In this paper,the statistical physics properties of sine-Gordon-Thirring model with impurity coupling are studied by means of functional integral method,and the effective potential is derived for weak coupling model by using a auxiliary Bose field technique.Free energy of the model is obtained from perturbation expansion of functional determinant.Moreover,the four-order statistical average values of two impurities and condensed term are calculated in weak coupling case.The diagrammatic presentations of the effective potential and statistical average values are also given.
In this paper we study the critical exponents of the quantum sine-Gordon model and U(1) Thirring models in the incommensurate phase. This phase appears when the chemical potential h exceeds a critical value and is characterized by a finite density of solitons. The low-energy sector of this phase is critical and is described by the Gaussian model (Tomonaga-Luttinger liquid) with the compactification radius dependent on the soliton density and the sine-Gordon model coupling constant $\ensuremath{\beta}.$ For a fixed value of $\ensuremath{\beta},$ we find that the Luttinger parameter K is equal to 1/2 at the commensurate-incommensurate transition point and approaches the asymptotic value ${\ensuremath{\beta}}^{2}/8\ensuremath{\pi}$ away from it. We describe a possible phase diagram of the model consisting of an array of weakly coupled chains. The possible phases are Fermi liquid, spin density wave, spin-Peierls, and Wigner crystal.
Motivated by the realization of coupled Luttinger liquids in a split condensate for quantum sine-Gordon (SG) model [Haller et al., Nature 466, 597 (2010); Schweiglet et al., Nature 545, 323 (2017); Nat. Phys. 17, 559 (2021)], here we study the phase transition in coupled condensates using the bosonization method. We show that when these condensates have different velocities
ui and Luttinger parameters
Ki, the physics is described by a generalized SG model, which is also exactly solvable. We use various approaches, including the Wilson renormalization group, correlation function, variational wave function, and the equivalent Coulomb gas methods, demonstrating the phase transition at
2=1/(4K1)+1/(4K2). The physical meaning of this boundary is explained by the density-phase uncertainty principle, showing how the many-body interactions drive the coupled condensates from two gapless phases to one gapless phase. The relative phase between the condensates may fundamentally affect the band structure, yet it will not affect this boundary. Furthermore, we also discuss the relevance of our result in the two-dimensional models for the Berezinskii-Kosterlitz-Thouless transition, from which the physical meaning of this boundary is self-evident. Our results can be used to study the phase transitions in one-dimensional coupled models with Fermions and Bosons, and even the Kane-Fisher model with impurities, and study their phase transitions, non-Gaussian dynamics, and thermalization, which may be of great interest in quantum simulation.
We examine a recently developed sine-Gordon model description of the non-N\'eel phase of quantum antiferromagnets on two-dimensional bipartite lattices. Using a spatial dimensionality d=1+\ensuremath{\epsilon} expansion we argue that the model always scales to its strong-coupling limit and displays spin Peierls or valence-bond-solid order. The structure of the theory in this strong-coupling limit bears a remarkable resemblance to a fermionic large-N limit of the nearest-neighbor SU(N) antiferromagnet.