Determine the critical singular-mode value for the entropy-density minimum

Determine analytically the critical value of the singular-mode parameter at which the minimum of the dimensionless entropy density disappears for a four-dimensional free scalar monodromy defect.

Background

For a monodromy defect with a singular mode parameterized by \xi, the paper observes numerically that the dimensionless entropy density changes from having a minimum to having no minimum as \xi increases. The corresponding critical value is distinct from the analytically estimated transition value for the energy density.

An analytic determination of this threshold would explain the thermodynamic transition and clarify how it depends on the monodromy parameter and the defect boundary condition.

References

We have not found a good way of estimating this value analytically but numerically we observe that it differs from the critical value \xi_* found for E(z).

— Boundaries and defects in conformal field theories at finite temperature  (2609.38130 - Artico et al., 29 Sep 2026) in Section 5, paragraph discussing Fig. s_mon_4d_xi