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Commuting quantum Ising and nonadditive phase countermodel

Physical scope

Two explicit finite-dimensional quantum models: a periodic diagonal one-dimensional Ising Hamiltonian and a global N-dependent three-sector Hamiltonian. The first has the exact specified one-dimensional classical correspondence; the second is deliberately non-product-additive. No generic noncommuting lattice theorem or short-range phase claim is asserted.

Statement

Periodic commuting Ising model. For $N \geq 2$ , let $H_{N} = (C^{2})^{\otimes N}$ with its product inner product, and let $Z_{i}$ act as $σ_{z}$ on tensor factor $i$ and as identity on the others; each $Z_{i}$ acts on all of $H_{N}$ . With periodic indexing $Z_{N + 1} = Z_{1}$ , count exactly the $N$ indexed bonds, so $N = 2$ has two bonds. For real energies $J, h$ and $β > 0$ in J $^{- 1}$ , adopt the finite matrix $H_{N}^{q} = - J \sum_{i = 1}^{N} Z_{i} Z_{i + 1} - h \sum_{i = 1}^{N} Z_{i}, D (H_{N}^{q}) = H_{N} .$ For each site set $∣ + 1 ⟩ : = e_{0}$ and $∣ - 1 ⟩ : = e_{1}$ , so $σ_{z} ∣ s_{i} ⟩ = s_{i} ∣ s_{i} ⟩$ ; define $∣ s_{1}, \dots, s_{N} ⟩ : = ∣ s_{1} ⟩ \otimes \dots \otimes ∣ s_{N} ⟩$ . Its energy equals $H_{N}^{c l} (s) = - J \sum_{i} s_{i} s_{i + 1} - h \sum_{i} s_{i}$ . Thus the quantum Gibbs density is diagonal in this basis, and its partition function and diagonal probabilities agree exactly with the specified classical finite ring. In particular, when $J \geq 0$ and $h = 0$ , its $σ_{z}$ correlations are those of Finite periodic Ising-ring correlations and exact decay. Use $\sinh x = (e^{x} - e^{- x}) / 2$ , $\cosh x = (e^{x} + e^{- x}) / 2$ , and $\tanh x = \sinh x / \cosh x$ from The six hyperbolic functions and their natural domains, where $\cosh x > 0$ for every real $x$ , and put $λ_{\pm} = e^{β J} \cosh (β h) \pm \sqrt{e^{2 β J} \sinh^{2} (β h) + e^{- 2 β J}} .$ Then $N^{- 1} \log Z_{N}^{q} \to \log λ_{+}$ , with every fixed finite order of derivatives converging locally uniformly on compact subsets of $(0, \infty) \times R^{2}$ . This is the exact commuting one-dimensional model, not a theorem for noncommuting interactions.

Global three-sector countermodel. For $N \geq 1$ and $ϵ > 0$ joules, let $K_{N} = C \oplus C \oplus (C^{2})^{\otimes 2 N}$ with its orthogonal direct-sum inner product, and write $I_{4^{N}}$ for the identity on the final $4^{N}$ -dimensional summand. Let $∣ A ⟩, ∣ B ⟩$ be the standard unit vectors in the first and second summands, and let $∣ s ⟩$ range over the standard product basis in the final summand. Define the finite matrix $H_{N}^{g} = 0 \oplus N ϵ \oplus 2 N ϵ I_{4^{N}}$ on the full domain $K_{N}$ , so the basis energies are $E (A) = 0$ , $E (B) = N ϵ$ , and $E (s) = 2 N ϵ$ . For $0 \leq Δ < N ϵ$ , define the spectral-shell basis subset $S_{N, Δ} : = {v : ∣ E (v) - N ϵ ∣ \leq Δ}$ and its uniform state $ρ_{N, Δ}^{s h e l l} : = ∣ S_{N, Δ} ∣^{- 1} \sum_{v \in S_{N, Δ}} ∣ v ⟩ ⟨ v ∣$ . For $β > 0$ this finite quantum model has $Z_{N}^{g} = 1 + e^{- β N ϵ} + 4^{N} e^{- 2 β N ϵ} .$ For every real $β$ as an algebraic finite-weight extension, the normalized exponential-tilt weight of the one-dimensional middle sector is at most $(2^{N + 1} + 1)^{- 1}$ ; the shell state above assigns that sector probability $1$ . The term “probability” here is the trace of the state against the middle-sector projection $∣ B ⟩ ⟨ B ∣$ . Also $\frac{1}{N} \log Z_{N}^{g} ⟶ \max {0, \log 4 - 2 β ϵ}, 0 \leq \frac{1}{N} \log Z_{N}^{g} - \max {0, \log 4 - 2 β ϵ} \leq \frac{\log 3}{N} .$ The limiting function has a corner at $β_{c} = (\log 2) / ϵ > 0$ J $^{- 1}$ . For $β > 0$ , define the canonical mean energy per member of the global model by

${\overset{ˉ}{E}}_{N} (β) : = \frac{N ϵ e^{- β N ϵ} + 2 N ϵ 4^{N} e^{- 2 β N ϵ}}{N Z_{N}^{g} (β)} .$

The limit $u (β) : = \lim_{N \to \infty} {\overset{ˉ}{E}}_{N} (β)$ exists for all $β > 0$ and equals $2 ϵ$ for $0 < β < β_{c}$ , $ϵ$ at $β = β_{c}$ , and $0$ for $β > β_{c}$ . Thus the one-sided limits of the thermodynamic-limit mean energy are $2 ϵ$ as $β ↑ β_{c}$ through positive values and $0$ as $β ↓ β_{c}$ from above. The finite partition expressions are real analytic as functions of $β \in R$ for fixed remaining model parameters; this is a regularity statement about the finite formulas, while the canonical preparation and mean-energy claims use only $β > 0$ . This global, size-dependent family is not a product of one fixed finite one-site alphabet and makes no claim about local or short-range matter.

Facts & Assumptions

Given: The finite Hilbert spaces, Hamiltonians, indexed-bond convention, and positive-temperature canonical preparations in the Statement. Energies and $ϵ$ are in joules; $β$ is in J $^{- 1}$ .

[A1]

Quantum statistics and model equilibrium preparation: use the normalized canonical Gibbs state only when its trace is positive and finite. For $β \leq 0$ in the three-sector bound, the finite exponential weights are only an algebraic extension, not a physical negative-temperature assumption.

[F1]

Pauli matrices and the two-qubit exchange operator: $σ_{z}$ is Hermitian with eigenvectors $e_{0}, e_{1}$ and eigenvalues $1, - 1$ .

[F2]

One-dimensional periodic Ising-chain limit: for the stated periodic two-state chain, with $\cosh x = (e^{x} + e^{- x}) / 2$ and $\sinh x = (e^{x} - e^{- x}) / 2$ , the transfer matrix has eigenvalues $λ_{\pm}$ with $λ_{+} > ∣ λ_{-} ∣$ , and $N^{- 1} \log Z_{N} \to \log λ_{+}$ with every fixed finite order of derivatives locally uniformly on compact parameter sets.

[F3]

Finite periodic Ising-ring correlations and exact decay (current draft Statement): for the finite periodic ring with $N \geq 2$ , $J \geq 0$ , $β > 0$ , and $h = 0$ , the one-site means and two-site correlations have the explicit finite-ring formulas stated there.

[F4]

Explicit nonadditive transition and ensemble inequivalence (current draft Statement and complete Proof): for the specified three-sector model, the canonical mean-energy limit is $2 ϵ$ for $0 < β < β_{c}$ , $ϵ$ at $β_{c}$ , and $0$ for $β > β_{c}$ , where $β_{c} = (\log 2) / ϵ > 0$ ; its finite partition, middle-sector bound, shell comparison, and limiting log partition are also as stated there. The canonical mean claim is restricted to positive $β$ .

[F5]

The six hyperbolic functions and their natural domains: $\sinh x = (e^{x} - e^{- x}) / 2$ , $\cosh x = (e^{x} + e^{- x}) / 2$ , $\tanh x = \sinh x / \cosh x$ , and $\cosh x > 0$ for every $x \in R$ .

[F6]

The real exponential function and the number $e$ by a power series, The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ , The natural logarithm as the inverse of the exponential function, and Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\exp$ is the everywhere-convergent real power series, is positive on $R$ , satisfies $e^{x + y} = e^{x} e^{y}$ , and $\log$ is its inverse on $(0, \infty)$ with $\log (x y) = \log x + \log y$ for positive $x, y$ .

[F7]

A real-analytic function on an open subset of $R$ is locally represented by a convergent real power series, The sum of a real power series is real analytic throughout the open interval determined by its radius, and Real-analytic functions are closed under sums, products and compositions, and under quotients where the denominator is nonzero: real analyticity means local representation by a convergent power series; power-series sums are real analytic, and finite sums and compatible compositions preserve real analyticity.

Proof

1.1givenA1F1F6algebra

Each $Z_{i}$ is diagonal in the orthonormal product basis $∣ s_{1}, \dots, s_{N} ⟩$ , with eigenvalue $s_{i}$ . Therefore $H_{N}^{q} ∣ s ⟩ = H_{N}^{c l} (s) ∣ s ⟩$ with real eigenvalues. This diagonal representation proves self-adjointness on the full domain $D (H_{N}^{q}) = H_{N}$ . The finite canonical trace is the sum of these positive weights by [F6], so $Z_{N}^{q} = \sum_{s} e^{- β H_{N}^{c l} (s)}$ and the diagonal Gibbs probability of each $∣ s ⟩$ equals the classical probability of $s$ . This proves the claimed finite correspondence without interpreting off-diagonal quantum observables classically.

1.2givenA1F4F6algebra

The direct sum in the second model has multiplicities $1, 1, 4^{N}$ , so summing its three energy weights gives $Z_{N}^{g} = 1 + x + 4^{N} x^{2}$ where $x = e^{- β N ϵ} > 0$ . Its real diagonal entries show that $H_{N}^{g}$ is self-adjoint on the stated full domain. Since $(1 - 2^{N} x)^{2} \geq 0$ , $1 + 4^{N} x^{2} \geq 2^{N + 1} x$ , whence the middle-sector probability $x / Z_{N}^{g} \leq (2^{N + 1} + 1)^{- 1}$ . For $0 \leq Δ < N ϵ$ , the sectors at energies $0$ and $2 N ϵ$ lie exactly $N ϵ$ from the middle energy; the only basis vector in the shell is $∣ B ⟩$ , so $ρ_{N, Δ}^{s h e l l} = ∣ B ⟩ ⟨ B ∣$ and $tr (ρ_{N, Δ}^{s h e l l} ∣ B ⟩ ⟨ B ∣) = 1$ . Identifying the three CSM sectors with $∣ A ⟩$ , $∣ B ⟩$ , and the final product-basis vectors preserves each energy and multiplicity; for every $β > 0$ it therefore preserves the canonical sector weights and mean energy per member. [F4] These are the same finite-model values, now realized on the displayed quantum Hilbert space.

2.1givenF2F3F5step 1.1

The definitions in [F5] agree with the exponential definitions of $\sinh$ and $\cosh$ in [F2] and of $\tanh$ in [F3]; in particular $\cosh (β J) > 0$ , so the zero-field correlation parameter $\tanh (β J)$ is defined. Applying [F2] to this identical finite energy function gives the displayed transfer eigenvalues, limiting normalized log partition, and compact-uniform derivative convergence. In the restricted case $J \geq 0, h = 0$ , the equality of diagonal probabilities from step 1.1 transports each $σ_{z}$ correlation to the classical spin correlation in [F3], under exactly that supplier's hypotheses.

2.2givenA1F4F6step 1.2algebra

Put $M_{N} = \max {1, 4^{N} x^{2}}$ . If $x \leq 1$ , then $x \leq M_{N}$ ; if $x > 1$ , then $4^{N} x^{2} \geq x$ for $N \geq 1$ . Hence $M_{N} \leq Z_{N}^{g} \leq 3 M_{N}$ . By the logarithm product law in [F6], $N^{- 1} \log M_{N} = \max {0, \log 4 - 2 β ϵ}$ , so taking logarithms gives the stated error bound and limit. Since $\log 4 = 2 \log 2$ , the two branches meet at $β_{c} = (\log 2) / ϵ$ . By [F4] and the exact finite-weight identification in step 1.2, the limits of the quantum mean energy per member are $2 ϵ$ for $0 < β < β_{c}$ , $ϵ$ at $β_{c}$ , and $0$ for $β > β_{c}$ . These are canonical mean-energy claims only for $β > 0$ , as required by [A1].

3.1givenF2F6F7algebra∎

Fix $N$ and the other model parameters. The Ising partition function is the finite sum $\sum_{s \in {- 1, 1}^{N}} \exp (- β H_{N}^{c l} (s))$ , and the global partition function is the finite sum of its three sector weights. Each energy in these sums is fixed as a function of $β$ , so the map $β \mapsto - β E$ is affine and real analytic. The exponential is a power-series sum and therefore real analytic by [F6]–[F7]; composing with the affine maps and taking finite sums preserves real analyticity by [F7]. Thus each fixed-parameter finite- $N$ partition function is real analytic on $R$ , in particular at every positive canonical parameter. The Ising limit is asserted here only with the derivative convergence supplied by [F2].

Depends on

Dependency tree · 2 levels

Sources

David Tong, Statistical Physics, §§5.3.1 and 5.4.2