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Exact quadratic reduction and Gibbs trace of the open Ising chain

Statement

Assume Countable Choice. Let $N \geq 1$ , use the open-chain Hilbert space and Hamiltonian $H_{N} (h, J)$ of the ordered Majorana definition, and let $β > 0$ have units $J^{- 1}$ while $h, J$ have units joules. Define the real skew matrix $A_{N} \in M_{2 N} (R)$ by $(A_{N})_{2 j - 1, 2 j} = 2 h$ for $1 \leq j \leq N$ , $(A_{N})_{2 j, 2 j + 1} = 2 J$ for $1 \leq j < N$ , the opposite entries by skew-symmetry, and all other entries by zero. Let $ϵ_{1}, \dots, ϵ_{N} \geq 0$ be its real orthogonal block energies, including zero values. Then $H_{N} (h, J) = \frac{i}{4} γ^{T} A_{N} γ = \sum_{r = 1}^{N} ϵ_{r} (b_{r}^{*} b_{r} - \frac{1}{2} I)$ for the transformed Majorana modes $b_{r}$ in the Proof, and its finite Gibbs trace is $Z_{N} (β; h, J) : = {Tr}_{H_{N}} (e^{- β H_{N} (h, J)}) = \prod_{r = 1}^{N} 2 \cosh ⁣ (\frac{β ϵ_{r}}{2}) .$ Every zero mode contributes a factor two. The chain is open throughout; no periodic Jordan–Wigner parity boundary term is included.

Statement references

Open transverse Ising chain and ordered Majorana operators

Hermiticity and CAR of the ordered Ising Majoranas

Orthogonal normal form of a finite real skew matrix

Facts & Assumptions

Given: Countable Choice; an integer $N \geq 1$ ; real $h, J$ in joules; and $β > 0$ in $J^{- 1}$ .

[A1]

Countable Choice is assumed exactly where the diagonal occupation-trace supplier is applied (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Open transverse Ising chain and ordered Majorana operators: $H_{N} = (C^{2})^{\otimes N}$ , $H_{N} = - h \sum_{j = 1}^{N} Z_{j} - J \sum_{j = 1}^{N - 1} X_{j} X_{j + 1}$ , and $γ_{2 j - 1} = Z_{1} \dots Z_{j - 1} X_{j}$ , $γ_{2 j} = Z_{1} \dots Z_{j - 1} Y_{j}$ .

[F2]

Hermiticity and CAR of the ordered Ising Majoranas: each $γ_{a}$ is Hermitian and ${γ_{a}, γ_{b}} = 2 δ_{a b} I$ .

[F3]

Orthogonal normal form of a finite real skew matrix: a real skew matrix of size $2 N$ has an orthogonal basis of two-dimensional blocks $(\begin{matrix} 0 & ϵ_{r} \\ - ϵ_{r} & 0 \end{matrix})$ with $ϵ_{r} \geq 0$ .

[F4]

Explicit diagonal Fock domains and occupation trace: under Countable Choice, for a finite fermionic mode set with energies $ϵ_{r}$ , $β > 0$ , and $q_{r} = e^{- β ϵ_{r}} > 0$ , the diagonal occupation trace is $\prod_{r} (1 + q_{r})$ ; its finite-mode case allows zero energies and hence $q_{r} = 1$ .

[F5]

Finite matrix spectral and state calculus: finite Hermitian matrices have orthonormal eigenbases, and the ordinary finite trace is basis-independent.

Proof

1.1givenF1F2algebra

For every site, $γ_{2 j - 1} γ_{2 j} = X_{j} Y_{j} = i Z_{j}$ , so $i h γ_{2 j - 1} γ_{2 j} = - h Z_{j}$ . For $j < N$ , $γ_{2 j} γ_{2 j + 1} = Y_{j} Z_{j} X_{j + 1} = i X_{j} X_{j + 1}$ , so $i J γ_{2 j} γ_{2 j + 1} = - J X_{j} X_{j + 1}$ . Summing exactly these terms gives the open-chain $H_{N}$ of [F1], with no $j = N$ bond.

2.1step 1.1F2algebra

For any paired indices $a < b$ with coefficient $c$ , setting $(A_{N})_{a b} = 2 c$ and $(A_{N})_{b a} = - 2 c$ contributes $(i / 4) (2 c γ_{a} γ_{b} - 2 c γ_{b} γ_{a}) = i c γ_{a} γ_{b}$ by [F2]. Applying this to $c = h$ on $(2 j - 1, 2 j)$ and $c = J$ on $(2 j, 2 j + 1)$ yields $H_{N} = (i / 4) γ^{T} A_{N} γ$ .

3.1givenF2F3step 2.1algebra

By [F3], choose a real orthogonal $O$ with $O A_{N} O^{T}$ in the displayed $2 \times 2$ blocks, and put $η = O γ$ . Since $O$ is real and orthogonal, [F2] gives ${η_{a}, η_{b}} = 2 δ_{a b} I$ . Define $b_{r} = (η_{2 r - 1} + i η_{2 r}) / 2$ . Direct expansion using these anticommutators gives ${b_{r}, b_{s}^{*}} = δ_{r s} I$ , ${b_{r}, b_{s}} = 0$ , and $b_{r}^{*} b_{r} = \frac{1}{2} I + \frac{i}{2} η_{2 r - 1} η_{2 r}$ . Using the matrix expression from step 2.1, substitution into the block expression gives $H_{N} = \sum_{r} ϵ_{r} (b_{r}^{*} b_{r} - \frac{1}{2} I)$ .

4.1step 3.1F5algebra

The projections $n_{r} = b_{r}^{*} b_{r}$ commute by the CAR in step 3.1 and are simultaneously diagonalizable: diagonalize $n_{1}$ using [F5], restrict the remaining commuting Hermitian matrices to its invariant eigenspaces, and continue through the finite list. Their joint eigenvalues lie in ${0, 1}^{N}$ . If a joint eigenspace has occupancy $0$ in coordinate $r$ , $b_{r}^{*}$ maps it isometrically into the eigenspace with that bit changed to $1$ , since $∥ b_{r}^{*} ψ ∥^{2} = ⟨ ψ, (I - n_{r}) ψ ⟩ = ∥ ψ ∥^{2}$ ; when the bit is $1$ , $b_{r}$ is the inverse isometry. Hence all $2^{N}$ joint eigenspaces have the same positive dimension. Their orthogonal sum is $H_{N}$ , whose dimension is $2^{N}$ , so each eigenspace is one-dimensional. The resulting ordered occupation basis identifies the $b_{r}$ representation unitarily with the finite fermionic Fock occupation basis.

5.1A1F4F5step 3.1step 4.1algebra∎

In that occupation basis, $\sum_{r} ϵ_{r} b_{r}^{*} b_{r}$ is the diagonal finite-mode Hamiltonian with energies $ϵ_{r}$ . Apply [F4] with mode set ${1, \dots, N}$ , chemical potential $0$ , and $q_{r} = e^{- β ϵ_{r}}$ to obtain trace $\prod_{r} (1 + e^{- β ϵ_{r}})$ . The scalar shift $- \frac{1}{2} \sum_{r} ϵ_{r} I$ multiplies this trace by $e^{β \sum_{r} ϵ_{r} / 2}$ , and [F5] permits the unitary change back to $H_{N}$ . Thus $Z_{N} = \prod_{r} (e^{β ϵ_{r} / 2} + e^{- β ϵ_{r} / 2}) = \prod_{r} 2 \cosh (β ϵ_{r} / 2)$ . If $ϵ_{r} = 0$ , its factor is $2$ .

Exact quadratic reduction and Gibbs trace of the open Ising chain

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources