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Infinite quantum spin quantities and physical adoption — Examples

1 · Prerequisites

2 · Summary

This example specializes the conditional diagonal Ising theorem to one-dimensional uniqueness and two-dimensional sufficient high- and low-temperature regimes. The bounds do not identify the two-dimensional critical temperature, and no behavior outside the stated conditions is inferred.

Nearest-neighbor square-lattice schematic

The dots represent sites in a finite window of $Z^{2}$ ; each solid edge represents one unordered nearest-neighbor pair in the Hamiltonian. Each site also has one field term. The picture shows only edges wholly inside the displayed window, matching the free-boundary finite-volume sum. It illustrates the convention and is not part of the proof.

\begin{tikzpicture}[scale=0.85]
  \foreach \y in {0,1,2} {
    \draw[thick] (0,\y) -- (1,\y) -- (2,\y);
  }
  \foreach \x in {0,1,2} {
    \draw[thick] (\x,0) -- (\x,1) -- (\x,2);
  }
  \foreach \x in {0,1,2} {
    \foreach \y in {0,1,2} {
      \fill[blue!70!black] (\x,\y) circle (2.5pt);
    }
  }
  \node[above] at (1,2.1) {$\Lambda$};
\end{tikzpicture}

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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Controlled diagonal quantum Ising regimes

Physical scope

Illustrative conditional regimes of the spin-1/2 diagonal nearest-neighbor Ising model at positive temperature. The example concerns model states and a dimensionless lattice pressure, not a measured material or mechanical pressure.

Example

Use the homogeneous nearest-neighbor spin- $\frac{1}{2}$ model $H_{Λ} = - J \sum_{{i, j} \subseteq Λ, ∣ i - j ∣_{1} = 1} Z_{i} Z_{j} - h \sum_{i \in Λ} Z_{i}$ , with $J$ and $h$ in joules, $Z_{i}$ the dimensionless Pauli matrices, and $T > 0$ . Here $∣ i - j ∣_{1} : = \sum_{k = 1}^{d} ∣ i_{k} - j_{k} ∣$ is the lattice $ℓ^{1}$ distance; each internal unordered nearest-neighbor pair occurs once. The relevant implications are conditional model predictions:

In one dimension, finite $J, h$ give exactly one full-algebra $β$ -KMS state for every $T > 0$ .
In two dimensions with $h = 0$ , the strict condition $4 \tanh (∣ J ∣ / (k_{B} T)) < 1$ is sufficient for uniqueness.
In two dimensions with $J > 0$ , $h = 0$ , and $T \leq J / (2 k_{B})$ , there are at least two translation-invariant $β$ -KMS states with opposite nonzero magnetizations. For the dimensionless pressure function $p_{β} (h)$ defined by the supplier, the secant slope from zero is at least $β m_{*}$ for every $h > 0$ and at most $- β m_{*}$ for every $h < 0$ , so it has a cusp at zero field.

The last inequality is a conservative sufficient coexistence range; it is not an identified critical temperature. The example does not classify parameters outside the displayed regimes or describe a transverse-field model.

Facts & Assumptions

Given: The physical units and positive-temperature convention of Infinite quantum spin physical quantities, and the conditional Ising model result of Conditional quantum Ising uniqueness and phase predictions.

[A1]

Infinite quantum spin physical quantities: $J, h$ are energy parameters in joules, $β = 1 / (k_{B} T)$ , and the dimensionless pressure per site is not a mechanical pressure in pascals.

[F1]

Conditional quantum Ising uniqueness and phase predictions: in the homogeneous nearest-neighbour model, the sufficient two-dimensional high-temperature row-sum condition is $4 \tanh (∣ J ∣ / (k_{B} T)) < 1$ ; the two-dimensional low-temperature sufficient condition is $T \leq J / (2 k_{B})$ ; the one-dimensional uniqueness and two-dimensional coexistence conclusions hold only under that theorem's stated assumptions. For the two-dimensional low-temperature model, $p_{β} (h) = \lim_{n \to \infty} ∣ Λ_{n} ∣^{- 1} \log {Tr}_{Λ_{n}} (e^{- β H_{Λ_{n}} (h)})$ , $Λ_{n} = [- n, n]^{2}$ , is dimensionless per site and has secant slopes at least $β m_{*}$ for $h > 0$ and at most $- β m_{*}$ for $h < 0$ .

Verification

technique · direct specialization of the earlier conditional result

Given: [A1], [F1], finite $J, h$ , and $T > 0$ .

1.1A1F1givenalgebra

The displayed Hamiltonian is the homogeneous nearest-neighbor specialization of the diagonal interaction in [F1], with each unordered internal edge counted once and the field term once per site. Its energy coefficients have units joules, so $β J = J / (k_{B} T)$ and $β h = h / (k_{B} T)$ are dimensionless.

2.1A1F1step 1.1algebra∎

In $d = 1$ , [F1] applies for every finite $J, h$ and $T > 0$ . In $d = 2$ , substitution of the $2 d = 4$ nearest-neighbor terms into the strict row-sum criterion yields the sufficient high-temperature condition in the Example, while $β J \geq 2$ is equivalent to $T \leq J / (2 k_{B})$ and gives the sufficient coexistence condition. Thus the three displayed regimes follow from [F1], and parameters outside these sufficient conditions receive no conclusion here.

Remarks

The example uses the physical-quantities item for the energy and temperature units only. Its separate AC-dependent entropy-density clause is not part of this specialization.