Transverse Ising Majorana reduction and bulk gap closing — Examples

At $h=0$, an $X$-basis configuration $s=(s_1,\dots ,s_N)\in \{-1,1{\}}^N$ has energy $-J{\sum }_{j=1}^{N-1}{s}_j{s}_{j+1}$. Each bond contributes at least $-J$, with equality exactly when $s_j=s_{j+1}$. Since the open chain is connected, all bonds attain equality exactly for the two constant configurations; for $N=1$ the empty bond sum leaves its two basis configurations degenerate as well.

At $J=0$, a $Z$-basis configuration $s\in \{-1,1{\}}^N$ has energy $-h{\sum }_j{s}_j$. If $h>0$, each site uniquely minimizes its term by taking $s_j=1$; if $h<0$, each site uniquely minimizes its term by taking $s_j=-1$. Thus the unique minimizing $Z$-configuration has the orientation stated in the Example.

For $\beta >0$, [F2] gives the stated product trace for the exact $H_N$ of [F1] and its open-chain energies. The trace itself is the ordinary trace of the finite matrix exponential on the $2^N$-dimensional space.

For an endomorphism $C$ of ${\mathcal{H}}_N$, use the operator norm $\mathrm{\parallel }C{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}:=\sup \{\mathrm{\parallel }C\psi {\mathrm{\parallel }}_{{\mathcal{H}}_N}:\mathrm{\parallel }\psi {\mathrm{\parallel }}_{{\mathcal{H}}_N}=1\}$ from the finite product inner product. The entries of $-\beta H_N(h,J)$ are polynomials in $\beta ,h,J$. On each compact set of complex parameter triples this matrix has a finite uniform operator-norm bound $C$. Define its complex-parameter exponential by $\exp (-\beta H_N)={\sum }_{m=0}^{\mathrm{\infty }}(-\beta H_N{)}^m\mathrm{/}m!$; the summands have norm at most $C^m\mathrm{/}m!$, so the series converges uniformly on each such compact set. The matrix exponential power series is holomorphic in its entries (that is, locally represented by convergent complex power series), so composing it with the polynomial matrix entries makes each matrix entry, and its finite trace, holomorphic in $(\beta ,h,J)$. Their restrictions to ${\mathbb{R}}^3$ are real-analytic in the sense defined in the Statement. For real parameters, [F3]'s spectral exponential agrees with this series by applying the scalar exponential series to each real eigenvalue ${\lambda }_r$ of the Hermitian $H_N$; hence $Z_N={\sum }_r{e}^{-\beta {\lambda }_r}>0$. This proves analyticity for each fixed finite $N$ only; it gives no conclusion about the existence or analyticity of a limit as $N\to \mathrm{\infty }$.