  The nondivergence of the generalized Gr\"uneisen ratio (GR) at a quantum
critical point (QCP) has been proposed to be a universal thermodynamic
signature of self-duality. In this work, we study how the Kramers-Wannier-type
self-duality manifests itself in the finite-size scaling behavior of
thermodynamic quantities in the quantum critical regime. While the self-duality
cannot be realized as a unitary transformation in the total Hilbert space for
the Hamiltonian with the periodic boundary condition, it can be implemented in
certain symmetry sectors with proper boundary conditions. Therefore, the GR and
the transverse magnetization of the one-dimensional transverse-field Ising
model exhibit different finite-size scaling behaviors in different sectors.
This implies that the numerical diagnosis of self-dual QCP requires identifying
the proper symmetry sectors.
