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Phase diagrams and thermodynamic study of the mixed spin-1/2 and spin-3 Blume-Capel model: renormalization group theory

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Abstract

In this work, we study the Blume-Capel model of a mixed spin system (\(\sigma =1/2\) and \(S=3\)) in a hypercubic lattice of dimension d. For this purpose, we choose to work with the renormalization group method, namely the Migdal-Kadanoff technique. The results show that there is a critical dimension \(d_{c}\approx 2.14\), above which the critical behavior of the system changes; therefore, we can distinguish two possible cases, one when \(d<d_{c}\) and the other when \(d\ge d_{c}\). Considering \(d=2\) \(\left( d<d_{c}\right)\) and \(d=3\) \(\left( d\ge d_{c}\right)\), we identify the stable and unstable fixed points where there is no tricritical point, determine the critical exponents and verify their universal behavior. For the same dimensions, we plot the phase diagrams in the \(\left( \Delta _{2}/J, 1/J\right)\) and \(\left( \Delta _{2}, J\right)\) planes with a second order transition. Furthermore, we show that at very low temperatures, and by using free energy derivation, three first-order phase transitions can take place.

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Data Availability Statement

No data were used for the research described in this manuscript, and all of the material is owned by the authors and/or no permissions are required.

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Authors and Affiliations

  1. LS2ME Laboratory, Polydisciplinary Faculty, Sultan Moulay Slimane University, 25000, Khouribga, Morocco

    T. Mouhrach, H. Zahir, A. Fathi & K. Sbiaai

  2. LPMC Laboratory, Theoretical Physics Group, Faculty of Sciences, Chouaïb Doukkali University, 24000, El Jadida, Morocco

    T. Mouhrach, H. Zahir & M. El Bouziani

Authors
  1. T. Mouhrach
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  2. H. Zahir
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  3. A. Fathi
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  4. K. Sbiaai
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  5. M. El Bouziani
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Corresponding author

Correspondence to M. El Bouziani.

Appendix

Appendix

The expressions of \(F_{i,j}\) obtained using the Eqs. (3), (4) and (5) are given by:

  • For \(\displaystyle \left( \sigma _{2}, S_{1}\right) =\left\{ \left( \frac{1}{2}, 0\right) ; \left( -\frac{1}{2}, 0\right) \right\}\)

    $$\begin{aligned} F_{\frac{1}{2},0}= \, & {} 2+2\cdot e^{\displaystyle 2\frac{\Delta _{2}}{z}+2\frac{\Delta _{4}}{z}+2\frac{\Delta _{6}}{z}}\times e^{\displaystyle \frac{1}{2}J+\frac{5}{16}C+\frac{17}{64}F}\times \cosh \left( \displaystyle \frac{1}{2}J+\frac{5}{16}C+\frac{17}{64}F\right) \nonumber \\{} & {} +2\cdot e^{\displaystyle 2\frac{\Delta _{2}}{z}+2\frac{\Delta _{4}}{z}+2\frac{\Delta _{6}}{z}}\times e^{\displaystyle -\frac{1}{2}J-\frac{5}{16}C-\frac{17}{64}F}\times \cosh \left( \displaystyle \frac{1}{2}J+\frac{5}{16}C+\frac{17}{64}F\right) \nonumber \\{} & {} +2\cdot e^{\displaystyle 8\frac{\Delta _{2}}{z}+32\frac{\Delta _{4}}{z}+128\frac{\Delta _{6}}{z}}\times e^{\displaystyle J+\frac{17}{8}C+\frac{257}{32}F}\times \cosh \left( \displaystyle J+\frac{17}{8}C+\frac{257}{32}F\right) \nonumber \\{} & {} +2\cdot e^{\displaystyle 8\frac{\Delta _{2}}{z}+32\frac{\Delta _{4}}{z}+128\frac{\Delta _{6}}{z}}\times e^{\displaystyle -J-\frac{17}{8}C-\frac{257}{32}F}\times \cosh \left( J+\frac{17}{8}C+\frac{257}{32}F\right) \nonumber \\{} & {} +2\cdot e^{\displaystyle 18\frac{\Delta _{2}}{z}+162\frac{\Delta _{4}}{z}+1458\frac{\Delta _{6}}{z}}\times e^{\displaystyle \frac{3}{2}J+\frac{111}{16}C+\frac{3891}{64}F}\times \cosh \left( \displaystyle \frac{3}{2}J +\frac{111}{16}C+\frac{3891}{64}F\right) \nonumber \\{} & {} +2\cdot e^{\displaystyle 18\frac{\Delta _{2}}{z}+162\frac{\Delta _{4}}{z}+1458\frac{\Delta _{6}}{z}}\times e^{\displaystyle -\frac{3}{2}J-\frac{111}{16}C- \frac{3891}{64}F}\times \cosh \left( \displaystyle \frac{3}{2}J+\frac{111}{16}C+\frac{3891}{64}F\right) \end{aligned}$$
    (16)

  • For \(\displaystyle \left( \sigma _{2}, S_{1}\right) =\left\{ \left( \frac{1}{2}, 1\right) ;\left( -\frac{1}{2}, -1\right) \right\}\)

    $$\begin{aligned} F_{\frac{1}{2},1}= \, & {} 2\cdot e^{\displaystyle \frac{\Delta _{2}}{z}+\frac{\Delta _{4}}{z}+\frac{\Delta _{6}}{z}} \times \cosh \left( \displaystyle \frac{1}{2}J+\frac{5}{16}C+\frac{17}{64}F\right) \nonumber \\{} & {} +2\cdot e^{\displaystyle 3\frac{\Delta _{2}}{z}+3\frac{\Delta _{4}}{z}+3\frac{\Delta _{6}}{z}}\times e^{\displaystyle \frac{1}{2}J+ \frac{5}{16}C+\frac{17}{64}F} \times \cosh \left( \displaystyle J+\frac{10}{16}C+\frac{34}{64}F\right) \nonumber \\{} & {} + 2\cdot e^{\displaystyle 3\frac{\Delta _{2}}{z}+3\frac{\Delta _{4}}{z}+3\frac{\Delta _{6}}{z}}\times e^{\displaystyle -\frac{1}{2}J-\frac{5}{16}C-\frac{17}{64}F}\nonumber \\{} & {} +2\cdot e^{\displaystyle 9\frac{\Delta _{2}}{z}+33\frac{\Delta _{4}}{z}+129\frac{\Delta _{6}}{z}}\times e^{\displaystyle J+\frac{17}{8}C+\frac{257}{32}F}\times \cosh \left( \displaystyle \frac{3}{2}J+\frac{39}{16}C+\frac{531}{64}F\right) \nonumber \\{} & {} +2\cdot e^{\displaystyle 9\frac{\Delta _{2}}{z}+33\frac{\Delta _{4}}{z}+129\frac{\Delta _{6}}{z}}\times e^{\displaystyle -J-\frac{17}{8}C-\frac{257}{32}F}\times \cosh \left( \displaystyle \frac{1}{2}J+\frac{29}{16}C+\frac{497}{64}F\right) \nonumber \\{} & {} +2\cdot e^{\displaystyle 19\frac{\Delta _{2}}{z}+163\frac{\Delta _{4}}{z}+1459\frac{\Delta _{6}}{z}}\times e^{\displaystyle \frac{3}{2}J+\frac{111}{16}C+\frac{3891}{64}F}\times \cosh \left( \displaystyle 2J+\frac{116}{16}C+\frac{3908}{64}F\right) \nonumber \\{} & {} +2\cdot e^{\displaystyle 19\frac{\Delta _{2}}{z}+163\frac{\Delta _{4}}{z}+1459\frac{\Delta _{6}}{z}}\times e^{\displaystyle -\frac{3}{2}J-\frac{111}{16}C-\frac{3891}{64}F}\times \cosh \left( \displaystyle J+\frac{106}{16}C+\frac{3874}{64}F\right) \end{aligned}$$
    (17)

  • For \(\displaystyle \left( \sigma _{2}, S_{1}\right) =\left\{ \left( -\frac{1}{2}, 1\right) ;\left( \frac{1}{2}, -1\right) \right\}\)

    $$\begin{aligned} F_{-\frac{1}{2},1}=F_{\frac{1}{2},-1}=F_{\frac{1}{2},1}\left( J=-J,C=-C, F=-F\right) \end{aligned}$$
    (18)

  • For \(\displaystyle \left( \sigma _{2}, S_{1}\right) =\left\{ \left( \frac{1}{2}, 2\right) ;\left( -\frac{1}{2}, -2\right) \right\}\)

    $$\begin{aligned} F_{\frac{1}{2},2}= \, & {} 2\cdot e^{\displaystyle 4\frac{\Delta _{2}}{z}+16\frac{\Delta _{4}}{z}+64\frac{\Delta _{6}}{z}}\times \cosh \left( \displaystyle J+ \frac{34}{16}C+\frac{514}{64}F\right) \nonumber \\{} & {} + 2\cdot e^{\displaystyle 6\frac{\Delta _{2}}{z}+18\frac{\Delta _{4}}{z}+66\frac{\Delta _{6}}{z}} \times e^{\displaystyle \frac{1}{2}J+ \frac{5}{16}C+\frac{17}{64}F}\times \cosh \left( \displaystyle \frac{3}{2}J+ \frac{39}{16}C+ \frac{531}{64}F\right) \nonumber \\{} & {} +2\cdot e^{\displaystyle 6\frac{\Delta _{2}}{z}+18\frac{\Delta _{4}}{z}+66\frac{\Delta _{6}}{z}} \times e^{\displaystyle -\frac{1}{2}J-\frac{5}{16}C-\frac{17}{64}F}\times \cosh \left( \displaystyle \frac{1}{2}J+ \frac{29}{16}C+ \frac{497}{64}F\right) \nonumber \\{} & {} + 2\cdot e^{\displaystyle 12\frac{\Delta _{2}}{z}+48\frac{\Delta _{4}}{z}+192\frac{\Delta _{6}}{z}}\times e^{\displaystyle J+\frac{17}{8}C+\frac{257}{32}F}\times \cosh \left( \displaystyle 2J+\frac{68}{16}C+\frac{1028}{64}F\right) \nonumber \\{} & {} + 2\cdot e^{\displaystyle 12\frac{\Delta _{2}}{z}+48\frac{\Delta _{4}}{z}+192\frac{\Delta _{6}}{z}}\times e^{\displaystyle -J-\frac{17}{8}C-\frac{257}{32}F} \nonumber \\{} & {} +2\cdot e^{\displaystyle 22\frac{\Delta _{2}}{z}+178\frac{\Delta _{4}}{z}+1522\frac{\Delta _{6}}{z}}\times e^{\displaystyle -\frac{3}{2}J-\frac{111}{16}C-\frac{3891}{64}F}\times \cosh \left( \displaystyle \frac{1}{2}J+\frac{77}{16}C+\frac{3377}{64}F\right) \nonumber \\{} & {} + e^{\displaystyle 22\frac{\Delta _{2}}{z}+178\frac{\Delta _{4}}{z}+1522\frac{\Delta _{6}}{z}}\times e^{\displaystyle \frac{3}{2}J+\frac{111}{16}C+\frac{3891}{64}F}\times \cosh \left( \displaystyle \frac{5}{2}J+\frac{145}{16}C+\frac{4405}{64}F\right) \end{aligned}$$
    (19)

  • For \(\left( \sigma _{2}, S_{1}\right) =\left\{ \left( -\frac{1}{2}, 2\right) ;\left( \frac{1}{2}, -2\right) \right\}\)

    $$\begin{aligned} F_{-\frac{1}{2},2}=F_{\frac{1}{2},-2}=F_{\frac{1}{2},2}\left( J=-J,C=-C, F=-F\right) \end{aligned}$$
    (20)

  • For \(\displaystyle \left( \sigma _{2}, S_{1}\right) =\left\{ \left( \frac{1}{2}, 3\right) ;\left( -\frac{1}{2}, -3\right) \right\}\)

    $$\begin{aligned} F_{\frac{1}{2},3}= \, & {} 2\cdot e^{\displaystyle 9\frac{\Delta _{2}}{z}+81\frac{\Delta _{4}}{z}+729\frac{\Delta _{6}}{z}}\times \cosh \left( \displaystyle \frac{3}{2}J+ \frac{111}{16}C+\frac{3891}{64}F\right) \nonumber \\{} & {} +2\cdot e^{\displaystyle 11\frac{\Delta _{2}}{z}+83\frac{\Delta _{4}}{z}+731\frac{\Delta _{6}}{z}} \times e^{\displaystyle \frac{1}{2}J+ \frac{5}{16}C+\frac{17}{64}F}\times \cosh \left( \displaystyle 2J+ \frac{116}{16}C+ \frac{3908}{64}F\right) \nonumber \\{} & {} +2\cdot e^{\displaystyle 11\frac{\displaystyle \Delta _{2}}{z}+83\frac{\Delta _{4}}{z}+731\frac{\Delta _{6}}{z}} \times e^{\displaystyle -\frac{1}{2}J-\frac{5}{16}C-\frac{17}{64}F}\times \cosh \left( \displaystyle J+ \frac{106}{16}C+ \frac{3874}{64}F\right) \nonumber \\{} & {} +2\cdot e^{\displaystyle 17\frac{\Delta _{2}}{z}+113\frac{\Delta _{4}}{z}+857\frac{\Delta _{6}}{z}}\times e^{\displaystyle -J-\frac{17}{8}C-\frac{257}{32}F}\times \cosh \left( \frac{1}{2}J+\frac{77}{16}C+\frac{3377}{64}F\right) \nonumber \\{} & {} + 2\cdot e^{\displaystyle 17\frac{\Delta _{2}}{z}+113\frac{\Delta _{4}}{z}+857\frac{\Delta _{6}}{z}}\times e^{\displaystyle J+\frac{17}{8}C+\frac{257}{32}F}\times \cosh \left( \displaystyle \frac{5}{2}J+\frac{145}{16}C+\frac{4405}{64}F\right) \nonumber \\{} & {} + 2\cdot e^{\displaystyle 27\frac{\Delta _{2}}{z}+243\frac{\Delta _{4}}{z}+2187\frac{\Delta _{6}}{z}}\times e^{\displaystyle \frac{3}{2}J+\frac{111}{16}C+\frac{3891}{64}F}\times \cosh \left( \displaystyle 3J+\frac{222}{16}C+\frac{7782}{64}F\right) \nonumber \\{} & {} + 2\cdot e^{\displaystyle 27\frac{\Delta _{2}}{z}+243\frac{\Delta _{4}}{z}+2187\frac{\Delta _{6}}{z}}\times e^{\displaystyle -\frac{3}{2}J-\frac{111}{16}C-\frac{3891}{64}F} \end{aligned}$$
    (21)

  • For \(\displaystyle \left( \sigma _{2}, S_{1}\right) =\left\{ \left( -\frac{1}{2}, 3\right) ; \left( \frac{1}{2}, -3\right) \right\}\)

    $$\begin{aligned} F_{-\frac{1}{2},3}=F_{\frac{1}{2},-3}=F_{\frac{1}{2},3}\left( J=-J,C=-C, F=-F\right) \end{aligned}$$
    (22)

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Mouhrach, T., Zahir, H., Fathi, A. et al. Phase diagrams and thermodynamic study of the mixed spin-1/2 and spin-3 Blume-Capel model: renormalization group theory. Eur. Phys. J. Plus 139, 388 (2024). https://doi.org/10.1140/epjp/s13360-024-05190-3

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