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The finite Potts Gibbs matrix is well-defined at every finite inverse temperature

Statement

For $Q \in {2, 3, 4}$ , $L \geq 1$ , real coefficients $J_{i}$ , and finite real $β$ , the matrix $H_{Q, L} (J)$ defined on the Potts-clock chain is self-adjoint. With the spectral definitions $B_{β}$ , $Z_{β}$ , and $ρ_{β}$ from that definition, $Z_{β} > 0$ and $ρ_{β}$ is a positive semidefinite matrix of trace one. This is a finite-dimensional mathematical fact; it makes no thermodynamic-limit or empirical claim.

Statement references

Finite Hermitian Potts-clock chain

Facts & Assumptions

Given: $Q \in {2, 3, 4}$ , integer $L \geq 1$ , real $J_{i}$ , and $β \in R$ .

[A1]

These finite-dimensional parameters and coefficients have the ranges just stated.

[F1]

Finite Hermitian Potts-clock chain: $P_{j} = ∣ + ⟩ ⟨ + ∣$ , $D_{j, j + 1} = \sum_{a} ∣ a a ⟩ ⟨ a a ∣$ , $e_{2 j - 1} = \sqrt{Q} P_{j}$ , $e_{2 j} = \sqrt{Q} D_{j, j + 1}$ , and $H_{Q, L} = - \sum_{i} J_{i} e_{i}$ ; the inner product is linear in its first argument; $Π_{λ}$ is the self-adjoint idempotent projection onto the $λ$ -eigenspace, the projections sum to the identity, and $B_{β}$ is the spectral sum with weights $e^{- β λ}$ . A self-adjoint $A$ is positive definite if $⟨ A v, v ⟩ > 0$ for every nonzero $v$ , and positive semidefinite if $⟨ A v, v ⟩ \geq 0$ for every $v$ .

[F2]

Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space: an endomorphism is self-adjoint when it equals its adjoint; a self-adjoint endomorphism is normal.

[F3]

Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely: a normal endomorphism of a finite-dimensional complex inner-product space has an orthonormal eigenbasis.

[F4]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $e^{x} > 0$ and $e^{- x} = 1 / e^{x}$ for every real $x$ .

[F5]

The basis-independent trace of an endomorphism of a finite-dimensional vector space: the trace is the basis-independent sum of diagonal entries.

Proof

1.1A1F1F2algebra

Each $P_{j}$ is an orthogonal projection because $⟨ + ∣ + ⟩ = 1$ . Each $D_{j, j + 1}$ is a sum of mutually orthogonal rank-one projections onto the computational vectors $∣ a a ⟩$ . Hence these operators are self-adjoint; since $\sqrt{Q}$ and each $J_{i}$ are real, every $e_{i}$ and then $H_{Q, L} = - \sum_{i} J_{i} e_{i}$ are self-adjoint.

2.1A1F1F2F3F4step 1.1algebra

By [F2] and [F3], $H_{Q, L}$ has an orthonormal eigenbasis. For a unit eigenvector $v$ with $H_{Q, L} v = λ v$ , self-adjointness and the linear-first inner-product convention give $λ = ⟨ H_{Q, L} v, v ⟩ = ⟨ v, H_{Q, L} v ⟩ = \overline{λ}$ , so every eigenvalue $λ$ is real. The distinct-eigenvalue projections $Π_{λ}$ are self-adjoint and sum to the identity by [F1]. Each weight $w_{λ} = e^{- β λ}$ is finite, real, and strictly positive by [F4]. Therefore $B_{β} = \sum_{λ} w_{λ} Π_{λ}$ is self-adjoint, and for every nonzero $v$ , $⟨ B_{β} v, v ⟩ = \sum_{λ} w_{λ} ∥ Π_{λ} v ∥^{2} > 0$ : at least one projected component is nonzero since $\sum_{λ} Π_{λ} = I$ . Thus $B_{β}$ is positive definite.

3.1A1F1F5step 2.1algebra

In an orthonormal eigenbasis, [F5] gives $Z_{β} = tr B_{β} = \sum_{λ} w_{λ} \dim (ran Π_{λ})$ . The space has finite positive dimension $Q^{L}$ , so this is a finite sum with at least one positive term; hence $Z_{β}$ is finite and strictly positive. It is a real scalar. Dividing the self-adjoint positive definite $B_{β}$ by $Z_{β}$ gives a self-adjoint positive semidefinite $ρ_{β}$ , and $tr ρ_{β} = Z_{β}^{- 1} Z_{β} = 1$ .

4.1A1F1F4F5algebra∎

If all $J_{i} = 0$ , then $H_{Q, L} = 0$ ; if $β = 0$ , all spectral weights also equal $e^{0}$ . By [F4] at $x = 0$ , $a : = e^{0} > 0$ and $a = 1 / a$ , so $a = 1$ . In either boundary case $B = I$ , $Z = Q^{L}$ by [F5], and $ρ = I / Q^{L}$ .

The finite Potts Gibbs matrix is well-defined at every finite inverse temperature

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources