Ising model: Difference between revisions

Revision as of 13:57, 21 November 2024

Revision as of 17:50, 28 November 2024

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==== Renormalization ====

When there is no external field, we can derive a functional equation that <math>f(\beta, 0) = f(\beta)</math> satisfies using renormalization.<ref>{{Cite journal |last1=Maris |first1=Humphrey J. |last2=Kadanoff |first2=Leo P. |date=June 1978 |title=Teaching the renormalization group |url=https://pubs.aip.org/aapt/ajp/article/46/6/652-657/1045608 |journal=American Journal of Physics |language=en |volume=46 |issue=6 |pages=652–657 |doi=10.1119/1.11224 |bibcode=1978AmJPh..46..652M |issn=0002-9505}}</ref> Specifically, let <math>Z_N(\beta, J)</math> be the partition function with <math>N</math> sites. Now we have:<math display="block">Z_N(\beta, J) = \sum_{\sigma} e^{K \sigma_2(\sigma_1 + \sigma_3)}e^{K \sigma_4(\sigma_3 + \sigma_5)}\cdots</math>where <math>K := \beta J</math>. We sum over each of <math>\sigma_2, \sigma_4, \cdots</math>, to obtain<math display="block">Z_N(\beta, J) = \sum_{\sigma} (2\cosh(K(\sigma_1 + \sigma_3))) \cdot (2\cosh(K(\sigma_3 + \sigma_5))) \cdots</math>Now, since the cosh function is even, we can solve <math>Ae^{K'\sigma_1\sigma_3} = 2\cosh(K(\sigma_1+\sigma_3))</math> as <math display="inline">A = 2\sqrt{\cosh(2K)}, K' = \frac 12 \ln\cosh(2K)</math>. Now we have a self-similarity relation:<math display="block">\frac 1N \ln Z_N(K) = \frac 12 \ln\left(2\sqrt{\cosh(2K)}\right) + \frac 12 \frac{1}{N/2} \ln Z_{N/2}(K')</math>Taking the limit, we obtain<math display="block">f(\beta) = \frac 12 \ln\left(2\sqrt{\cosh(2K)}\right) + \frac 12 f(\beta')</math>where <math>\beta' J = \frac 12 \ln\cosh(2\beta J)</math>.

When <math>\beta</math> is small, we have <math>f(\beta)\approx \ln 2</math>, so we can numerically evaluate <math>f(\beta)</math> by iterating the functional equation until <math>K</math> is small.

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In three as in two dimensions, the most studied case of the Ising model is the translation-invariant model on a cubic lattice with nearest-neighbor coupling in the zero magnetic field. Many theoreticians searched for an analytical three-dimensional solution for many decades, which would be analogous to Onsager's solution in the two-dimensional case.'''<ref>{{Cite web|last=Wood|first=Charlie|title=The Cartoon Picture of Magnets That Has Transformed Science|url=https://www.quantamagazine.org/the-cartoon-picture-of-magnets-that-has-transformed-science-20200624/|access-date=2020-06-26|website=Quanta Magazine|date=24 June 2020|language=en}}</ref>''' <ref>{{Cite web |title=Ken Wilson recalls how Murray Gell-Mann suggested that he solve the three-dimensional Ising model |url=https://authors.library.caltech.edu/5456/1/hrst.mit.edu/hrs/renormalization/Wilson/index.htm}}</ref> Such a solution has not been found until now, although there is no proof that it may not exist. In three dimensions, the Ising model was shown to have a representation in terms of non-interacting fermionic strings by [[Alexander Markovich Polyakov|Alexander Polyakov]] and [[Vladimir Dotsenko]]. This construction has been carried on the lattice, and the [[continuum limit]], conjecturally describing the critical point, is unknown.

In three as in two dimensions, Peierls' argument shows that there is a phase transition. This phase transition is rigorously known to be continuous (in the sense that correlation length diverges and the magnetization goes to zero), and is called the [[Critical point (thermodynamics)|critical point]]. It is believed that the critical point can be described by a renormalization group fixed point of the Wilson-Kadanoff renormalization group transformation. It is also believed that the phase transition can be described by a three-dimensional unitary conformal field theory, as evidenced by [[Metropolis–Hastings algorithm|Monte Carlo]] simulations,<ref>{{Cite journal|last1=Billó|first1=M.|last2=Caselle|first2=M.|last3=Gaiotto|first3=D.|last4=Gliozzi|first4=F.|last5=Meineri|first5=M.|last6=others|date=2013|title=Line defects in the 3d Ising model|journal=JHEP|volume=1307|issue=7|pages=055|arxiv=1304.4110|bibcode=2013JHEP...07..055B|doi=10.1007/JHEP07(2013)055|s2cid=119226610}}</ref><ref>{{Cite journal|last1=Cosme|first1=Catarina|last2=Lopes|first2=J. M. Viana Parente|last3=Penedones|first3=Joao|date=2015|title=Conformal symmetry of the critical 3D Ising model inside a sphere|journal=Journal of High Energy Physics|volume=2015|issue=8|pages=22|arxiv=1503.02011|bibcode=2015JHEP...08..022C|doi=10.1007/JHEP08(2015)022|s2cid=53710971}}</ref> exact diagonalization results in quantum models,<ref>{{Cite journal |last1=Zhu |first1=Wei |last2=Han |first2=Chao |last3=Huffman |first3=Emilie |last4=Hofmann |first4=Johannes S. |last5=He |first5=Yin-Chen |date=2023 |title=Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization |journal=Physical Review X |volume=13 |issue=2 |page=021009 |doi=10.1103/PhysRevX.13.021009 |arxiv=2210.13482|bibcode=2023PhRvX..13b1009Z |s2cid=253107625 }}</ref> and quantum field theoretical arguments.<ref>{{Cite journal|last1=Delamotte|first1=Bertrand|last2=Tissier|first2=Matthieu|last3=Wschebor|first3=Nicolás|year=2016|title=Scale invariance implies conformal invariance for the three-dimensional Ising model|journal=Physical Review E|volume=93|issue=12144|pages=012144|arxiv=1501.01776|bibcode=2016PhRvE..93a2144D|doi=10.1103/PhysRevE.93.012144|pmid=26871060|s2cid=14538564}}</ref> Although it is an open problem to establish rigorously the renormalization group picture or the conformal field theory picture, theoretical physicists have used these two methods to compute the [[critical exponents]] of the phase transition, which agree with the experiments and with the Monte Carlo simulations. This conformal field theory describing the three-dimensional Ising critical point is under active investigation using the method of the [[conformal bootstrap]].<ref>{{Cite journal|last1=El-Showk|first1=Sheer|last2=Paulos|first2=Miguel F.|last3=Poland|first3=David|last4=Rychkov|first4=Slava|last5=Simmons-Duffin|first5=David|last6=Vichi|first6=Alessandro|date=2012|title=Solving the 3D Ising Model with the Conformal Bootstrap|journal=Phys. Rev.|volume=D86|issue=2|pages=025022|arxiv=1203.6064|bibcode=2012PhRvD..86b5022E|doi=10.1103/PhysRevD.86.025022|s2cid=39692193}}</ref><ref name="cmin">{{Cite journal|last1=El-Showk|first1=Sheer|last2=Paulos|first2=Miguel F.|last3=Poland|first3=David|last4=Rychkov|first4=Slava|last5=Simmons-Duffin|first5=David|last6=Vichi|first6=Alessandro|date=2014|title=Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents|journal=Journal of Statistical Physics|volume=157|issue=4–5|pages=869–914|arxiv=1403.4545|bibcode=2014JSP...157..869E|doi=10.1007/s10955-014-1042-7|s2cid=119627708}}</ref><ref name="SDPB">{{Cite journal|last=Simmons-Duffin|first=David|date=2015|title=A semidefinite program solver for the conformal bootstrap|journal=Journal of High Energy Physics|volume=2015|issue=6|pages=174|arxiv=1502.02033|bibcode=2015JHEP...06..174S|doi=10.1007/JHEP06(2015)174|issn=1029-8479|s2cid=35625559}}</ref><ref name="Kadanoff">{{cite journal |last=Kadanoff|first=Leo P.|date=April 30, 2014|title=Deep Understanding Achieved on the 3d Ising Model|url=http://www.condmatjournalclub.org/?p=2384|url-status=dead|archive-url=https://web.archive.org/web/20150722062827/http://www.condmatjournalclub.org/?p=2384|archive-date=July 22, 2015|access-date=July 19, 2015|journal=Journal Club for Condensed Matter Physics}}</ref> This method currently yields the most precise information about the structure of the critical theory (see [[Ising critical exponents]]).

In 2000, [[Sorin Istrail]] of [[Sandia National Laboratories]] proved that the spin glass Ising model on a [[nonplanar]] lattice is [[NP-completeness|NP-complete]]. That is, assuming '''P''' ≠ '''NP,''' the general spin glass Ising model is exactly solvable only in [[Planar graph|planar]] cases, so solutions for dimensions higher than two are also intractable.<ref>{{cite journal |last=Cipra |first=Barry A. |year=2000 |title=The Ising Model Is NP-Complete |url=https://archive.siam.org/pdf/news/654.pdf |journal=SIAM News |volume=33 |issue=6}}</ref> Istrail's result only concerns the spin glass model with spatially varying couplings, and tells nothing about Ising's original ferromagnetic model with equal couplings.

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* {{Citation | last1=Krizan | first1=J. E. |last2=Barth | first2=P. F. | author-link2=Peter F. Barth | last3=Glasser | first3=M.L.| year=1983 | title= Exact Phase Transitions for the Ising Model on the Closed Cayley Tree| journal=Physica | volume=119A | pages=230–242 | publisher= North-Holland Publishing Co.| doi=10.1016/0378-4371(83)90157-7 }}

*{{Citation | last1=Glasser | first1=M. L. | last2=Goldberg | first2=M. | year=1983| title= The Ising model on a closed Cayley tree | journal=Physica | volume=117A | issue=2 | pages=670–672 | doi=10.1016/0378-4371(83)90138-3 | bibcode=1983PhyA..117..670G }}

*{{Citation | last1=Süzen | first1=Mehmet | year=2014 | title= Effective ergodicity in single-spin-flip dynamics | journal=Physical Review E | volume=90 | issue=3 | pages=032141 | doi=10.1103/PhysRevE.90.032141 | pmid=25314429 | arxiv=1405.4497 | bibcode=2014PhRvE..90c2141S }}

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