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Finite periodic Ising-ring correlations and exact decay

Physical scope

Finite periodic one-dimensional nearest-neighbor Ising ring with N>=2, exactly N indexed bonds, J>=0, zero field, and finite beta>0; no general long-range or higher-dimensional claim.

Statement

Use the real exponential $\exp (x) = \sum_{n \geq 0} x^{n} / n!$ of The real exponential function and the number $e$ by a power series and the hyperbolic functions $\cosh x = (\exp x + \exp (- x)) / 2$ , $\sinh x = (\exp x - \exp (- x)) / 2$ . Define $\tanh x : = \sinh x / \cosh x$ ; the denominator is positive for every real $x$ . Let $N \geq 2$ , let $Ω_{N} = {- 1, 1}^{N}$ with cyclic indexing $s_{N + 1} = s_{1}$ , and count exactly the $N$ indexed periodic bonds (so $N = 2$ has two bonds). For coupling $J \geq 0$ in joules, zero field, and $β > 0$ in J $^{- 1}$ , adopt the finite canonical law $P_{N} (s) = Z_{N}^{- 1} \exp ⁣ (β J \sum_{i = 1}^{N} s_{i} s_{i + 1}) .$ Set $t = \tanh (β J) \in [0, 1)$ . For $1 \leq r \leq N - 1$ , $E_{N} [s_{i}] = 0, E_{N} [s_{i} s_{i + r}] = \frac{t^{r} + t^{N - r}}{1 + t^{N}} .$ For each fixed positive integer separation $r$ and $N \to \infty$ , the correlation tends to $t^{r}$ , with absolute error at most $t^{N - r} + t^{N + r}$ . When $J > 0$ , define the limiting correlation length $ξ > 0$ in lattice spacings as the unique number satisfying $t^{r} = e^{- r / ξ}$ for every positive integer separation $r$ ; equivalently, $ξ = - 1 / \log t$ . The zero-field susceptibility per spin, defined by $χ_{N} : = N^{- 1} \partial_{h} E_{N, h} [\sum_{i} s_{i}] ∣_{h = 0}$ for the field-coupled law with energy $- ⁣ J \sum_{i} s_{i} s_{i + 1} - h \sum_{i} s_{i}$ , where the field parameter $h \in R$ is in joules, has limit $χ_{\infty} = β \frac{1 + t}{1 - t} .$ Both $χ_{N}$ and $χ_{\infty}$ have units J $^{- 1}$ per spin (the spin variables are dimensionless). Define the finite-ring model free energy by $F_{N} : = - β^{- 1} \log Z_{N}$ and its limiting free energy per spin, when the limit exists, by $f_{\infty} : = \lim_{N \to \infty} F_{N} / N$ . Then $f_{\infty} = - β^{- 1} \log (2 \cosh (β J))$ , analytic at every finite positive temperature. No statement is made for arbitrary one-dimensional interactions.

Facts & Assumptions

Given: The finite spin space $Ω_{N}$ , the stated bond convention, $N \geq 2$ , $J \geq 0$ , and finite $β > 0$ .

[F1]

Declared example preparations and thermodynamic interpretation: each finite model uses its explicitly adopted canonical probability on the stated reference/counting space; the preparation is a model assumption.

[F2]

One-dimensional periodic Ising-chain limit: for the periodic nearest-neighbor ring with $N \geq 2$ , finite real $J, h$ and $β > 0$ , the same $\cosh$ and $\sinh$ definitions give $T_{s, t} = \exp (β J s t + β h (s + t) / 2)$ and $Z_{N} = tr (T^{N})$ . Its explicit eigenvalues satisfy $λ_{+} > ∣ λ_{-} ∣$ , and the logarithmic partition limit and each fixed finite order of parameter derivatives converge uniformly on compact subsets of $(0, \infty) \times R^{2}$ . This is the current published mathematical supplier for the exact indexed-bond model used here.

[F3]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $\exp (x) > 0$ for every real $x$ , hence the defined $\cosh x$ is positive and $\tanh x$ is well-defined.

[F4]

The exponential function is strictly increasing: $\exp$ is continuous and $x \leq y$ implies $\exp (x) \leq \exp (y)$ .

[F5]

A real-analytic function on an open subset of $R$ is locally represented by a convergent real power series: a real-valued function on an open interval is real analytic when each point has a neighborhood on which the function equals a convergent real power series.

[F6]

The sum of a real power series is real analytic throughout the open interval determined by its radius: the sum of a real power series is real analytic on the interior of its radius of convergence; the exponential series has infinite radius.

[F7]

Real-analytic functions are closed under sums, products and compositions, and under quotients where the denominator is nonzero: sums, products, quotients with nonzero denominator, and compositions of real-analytic functions on their open domains are real analytic.

[F8]

[The power series for log(1+x) on (-1,1], including the Abel endpoint](/item/thm-log-one-plus-x-power-series): for $∣ u ∣ < 1$ , $\log (1 + u) = \sum_{n = 1}^{\infty} (- 1)^{n + 1} u^{n} / n$ , so $u \mapsto \log (1 + u)$ is real analytic near $u = 0$ .

[F9]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: for positive $x, y$ , $\log (x y) = \log x + \log y$ .

[A1]

For the response clause, the field-coupled finite law is $P_{N, h} (s) \propto \exp (β J \sum_{i} s_{i} s_{i + 1} + β h \sum_{i} s_{i})$ for real field parameter $h$ in joules; this is a finite model family, not an empirical magnetic law.

Proof

1.1givenF2F4algebra

In the zero-field case the transfer matrix in [F2] is $T_{s, t} = e^{K s t}$ with $K = β J$ , so $T = (\begin{array}{cc} e^{K} & e^{- K} \\ e^{- K} & e^{K} \end{array})$ . Direct multiplication gives orthonormal eigenvectors $u_{+} = (1, 1) / \sqrt{2}$ , $u_{-} = (1, - 1) / \sqrt{2}$ with eigenvalues $λ_{+} = 2 \cosh K$ and $λ_{-} = 2 \sinh K$ . Hence $Z_{N} = tr (T^{N}) = λ_{+}^{N} + λ_{-}^{N} = λ_{+}^{N} (1 + t^{N})$ , where $t = λ_{-} / λ_{+} = \tanh K$ . Since $K \geq 0$ , [F4] gives $e^{K} \geq e^{- K}$ , so $λ_{-} = e^{K} - e^{- K} \geq 0$ ; [F2] gives $λ_{+} > ∣ λ_{-} ∣$ . Thus $t \in [0, 1)$ .

2.1A1F1F2step 1.1algebra

By the finite preparation [F1], the partition sum is the normalization for the adopted law. Put $D_{0} = diag (1, - 1)$ ; then $D_{0} u_{+} = u_{-}$ and $D_{0} u_{-} = u_{+}$ . Expanding the finite canonical sum over the $N$ cyclic indices with $s_{i} s_{i + r}$ inserted gives numerator $tr (D_{0} T^{r} D_{0} T^{N - r})$ . Evaluating this trace in the eigenbasis yields $λ_{-}^{r} λ_{+}^{N - r} + λ_{+}^{r} λ_{-}^{N - r}$ . Dividing by $Z_{N}$ from step 1.1 gives the stated correlation formula.

3.1step 2.1algebra

The map $s \mapsto - s$ preserves every bond product and reverses $\sum_{i} s_{i}$ , so at zero field $E_{N} [s_{i}] = 0$ . For fixed $r$ , subtracting $t^{r}$ from the formula of step 2.1 gives $(t^{N - r} - t^{N + r}) / (1 + t^{N})$ ; its absolute value is at most $t^{N - r} + t^{N + r}$ because $0 \leq t < 1$ . Thus the fixed-distance limit is $t^{r}$ . When $0 < t < 1$ , writing $t^{r} = e^{- r / ξ}$ gives $ξ = - 1 / \log t$ ; for $J = 0$ , $t = 0$ and all nonzero-separation correlations vanish.

3.2A1step 2.1algebra

For the field-coupled finite law [A1], differentiating its finite numerator and denominator sums gives $\partial_{h} E_{N, h} [M] = β (E_{N, h} [M^{2}] - E_{N, h} [M]^{2})$ , where $M = \sum_{i} s_{i}$ . At $h = 0$ , spin-flip symmetry gives $E M = 0$ and translation invariance gives $\frac{1}{N} E [M^{2}] = 1 + \sum_{r = 1}^{N - 1} E [s_{1} s_{1 + r}] = 1 + \frac{2 \sum_{r = 1}^{N - 1} t^{r}}{1 + t^{N}} .$ The finite geometric sum tends to $2 t / (1 - t)$ as $N \to \infty$ , so $χ_{N} \to β (1 + t) / (1 - t)$ . This also gives $χ_{N} = β$ when $J = 0$ .

4.1step 1.1step 3.1algebra

From step 1.1, $N^{- 1} \log Z_{N} = \log (2 \cosh (β J)) + N^{- 1} \log (1 + t^{N})$ . Since $1 \leq 1 + t^{N} \leq 2$ , the second term tends to zero, so $f_{\infty} = - β^{- 1} \log (2 \cosh (β J))$ . The correlation length diverges only in the excluded zero-temperature limit $β J \to \infty$ .

5.1F3F4F5F6F7F8F9step 4.1∎

Fix any $β_{0} > 0$ and put $A (β) : = 2 \cosh (β J)$ and $u (β) : = A (β) / A (β_{0}) - 1$ . By [F5, F6, F7], $A$ and $u$ are real analytic near $β_{0}$ ; $A (β_{0}) > 0$ by [F3] and $u (β_{0}) = 0$ . Continuity from [F4] allows a smaller neighborhood where $∣ u (β) ∣ < 1$ . By [F8], [F6], and the product law [F9], $\log A (β) = \log A (β_{0}) + \log (1 + u (β))$ is real analytic there, and [F7] makes $f_{\infty} (β) = - β^{- 1} \log A (β)$ real analytic since $β \neq 0$ . The relation $β = (k_{B} T)^{- 1}$ is real analytic for $T > 0$ , so the limit is analytic at every finite positive temperature as well.

Remarks

The transfer-matrix premise uses the published exact periodic-chain theorem. The local spin-insertion and finite-response arguments prove the additional correlation and susceptibility formulas. No Gaussian-tail research token is used as a supplier.

Finite periodic Ising-ring correlations and exact decay

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources