Example. Antiperiodic spatial Dirac modes [fdip-006J]
Example. Antiperiodic spatial Dirac modes [fdip-006J]
Keep the spatial Clifford module \(S=(\mathbb C^2)^{\otimes d}\), with \(\dim _{\mathbb C}S=2^d\), and periods \(N_j\ge 3\). Give scalar fields on the fundamental box \(0\le x_j<N_j\) the antiperiodic extension \[f(x+N_je_j)=-f(x)\qquad (1\le j\le d).\] The forward shift \(T_j^-\) uses this extension at a seam. It is a signed permutation, hence unitary in the counting pairing; shifts in different coordinates commute. They act on the lattice factor only, so they commute with the Clifford matrices on \(S\). Define \[D^- =\sum _{j=1}^d\gamma _j\otimes \frac {T_j^--(T_j^-)^{-1}}2.\] These are fixed flat line twists, each with holonomy minus one.
For labels \(0\le r_j<N_j\), set \[p_j=\frac {(2r_j+1)\pi }{N_j},\qquad \chi _r(x)=\left (\prod _jN_j\right )^{-1/2} e^{i\sum _jp_jx_j}.\] The exponential obeys the declared antiperiodic extension. Differences of two allowed momenta are periodic Fourier momenta, so the same geometric-sum proof as Theorem [fdip-000S] makes these functions a complete orthonormal basis of fields on the box. On its momentum block, \[\begin {aligned} D^-(p)&=i\sum _j\gamma _j\sin p_j,\\ -D^-(p)^2&=\left (\sum _j\sin ^2p_j\right )I_S. \end {aligned}\] The square follows from the Clifford anticommutators and commuting coordinate shifts, exactly as in the untwisted calculation.
A block is zero precisely when every \(\sin p_j=0\); otherwise its inverse is \(-D^-(p)/(\sum _j\sin ^2p_j)\). Since \(0<p_j<2\pi \), the only possible zero is \(p_j=\pi \), which requires \(2r_j+1=N_j\). It occurs once when \(N_j\) is odd and never when it is even. Hence \[\dim _{\mathbb C}\ker D^-= \begin {cases} 2^d,&\text {all periods are odd},\\ 0,&\text {at least one period is even}. \end {cases}\] For two directions the comparison with Proposition [fdip-000T] is \[\begin {array}{c|ccc} (N_1,N_2)&(4,4)&(4,3)&(3,3)\\\hline \dim \ker D&16&8&4\\ \dim \ker D^-&0&0&4 \end {array}\] These are dimensions in the specified reducible module, not physical species or chirality counts.
The twist changes exact zeros without removing the centered symbol's extra low-energy branches. For large \(N_j\), an allowed momentum lies within \(\pi /N_j\) of each corner coordinate \(0\) or \(\pi \), with distance to \(0\) understood modulo \(2\pi \). Thus \(|\sin p_j|\le \pi /N_j\) near every such corner. After restoring a fixed-length mesh spacing \(a_j=L_j/N_j\), the corresponding \(|\sin p_j|/a_j\) is bounded by \(\pi /L_j\); branches near the \(\pi \) corners remain at finite energy alongside the branch near zero. This does not say that the rescaled antiperiodic eigenvalues tend to zero at fixed circle lengths. The additional branches are the centered-difference doubling problem, whose sine symbol is discussed in [wingate2025lattice, Section 9.1, Equations (9.6)–(9.7), p. 20]. A boundary twist alone does not supply its Wilson-operator remedy.