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 $$h=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . Notably, two half rings including the minima $$(R_M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>M</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and maxima $$(R_P)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>P</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of R in the ferromagnetic phase are observed below the $$R=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> line. We also find that the separate $$R_M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:math> and $$R_P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> lines exist at higher temperatures within the paramagnetic phase. Under the effect of external magnetic field $$(h \ne 0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>h</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , these lines disappear and an additional $$R_P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> curve appears just above the second-order phase transition line in the phase diagram.