diff options
| -rw-r--r-- | essential-ising.bib | 61 | ||||
| -rw-r--r-- | essential-ising.tex | 34 | ||||
| -rw-r--r-- | figs/fig-susmag.gplot | 71 |
3 files changed, 143 insertions, 23 deletions
diff --git a/essential-ising.bib b/essential-ising.bib index df4a96b..16a0883 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -1,4 +1,22 @@ +@article{ambegaokar.1978.dissipation, + author = "V. Ambegaokar and B. I. Halperin and D. Nelson and E. Siggia", + year = "1978", + journal = "Physical Review Letters", + volume = "40", + pages = "783--6", + title = "Dissipation in two-dimensional superfluids", +} + +@article{ambegaokar.1980.dynamics, + author = "V. Ambegaokar and B. I. Halperin and D. Nelson and E. Siggia", + year = "1980", + journal = "Physical Review B", + volume = "21", + pages = "1806--26", + title = "Dynamics of superfluid films", +} + @article{aharony.1983.fields, title={Nonlinear scaling fields and corrections to scaling near criticality}, author={Aharony, Amnon and Fisher, Michael E}, @@ -90,6 +108,14 @@ publisher={IOP Publishing} } +@article{chen.2013.universal, + title={Universal scaling function for the two-dimensional Ising model in an external field: A pragmatic approach}, + author={Chen, Yan-Jiun and Paquette, Natalie M and Machta, Benjamin B and Sethna, James P}, + journal={arXiv preprint arXiv:1307.6899}, + year={2013} +} + + @article{dimitrovic.1991.finite, title={Finite-size effects, goldstone bosons and critical exponents in the d= 3 Heisenberg model}, author={Dimitrovi{\'c}, I and Hasenfratz, P and Nager, J and Niedermayer, Ferenc}, @@ -294,6 +320,28 @@ publisher={IOP Publishing} } +@article{mangazeev.2008.variational, + title={Variational approach to the scaling function of the 2D Ising model in a magnetic field}, + author={Mangazeev, Vladimir V and Batchelor, Murray T and Bazhanov, Vladimir V and Dudalev, Michael Yu}, + journal={Journal of Physics A: Mathematical and Theoretical}, + volume={42}, + number={4}, + pages={042005}, + year={2008}, + publisher={IOP Publishing} +} + +@article{mangazeev.2010.scaling, + title={Scaling and universality in the two-dimensional Ising model with a magnetic field}, + author={Mangazeev, Vladimir V and Dudalev, Michael Yu and Bazhanov, Vladimir V and Batchelor, Murray T}, + journal={Physical Review E}, + volume={81}, + number={6}, + pages={060103}, + year={2010}, + publisher={APS} +} + @article{mccraw.1978.metastability, title={Metastability in the two-dimensional Ising model}, author={McCraw, RJ and Schulman, LS}, @@ -413,6 +461,18 @@ publisher={APS} } +@article{wu.1976.spin, + title={Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region}, + author={Wu, Tai Tsun and McCoy, Barry M and Tracy, Craig A and Barouch, Eytan}, + journal={Physical Review B}, + volume={13}, + number={1}, + pages={316}, + year={1976}, + publisher={APS} +} + + @article{zinn.1996.universal, title={Universal surface-tension and critical-isotherm amplitude ratios in three dimensions}, author={Zinn, Shun-yong and Fisher, Michael E}, @@ -424,3 +484,4 @@ publisher={Elsevier} } + diff --git a/essential-ising.tex b/essential-ising.tex index ea8e9eb..c98e938 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -202,7 +202,7 @@ same singular behavior as the real part of the equilibrium free energy, and that for small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$, where \[ - \fiF(X)=-A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}} + \fiF(X)=A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}} \label{eq:im.scaling} \] and $\Theta$ is the Heaviside function. Results from combining an analysis of @@ -224,14 +224,14 @@ given our scaling ansatz, yielding \def\eqthreedeeone{ \fF^\threedee(Y/B)&= \frac{A}{12}\frac{e^{-1/Y^2}}{Y^2} - \bigg[4Y\Gamma(\tfrac23)E_{5/3}(-Y^{-2}) + \bigg[\Gamma(\tfrac16)E_{7/6}(-Y^{-2}) } \def\eqthreedeetwo{ - -\Gamma(\tfrac16)E_{7/6}(-Y^{-2})\bigg] + -4Y\Gamma(\tfrac23)E_{5/3}(-Y^{-2})\bigg] } \def\eqfourdeeone{ \fF^\fourdee(Y/B)&= - \frac{A}{9\pi}\frac{e^{1/Y^3}}{Y^2} + -\frac{A}{9\pi}\frac{e^{1/Y^3}}{Y^2} \Big[3\ei(-Y^{-3}) } \def\eqfourdeetwo{ @@ -242,7 +242,7 @@ given our scaling ansatz, yielding \begin{align} &\begin{aligned} \eqthreedeeone\\ - &\hspace{8em} + &\hspace{7em} \eqthreedeetwo \end{aligned} \\ @@ -264,13 +264,13 @@ At the level of truncation of \eqref{eq:im.scaling} at which we are working the Kramers--Kronig relation does not converge in \twodee. However, higher moments can still be extracted, e.g., the susceptibility, by taking \[ - \chi=\pd[2]Fh - =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. + \chi=\pd MH=-\frac1{T_\c}\pd[2]Fh + =-\frac2{\pi T_\c}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. \] With a scaling form defined by $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, this yields \[ - \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] + \fX^\twodee(Y/B)=\frac{AB^2}{\pi T_\c Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] \label{eq:sus_scaling} \] Scaling forms for the free energy can then be extracted by direct integration @@ -278,9 +278,9 @@ and their constants of integration fixed by known zero field values, yielding \begin{align} \label{eq:mag_scaling} \fM^\twodee(Y/B) - &=\fM(0)+\frac{ABT_\c}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ + &=\fM(0)+\frac{AB}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ \fF^\twodee(Y/B) - &=\fF(0)+T_\c Y\bigg(\frac{\fM(0)}B+\frac{AT_\c}\pi e^{1/Y}\ei(-1/Y)\bigg) + &=-Y\bigg(\frac{\fM(0)}B-\frac{A}\pi e^{1/Y}\ei(-1/Y)\bigg) \end{align} with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in two dimensions. @@ -325,8 +325,8 @@ $C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of the susceptibility scaling function, then -$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant -$A$ is fixed to $A=\pi\fX(0)/2B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_0^-$. The +$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi T_\c$ and the constant +$A$ is fixed to $A=\pi T_\c\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. The resulting scaling functions $\fX$ and $\fM$ are plotted as solid lines in Fig.~\ref{fig:scaling_fits}. As can be seen, there is very good agreement between our proposed functional forms and what is measured. However, there @@ -349,8 +349,10 @@ where $F_n'(Y)=f_n(Y)$ and The functions $f_n$ have been chosen to be pure integer power laws for small argument, but vanish appropriately at large argument. This is necessary because the susceptibility vanishes with $h|t|^{-\beta\delta}$ while bare -polynomial corrections would not. We fit these functions to our numeric data -for $N=0$ while requiring that $C_0^-/T_\c=\fX'(0)=c_0+2AB^2/\pi$. The +polynomial corrections would not. We fit these functions to known moments of +the free energy's scaling function +\cite{mangazeev.2008.variational,mangazeev.2010.scaling} our numeric data +for $N=0$. The resulting curves are also plotted as dashed lines in Fig.~\ref{fig:scaling_fits}. Our singular scaling function with one low-order correction appears to match data quite well. @@ -385,10 +387,10 @@ into the scaling function gives good convergence to the simulations in \twodee. Our results should allow improved high-precision functional forms for the free energy~\cite{caselle.2001.critical}, and should have implications for the scaling -of correlation functions~\cite{YJXXX,XXX}. Our methods might be generalized +of correlation functions~\cite{chen.2013.universal,wu.1976.spin}. Our methods might be generalized to predict similar singularities in systems where nucleation and metastability are proximate to continuous phase transitions, such as 2D superfluid -transitions~\cite{ALHN}, the melting of 2D crystals~\cite{XXX}, and +transitions~\cite{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{XXX}, and freezing transitions in glasses, spin glasses, and other disordered systems. diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index 9682953..a2f4584 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -7,6 +7,8 @@ Delta = 15. / 8 gamma = 7. / 4 beta = 1. / 8 +Ch = 0.838677624411 + t(T) = abs((T - Tc) / T) h(T, H) = H / T X(T, H) = h(T, H) * t(T)**(-Delta) @@ -14,14 +16,69 @@ X(T, H) = h(T, H) * t(T)**(-Delta) poly(A, l, m, X) = A * X**m / (1 + (l * X)**(m + 1)) polyint(A, l, m, X) = A * l**(-(m + 1)) * log(1 + (l * X)**(m + 1)) / (m + 1) +G(i) = i == 1 ? -1.3578383417066 : \ + i == 2 ? -0.048953289720 : \ + i == 3 ? 0.038863932 : \ + i == 4 ? -0.068362119 : \ + i == 5 ? 0.18388370 : \ + i == 6 ? -0.6591714 : \ + i == 7 ? 2.937665 : \ + i == 8 ? -15.61 : 0 +GC(i) = G(i) * (2 * asinh(1))**2 * (Ch * (2 * asinh(1))**(-Delta))**i + M0 = (2**2.5 * asinh(1))**0.125 B = Tc**2 * M0 / (16 * pi) C0 = 0.0255369719 A = pi / 2 * C0 / (B**2 * Tc) -c0 = -0.012384 -lamb = 1.76962 -A2 = pi / 2 * (C0 / Tc - c0) / B**2 +#c0 = -0.012384 +#lamb = 1.76962 +#A2 = pi / 2 * (C0 / Tc - c0) / B**2 + +#c0 = 0 +#lamb = 1 +#B2 = 0.521944 +#A2 = pi / 2 * C0 / (B2**2 * Tc) + +n = 1 +c(i) = i == 1 ? 0.0037735 : 0 +lamb = 10.487 +B2 = B +A2 = 0.749317 + +#n = 2 +#c(i) = i == 1 ? -0.177238 : \ +# i == 2 ? -0.0545988 : 0 +#lamb = 3.14865 +#B2 = B +#A2 = 18.8816 + +#n = 3 +#c(i) = i == 1 ? 0.00280714 : \ +# i == 2 ? -0.00243938 : \ +# i == 3 ? -0.0140978 : 0 +#lamb = 12.1954 +#B2 = B +#A2 = 0.846118 + +n2 = 4 +c2(i) = i == 1 ? 0.00245324 : \ + i == 2 ? -0.00468448 : \ + i == 3 ? 0.0092602 : \ + i == 4 ? -0.536727 : 0 +lamb2 = 12.6068 +B3 = B +A3 = 0.88157 + +#n2 = 5 +#c2(i) = i == 1 ? 0.00281829 : \ +# i == 2 ? 0.000632215 : \ +# i == 3 ? -0.0911689 : \ +# i == 4 ? 1.93584 : \ +# i == 5 ? -24.7397 : 0 +#lamb2 = 13.2489 +#B3 = B +#A3 = 0.845002 susfunc = "figs/fig-sus_scaling-func.dat" magfunc = "figs/fig-mag_scaling-func.dat" @@ -42,7 +99,7 @@ set mxtics 5 set mytics 5 set bmargin 0.2 -plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B):(10**(3+$2) * A2 * B**2 + 10**3 * poly(c0, lamb, 0, 10**$1)) with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**(3+$2) * A2 * B2**2 + 10**3 * (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B2):(10**(3+$2) * A3 * B3**2 + 10**3 * (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B2):(-10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (10**$1 / B2)**(i-1) / Tc)) with lines dt 5 lw 2 lc black set bmargin -1 set tmargin 0.2 @@ -53,7 +110,7 @@ set ylabel offset 1,0 '$M|t|^{-\beta}$' set xlabel '$h|t|^{-\beta\delta}$' set xtics format '%g' -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A2 * B + polyint(Tc * c0, lamb, 0, 10**$1) / B) smooth csplines with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * Tc * A2 * B2 + (sum[i=1:n] polyint(Tc * c(i), lamb, i-1, 10**$1)) / B2) smooth csplines with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * Tc * A3 * B3 + (sum[i=1:n2] polyint(Tc * c2(i), lamb2, i-1, 10**$1)) / B3) smooth csplines with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10**$1 / B)**(i-1)) with lines dt 5 lw 2 lc black set logscale xy set tmargin -1 @@ -70,7 +127,7 @@ set xtics add ('$\footnotesize10^{-2}$' 10**(-2), "" 0.1, '$\footnotesize10^0$' set mytics 5 set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 -plot num using (X($2, $3)):($10 * t($2)**gamma):($11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B):(10**$2 * A2 * B**2 + poly(c0, lamb, 0, 10**$1)) with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):($10 * t($2)**gamma):($11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**$2 * A2 * B2**2 + (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B3):(10**$2 * A3 * B3**2 + (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B):(-sum[i=2:8] GC(i) * i * (i-1) * (10**$1 / B)**(i-2) / Tc) with lines dt 5 lw 2 lc black unset logscale xy set logscale x @@ -81,5 +138,5 @@ set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' set ytics format '\footnotesize {%g}' 1.2,0.2,1.8 set mytics 5 -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A2 * B + polyint(Tc * c0, lamb, 0, 10**$1) / B) with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * Tc * A2 * B2 + (sum[i=1:n] polyint(Tc * c(i), lamb, i-1, 10**$1)) / B2) with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * Tc * A3 * B3 + (sum[i=1:n2] polyint(Tc * c2(i), lamb2, i-1, 10**$1)) / B2) with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10 **$1 / B)**(i-1)) with lines dt 5 lw 2 lc black |