We propose a new approach to compute exact $g$ -function for integrable quantum field theories with non-diagonal scattering S-matrices. The approach is based on an integrable lattice regularization of the quantum field theory. The exact $g$ -function is encoded in the overlap of the integrable boundary state and the ground state on the lattice, which can be computed exactly by Bethe ansatz. In the continuum limit, after subtracting the contribution proportional to the volume of the closed channel, we obtain the exact $g$ -function, given in terms of the counting function which is the solution of a nonlinear integral equation. The resulting $g$ -function contains two parts, the scalar part, which depends on the boundary parameters and the ratio of Fredholm determinants, which is universal. This approach bypasses the difficulties of dealing with magnetic excitations for non-diagonal scattering theories in the framework of thermodynamic Bethe ansatz. We obtain numerical and analytical results of the exact $g$ -function for the prototypical sine-Gordon theory with various integrable boundary conditions.We propose a new approach to compute exact $g$-function for integrable quantum field theories with non-diagonal scattering S-matrices. The approach is based on an integrable lattice regularization of the quantum field theory. The exact $g$-function is encoded in the overlap of the integrable boundary state and the ground state on the lattice, which can be computed exactly by Bethe ansatz. In the continuum limit, after subtracting the contribution proportional to the volume of the closed channel, we obtain the exact $g$-function, given in terms of the counting function which is the solution of a nonlinear integral equation. The resulting $g$-function contains two parts, the scalar part, which depends on the boundary parameters and the ratio of Fredholm determinants, which is universal. This approach bypasses the difficulties of dealing with magnetic excitations for non-diagonal scattering theories in the framework of thermodynamic Bethe ansatz. We obtain numerical and analytical results of the exact $g$-function for the prototypical sine-Gordon theory with various integrable boundary conditions.

arXiv:2412.12869Dec 17, 20241 citation

Boundary energy and boundary states in integrable quantum field theories

André LeClair, Giuseppe Mussardo, Hubert Saleur +1 more

We study the ground-state energy of integrable 1 + 1 quantum field theories with boundaries (the genuine Casimir effect). In the scalar case, this is done by introducing a new “R-channel TBA”, where the boundary is represented by a boundary state, and the thermodynamics involves evaluating scalar products of boundary states with all the states of the theory. In the non-scalar, sine-Gordon case, this is done by generalizing the method of Destri and De Vega. The two approaches are compared. Miscellaneous other results are obtained, in particular formulas for the overall normalization and scalar products of boundary states, exact partition functions for the critical Ising model in a boundary magnetic field, and also results for the energy, excited states and boundary S-matrix of O(n) and minimal models.We study the ground-state energy of integrable 1 + 1 quantum field theories with boundaries (the genuine Casimir effect). In the scalar case, this is done by introducing a new “R-channel TBA”, where the boundary is represented by a boundary state, and the thermodynamics involves evaluating scalar products of boundary states with all the states of the theory. In the non-scalar, sine-Gordon case, this is done by generalizing the method of Destri and De Vega. The two approaches are compared. Miscellaneous other results are obtained, in particular formulas for the overall normalization and scalar products of boundary states, exact partition functions for the critical Ising model in a boundary magnetic field, and also results for the energy, excited states and boundary S-matrix of O(n) and minimal models.

arXiv:hep-th/9503227Nuclear Physics BOct 1, 1995207 citations

Integrable Boundary Flows and the g-function

Ruth Margaret Wilbourne

This thesis explores renormalisation group flows in integrable quantum field theories with boundaries, as described by the g-function. The main focus is on the g-function in the staircase model, the renormalisation group flow of which passes close to the unitary minimal models. This g-function is used to identify flows between boundary conditions both within and between the minimal models. In certain limits the $\\mathcal{M}A_m^{(+)}$ theories which interpolate between pairs of minimal models emerge from the staircase model, and exact expressions for the g-function in these models are extracted from the staircase g-function. Perturbative tests on the $\\mathcal{M}A_4^{(+)}$ g-function are discussed, as is initial work on the g-function for the $\\mathcal{M}A_4^{(-)}$ theory, which describes flows that emerge when the bulk coupling is taken to have the opposite sign to that in $\\mathcal{M}A_4^{(+)}$ . Expressions are also found for excited state versions of the $\\mathcal{M}A_m^{(+)}$ g-function, and these allow the unique identification of certain boundary flows.This thesis explores renormalisation group flows in integrable quantum field theories with boundaries, as described by the g-function. The main focus is on the g-function in the staircase model, the renormalisation group flow of which passes close to the unitary minimal models. This g-function is used to identify flows between boundary conditions both within and between the minimal models. In certain limits the $\\mathcal{M}A_m^{(+)}$ theories which interpolate between pairs of minimal models emerge from the staircase model, and exact expressions for the g-function in these models are extracted from the staircase g-function. Perturbative tests on the $\\mathcal{M}A_4^{(+)}$ g-function are discussed, as is initial work on the g-function for the $\\mathcal{M}A_4^{(-)}$ theory, which describes flows that emerge when the bulk coupling is taken to have the opposite sign to that in $\\mathcal{M}A_4^{(+)}$. Expressions are also found for excited state versions of the $\\mathcal{M}A_m^{(+)}$ g-function, and these allow the unique identification of certain boundary flows.

Jan 1, 20120 citations

Quantum field theory, integrable models and beyond

武夫稲見, 隆佐々木

Progress of Theoretical Physics eBooksJan 1, 19959 citations

We propose a new approach to compute exact

g

-function for integrable quantum field theories with non-diagonal scattering S-matrices. The approach is based on an integrable lattice regularization of the quantum field theory. The exact

g

-function is encoded in the overlap of the integrable boundary state and the ground state on the lattice, which can be computed exactly by Bethe ansatz. In the continuum limit, after subtracting the contribution proportional to the volume of the closed channel, we obtain the exact

g

-function, given in terms of the counting function which is the solution of a nonlinear integral equation. The resulting

g

-function contains two parts, the scalar part, which depends on the boundary parameters and the ratio of Fredholm determinants, which is universal. This approach bypasses the difficulties of dealing with magnetic excitations for non-diagonal scattering theories in the framework of thermodynamic Bethe ansatz. We obtain numerical and analytical results of the exact

g

-function for the prototypical sine-Gordon theory with various integrable boundary conditions.

\\mathcal{M}A_m^{(+)}

theories which interpolate between pairs of minimal models emerge from the staircase model, and exact expressions for the g-function in these models are extracted from the staircase g-function. Perturbative tests on the

\\mathcal{M}A_4^{(+)}

g-function are discussed, as is initial work on the g-function for the

\\mathcal{M}A_4^{(-)}

theory, which describes flows that emerge when the bulk coupling is taken to have the opposite sign to that in

\\mathcal{M}A_4^{(+)}

. Expressions are also found for excited state versions of the

\\mathcal{M}A_m^{(+)}

g-function, and these allow the unique identification of certain boundary flows.