Geometric characterization of the phase transitions in a Blume-Capel model via the Ruppeiner metric

Abstract The thermodynamic geometry based on the Ruppeiner formalism is utilized to investigate the spin-1 Blume-Capel model in the absence and presence of the external magnetic field ( h ). An explicit expression for the thermodynamic curvature or Ricci scalar ( R ) is derived. Its behaviour near the critical and tricritical points and a geometric phase diagram highlighting the characteristics of R are presented when $$h=0$$ h = 0 . Notably, two half rings including the minima $$(R_M)$$ ( R M ) and maxima $$(R_P)$$ ( R P ) of R in the ferromagnetic phase are observed below the $$R=0$$ R = 0 line. We also find that the separate $$R_M$$ R M and $$R_P$$ R P lines exist at higher temperatures within the paramagnetic phase. Under the effect of external magnetic field $$(h \ne 0)$$ ( h ≠ 0 ) , these lines disappear and an additional $$R_P$$ R P curve appears just above the second-order phase transition line in the phase diagram.