Microcanonical finite-size scaling in second-order phase transitions with diverging specific heat

A microcanonical finite-size ansatz in terms of quantities measurable in a finite lattice allows extending phenomenological renormalization (the so-called quotients method) to the microcanonical ensemble. The ansatz is tested numerically in two models where the canonical specific heat diverges at criticality, thus implying Fisher renormalization of the critical exponents: the three-dimensional ferromagnetic Ising model and the two-dimensional four-state Potts model (where large logarithmic corrections are known to occur in the canonical ensemble). A recently proposed microcanonical cluster method allows simulating systems as large as $L=1024$ (Potts) or $L=128$ (Ising). The quotients method provides accurate determinations of the anomalous dimension, $\eta$, and of the (Fisher-renormalized) thermal $\nu$ exponent. While in the Ising model the numerical agreement with our theoretical expectations is very good, in the Potts case, we need to carefully incorporate logarithmic corrections to the microcanonical ansatz in order to rationalize our data.

Figure 1

(Color online) Scaling plot of the correlation length (in lattice size units) and the scaled susceptibility for the three-dimensional Ising model. We used the critical values $e_c=-0.867433$ and ${\nu }_m=0.7077$. Notice the strong scaling corrections for the small systems, as well as the data collapse for the largest lattices.

Figure 2

(Color online) Crossing points of the correlation length in lattice size units for the three-dimensional Ising model. The error bars are in every case smaller than the point sizes. The values of the different quantities at the crossing as well as the critical exponents are shown in Table .

Figure 3

(Color online) Microcanonical estimate of the specific heat, $C_m\left(L,e\right)$, at $e_{c,L,2L}$ for the $D=3$ Ising model, as a function of the system size. The numerical estimates of exponents $\alpha /\nu$ and $\omega$ were taken from Ref. [41]. The error bars are in every case smaller than the point sizes. The solid line is a fit to Eq. (48) (fitting parameters: $A_0$, $A_1$, and $B$) and the dashed one is obtained by constraining the fit to $A_1=0$.

Figure 4

(Color online) Graphical demonstration of Eq. (55) as applied to the microcanonical $D=2$, $Q=4$ Potts model. Both the correlation length in units of the lattice size (top) and the scaled susceptibility, $\overline{\chi }$ in Eq. (56) (bottom), are functions of the scaling variable $\left(e-e_c\right)L^{1/2}{\left(\text{log}L\right)}^{3/4}$.

Figure 5

(Color online) Comparison of the scaling for the naively scaled susceptibility $\chi L^{-7/4}$ (top) and for $\overline{\chi }$ (bottom) as a function of the correlation length in units of the lattice size, for the microcanonical $D=2$, $Q=4$ Potts model.

Figure 6

(Color online) Logarithmic scaling behavior of the susceptibility at the critical point. The error bars are in every case smaller than the point sizes. Dashed line does not include the subleading additive terms of Eq. (57) while the solid line does.

Figure 7

(Color online) Correlation length in lattice size units for the two-dimensional $Q=4$ Potts model. The values of the different quantities on the crossings for lattices $L$ and $2L$, as well as the corresponding estimate for critical exponents, are in Table . The inset is a magnification of the critical region.

Figure 8

(Color online) Canonical probability distribution function for the energy density, $P_{\beta }^{\left(L\right)}\left(e\right)$, as reconstructed from microcanonical simulations of the $D=2$, $Q=4$ Potts model and different system sizes. The $L$-dependent critical point ${\beta }_{c,L}$ is computed using the Maxwell’s rule, Sec. 2B (note the equal height of the two peaks enforced by Maxwell’s construction). The system displays an apparent latent heat that becomes smaller for growing $L$ and vanishes in the large $L$ limit.

Figure 9

(Color online) (Top) From the microcanonical mean values ${⟨\widehat{\beta }⟩}_{e,L}$ for the $D=2$, $Q=4$ Potts model, we estimate the size-dependent canonical inverse critical temperature ${\beta }_{c,L}$ (horizontal lines) for all the simulated lattice sizes, ranging from $L=32$ (lower) to $L=1024$ (upper). We show as well the analytical prediction (upper horizontal line). (Bottom left) Example of Maxwell’s construction for our $L=32$ data. The $e$ integral of ${⟨\widehat{\beta }⟩}_{e,L}-{\beta }_{c,L}$ from $e_o$ to $e_d$ vanishes. (Bottom right) Zoom of upper panel showing only data for lattice sizes $L=256$ (lower curve), $L=512$ (medium curve), and $L=1024$ (upper curve).

Figure 10

(Color online) System-size-dependent estimates for the energies of the “coexisting” ordered ($e_o$, blue squares) and disordered ($e_d$, red triangles) phases of the $D=2$, $Q=4$ Potts model as a function of $L^{-1/2}$. Lines are fits to the expected analytical behavior Eq. (60). Horizontal line corresponds to the asymptotic value, $e_c$.