Specific heats of finite Ising rectangles at critical coupling

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Numbers

$m$

$n$

$c_{m,n}$

0.2157831666376933281946040861359496957241827261972096970025065696551868527854881909168439145184254566

0.2706320002842541103290293605386992043533140975648563556073647856419379890812416752725781447093476186

0.2996405544033801196400168343775828711101600333927821035729279051992834092240493280316426518645456477

0.3172966356426459624784187396056319930965843667682032172762931499738005950299653384060338132460553482

0.3291063115016030434116774629152506552717333136072156202506333055289148410968041093150733213700920900

0.3453754935525310123463312017846044906140822980656708469001842798588506262614465488827569841452065270

0.3877742202543160040374944383032611368287290868266506668572262258849196363137128570553045567302239630

0.4144530076583395505853848241643533073520299487301042681047508116375682851397935715136713764746035557

0.4325402225624607997782860977984645594745613091430397087093598186719173770181898102517678051211012971

0.4408498357294446919755792360738487955505412562934869095192835726127152481319122813842473587416267999

0.4756114002141278499885212008038966733116039530098632359608006064373566080612767806427808655505302799

0.4997263102351443117554132830410410410497289548400672232736360269539892605806045836204975133812120871

0.5171605116052377268356063032917556027632247382178696247438293262532427363651783352698847843307510992

0.5467488780900944149632054135055402387419304923472886518246336801760892825535527387072844332283900600

0.5810716307097933472443896471972355015231029732870898531628350345241978991505752874309836907572130906

Definition

The dimensionless specific heat $c_{m,n}=C_{m,n}/(mn k_B)$ is the heat capacity per spin at $K_c=\tfrac12\log(1+\sqrt2)$ of the zero-field ferromagnetic Ising model on an $m$ by $n$ vertex grid with free boundaries [1].

Parameters

$m$

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

$n$

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

Formulas

(1)

$c_{m,n}=\left.\dfrac{K^2}{N}\dfrac{\partial^2\log Z_{m,n}(K,0)}{\partial K^2}\right|_{K=K_c}=\dfrac{K_c^2}{N}\bigl(\langle B^2\rangle-\langle B\rangle^2\bigr)$, where $Z_{m,n}(K,0)=\sum_\sigma e^{KB(\sigma)}$ and $\langle\cdot\rangle$ is expectation with probabilities $e^{K_c B}/Z_{m,n}(K_c,0)$ [1].

(2)

$P_{m,n}(z,b)=\sum_{S\subseteq V(G_{m,n})}z^{|S|}b^{d(S)}$, $F_{m,n}(b)=P_{m,n}(1,b)$ and $Z_{m,n}(K,0)=e^{KE}F_{m,n}(e^{-2K})$, where $d(S)$ counts bonds with one endpoint in $S$, $z=e^{-2H}$ is field fugacity, $H=h/(k_BT)$ is dimensionless external field, and $b=e^{-2K}$ [2].

(3)

$c_{m,n}=\dfrac{4K_c^2}{N}\left[\dfrac{bF_{m,n}^\prime(b)+b^2F_{m,n}^{\prime\prime}(b)}{F_{m,n}(b)}-\left(\dfrac{bF_{m,n}^\prime(b)}{F_{m,n}(b)}\right)^2\right]_{b=\sqrt2-1}$.

(4)

$c_{2,2}=\dfrac{5}{18}\log^2(1+\sqrt2)$. The four-cycle has $F_{2,2}(b)=2(1+6b^2+b^4)$.

Comments

(5)

The graph is $G_{m,n}=P_m\square P_n$, where $P_m$ is a path on $m$ vertices. There are $N=mn$ spins $\sigma_v\in\{-1,1\}$ and $E=2mn-m-n$ bonds. The energy is $\mathcal H=-JB$, where $B=\sum_{uv\in E(G_{m,n})}\sigma_u\sigma_v$ counts each bond once. There are no wraparound bonds, fixed boundary spins, diagonal interactions or external field. The heat capacity is $C_{m,n}=\partial\langle\mathcal H\rangle/\partial T$ at fixed $J$ and zero field; division by $Nk_B$ gives the stored dimensionless quantity. The dimensionless coupling is $K=J/(k_BT)$ for uniform $J>0$.

(6)

The coupling is the infinite square lattice's bulk critical coupling. A finite rectangle has no thermodynamic singularity; this evaluation need not occur at its heat-capacity maximum. Rectangular shapes and their aspect ratios are used to study boundary contributions and finite-size behavior [1].

(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 SpecificHeats
g = SpecificHeats()
g.value({'m': 6, 'n': 6}, 100)
list(g.enumerate())  # all rectangles 2 <= m <= n <= 6

References

[1]

X. Wu, R. Zheng, N. Izmailian and W. Guo, Accurate expansions of internal energy and specific heat of critical two-dimensional Ising model with free boundaries, Journal of Statistical Physics 155 (2014), 106-124, Section II, equations (1)-(5), and Section III. (arXiv) (doi)

[2]

H. Peters and G. Regts, Location of zeros for the partition function of the Ising model on bounded degree graphs, 2019, 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 zero-field partition function in the bond coordinate.

Lee–Yang zeros of the Ising model on finite rectangles at critical coupling — Zeros of the same finite-rectangle partition function in field fugacity 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 the thermodynamic limit, 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 lattice 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 bond polynomial degree to $60$)

How they were obtained:

Exact integer cut counts are constructed by row transfer. The logarithm in $K_c$ and the partition-function moments are evaluated with real ball arithmetic [3] at 429 bits, including 96 guard bits. Every entry retains at least 416 bits of relative accuracy, about 125 decimal digits, before serialization to 100 significant digits. The generator recomputes every count and does not use the attached numerical outputs as inputs.

The independent checks enumerate the cycle space of each rectangle to construct its even-subgraph polynomial, then recover every coefficient of the zero-field cut polynomial by the high-temperature expansion. Bond-energy variances are recomputed exactly in $\mathbb Q(\sqrt2)$ and combined with a 600-bit ball logarithm. Each resulting enclosure lies strictly inside the interval represented by the stored decimal. The logarithmic-derivative formula is also evaluated independently of the generator. Direct enumeration checks the full two-variable cut counts for all nine rectangles with at most 16 spins. For every rectangle, separate two-variable transfer checks spin reversal and the specializations at $b=0$ and $b=1$; additional checks verify rotational symmetry and the zero-coupling mean and variance of $B$. The single bond and the four-cycle give known-answer controls. Reproducible checks, exact counting data and numerical evidence are attached.