Binder cumulants of finite Ising rectangles at critical coupling

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Numbers

$m$

$n$

$U_{m,n}$

0.4785533905932737622004221810524245196424179688442370182941699344976831196155267597125968835819103932

0.4473912100943572679981516837578052044019905825563313938289322483437751624763010973545651937535411495

0.4159307498819930395095110065732994791903873701684008543673867965010703209695381830457412137924387896

0.3851100717448640058669998526231721423947526116613637332766541261196857311283798566116665774183037608

0.3560097172775295757567069515731824545748690756436373147080235434364662636177329187143578242860537559

0.4412972572857605172251506869250500189737723126532583258093906067924368588332623683730965011213278410

0.4273015406636573110574244253828673138964558689159098770075233392796953164393488879922425416148235258

0.4091998805046454314505728121572594054723616242843000196037200496349997874976546166881585718243993190

0.3893759966540998944161150518621469582701788529629731773496408772753732428373556048948605294597886249

0.4262826702955061976323570994317348192152590179857716999712041393495510591025264571924502327485612543

0.4182475382249420205472775938643420834967170849811560868828003777152699265102075425883794408469382073

0.4063678876149378648094285456511948849244669640045465502212030313838954144090869055644191130654961622

0.4183262715539310468713472281717088513619311418459038030203503394080338064828644325188261413519104684

0.4130955467536915753517341518385014982151160238077407459101866111255522123384068204080169181795628358

0.4134560756760306758087675006746955537347205526863607055705388303066104370379709006257337442477521374

Definition

The Binder cumulant $U_{m,n}=1-\langle M^4\rangle/(3\langle M^2\rangle^2)$ [1] is evaluated for the zero-field ferromagnetic Ising model on an $m$ by $n$ vertex grid with free boundaries at $K_c=\tfrac12\log(1+\sqrt2)$, where $M$ is total magnetization and brackets denote thermal expectation.

Parameters

$m$

— shorter side length in vertices ($2\le m\le n$)

$n$

— longer side length in vertices ($n\ge m$)

Formulas

(1)

$P_{m,n}(z,b)=\sum_{S\subseteq V(G_{m,n})}z^{|S|}b^{d(S)}$, where $d(S)$ counts bonds crossing from $S$ to its complement. With dimensionless field $H=h/(k_BT)$, field fugacity $z=e^{-2H}$ and bond weight $b=e^{-2K}$, $Z_{m,n}(K,H)=e^{KE+HN}P_{m,n}(z,b)$ [2]. The critical zero-field slice is $z=1$, $b_c=\sqrt2-1$.

(2)

Let $c_{k,d}=[z^kb^d]P_{m,n}(z,b)$ and $S_r=\sum_{k,d}c_{k,d}(N-2k)^r b_c^d$ for $r=0,2,4$. Then $\langle M^r\rangle=S_r/S_0$ and $U_{m,n}=1-S_0S_4/(3S_2^2)$. In particular, $U_{m,n}\in\mathbb Q(\sqrt2)$.

(3)

$U_{2,2}=\dfrac18+\dfrac{\sqrt2}{4}$. The $2$ by $2$ rectangle is a four-cycle, with zero-field cut polynomial $P_{2,2}(1,b)=2(1+6b^2+b^4)$.

Comments

(4)

The graph is $G_{m,n}=P_m\square P_n$, where $P_m$ has $m$ vertices, with $N=mn$ sites and $E=2mn-m-n$ bonds. Spins are $\sigma_v\in\{-1,1\}$, $M=\sum_v\sigma_v$, and $B=\sum_{uv\in E(G_{m,n})}\sigma_u\sigma_v$. Every bond is counted once with the same coupling $J>0$. The Hamiltonian is $\mathcal H=-JB$ and $K=J/(k_BT)$; at $K_c$ each configuration has probability $e^{K_cB}/\sum_\sigma e^{K_cB}$. There are no wraparound bonds, fixed boundary spins, diagonal interactions or external field.

(5)

The coupling is the infinite square lattice's bulk critical coupling. A finite rectangle has no thermodynamic singularity, so the entries are finite-size observables evaluated at this fixed bulk value. Boundary conditions and shape affect the Binder cumulant [1].

(6)

The stored normalization is $U=1-\langle M^4\rangle/(3\langle M^2\rangle^2)$, rather than the unshifted moment ratio $\langle M^4\rangle/\langle M^2\rangle^2$. Replacing total magnetization $M$ by magnetization per spin $M/N$ leaves $U$ unchanged. Spin reversal at zero field gives $\langle M\rangle=0$, so the raw second and fourth moments equal the centered moments.

(7)

Rectangles are listed lexicographically by $(m,n)$ with $m\le n$, omitting rotated copies.

Programs

(P1)

Sage

# With the attached generate.py in the current directory:
from generate import BinderCumulants
g = BinderCumulants()
g.value({'m': 6, 'n': 6}, 100)
list(g.enumerate())  # every rectangle 2 <= m <= n <= 6

References

[1]

W. Selke, Critical Binder cumulant for isotropic Ising models on square and triangular lattices, Journal of Statistical Mechanics: Theory and Experiment (2007), P04008, Section 2, equations (1), (2), and (4). (arXiv) (doi)

[2]

H. Peters and G. Regts, Location of zeros for the partition function of the Ising model on bounded degree graphs, Journal of the London Mathematical Society 101 (2020), 765-785, equations (1.1)-(1.2). (arXiv) (doi)

Links

[3]

SageMath: real ball arithmetic

Similar tables

Fisher zeros of the Ising model on finite rectangles with free boundaries — Zeros of the same finite-rectangle partition function in the bond coordinate at zero field.

Lee–Yang zeros of the Ising model on finite rectangles at critical coupling — Zeros of the same partition function in field fugacity at the same bulk critical coupling.

Specific heats of finite Ising rectangles at critical coupling — Energy fluctuations of the same finite rectangles at the same critical coupling.

Critical couplings of the Ising model on lattices — Bulk transition couplings, including the square-lattice coupling fixed here.

Critical exponents of the two-dimensional universality classes — Critical exponents for thermodynamic limits, including the two-dimensional Ising universality class.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every rectangle with $2\le m\le n\le6$, including five squares and ten nonsquare rectangles; these discrete sizes cover ladders and intermediate aspect ratios before extending only the square sequence, and the practical, noncanonical cutoff limits transfer to $64$ row states and the cut polynomial to degree $60$ in the bond weight)

How they were obtained:

Exact integer counts of down spins and disagreement bonds are computed by row transfer, and each cumulant is evaluated exactly in $\mathbb Q(\sqrt2)$. Its rational coefficients are enclosed at the positive square root of $2$ using real ball arithmetic [3]. Computation at 429 bits, including 96 guard bits, retains at least 418 bits of relative accuracy, about 125 decimal digits, before serialization to 100 significant digits. The generator recomputes the counts; the attached count data and numerical evidence are outputs, not inputs.

An independent edge-subgraph expansion counts subgraphs with zero, two or four odd-degree vertices, giving separate second and fourth magnetization moments for every rectangle. Its even-subgraph polynomial also reproduces every coefficient of the zero-field cut polynomial. Direct spin enumeration checks the full two-variable counts for all nine rectangles with at most 16 sites. Checks cover spin reversal, rotational symmetry, the specializations at $b=0$ and $b=1$, and the corresponding magnetization moments. The single spin, the single bond and the four-cycle provide known-answer controls. Every stored decimal interval contains a 600-bit enclosure of the independent exact cumulant, and every entry passes generator verification. Reproducible checks, exact counts and numerical evidence are attached.