Speaker
Description
In this talk, we discuss some aspects of the connection between the one-dimensional XXZ chain and two-dimensional conformal field theories. Namely, we consider the XXZ spin chain in the scaling limit in the Matsubara direction. Our approach is based on the fermionic basis construction developed in [1, 2, 3, 4]. The main feature of the fermionic basis is the factorization of the algebraic and physical parts of the model: the basis itself is constructed using the representation theory of the quantum group $U_q(\widehat{\mathfrak{sl}_2})$ and is independent of any physical data such as magnetic field, temperature, or boundary conditions. The physical properties of the model are encoded in two transcendental functions: $\rho(\zeta, \kappa)$ and $\omega(\zeta, \xi; \kappa, \kappa')$.
The main object of interest in any quantum field theory is the correlation function of local operators. The fermionic basis approach allows for a simple determinant description of all correlation functions, due to the so-called Jimbo-Miwa-Smirnov (JMS) theorem [3]. The partition function of fermionic generators can be calculated using the function $\omega(\zeta, \xi; \kappa, \kappa')$ mentioned above. A recursive procedure for computing the $\kappa$-asymptotics of this function in the case of identical boundary conditions $\kappa = \kappa'$ was established in [4]. It is based on a linear integral equation for the function $\omega(\zeta, \xi; \kappa, \kappa')$ and employs the standard Wiener-Hopf method. The main limitation of this approach is that the correlation functions of the integrals of motion vanish when the boundary conditions are identical. To incorporate the action of the integrals of motion, it is therefore necessary to consider the case $\kappa \neq \kappa'$ as well. Unfortunately, this presents significant technical challenges.
In this talk, we discuss an alternative description for the function $\omega^{\text{sc}}(\lambda, \mu; \kappa, \kappa', \alpha)$. Instead of working with the integral equation, we use the master function approach developed in [5]. In the free fermion case, we determine the explicit form of the master function and, using the scaling limit of its functional relation with $\omega(\zeta, \xi; \kappa, \kappa')$, we calculate it explicitly, incorporating the action of the integrals of motion.
Finally, we discuss possible ways of lifting the result outside of the free fermion point.
[1] Boos H., Jimbo M., Miwa T., Smirnov F., Takeyama Y., Hidden Grassmann structure in the XXZ model, Comm. Math. Phys. (2007)
[2] Boos H., Jimbo M., Miwa T., Smirnov F., Takeyama Y., Hidden Grassmann structure in the XXZ model. II. Creation operators,Comm. Math. Phys.(2009)
[3] Jimbo M., Miwa T., Smirnov F., Hidden Grassmann structure in the XXZ model. III. Introducing Matsubara direction, J. Phys. A: Math. Theor. (2009)
[4] Boos H., Jimbo M., Miwa T., Smirnov F., Hidden Grassmann structure in the XXZ model. IV. CFT limit, Comm. Math. Phys. (2010)
[5] Boos H., Göhmann F., Properties of linear integral equations related to the six-vertex model with disorder parameter II, J. Phys. A: Math. Theor. (2012)
