Complex-Temperature Ising Model

• The complex-temperature Ising model is an analytic continuation of the classical Ising model, extending temperature and coupling parameters into the complex plane.
• It uncovers unique features such as Fisher zeros, phase boundaries, and helimagnetic, spatially modulated correlations that impact computational complexity.
• The analytic structure informs algorithmic tractability and quantum measurement protocols through unitary points and transfer matrix analyses.

The complex-temperature Ising model refers to the analytic continuation of the classical Ising model’s partition function, magnetization, and correlation functions to complex values of the inverse temperature $\beta = J/(k_B T)$ or, equivalently, to complex coupling parameters such as $\tanh\beta$. This extension uncovers rich mathematical and physical structures, including new “phases” in the complex plane, modulated correlations ("helimagnetism"), connections to quantum dynamics, and intricate relations among zeros of the partition function, analytic singularities, computational intractability, and the emergence of long-range order.

1. Analytic Continuation and Partition Function Structure

The Ising model on a lattice (or general graph) is defined by the Hamiltonian

$$H[\{\sigma\}] = -\sum_{\langle i,j\rangle} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i$$

with $\sigma_i \in \{\pm 1\}$. The partition function at complex $\beta$ is

$$Z(\beta) = \sum_{\{\sigma\}} \exp\left[-\beta H(\{\sigma\})\right], \quad \beta \in \mathbb{C}.$$

On the square lattice, Onsager’s solution and its generalizations apply, with the free-energy density for complex $\beta$ in the isotropic case ($J_x = J_y = J$)

$$-\beta f(\beta) = \ln 2 + \frac{1}{2\pi} \int_0^\pi d\theta \,\ln [C^2 - 2CS\cos\theta + 1]$$

where $S = \sinh 2\beta$, $\tanh\beta$0 (Basu et al., 2020). For anisotropic couplings, further analytic continuation is required (Basu et al., 2020).

For general graphs of maximum degree $\tanh\beta$1, the complex-temperature partition function can be recast as

$\tanh\beta$2

allowing deep combinatorial and algorithmic analyses (Galanis et al., 2021).

2. Fisher Zeros and Phase Boundaries in the Complex Plane

The zeros of $\tanh\beta$3 in the complex $\tanh\beta$4-plane (“Fisher zeros”) are central to delineating phase boundaries, generalizing the role of critical points at real temperature. For the 2D isotropic Ising model, Fisher showed the zeros condense on two circles in the complex $\tanh\beta$5-plane: $\tanh\beta$6 with intersections at $\tanh\beta$7 (Basu et al., 2020). In the anisotropic model, these loci deform into arcs that define four regions: a ferromagnetic (FM) bubble, a paramagnetic (PM) region, and two non-ferromagnetic (NFM1, NFM2) regions. The Fisher zero structure defines bona fide phase boundaries in the thermodynamic limit (Basu et al., 2020, Beichert et al., 2013).

For finite-size 1D chains, zeros cluster differently: with open boundaries, all zeros stay at $\tanh\beta$8 in the $\tanh\beta$9 plane, while for periodic boundary conditions, zeros condense towards the origin along rays as system size increases, directly impacting low-temperature expansions (Lee, 2014).

3. Correlation Functions and Spatially Modulated Order

In regions of the complex-temperature plane where the two leading eigenvalues of the transfer matrix become equimodular, the system exhibits spatially modulated (“helimagnetic”) correlations: $$H[\{\sigma\}] = -\sum_{\langle i,j\rangle} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i$$0 with $$H[\{\sigma\}] = -\sum_{\langle i,j\rangle} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i$$1 (Beichert et al., 2013, Basu et al., 2020). In the 1D limit, $$H[\{\sigma\}] = -\sum_{\langle i,j\rangle} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i$$2. In the 2D anisotropic model, spatial modulation and long-range order can persist throughout distinct NFM regions of the plane; the FM phase maintains nonzero uniform magnetization only within a central “bubble.”

The magnetization itself generalizes to

$$H[\{\sigma\}] = -\sum_{\langle i,j\rangle} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i$$3

with the branch cut chosen for continuity with real-temperature results (Basu et al., 2020).

4. Dimensional Crossover: Ladders, Boundary Conditions, and Zeros

Finite-width Ising ladders interpolate between 1D chains and 2D planes. The ladder partition function $$H[\{\sigma\}] = -\sum_{\langle i,j\rangle} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i$$4 (for $$H[\{\sigma\}] = -\sum_{\langle i,j\rangle} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i$$5 legs) is determined by the eigenvalue spectrum of a $$H[\{\sigma\}] = -\sum_{\langle i,j\rangle} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i$$6 transfer matrix. Zeros of $$H[\{\sigma\}] = -\sum_{\langle i,j\rangle} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i$$7 arise where two eigenvalues become equimodular $$H[\{\sigma\}] = -\sum_{\langle i,j\rangle} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i$$8, marking phases with persistent spatial modulation (Basu et al., 2020, Beichert et al., 2013). As $$H[\{\sigma\}] = -\sum_{\langle i,j\rangle} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i$$9, the loci of zeros fill extended areas between Fisher circles, resulting in a phase diagram with regions of spatially modulated order.

Boundary conditions dramatically affect the analytic structure of zeros. For 1D periodic chains, zeros approach the real axis as $\sigma_i \in \{\pm 1\}$0 and strongly alter low-temperature behavior compared to open chains, where zeros remain static (Lee, 2014). Thus, both boundary conditions and finite-size scaling encode subtle analytic information even in the absence of finite-temperature transitions.

5. Algorithmic and Quantum Information Aspects

Complex-temperature Ising partition functions connect directly to computational complexity and quantum measurement protocols. For bounded-degree graphs, there exists a large zero-free region in the complex $\sigma_i \in \{\pm 1\}$1-plane,

$\sigma_i \in \{\pm 1\}$2

within which a deterministic fully polynomial-time approximation scheme (FPTAS) is provably available; outside a strictly larger circle, approximation becomes $\sigma_i \in \{\pm 1\}$3-hard (Galanis et al., 2021). The existence of zeros directly implies algorithmic intractability.

Quantum circuits of constant depth enable the measurement of complex-temperature Ising partition functions, exploiting commuting entangling gates and analytic continuation from imaginary time protocols. Wick rotations relate measured amplitudes $\sigma_i \in \{\pm 1\}$4 for imaginary “Hamiltonian time” to $\sigma_i \in \{\pm 1\}$5, with precise statistical error bounds achievable via central-limit theorems (Iblisdir et al., 2012). This approach allows practical probing of partition functions and the associated Jones polynomial evaluations in moderate-sized systems.

Table: Zones of Algorithmic Tractability for $\sigma_i \in \{\pm 1\}$6 (Galanis et al., 2021)

Region in $\sigma_i \in \{\pm 1\}$7-plane	Approximation Complexity	Zeros Present?
$\sigma_i \in \{\pm 1\}$8	FPTAS	Zero-free
$\sigma_i \in \{\pm 1\}$9	$\beta$0-hard	Zeros occur
Intermediate annulus	No FPTAS, evidence of zeros	Zeros numerically

Within the zero-free disc, Taylor expansion and Barvinok–Patel–Regts cluster expansion strategies guarantee efficient computation, while zeros and associated Fisher phase boundaries in the complex plane demarcate the computational transition.

6. Special Unitary Points and Quantum Dynamical Connections

Points in the complex $\beta$1-plane where $\beta$2 are “special unitary points.” Here, the transfer matrix $\beta$3 is (after normalization) unitary, and all eigenvalues reside on the unit circle. Physically, these points admit a quantum-dynamics interpretation: $\beta$4 realizes a unitary quantum evolution, with the partition function representing a perfectly coherent quantum circuit (Basu et al., 2020). In tensor-network language, each Boltzmann weight becomes a unitary gate and the classical partition sum generalizes to a quantum circuit amplitude.

These unitary points coincide with multicritical junctions of the phase diagram (Fisher circles' intersection points) and are crucial for mapping between equilibrium statistical mechanics at complex temperature and protocols in quantum information science.

7. Thermodynamic and Physical Consequences

Analytic continuation endows the Ising model with a more intricate phase diagram, defined not by thermodynamic phases but by regions of holomorphy, loci of degenerate transfer-matrix eigenvalues, and modulated correlation functions (“helimagnetism”). The partition function's zeros correspond to both analytic singularities and computational hardness boundaries. Thermodynamic observables such as free energy, internal energy, and specific heat become multivalued and may possess complex branch cuts and discontinuities across these boundaries (Beichert et al., 2013, Basu et al., 2020).

In the 2D limit, only the Fisher circles (specific loci in the complex $\beta$5-plane) are associated with singularities in a given topological sector; for finite-width ladders and 1D chains with periodic boundary conditions, zeros fill extended lines or regions, fundamentally altering physical and analytic behavior even in the absence of true transitions at positive real temperature (Lee, 2014).

The complex-temperature Ising model thus serves as a rich theoretical testbed, connecting statistical mechanics, complex analysis, computational complexity theory, and quantum information science.