Fisher-zero accumulation geometry and free-energy analyticity of the square-lattice Ising model at generic nonzero magnetic field

For the square-lattice Ising model, the zeros of the partition function in the complex-temperature plane (Fisher zeros) are rigorously understood at only two field values: $H=0$ and the imaginary field $H/k_{B}T=i\pi/2$, where under Brascamp-Kunz boundary conditions every finite-lattice zero lies on a one-dimensional set. Lyberg's 2008 exact solution with these boundary conditions left open whether such one-dimensional condensation persists at generic nonzero field (a then-current conjecture of Jensen, Maillard, McCoy and Shrock held it does not), whether any theorem on the analyticity of the free energy in the complex-temperature plane exists at $H\neq 0$, and why the zero-field susceptibility alone develops a dense natural boundary on $|\sinh 2K|=1$. An exact finite-lattice study by Assis, Jacobsen, Jensen, Maillard and McCoy (2017) produced a refined conjectural picture — zeros condensing on a curve on the ferromagnetic branch while filling a two-dimensional region on the antiferromagnetic branch, with a proposed $H^{2}$-amplitude mechanism reconciling the Nickel singularities with the analyticity assumed by scaling field theory — while stating explicitly that no analytic argument for this scenario exists. Establishing the Fisher-zero accumulation geometry and the resulting analyticity structure of the free energy at generic nonzero field, and the mechanism by which the Nickel singularities emerge as $H\to 0$, remains open.

Background

The Ising model on the square lattice assigns to each site $j,k$ a spin $\sigma_{j,k}=\pm 1$ interacting with its nearest neighbours through a coupling $K=E/(k_{B}T)$ and with an external magnetic field $H$ through the energy $\mathcal{E}=-E\sum(\sigma_{j,k}\sigma_{j+1,k}+\sigma_{j,k}\sigma_{j,k+1})-H\sum\sigma_{j,k}$. Two complex extensions of the model are studied through the zeros of the finite-lattice partition function $Z$: zeros in the complex field variable $x=e^{-2H/k_{B}T}$ (Lee-Yang zeros, which for ferromagnetic couplings all lie on the unit circle $|x|=1$ by the Lee-Yang circle theorem), and zeros in the complex-temperature variable, for instance $u=e^{-2K}$ or $s=\sinh 2K$ (customarily called Fisher zeros). In the thermodynamic limit, the accumulation set of these zeros is exactly the locus of possible non-analyticities of the free energy per site, so locating it is the standard route to the singularity structure of the model.

At zero field the free energy is known exactly, and its only singularity is the physical critical point at $s=1$. Brascamp and Kunz (1974) introduced special boundary conditions — a finite cylinder periodic in one direction, with one boundary row of fixed $+$ spins and the other row alternating $+,-,+,-$ — under which every zero of every finite-lattice partition function lies exactly on the unit circle $|s|=1$. This makes the approach of the Fisher zeros to their limiting locus exactly controllable. Lyberg (2008) solved the same problem in an external field for exactly two field values, $H=0$ (recovering the Brascamp-Kunz result) and $H/k_{B}T=i\pi/2$, and found the zeros on a one-dimensional connected set in both cases.

Zero field also holds a famous mystery. Nickel (1999, 2000) discovered that the zero-field magnetic susceptibility $\chi$ — unlike the free energy and the spontaneous magnetization, whose complex-temperature singularities are confined to isolated points — has, both above and below $T_{c}$, an infinite set of singularities dense on the entire circle $|\sinh 2K|=1$ (at positions $s_{j}$ with $s_{j}+s_{j}^{-1}=\cos(2\pi m/j)+\cos(2\pi n/j)$), which is expected to make that circle a natural boundary for $\chi$: a curve across which no analytic continuation of $\chi$ is possible. Guttmann and Enting (1996) found that the free energy and magnetization are D-finite (solutions of linear ordinary differential equations of finite order with polynomial coefficients) while the susceptibility shows strong evidence of not being so, and Orrick, Nickel, Guttmann and Perk (2001) strengthened the natural-boundary evidence. Tracy and Widom (2013, 2014, 2017) later proved rigorously, for the diagonal susceptibility below $T_{c}$ (the exactly summable component of $\chi$ built from correlations along lattice diagonals), that the unit circle is indeed a natural boundary, and that Nickel's singularities are the only ones there.

At nonzero field almost nothing is rigorous: the model is not exactly solvable for generic $H$, and Lee-Yang theory controls only the field plane at fixed temperature. Numerically, Matveev and Shrock (1996) found on small lattices that for nonzero real $H$ the complex-temperature zeros leave curves and fill two-dimensional regions; for imaginary field, exact quasi-one-dimensional results together with two-dimensional zero computations indicate a phase boundary containing a real line segment ending at the Yang-Lee edge (the endpoint of the Lee-Yang zero arc, at which a non-physical edge singularity sits). In the scaling limit, Fonseca and Zamolodchikov (2003) developed Ising field theory in a magnetic field assuming substantial analyticity of the free energy — an assumption in visible tension with a dense set of singularities approaching the critical point as $H\to 0$, a tension reviewed by McCoy and Maillard (2012).

The most direct subsequent study is that of Assis, Jacobsen, Jensen, Maillard and McCoy (2017), who computed exactly the Brascamp-Kunz partition-function zeros at nonzero real field on lattices up to $22\times 22$ (in the rescaled variable $y=u^{2}x^{1/2}$) and the equimodular curves of the transfer-matrix eigenvalues (curves on which two dominant eigenvalues have equal modulus; in the thermodynamic limit these are the candidate singular loci). Their picture: on the ferromagnetic branch the zeros approach a smooth curve even for $H\neq 0$ (with a zero-spacing limit that cannot be uniform in $H$), while on the antiferromagnetic branch the zeros spread into a two-dimensional area, with special bifurcation points (at $u=\pm i$ for $H=0$, becoming $z=-1$ in the hard-square lattice-gas limit — the model of particles on a lattice that exclude occupation of adjacent sites, reached as $H\to\infty$) conjectured singular for all $H$. They reconcile the Nickel singularities with scaling-theory analyticity by conjecturing that for fixed $H\neq 0$ the free energy is analytic on the zero locus while the singularities lie beyond the locus with amplitudes vanishing as $H^{2}$, so that taking two field-derivatives — the susceptibility — exposes them exactly at $H=0$; they stress that the limits $H\to 0$ and lattice size $N\to\infty$ do not commute, and that no analytic argument for this scenario exists.

Problem Statement

For the square-lattice Ising model in a fixed nonzero magnetic field $H$ (real or complex), determine the accumulation geometry of the Fisher zeros — the zeros of the finite-lattice partition function in the complex-temperature plane, under the Brascamp-Kunz boundary conditions for which the exact zero loci are known at $H=0$ and $H/k_{B}T=i\pi/2$ — in the thermodynamic limit, and through it the analyticity structure of the free energy per site as a function of complex temperature at fixed $H\neq 0$. The single scientific objective is the singularity structure of the two-dimensional Ising model at generic magnetic field, the regime between the exactly solved $H=0$ model and its solvable imaginary-field point $H/k_{B}T=i\pi/2$. The cited sources pose the following distinct open questions:

(i) For which fields $H$ do the zeros condense on one-dimensional sets, as they do exactly at $H=0$ and $H/k_{B}T=i\pi/2$ under Brascamp-Kunz boundary conditions, and for which do they fill two-dimensional regions? In particular, is the branch-dependent picture suggested by exact finite-lattice computations — one-dimensional condensation on the ferromagnetic branch, two-dimensional spreading on the antiferromagnetic branch, with singular bifurcation points — correct in the thermodynamic limit, and does it extend from real to complex $H$?

(ii) Where, relative to the zero locus, is the thermodynamic-limit free energy analytic at fixed $H\neq 0$? Equivalently: is there a theorem for the complex-temperature plane at nonzero field analogous to the Lee-Yang circle theorem for the complex-field plane, or an exact computation establishing the same?

(iii) By what mechanism do the dense Nickel singularities of the zero-field susceptibility emerge as $H\to 0$? In particular, is the conjectured scenario correct — singularities of the free energy lying beyond the zero locus at $H\neq 0$ with amplitudes vanishing as $H^{2}$, so that the $H\to 0$ and thermodynamic limits fail to commute and the susceptibility's natural boundary is a singular limit of an otherwise analytic structure?

Questions (i)-(iii) are the facets of this single objective: (i) the general classification of fields by condensation geometry, (ii) the free-energy analyticity structure at fixed $H\neq 0$, and (iii) the $H\to 0$ mechanism behind the Nickel singularities. A result established on a class of fields strictly larger than the two known ones covers the accumulation geometry (i) and the analyticity structure (ii) over that class, but does not by itself yield the general field classification of (i) beyond that class, nor the $H\to 0$ mechanism (iii).

References

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