether FRG also reproduces this universal behaviour and it is concluded that no such universal behaviour exists in the FRG framework.
We use bosonic tensor networks to connect the lattice sine-Gordon model in the Hamiltonian formulation quantitatively to its continuum theory. Matching the lattice vertex operator to its conformal normalization at the free-boson ultraviolet fixed point yields the exact relation between the bare lattice coupling and the renormalized continuum mass parameter. The soliton mass then approaches Zamolodchikov's exact continuum prediction throughout the studied parameter range, without adjustable parameters. Using uniform matrix product states and a quasiparticle ansatz, we also recover the relativistic soliton dispersion and the two lightest breather masses at the percent level. We simulate real-time collisions of Gaussian soliton-antisoliton wave packets near a reflectionless point and extract the Wigner spatial displacement. We compare this displacement with the exact continuum prediction obtained from the momentum derivative of the transmission phase, recovering its characteristic rapidity dependence. Our bosonic simulations provide a foundation for nonintegrable extensions, offer lessons for renormalization in other Hamiltonian lattice field theories, including gauge theories, and provide benchmarks for continuous-variable quantum simulations.
In the quantum inverse scattering method, the sine-Gordon model is solved on a lattice, and its continuum and infinite-volume limits become uniform only after the length of the box is rescaled. We show that this rescaling, supplemented by an on-shell condition, defines a renormalization scheme. The whole dependence on the cut-off is absorbed into a factor
ZL multiplying the length of the box, with an exponent fixed by the scaling dimension of the vertex operator; neither the field nor the coupling
β is renormalized, no subtraction scale is introduced, and the mass parameter receives a finite renormalization, fixed by requiring that the lightest breather has the mass of the boson in the linearized theory. The soliton and breather masses follow from the eigenvalues of the monodromy operator as functions of parameters that do not run, and reduce to the classical and semiclassical results as
β→0. In particular, the mass ratios are produced directly from the regularized construction, while the relation between the lattice parameters and the physical scale is absorbed into
ZL and never needs to be computed. The scheme is a reparameterization of the theory which makes explicit the interplay between integrability and renormalization. We compare it with Coleman's normal ordering and with conformal perturbation theory, and we argue that it extends to ultralocal integrable theories with a massive spectrum, possibly including asymptotically free models that admit an ultralocal lattice regularization.
Using the methods of the "form factor program" exact expressions of all matrix elements are obtained for several operators of the quantum sine Gordon model: all powers of the fundamental bose field, general exponentials of it, the energy momentum tensor and all higher currents. It is found that the quantum sineGordon field equation holds with an exact relation between the "bare" mass and the renormalized mass. Also a relation for the trace of the energy momentum is obtained. The eigenvalue problem for all higher conserved charges is solved. All results are compared with Feynman graph expansions and full agreement is found. PACS: 11.10.-z; 11.10.Kk; 11.55.Ds Keywords: Integrable quantum field theory, Form factors The classical sine-Gordon model is given by the wave equation 2'(t; x) + ff fi sin fi'(t; x) = 0: Since Coleman [1] found the wonderful duality between the quantum sine-Gordon and the massive Thirring model a lot of effort has been made to understand this quantum field theo...