History of Critical couplings of the Ising model on lattices

 Display properties:   number-header: $K_c$ or $k_BT_c/J$-Numbers: []+Numbers:+- params:+    lattice: square+    expression: coupling+  number: '0.4406867935097715126163046624898961545140801641308177053766478043266885921110130439168534459551280214'+  comment: $K_c=\frac12\ln(1+\sqrt2)=\frac12\operatorname{arsinh}1$, found by Kramers+    and Wannier from self-duality CITE{KW} and confirmed by Onsager\'s solution CITE{Onsager},+    OEIS A245592 CITE{OEISsq}; $\tanh K_c=\sqrt2-1$ is in the HREF{Algebraic_numbers_of_degree_2#1,2,-1,2}[table+    of quadratic irrationals], and $K_c$ is half the HREF{Regulators_of_real_quadratic_fields#8}[regulator+    of $\mathbb{Q}(\sqrt2)$].+- params:+    lattice: square+    expression: temperature+  number: '2.269185314213021968114490810306572475725981585504001350050658222360278655793768820765443549101251894'+  comment: $k_BT_c/J=2/\ln(1+\sqrt2)$, OEIS A169800 CITE{OEISsqT}.+- params:+    lattice: triangular+    expression: coupling+  number: '0.2746530721670274228488113092306314261618726394556873629336735834093735733046522417184039387034330222'+  comment: $K_c=\frac14\ln3$ CITE{Wannier} CITE{Houtappel}; $\tanh K_c=2-\sqrt3$ is+    in the HREF{Algebraic_numbers_of_degree_2#1,-4,1,1}[table of quadratic irrationals].+- params:+    lattice: triangular+    expression: temperature+  number: '3.640956906507349574456960662944428002450544229020846978905208253181124332775174986589087123352348593'+  comment: $k_BT_c/J=4/\ln3$.+- params:+    lattice: honeycomb+    expression: coupling+  number: '0.6584789484624083543125231736539842220134909857337582398842361284602300927082219880371095067250508918'+  comment: $K_c=\frac12\ln(2+\sqrt3)=\frac12\operatorname{arcosh}2$ CITE{Houtappel},+    OEIS A329247 CITE{OEIShc}; $\tanh K_c=1/\sqrt3$ is in the HREF{Algebraic_numbers_of_degree_2#3,0,-1,2}[table+    of quadratic irrationals], and $K_c$ is half the HREF{Regulators_of_real_quadratic_fields#12}[regulator+    of $\mathbb{Q}(\sqrt3)$].+- params:+    lattice: honeycomb+    expression: temperature+  number: '1.518651435000413846437968245887344103041271466917448195478590695574848139875269888846784352370915275'+  comment: $k_BT_c/J=2/\ln(2+\sqrt3)$; the honeycomb lattice is also called the hexagonal+    lattice.+- params:+    lattice: kagome+    expression: coupling+  number: '0.4665660103147178885806672414423078240876818125947228014089548559348018330064371148777567227142419570'+  comment: $K_c=\frac14\ln(3+2\sqrt3)$ CITE{KanoNaya}, so that $e^{4K_c}=3+2\sqrt3$+    and $\tanh K_c=\frac12-\sqrt{\frac{\sqrt3}{2}}+\frac{\sqrt3}{2}$; the same value+    as on $(3,4,6,4)$ CITE{Codello}.+- params:+    lattice: kagome+    expression: temperature+  number: '2.143319440105504109174386412695560284458375241995927334414522851526132920306042084354723923811816110'+  comment: $k_BT_c/J=4/\ln(3+2\sqrt3)$.+- params:+    lattice: 3-12-12+    expression: coupling+  number: '0.8120101489313014796912365819731105173217604339776951703562970332677298468415634852937821943970421005'+  comment: $\tanh K_c=-\frac14-\frac{\sqrt3}{4}+\frac12\sqrt{3+\frac{5\sqrt3}{2}}$,+    the root in $(0,1)$ of $1-2v+3v^2-2v^3-2v^4$ CITE{Syozi} CITE{Codello}; the lattice+    is also called the extended kagome or three-twelve lattice.+- params:+    lattice: 3-12-12+    expression: temperature+  number: '1.231511701320623557407845460452390941186467384580393515398521560871869407241301849210260501292669526'+  comment: $k_BT_c/J=1/K_c$; Codello gives $1.2315$ CITE{Codello}.+- params:+    lattice: 4-6-12+    expression: coupling+  number: '0.7195101852066440467044870022370488119532865358236159698876517491761780146945626285099485013492563683'+  comment: $\tanh K_c=\sqrt{\frac{5+3\sqrt3-\sqrt{44+26\sqrt3}}{2}}$, the root in+    $(0,1)$ of $1-2v^2+2v^4-10v^6+v^8$, found by Codello CITE{Codello}; the Monte+    Carlo estimate $k_BT_c/J\approx1.40$ CITE{Malarz} preceded it.+- params:+    lattice: 4-6-12+    expression: temperature+  number: '1.389834390895799476717937172881279231512412462487970289867818109378160850959662939035294407564886543'+  comment: $k_BT_c/J=1/K_c$; Codello gives $1.3898$ CITE{Codello}.+- params:+    lattice: 4-8-8+    expression: coupling+  number: '0.6950741361555963439192897658050875248116833553318644978123043633930542539166147589289400385621785901'+  comment: $\tanh K_c=-1-\frac1{\sqrt2}+\sqrt{\frac{5+4\sqrt2}{2}}$, the root in $(0,1)$+    of $1-4v^3-v^4$ CITE{Utiyama} CITE{Codello}, so that $e^{2K_c}=1+\frac{1+\sqrt{5+4\sqrt2}}{\sqrt2}$+    CITE{Jacobsen}; the lattice is also called the bathroom-tile or four-eight lattice.+- params:+    lattice: 4-8-8+    expression: temperature+  number: '1.438695454172595263014421930849493816248894934958647352801812117566604716754886032830748169452580505'+  comment: $k_BT_c/J=1/K_c$; Codello gives $1.4387$ CITE{Codello}.+- params:+    lattice: 3-4-6-4+    expression: coupling+  number: '0.4665660103147178885806672414423078240876818125947228014089548559348018330064371148777567227142419570'+  comment: $K_c=\frac14\ln(3+2\sqrt3)$, the same value as on the kagome lattice, because+    the two polynomials $P(v)$ share the factor $1-4v^2-6v^4-4v^6+v^8$ CITE{Codello};+    found by Codello, and confirmed by Jacobsen\'s critical polynomials CITE{Jacobsen}+    and by the Monte Carlo estimates $k_BT_c/J\approx2.15$ CITE{Malarz} and $2.145(3)$+    CITE{Lima}. The lattice is also called the ruby lattice.+- params:+    lattice: 3-4-6-4+    expression: temperature+  number: '2.143319440105504109174386412695560284458375241995927334414522851526132920306042084354723923811816110'+  comment: $k_BT_c/J=4/\ln(3+2\sqrt3)$.+- params:+    lattice: 3-3-3-3-6+    expression: coupling+  number: '0.3589577725782364467842695651131984423364824609236030345211485450723445333162067234374372538501900104'+  comment: $\tanh K_c$ is the root in $(0,1)$ of $1-4v+7v^2-12v^3+3v^4-3v^6$, found+    by Codello CITE{Codello}, and $e^{2K_c}=1+\frac13\left(\omega^{1/3}-2\omega^{-1/3}-2\right)$+    with $\omega=37+27\sqrt3+3\sqrt{6(66+37\sqrt3)}$ CITE{Jacobsen}; the Monte Carlo+    estimates are $k_BT_c/J\approx2.80$ CITE{Malarz} and $2.784(3)$ CITE{Lima}. The+    lattice is also called the snub hexagonal or maple-leaf lattice.+- params:+    lattice: 3-3-3-3-6+    expression: temperature+  number: '2.785843005480667066353556126542107747073489926146577269636425973872328606997328301554362484137093055'+  comment: $k_BT_c/J=1/K_c$; Codello gives $2.7858$ CITE{Codello}.+- params:+    lattice: 3-3-4-3-4+    expression: coupling+  number: '0.3417329500683170613045125962898835835915161001643820943284039917216615660799083247143281192336943120'+  comment: $\tanh K_c$ is the root in $(0,1)$ of $1-2v-v^2-4v^3-9v^4+6v^5-7v^6$ CITE{ThompsonWardrop}+    CITE{Codello}, and $e^{2K_c}=1+\frac13\left(\omega^{1/3}-2\omega^{-1/3}-2\right)$+    with $\omega=37+27\sqrt2+3\sqrt{315+222\sqrt2}$ CITE{Jacobsen}; the lattice is+    also called the snub square or Shastry–Sutherland lattice.+- params:+    lattice: 3-3-4-3-4+    expression: temperature+  number: '2.926261572962415268199590400659267850457970596702099179458458506052177209334078461157131515147511805'+  comment: $k_BT_c/J=1/K_c$; Codello gives $2.9263$ CITE{Codello}.+- params:+    lattice: 3-3-3-4-4+    expression: coupling+  number: '0.3465735902799726547086160607290882840377500671801276270603400047466968109848473578029316634982093438'+  comment: $K_c=\frac12\ln2$, since $\tanh K_c=\frac13$ CITE{ThompsonWardrop} CITE{Codello};+    the lattice is also called the elongated triangular or trellis lattice.+- params:+    lattice: 3-3-3-4-4+    expression: temperature+  number: '2.885390081777926814719849362003784274853291908305971868270898813862218438362370159771053245787012689'+  comment: $k_BT_c/J=2/\ln2$.+- params:+    lattice: D-3-3-4-3-4+    expression: coupling+  number: '0.5558128660438369486574652462226086763735895102657953480425065118063065606771666890266860576762944275'+  comment: $K_c=-\frac12\ln\tanh K_c(L)$ with $L$ the snub square lattice $(3^2,4,3,4)$,+    its planar dual, by Kramers–Wannier duality CITE{KW} CITE{formula-duality}.+- params:+    lattice: D-3-3-4-3-4+    expression: temperature+  number: '1.799166699968276772037280389447538423350234027213115407459438190912337154256637330923237279597079754'+  comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the snub square lattice $(3^2,4,3,4)$;+    Codello gives $1.7992$ CITE{Codello}.+- params:+    lattice: D-3-3-3-4-4+    expression: coupling+  number: '0.5493061443340548456976226184612628523237452789113747258673471668187471466093044834368078774068660444'+  comment: $K_c=-\frac12\ln\tanh K_c(L)$ with $L$ the elongated triangular lattice+    $(3^3,4^2)$, its planar dual, by Kramers–Wannier duality CITE{KW} CITE{formula-duality}.+- params:+    lattice: D-3-3-3-4-4+    expression: temperature+  number: '1.820478453253674787228480331472214001225272114510423489452604126590562166387587493294543561676174297'+  comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the elongated triangular lattice+    $(3^3,4^2)$; Codello gives $1.8205$ CITE{Codello}.+- params:+    lattice: D-3-3-3-3-6+    expression: coupling+  number: '0.5331272254628337597063019699733480686100615765409042935014319036009767437888698644578391008612256067'+  comment: $K_c=-\frac12\ln\tanh K_c(L)$ with $L$ the snub hexagonal lattice $(3^4,6)$,+    its planar dual, by Kramers–Wannier duality CITE{KW} CITE{formula-duality}.+- params:+    lattice: D-3-3-3-3-6+    expression: temperature+  number: '1.875724878113008020718340719580668062135483952230290917322690971388098778239229532950218561486752733'+  comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the snub hexagonal lattice $(3^4,6)$;+    Codello gives $1.8757$ CITE{Codello}.+- params:+    lattice: D-3-6-3-6+    expression: coupling+  number: '0.4157214727646552689131212597698515148831868219812646366365833475424329448616883456099635079001645774'+  comment: $K_c=-\frac12\ln\tanh K_c(L)$ with $L$ the kagome lattice $(3,6,3,6)$,+    its planar dual, by Kramers–Wannier duality CITE{KW} CITE{formula-duality}.+- params:+    lattice: D-3-6-3-6+    expression: temperature+  number: '2.405456695199652482403511189622525572309295480126501442678044871863409427962858588446396057774563706'+  comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the kagome lattice $(3,6,3,6)$;+    Codello gives $2.4055$ CITE{Codello}.+- params:+    lattice: D-3-4-6-4+    expression: coupling+  number: '0.4157214727646552689131212597698515148831868219812646366365833475424329448616883456099635079001645774'+  comment: $K_c=-\frac12\ln\tanh K_c(L)$ with $L$ the rhombitrihexagonal lattice $(3,4,6,4)$,+    its planar dual, by Kramers–Wannier duality CITE{KW} CITE{formula-duality}.+- params:+    lattice: D-3-4-6-4+    expression: temperature+  number: '2.405456695199652482403511189622525572309295480126501442678044871863409427962858588446396057774563706'+  comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the rhombitrihexagonal lattice $(3,4,6,4)$;+    Codello gives $2.4055$ CITE{Codello}.+- params:+    lattice: D-4-8-8+    expression: coupling+  number: '0.2543873426458248313029851033151913702976031912010467924356565590663656618056017150120865926070505687'+  comment: $K_c=-\frac12\ln\tanh K_c(L)$ with $L$ the truncated square lattice $(4,8^2)$,+    its planar dual, by Kramers–Wannier duality CITE{KW} CITE{formula-duality}.+- params:+    lattice: D-4-8-8+    expression: temperature+  number: '3.931013192713236761466555198077718633278986020721122796968528752736638611782418221968043984880533404'+  comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the truncated square lattice $(4,8^2)$;+    Codello gives $3.9310$ CITE{Codello}.+- params:+    lattice: D-4-6-12+    expression: coupling+  number: '0.2417626715306217970659860817512747164185895141771871485606557192925429341151241652063348810373633501'+  comment: $K_c=-\frac12\ln\tanh K_c(L)$ with $L$ the truncated trihexagonal lattice+    $(4,6,12)$, its planar dual, by Kramers–Wannier duality CITE{KW} CITE{formula-duality}.+- params:+    lattice: D-4-6-12+    expression: temperature+  number: '4.136287846543503441376557827954602075902077810769503672963505253426655316585139688103001627531291362'+  comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the truncated trihexagonal lattice+    $(4,6,12)$; Codello gives $4.1363$ CITE{Codello}.+- params:+    lattice: D-3-12-12+    expression: coupling+  number: '0.1997184094036070670041971106133048150765922573356981390283571761785894600555537524198709744354849601'+  comment: $K_c=-\frac12\ln\tanh K_c(L)$ with $L$ the truncated hexagonal lattice+    $(3,12^2)$, its planar dual, by Kramers–Wannier duality CITE{KW} CITE{formula-duality}.+- params:+    lattice: D-3-12-12+    expression: temperature+  number: '5.007049690542644769695768829615897531217574510422890181238833802977825367061835482450912783886526795'+  comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the truncated hexagonal lattice+    $(3,12^2)$; Codello gives $5.0071$ CITE{Codello}.+- params:+    lattice: sc+    expression: coupling+  number: 0.221654626 +/- 5e-9+  comment: $0.221654626(5)$ CITE{FXL}, from Monte Carlo simulations with the Wolff+    cluster algorithm on lattices of up to $1024^3$ sites; the high-temperature-series+    value $0.221655(2)$ CITE{ButeraComi} agrees.+- params:+    lattice: sc+    expression: temperature+  number: 4.51152326 +/- 1.1e-7+  comment: $k_BT_c/J=1/K_c$, the reciprocal of the entry for $K_c$ with its uncertainty+    propagated.+- params:+    lattice: bcc+    expression: coupling+  number: 0.1573725 +/- 1e-6+  comment: $0.1573725(10)$ CITE{ButeraComi}, from high-temperature series; Lundow+    and Campbell CITE{LundowCampbell} quote $0.1573725(5)$ from CITE{LMR} and CITE{MuraseIto}.+- params:+    lattice: bcc+    expression: temperature+  number: 6.354350 +/- 0.000041+  comment: $k_BT_c/J=1/K_c$, the reciprocal of the entry for $K_c$ with its uncertainty+    propagated.+- params:+    lattice: fcc+    expression: coupling+  number: 0.102069 +/- 1e-6+  comment: $0.102069(1)$ from Monte Carlo simulations CITE{LMR} CITE{MuraseIto}, as+    quoted by Lundow and Campbell CITE{LundowCampbell}.+- params:+    lattice: fcc+    expression: temperature+  number: 9.797294 +/- 0.000096+  comment: $k_BT_c/J=1/K_c$, the reciprocal of the entry for $K_c$ with its uncertainty+    propagated.+- params:+    lattice: diamond+    expression: coupling+  number: 0.3697398 +/- 1e-7+  comment: $0.3697398(1)$ from Monte Carlo simulations CITE{DengBlote} CITE{LMR},+    as quoted by Lundow and Campbell CITE{LundowCampbell}.+- params:+    lattice: diamond+    expression: temperature+  number: 2.70460470 +/- 7.4e-7+  comment: $k_BT_c/J=1/K_c$, the reciprocal of the entry for $K_c$ with its uncertainty+    propagated.