Exact boundary free energy of the open XXZ chain with arbitrary boundary conditions

We derive an exact formula for the boundary free energy of the open Heisenberg XXZ spin chain. We allow for arbitrary boundary magnetic fields, but assume zero bulk magnetization. The result is completely analogous to earlier formulas for the so-called $g$-function: it is expressed as a combination of single integrals and two simple Fredholm determinants. Our expressions can be evaluated easily using numerical algorithms with arbitrary precision. We demonstrate that the boundary free energy can show a wide variety of behaviour as a function of the temperature, depending on the anisotropy and the boundary fields. We also compute the low temperature limit of the boundary free energy, and reproduce the known results for the ground state boundary energy, including the case of non-diagonal fields.