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Sufficient Ising uniqueness and coexistence regimes

Physical scope

Two-dimensional classical nearest-neighbor ferromagnetic Ising with J>0 and positive temperature: the sufficient uniqueness bound holds at every constant finite field energy h, while the stated coexistence bound requires h=0. Conditional mathematical regimes only; no empirical threshold claim.

Example

For the two-dimensional nearest-neighbor ferromagnetic Ising model, take coupling energy $J = J_{I} > 0$ joules, a constant finite field energy $h \in R$ in joules per dimensionless spin, and temperature $T > 0$ kelvin, set $β = (k_{B} T)^{- 1}$ in $J^{- 1}$ , and write $K = β J$ (dimensionless). The results on this page give a sufficient high-temperature uniqueness regime $4 \tanh K < 1$ at every such $h$ , and a sufficient low-temperature coexistence regime $K \geq 2$ at $h = 0$ . Parameters outside these sufficient hypotheses are not classified by these bounds. These are conditional model predictions, not measured transition temperatures.

Facts & Assumptions

Given: A two-dimensional nearest-neighbor classical Ising model with $J > 0$ , $β > 0$ , a constant finite field energy $h$ , and the adopted Gibbs-equilibrium interpretation; the coexistence branch additionally assumes $h = 0$ .

[A1]

Gibbs/DLR states are adopted as the equilibrium model for the chosen Hamiltonian and reference (Interacting classical Gibbs equilibrium adoption).

[A2]

Countable Choice is retained in the DLR construction (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

If $4 \tanh K < 1$ , the influence row sum is below one and the DLR state is unique (Conditional Ising equilibrium phase predictions).

[F2]

For the zero-field model $h = 0$ , if $K \geq 2$ , there are distinct stationary DLR states with opposite nonzero magnetizations (Conditional Ising equilibrium phase predictions).

Verification

1.1A1A2F1algebra

For nearest-neighbor interactions in two dimensions, each site has four neighbors, so the sufficient row sum is $4 \tanh K$ . Thus $4 \tanh K < 1$ is a sufficient high-temperature uniqueness condition; [F1] is applicable under the stated Countable Choice hypothesis [A2].

2.1A2F2step 1.1algebra

For $h = 0$ , the low-temperature theorem [F2] applies for $K \geq 2$ under [A2]. At other parameter values outside the high-temperature inequality and this zero-field coexistence regime, neither sufficient conclusion applies in this comparison.

3.1A1A2step 1.1step 2.1∎

The postulate [A1] makes these mathematical DLR conclusions conditional equilibrium-model predictions only. The thresholds are not empirical measurements, and no exact critical point is asserted.

Sufficient Ising uniqueness and coexistence regimes

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources