Continuous phase transitions with a convex dip in the microcanonical entropy

The appearance of a convex dip in the microcanonical entropy of finite systems usually signals a first order transition. However, a convex dip also shows up in some systems with a continuous transition as, for example, in the Baxter-Wu model and in the four-state Potts model in two dimensions. We demonstrate that the appearance of a convex dip in those cases can be traced back to a finite-size effect. The properties of the dip are markedly different from those associated with a first order transition and can be understood within a microcanonical finite-size scaling theory for continuous phase transitions. Results obtained from numerical simulations corroborate the predictions of the scaling theory.

Figure 1

Schematic depiction of the convex dip in the entropy of a finite system whose corresponding infinite system undergoes a discontinuous phase transition. The double-tangent with slope ${\beta }_{\mathrm{t}}$ is indicated by a dashed line (top). For the function $\tilde{s}=s-{\beta }_{\mathrm{t}}e$ the definitions of the width $\omega$ and of the depth $\delta$ of the dip are shown (bottom).

Figure 2

The depth $\delta \left(l\right)$ and width $\omega \left(l\right)$ of the convex dip in the microcanonical entropy of the three-state Potts model in three dimensions as a function of the inverse system size. The dashed lines are linear fits to the data. Error bars are comparable to the size of the symbols.

Figure 3

The derivative ${\beta }_{\mu }^{\prime }(e_{\mathrm{pc}},l)=d{\beta }_{\mu }(e_{\mathrm{pc}},l)∕de$ of the Baxter-Wu model as a function of the inverse system size $l$ for open boundary conditions. The errors are approximately as large as the symbol size. The dashed line shows a linear fit. The data extrapolate to zero in the limit $l\to 0$, corresponding to a divergent specific heat in the infinite system.

Figure 4

The function $\tilde{s}$ for the finite Baxter-Wu system with $L=60$ and periodic boundary conditions (here ${\beta }_{\mathrm{t}}=0.43945$). Note that $\tilde{s}$ is only obtained from a numerical simulation up to an additive constant. The data show a double-peak structure typical of finite systems with a discontinuous transition in the macroscopic limit.

Figure 5

The depth $\delta \left(l\right)$ and width $\omega \left(l\right)$ as functions of the inverse system size $l$ for Baxter-Wu systems with periodic boundary conditions. The errors are of the order of the symbol size. The full lines show fits to the theoretically predicted behavior [Eqs. (11, 12)] using the six largest system sizes. The dashed lines include in addition the first correction term coming from the regular part $s_{\mathrm{r}}$ of the microcanonical entropy. The inset shows a double-logarithmic plot of the width together with a straight line of slope 1/2. From the six largest systems one gets a slope of 0.481(3).

Figure 6

The derivative ${\beta }_{\mu }^{\prime }(e_{\mathrm{pc}},l)=d{\beta }_{\mu }(e_{\mathrm{pc}},l)∕de$ of the four-state Potts model as a function of the inverse system size $l$ for open boundary conditions. The errors are approximately of the order of the symbol size. The dashed line shows a fit with an asymptotic linear term and corrections including a logarithmic term. The data extrapolate to zero for the limit $l\to 0$, yielding a divergent specific heat in the infinite system. The inset shows a fit without the logarithmic correction term (see main text for further details).

Figure 7

The depth $\delta \left(l\right)$ and width $\omega \left(l\right)$ as functions of the inverse system size $l$ for four-state Potts systems with periodic boundary conditions. The errors are approximately as large as the symbol size. The dashed lines show fits of the expected scaling relations (11, 12) to the data with a logarithmic correction term.