Critical couplings of the Ising model on lattices

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Numbers

lattice

expression

$K_c$ or $k_BT_c/J$

square $(4^4)$

$K_c=J/(k_BT_c)$:

0.4406867935097715126163046624898961545140801641308177053766478043266885921110130439168534459551280214

comment: $K_c=\frac12\ln(1+\sqrt2)=\frac12\operatorname{arsinh}1$, found by Kramers and Wannier from self-duality [1] and confirmed by Onsager's solution [2], OEIS A245592 [22]; $\tanh K_c=\sqrt2-1$ is in the table of quadratic irrationals, and $K_c$ is half the regulator of $\mathbb{Q}(\sqrt2)$.

square $(4^4)$

$k_BT_c/J$:

2.269185314213021968114490810306572475725981585504001350050658222360278655793768820765443549101251894

comment: $k_BT_c/J=2/\ln(1+\sqrt2)$, OEIS A169800 [23].

triangular $(3^6)$

$K_c=J/(k_BT_c)$:

0.2746530721670274228488113092306314261618726394556873629336735834093735733046522417184039387034330222

comment: $K_c=\frac14\ln3$ [3] [4]; $\tanh K_c=2-\sqrt3$ is in the table of quadratic irrationals.

triangular $(3^6)$

$k_BT_c/J$:

3.640956906507349574456960662944428002450544229020846978905208253181124332775174986589087123352348593

comment: $k_BT_c/J=4/\ln3$.

honeycomb $(6^3)$

$K_c=J/(k_BT_c)$:

0.6584789484624083543125231736539842220134909857337582398842361284602300927082219880371095067250508918

comment: $K_c=\frac12\ln(2+\sqrt3)=\frac12\operatorname{arcosh}2$ [4], OEIS A329247 [24]; $\tanh K_c=1/\sqrt3$ is in the table of quadratic irrationals, and $K_c$ is half the regulator of $\mathbb{Q}(\sqrt3)$.

honeycomb $(6^3)$

$k_BT_c/J$:

1.518651435000413846437968245887344103041271466917448195478590695574848139875269888846784352370915275

comment: $k_BT_c/J=2/\ln(2+\sqrt3)$; the honeycomb lattice is also called the hexagonal lattice, a name that in crystallography belongs to the triangular lattice.

kagome $(3,6,3,6)$

$K_c=J/(k_BT_c)$:

0.4665660103147178885806672414423078240876818125947228014089548559348018330064371148777567227142419570

comment: $K_c=\frac14\ln(3+2\sqrt3)$ [5], 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 $(3,4,6,4)$ [9].

kagome $(3,6,3,6)$

$k_BT_c/J$:

2.143319440105504109174386412695560284458375241995927334414522851526132920306042084354723923811816110

comment: $k_BT_c/J=4/\ln(3+2\sqrt3)$.

$(3,12^2)$

$K_c=J/(k_BT_c)$:

0.8120101489313014796912365819731105173217604339776951703562970332677298468415634852937821943970421005

comment: $\tanh K_c=-\frac14-\frac{\sqrt3}{4}+\frac12\sqrt{3+\frac{5\sqrt3}{2}}$, the root in $(0,1)$ of $1-2v+3v^2-2v^3-2v^4$ [7] [9]; the lattice is also called the extended kagome or three-twelve lattice.

$(3,12^2)$

$k_BT_c/J$:

1.231511701320623557407845460452390941186467384580393515398521560871869407241301849210260501292669526

comment: $k_BT_c/J=1/K_c$; Codello gives $1.2315$ [9].

$(4,6,12)$

$K_c=J/(k_BT_c)$:

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 [9]; the Monte Carlo estimate $k_BT_c/J\approx1.40$ [11] preceded it.

$(4,6,12)$

$k_BT_c/J$:

1.389834390895799476717937172881279231512412462487970289867818109378160850959662939035294407564886543

comment: $k_BT_c/J=1/K_c$; Codello gives $1.3898$ [9].

$(4,8^2)$

$K_c=J/(k_BT_c)$:

0.6950741361555963439192897658050875248116833553318644978123043633930542539166147589289400385621785901

comment: $\tanh K_c=-1-\frac1{\sqrt2}+\sqrt{\frac{5+4\sqrt2}{2}}$, the root in $(0,1)$ of $1-4v^3-v^4$ [6] [9], so that $e^{2K_c}=1+\frac{1+\sqrt{5+4\sqrt2}}{\sqrt2}$ [10]; the lattice is also called the bathroom-tile or four-eight lattice.

$(4,8^2)$

$k_BT_c/J$:

1.438695454172595263014421930849493816248894934958647352801812117566604716754886032830748169452580505

comment: $k_BT_c/J=1/K_c$; Codello gives $1.4387$ [9].

$(3,4,6,4)$

$K_c=J/(k_BT_c)$:

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 $1-4v^2-6v^4-4v^6+v^8$ [9]; found by Codello, and confirmed by Jacobsen's critical polynomials [10] and by the Monte Carlo estimates $k_BT_c/J\approx2.15$ [11] and $2.145(3)$ [12]. The lattice is also called the ruby lattice.

$(3,4,6,4)$

$k_BT_c/J$:

2.143319440105504109174386412695560284458375241995927334414522851526132920306042084354723923811816110

comment: $k_BT_c/J=4/\ln(3+2\sqrt3)$.

$(3^4,6)$

$K_c=J/(k_BT_c)$:

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 [9], and $e^{2K_c}=1+\frac13\left(\omega^{1/3}-2\omega^{-1/3}-2\right)$ with $\omega=37+27\sqrt3+3\sqrt{6(66+37\sqrt3)}$ [10]; the Monte Carlo estimates are $k_BT_c/J\approx2.80$ [11] and $2.784(3)$ [12]. The lattice is also called the snub hexagonal or maple-leaf lattice.

$(3^4,6)$

$k_BT_c/J$:

2.785843005480667066353556126542107747073489926146577269636425973872328606997328301554362484137093055

comment: $k_BT_c/J=1/K_c$; Codello gives $2.7858$ [9].

$(3^2,4,3,4)$

$K_c=J/(k_BT_c)$:

0.3417329500683170613045125962898835835915161001643820943284039917216615660799083247143281192336943120

comment: $\tanh K_c$ is the root in $(0,1)$ of $1-2v-v^2-4v^3-9v^4+6v^5-7v^6$ [8] [9], and $e^{2K_c}=1+\frac13\left(\omega^{1/3}-2\omega^{-1/3}-2\right)$ with $\omega=37+27\sqrt2+3\sqrt{315+222\sqrt2}$ [10]; the lattice is also called the snub square or Shastry–Sutherland lattice.

$(3^2,4,3,4)$

$k_BT_c/J$:

2.926261572962415268199590400659267850457970596702099179458458506052177209334078461157131515147511805

comment: $k_BT_c/J=1/K_c$; Codello gives $2.9263$ [9].

$(3^3,4^2)$

$K_c=J/(k_BT_c)$:

0.3465735902799726547086160607290882840377500671801276270603400047466968109848473578029316634982093438

comment: $K_c=\frac12\ln2$, since $\tanh K_c=\frac13$ [8] [9]; the lattice is also called the elongated triangular or trellis lattice.

$(3^3,4^2)$

$k_BT_c/J$:

2.885390081777926814719849362003784274853291908305971868270898813862218438362370159771053245787012689

comment: $k_BT_c/J=2/\ln2$.

Cairo pentagonal $D(3^2,4,3,4)$

$K_c=J/(k_BT_c)$:

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 [1] (4).

Cairo pentagonal $D(3^2,4,3,4)$

$k_BT_c/J$:

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 $1.7992$ [9].

prismatic pentagonal $D(3^3,4^2)$

$K_c=J/(k_BT_c)$:

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 [1] (4).

prismatic pentagonal $D(3^3,4^2)$

$k_BT_c/J$:

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 $1.8205$ [9].

floret pentagonal $D(3^4,6)$

$K_c=J/(k_BT_c)$:

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 [1] (4).

floret pentagonal $D(3^4,6)$

$k_BT_c/J$:

1.875724878113008020718340719580668062135483952230290917322690971388098778239229532950218561486752733

comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the snub hexagonal lattice $(3^4,6)$; Codello gives $1.8757$ [9].

rhombille $D(3,6,3,6)$

$K_c=J/(k_BT_c)$:

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 [1] (4).

rhombille $D(3,6,3,6)$

$k_BT_c/J$:

2.405456695199652482403511189622525572309295480126501442678044871863409427962858588446396057774563706

comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the kagome lattice $(3,6,3,6)$; Codello gives $2.4055$ [9].

deltoidal trihexagonal $D(3,4,6,4)$

$K_c=J/(k_BT_c)$:

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 [1] (4).

deltoidal trihexagonal $D(3,4,6,4)$

$k_BT_c/J$:

2.405456695199652482403511189622525572309295480126501442678044871863409427962858588446396057774563706

comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the rhombitrihexagonal lattice $(3,4,6,4)$; Codello gives $2.4055$ [9].

tetrakis square $D(4,8^2)$

$K_c=J/(k_BT_c)$:

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 [1] (4).

tetrakis square $D(4,8^2)$

$k_BT_c/J$:

3.931013192713236761466555198077718633278986020721122796968528752736638611782418221968043984880533404

comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the truncated square lattice $(4,8^2)$; Codello gives $3.9310$ [9].

kisrhombille $D(4,6,12)$

$K_c=J/(k_BT_c)$:

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 [1] (4).

kisrhombille $D(4,6,12)$

$k_BT_c/J$:

4.136287846543503441376557827954602075902077810769503672963505253426655316585139688103001627531291362

comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the truncated trihexagonal lattice $(4,6,12)$; Codello gives $4.1363$ [9].

triakis triangular $D(3,12^2)$

$K_c=J/(k_BT_c)$:

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 [1] (4).

triakis triangular $D(3,12^2)$

$k_BT_c/J$:

5.007049690542644769695768829615897531217574510422890181238833802977825367061835482450912783886526795

comment: $k_BT_c/J=-2/\ln\tanh K_c(L)$ with $L$ the truncated hexagonal lattice $(3,12^2)$; Codello gives $5.0071$ [9].

simple cubic

$K_c=J/(k_BT_c)$:

0.221654626 +/- 5e-9

comment: $0.221654626(5)$ [13], from Monte Carlo simulations with the Wolff cluster algorithm on lattices of up to $1024^3$ sites; the high-temperature-series value $0.221655(2)$ [14] agrees.

simple cubic

$k_BT_c/J$:

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.

body-centred cubic

$K_c=J/(k_BT_c)$:

0.1573725 +/- 5e-7

comment: $0.1573725(5)$ as quoted by Lundow and Campbell [18] from the high-temperature series of Butera and Comi [14] and the Monte Carlo simulations [16] [17]; Butera and Comi's own Table II prints $0.1573725(10)$.

body-centred cubic

$k_BT_c/J$:

6.354350 +/- 0.000021

comment: $k_BT_c/J=1/K_c$, the reciprocal of the entry for $K_c$ with its uncertainty propagated.

face-centred cubic

$K_c=J/(k_BT_c)$:

0.102069 +/- 1e-6 †

comment: $0.102069(1)$ from Monte Carlo simulations [16] [17], as quoted by Lundow and Campbell [18].

face-centred cubic

$k_BT_c/J$:

9.797294 +/- 0.000096 †

comment: $k_BT_c/J=1/K_c$, the reciprocal of the entry for $K_c$ with its uncertainty propagated.

diamond

$K_c=J/(k_BT_c)$:

0.3697398 +/- 1e-7

comment: $0.3697398(1)$ from Monte Carlo simulations [15] [16], as quoted by Lundow and Campbell [18].

diamond

$k_BT_c/J$:

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.

Definition

For the nearest-neighbour ferromagnetic Ising model [19] on an infinite lattice $L$, with energy $E=-J\sum_{\langle ij\rangle}s_is_j$ over the edges $\langle ij\rangle$ of $L$ and spins $s_i=\pm1$, the critical coupling is $K_c=J/(k_BT_c)$, where $T_c$ is the Curie temperature at which the model orders.

Parameters

lattice

— lattice, named by its tiling, crystal structure or vertex configuration

expression

— the coupling $K_c$ or the temperature $k_BT_c/J=1/K_c$

Formulas

(1)

$K_c(4^4)=\frac12\ln(1+\sqrt2)=\frac12\operatorname{arsinh}1$, equivalently $\sinh2K_c=1$ and $\tanh K_c=e^{-2K_c}=\sqrt2-1$ [1] [2] [20], so that $k_BT_c/J=2/\ln(1+\sqrt2)=2.2691853142\ldots$ [23].

(2)

$K_c(3^6)=\frac14\ln3$, $K_c(6^3)=\frac12\ln(2+\sqrt3)$ [4], $K_c(3,6,3,6)=K_c(3,4,6,4)=\frac14\ln(3+2\sqrt3)$ [5] [9], $K_c(3^3,4^2)=\frac12\ln2$ [8], and, with $\omega=37+27\sqrt2+3\sqrt{315+222\sqrt2}$ for $(3^2,4,3,4)$ and $\omega=37+27\sqrt3+3\sqrt{6(66+37\sqrt3)}$ for $(3^4,6)$, $e^{2K_c}=1+\frac13\left(\omega^{1/3}-2\omega^{-1/3}-2\right)$ [10]; for the remaining Archimedean lattices $\tanh K_c$ is $-\frac14-\frac{\sqrt3}{4}+\frac12\sqrt{3+\frac{5\sqrt3}{2}}$ on $(3,12^2)$, $\sqrt{\frac{5+3\sqrt3-\sqrt{44+26\sqrt3}}{2}}$ on $(4,6,12)$ and $-1-\frac1{\sqrt2}+\sqrt{\frac{5+4\sqrt2}{2}}$ on $(4,8^2)$ [9].

(3)

$v_c=\tanh K_c$ is the unique root in $(0,1)$ of $P(v)$ [9], where $P$ is $1-2v-v^2$ for $(4^4)$, $1-4v+v^2$ for $(3^6)$, $1-3v^2$ for $(6^3)$, $1-4v^2-6v^4-4v^6+v^8$ for $(3,6,3,6)$ and for $(3,4,6,4)$, $1-2v+3v^2-2v^3-2v^4$ for $(3,12^2)$, $1-2v^2+2v^4-10v^6+v^8$ for $(4,6,12)$, $1-4v^3-v^4$ for $(4,8^2)$, $1-4v+7v^2-12v^3+3v^4-3v^6$ for $(3^4,6)$, $1-2v-v^2-4v^3-9v^4+6v^5-7v^6$ for $(3^2,4,3,4)$ and $1-3v$ for $(3^3,4^2)$; each is the factor of $\det(1-v\mathbf{W}_\Lambda(0,0))$, $\mathbf{W}_\Lambda$ the Feynman–Vdovichenko random-walk matrix of the lattice $\Lambda$, that vanishes in $(0,1)$.

(4)

$\sinh2K_c\,\sinh2K_c^{*}=1$, that is $K_c^{*}=-\frac12\ln\tanh K_c=\frac12\ln\coth K_c$, for a planar lattice and its dual [1] [21]; in particular for each Archimedean lattice and its Laves dual, for the triangular and honeycomb lattices, and for the self-dual square lattice, where it gives $\sinh2K_c=1$.

Comments

(5)

The energy is $-J\sum s_is_j$ with $s_i=\pm1$ and $J>0$, summed once over each edge, so that $K=J/(k_BT)$ is the coupling in units of the temperature and $k_BT_c/J$ is the Curie temperature in units of $J/k_B$; the values are read as $\beta_cJ$ in a paper that writes $\beta=1/(k_BT)$ and as $T_c/J$ in one that sets $k_B=1$. With spins $\pm\frac12$ and the same energy the transition is at $J/(k_BT_c)=4K_c$; with the energy written as $-2J\sum s_is_j$ it is at $K_c/2$ for spins $\pm1$ and at $2K_c$ for spins $\pm\frac12$. The lattices are the nearest-neighbour graphs of the tilings and crystal structures named, under the names of the table of percolation thresholds: an Archimedean lattice by its vertex configuration in the notation of Grünbaum and Shephard, and each Laves lattice as $D(\ldots)$ of the Archimedean lattice it is the planar dual of; the square lattice is self-dual and the triangular and honeycomb lattices are dual to each other. $(3,12^2)$ is also called the truncated hexagonal, extended kagome or three-twelve lattice, $(4,8^2)$ the truncated square, bathroom-tile or four-eight lattice, $(4,6,12)$ the truncated trihexagonal lattice, $(3,4,6,4)$ the rhombitrihexagonal or ruby lattice, $(3^4,6)$ the snub hexagonal or maple-leaf lattice, $(3^2,4,3,4)$ the snub square or Shastry–Sutherland lattice, $(3^3,4^2)$ the elongated triangular or trellis lattice, $D(3,6,3,6)$ the dice lattice and $D(4,8^2)$ the union-jack lattice.

(6)

The nineteen planar couplings are known exactly. Kramers and Wannier [1] located the square-lattice transition at $\sinh2K_c=1$ from self-duality, and Onsager's solution [2] confirmed it; Wannier [3] and Houtappel [4] solved the triangular and honeycomb lattices, Kano and Naya [5] the kagome lattice, Utiyama [6] the lattice $(4,8^2)$, Syozi [7] the lattice $(3,12^2)$ by decoration, and Thompson and Wardrop [8] the lattices $(3^3,4^2)$ and $(3^2,4,3,4)$. Codello [9] treated all eleven Archimedean lattices at once by the Feynman–Vdovichenko random-walk method, which gives $\tanh K_c$ as the root in $(0,1)$ of a polynomial (3), and so found the couplings of $(4,6,12)$, $(3,4,6,4)$ and $(3^4,6)$, known until then from Monte Carlo simulations [11]; the polynomials of the kagome lattice and of $(3,4,6,4)$ share the factor that carries the root, so the two lattices have the same Curie temperature. Jacobsen's critical polynomials [10], computed by transfer matrices, reproduce every one of these values at $q=2$ and give the cube-root closed forms (2) of the two couplings that Codello left as decimals. The eight Laves lattices follow from their Archimedean duals by Kramers–Wannier duality [21] (4). Each exact $K_c$ is the logarithm of an algebraic number other than $1$, hence transcendental by the Lindemann–Weierstrass theorem, and the square and honeycomb couplings are half the regulators of $\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}(\sqrt3)$, since $1+\sqrt2$ and $2+\sqrt3$ are the fundamental units, both in the table of regulators.

(7)

No three-dimensional lattice has been solved, and the four cubic couplings are numerical estimates, each written as $K_c\pm\varepsilon$ with $\varepsilon$ the uncertainty stated in the source named in the entry; in the notation $0.221654626(5)$ the digits in parentheses are that uncertainty in the last digits written. They are the most precise published values as of 6 September 2026: the simple cubic coupling from the Monte Carlo study of Ferrenberg, Xu and Landau [13], and the three others as quoted by Lundow and Campbell [18]. They quote the body-centred cubic coupling from the high-temperature series of Butera and Comi [14] and the Monte Carlo studies of Lundow, Markström and Rosengren [16] and of Murase and Ito [17], the face-centred cubic coupling from those two Monte Carlo studies, and the diamond coupling from the Monte Carlo studies of Deng and Blöte [15] and of Lundow, Markström and Rosengren. The reciprocal $k_BT_c/J$ of each carries the uncertainty of $K_c$ propagated through the reciprocal.

Programs

(P1)

Sage

N(log(1 + sqrt(2))/2, digits=30)                       # K_c on the square lattice
v = 1/2 - sqrt(sqrt(3)/2) + sqrt(3)/2                  # tanh K_c on the kagome lattice
N(atanh(v), digits=30), N(-log(v)/2, digits=30)        # K_c there and on its dual, the rhombille lattice
R.<v> = QQ[]
(1 - 4*v + 7*v^2 - 12*v^3 + 3*v^4 - 3*v^6).roots(RealIntervalField(200))   # tanh K_c on (3^4,6) is the root in (0, 1)

References

[1]

H. A. Kramers and G. H. Wannier, Statistics of the two-dimensional ferromagnet. Part I, Physical Review 60 (1941), 252–262.

[2]

L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Physical Review 65 (1944), 117–149.

[3]

G. H. Wannier, Antiferromagnetism. The triangular Ising net, Physical Review 79 (1950), 357–364.

[4]

R. M. F. Houtappel, Order-disorder in hexagonal lattices, Physica 16 (1950), 425–455.

[5]

K. Kano and S. Naya, Antiferromagnetism. The kagome Ising net, Progress of Theoretical Physics 10 (1953), 158–172.

[6]

T. Utiyama, Statistics of two-dimensional Ising lattices of chequered types, Progress of Theoretical Physics 6 (1951), 907–909.

[7]

I. Syozi, Transformation of Ising models, in Phase Transitions and Critical Phenomena, volume 1, eds C. Domb and M. S. Green, Academic Press, London, 1972, 269–329.

[8]

C. J. Thompson and M. J. Wardrop, Critical points of two-dimensional Ising models, Journal of Physics A 7 (1974), L65–L69.

[9]

A. Codello, Exact Curie temperature for the Ising model on Archimedean and Laves lattices, Journal of Physics A 43 (2010), 385002. (arXiv)

[10]

J. L. Jacobsen, High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials, Journal of Physics A 47 (2014), 135001. (arXiv)

[11]

K. Malarz, M. Zborek and B. Wróbel, Curie temperature for an Ising model on Archimedean lattices, TASK Quarterly 9 (2005), 475–480. (arXiv)

[12]

F. W. S. Lima, J. Mostowicz and K. Malarz, Critical behaviour of the Ising S=1/2 and S=1 model on (3,4,6,4) and (3,3,3,3,6) Archimedean lattices, Communications in Computational Physics 10 (2011), 912–919. (arXiv)

[13]

A. M. Ferrenberg, J. Xu and D. P. Landau, Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model, Physical Review E 97 (2018), 043301. (arXiv)

[14]

P. Butera and M. Comi, Critical universality and hyperscaling revisited for Ising models of general spin using extended high-temperature series, Physical Review B 65 (2002), 144431, Table II. (arXiv)

[15]

Y. Deng and H. W. J. Blöte, Simultaneous analysis of several models in the three-dimensional Ising universality class, Physical Review E 68 (2003), 036125.

[16]

P. H. Lundow, K. Markström and A. Rosengren, The Ising model for the bcc, fcc and diamond lattices: a comparison, Philosophical Magazine 89 (2009), 2009–2042.

[17]

Y. Murase and N. Ito, Dynamic critical exponents of three-dimensional Ising models and two-dimensional three-states Potts models, Journal of the Physical Society of Japan 77 (2008), 014002.

[18]

P. H. Lundow and I. A. Campbell, The Ising universality class in dimension three: corrections to scaling, Physica A 511 (2018), 40–53. (arXiv)

Links

[19]

Wikipedia: Ising model

[20]

Wikipedia: Square lattice Ising model

[21]

Wikipedia: Kramers–Wannier duality

[22]

OEIS A245592: the Ising constant $K_c=\frac12\ln(1+\sqrt2)$

[23]

OEIS A169800: $2/\ln(1+\sqrt2)$, the critical temperature of the two-dimensional Ising model

[24]

OEIS A329247: $\frac12\ln(2+\sqrt3)$

Similar tables

Site and bond percolation thresholds of lattices — the same lattices, under the same names, and the other critical points of lattice statistical mechanics

Entropy constants of lattice models — the entropies of the hard-core, ice, dimer and spanning-tree models on several of the same lattices

Regulators of real quadratic fields — holds $\ln(1+\sqrt2)$ and $\ln(2+\sqrt3)$, twice the couplings of the square and honeycomb lattices, which link them

Algebraic numbers of degree 2 — holds $\sqrt2-1$, $2-\sqrt3$ and $1/\sqrt3$, the values of $\tanh K_c$ on the square, triangular and honeycomb lattices, which link them

Data properties

Entries are of type: real number

How well the digits are known: heuristic

Sources of data: [9], [13], [18]

Table is complete: no (it holds $K_c$ and $k_BT_c/J$ for the eleven Archimedean lattices, the eight Laves lattices and the simple cubic, body-centred cubic, face-centred cubic and diamond lattices, the cubic values being the most precise published estimates as of 6 September 2026)

How they were obtained:

Mixed, and labelled by its weakest entries.

The 38 entries of the planar lattices are enclosures computed in ball arithmetic in arb at 100 digits with 64 guard bits: for each Archimedean lattice $v_c=\tanh K_c$ is enclosed in an interval of half-width $10^{-105}$ around the value of its closed form, proved to contain the root by a sign change of Codello's polynomial $P$ across the ends and proved to be the only root in $(0,1)$ by $P$ keeping one sign on balls covering the rest of the interval; $K_c=\operatorname{artanh}v_c$ and $k_BT_c/J=1/K_c$ follow in balls, and each Laves lattice takes $K_c=-\frac12\ln v_c$ of its Archimedean dual. The 8 entries of the cubic lattices were not computed: each $K_c$ is a published estimate, written as centre and radius with the paper's stated uncertainty as the radius, and each $k_BT_c/J$ is the reciprocal of that interval in exact decimal arithmetic, its centre rounded to a tenth of its width and its radius rounded up. Before any entry was written, every planar value was compared with the four decimals of $k_BT_c/J$ and $k_BT_c^{*}/J$ that Codello prints; every Archimedean value with the Ising critical polynomial of Jacobsen for that lattice, a transfer-matrix computation sharing no method with Codello's, which changes sign across the stored enclosure of $e^{2K_c}-1$; the square and honeycomb couplings with OEIS A245592, A169800 and A329247 to every digit listed there and with half the regulators of $\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}(\sqrt3)$ stored in the table of regulators; the three values Codello found first with the Monte Carlo estimates of Malarz, Zborek and Wróbel and of Lima, Mostowicz and Malarz; the duality $\sinh2K_c\sinh2K_c^{*}=1$ on every dual pair; the mean-field bound $K_c>1/z$, $z$ the coordination number, on every lattice; and the simple cubic estimate with the text of the paper cited. The body-centred cubic, face-centred cubic and diamond values were taken from Lundow and Campbell; of the papers they quote, Butera and Comi's was read here and prints the body-centred cubic value with an uncertainty of $10$ in the last digit where Lundow and Campbell quote $5$, and the others have no arXiv version and were not read.