From c1fd415bd5dcee285c95e36e4def5fa412a87bb1 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 22 Aug 2017 22:28:01 -0400 Subject: new citations, some small fixes --- essential-ising.bib | 73 +++++++++++++++++++++++++++++++++++++++++++++++++++++ essential-ising.tex | 19 ++++++++------ 2 files changed, 84 insertions(+), 8 deletions(-) diff --git a/essential-ising.bib b/essential-ising.bib index f490dd9..1de2478 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -28,6 +28,13 @@ publisher={APS} } +@article{assis.2017.analyticity, + title={Analyticity of the Ising susceptibility: An interpretation}, + author={Assis, M and Jacobsen, JL and Jensen, I and Maillard, JM and McCoy, BM}, + journal={arXiv preprint arXiv:1705.02541}, + year={2017} +} + @article{barouch.1973.susceptibility, title={Zero-field susceptibility of the two-dimensional Ising model near ${T}_c$}, author={Barouch, Eytan and McCoy, Barry M and Wu, Tai Tsun}, @@ -108,6 +115,17 @@ publisher={IOP Publishing} } +@article{chan.2011.ising, + title={The Ising susceptibility scaling function}, + author={Chan, Y and Guttmann, Anthony J and Nickel, BG and Perk, JHH}, + journal={Journal of Statistical Physics}, + volume={145}, + number={3}, + pages={549--590}, + year={2011}, + publisher={Springer} +} + @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}, @@ -115,6 +133,17 @@ year={2013} } +@article{dahm.1989.dynamics, + title={Dynamics of dislocation-mediated melting in a two-dimensional lattice in the presence of an oscillatory applied strain}, + author={Dahm, AJ and Stan, MA and Petschek, RG}, + journal={Physical Review B}, + volume={40}, + number={13}, + pages={9006}, + year={1989}, + publisher={APS} +} + @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}, @@ -221,6 +250,17 @@ publisher={APS} } +@article{guttmann.1996.solvability, + title={Solvability of some statistical mechanical systems}, + author={Guttmann, AJ and Enting, IG}, + journal={Physical review letters}, + volume={76}, + number={3}, + pages={344}, + year={1996}, + publisher={APS} +} + @article{harris.1984.metastability, title={Metastability in the (1+ 1) D Ising model: a primitive droplet model calculation}, author={Harris, CK}, @@ -363,6 +403,28 @@ publisher={AIP} } +@article{nickel.1999.singularity, + title={On the singularity structure of the 2D Ising model susceptibility}, + author={Nickel, Bernie}, + journal={Journal of Physics A: Mathematical and General}, + volume={32}, + number={21}, + pages={3889}, + year={1999}, + publisher={IOP Publishing} +} + +@article{nickel.2000.addendum, + title={Addendum toOn the singularity structure of the 2D Ising model susceptibility'}, + author={Nickel, Bernie}, + journal={Journal of Physics A: Mathematical and General}, + volume={33}, + number={8}, + pages={1693}, + year={2000}, + publisher={IOP Publishing} +} + @article{onsager.1944.crystal, title={Crystal statistics. I. A two-dimensional model with an order-disorder transition}, author={Onsager, Lars}, @@ -374,6 +436,17 @@ publisher={APS} } +@article{orrick.2001.susceptibility, + title={The susceptibility of the square lattice Ising model: new developments}, + author={Orrick, WP and Nickel, B and Guttmann, AJ and Perk, JHH}, + journal={Journal of Statistical Physics}, + volume={102}, + number={3}, + pages={795--841}, + year={2001}, + publisher={Springer} +} + @article{parisi.1977.asymptotic, title={Asymptotic estimates in perturbation theory}, author={Parisi, Giorgio}, diff --git a/essential-ising.tex b/essential-ising.tex index ba8e452..d641a57 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -119,8 +119,10 @@ methods of critical droplet theory for the decay of an Ising system in a metastable state, i.e., an equilibrium Ising state for $T0$ subjected to a small negative external field $H<0$. The existence of an essential singularity has also been suggested by transfer matrix -\cite{mccraw.1978.metastability,enting.1980.investigation} and \textsc{rg} -methods \cite{klein.1976.essential}. It has long been known that the decay +\cite{mccraw.1978.metastability,enting.1980.investigation,mangazeev.2008.variational,mangazeev.2010.scaling} and \textsc{rg} +methods \cite{klein.1976.essential}, and a different kind of essential +singularity is known to exist in the zero-temperature susceptibility +\cite{orrick.2001.susceptibility,chan.2011.ising,guttmann.1996.solvability,nickel.1999.singularity,nickel.2000.addendum,assis.2017.analyticity}. It has long been known that the decay rate $\Gamma$ of metastable states in statistical mechanics is often related to the metastable free energy $F$ by $\Gamma\propto\im F$ \cite{langer.1969.metastable,penrose.1987.rigorous,gaveau.1989.analytic,privman.1982.analytic}. @@ -264,10 +266,10 @@ 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 \[ - \chi=\pd MH=-\frac1{T_\c}\pd[2]Fh - =-\frac2{\pi T_\c}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. + \chi=\pd MH=-\frac1{T}\pd[2]Fh + =-\frac2{\pi T}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. \] -With a scaling form defined by $T_\c\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, +With a scaling form defined by $T\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, this yields \[ \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] @@ -325,7 +327,7 @@ temperature have $\fS(0)=4/T_\c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of the susceptibility scaling function, then -$T_\c\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant +$T\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant $A$ is fixed to $A=\pi\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. 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 @@ -405,7 +407,7 @@ energy~\cite{caselle.2001.critical}, and should have implications for the scalin 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{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{XXX}, and +transitions~\cite{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{dahm.1989.dynamics}, and freezing transitions in glasses, spin glasses, and other disordered systems. @@ -424,7 +426,8 @@ freezing transitions in glasses, spin glasses, and other disordered systems. \begin{acknowledgments} The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien - for helpful conversations. JPS thanks Jim Langer for past inspiration, + for helpful conversations. The authors would also like to think Jacques Perk + for pointing us to several insightful studies. JPS thanks Jim Langer for past inspiration, guidance, and encouragement. This work was supported by NSF grants DMR-1312160 and DMR-1719490. \end{acknowledgments} -- cgit v1.2.3-70-g09d2