Published October 28, 2026
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The principal Lee--Yang zeros of the Ising model need not decrease with the coupling
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Description
The Lee--Yang circle theorem states that, as a function of the external field $h$, the
partition function of a ferromagnetic Ising model vanishes only on the imaginary axis $h=ix$.
Consider a connected graph $G$ on $n$ vertices whose every edge carries the same coupling
$t\ge0$, the interaction divided by the temperature. For $t>0$, its partition function has $K=\lfloor n/2\rfloor$
zeros $x_1<\dots<x_K$ in $(0,\pi/2)$, the principal zeros, and $x_k$ tends to $\pi/2$ as $t\to0$
and to $(2k-1)\pi/(2n)$ as $t\to\infty$. Nishimori and Griffiths conjectured in 1983 that every
principal zero decreases strictly in $t$. The conjecture holds for the complete graph, as Hou,
Jiang and Newman proved in 2023, but it is false in general. An expansion of the partition function for
large $t$ shows that $x_k$ exceeds its limit by $S_k(G)e^{-2\lambda t}/(2n)$ to leading order.
Here $\lambda$ is the size of the minimum edge cuts of $G$, the smallest sets of edges whose
removal disconnects $G$, and the coefficient $S_k(G)$ is $(-1)^{k+1}$ times a sum of cosines with two
terms for each minimum edge cut. A negative $S_k(G)$ therefore makes $x_k$ increase for some $t$. The coefficient $S_K(G)$ is negative on two families of trees, which together give a tree
for every number $n\ge6$ of vertices. The coefficient $S_k(G)$ is negative for every even $k$ when
every minimum edge cut splits the vertices into halves. On all 995 connected graphs with
at most 7 vertices, a principal zero increases on a grid of couplings exactly when its coefficient
is negative.