We study the quantum sine-Gordon model within a nonperturbative functional renormalizationgroup approach (FRG). This approach is benchmarked by comparing our findings for the soliton and lightest breather (soliton-antisoliton bound state) masses to exact results. We then examine the validity of the Lukyanov-Zamolodchikov conjecture for the expectation value e i 2 nβϕ of the exponential fields in the massive phase (n is integer and 2π/β denotes the periodicity of the potential in the sine-Gordon model). We find that the minimum of the relative and absolute disagreements between the FRG results and the conjecture is smaller than 0.01.
Bosonization allows one to describe the low-energy physics of
one-dimensional quantum fluids within a bosonic effective field theory
formulated in terms of two fields: the ``density’’ field
\varphiφ
and its conjugate partner, the phase \varthetaϑ
of the superfluid order parameter. We discuss the implementation of the
nonperturbative functional renormalization group in this formalism,
considering a Luttinger liquid in a periodic potential as an example. We
show that in order for \varthetaϑ
and \varphiφ
to remain conjugate variables at all energy scales, one must dynamically
redefine the field \varthetaϑ
along the renormalization-group flow. We derive explicit flow equations
using a derivative expansion of the scale-dependent effective action to
second order and show that they reproduce the flow equations of the
sine-Gordon model (obtained by integrating out the field
\varthetaϑ
from the outset) derived within the same approximation. Only with the
scale-dependent (flowing) reparametrization of the phase field
\varthetaϑ
do we obtain the standard phenomenology of the Luttinger liquid (when
the periodic potential is sufficiently weak so as to avoid the
Mott-insulating phase) characterized by two low-energy parameters, the
velocity of the sound mode and the renormalized Luttinger parameter.
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