  We solve for the elementary excitation in infinite quasi-1D quantum lattices
by extending the recently developed infinite quasi-1D entanglement perturbation
theory. The wave function of an excited state is variationally determined by
optimizing superposition of cluster operation, each of which is composed of
simultaneous on-site operation inside a block of lattice sites, on the ground
state in a form of plane wave. The excitation energy with respect to the wave
number gives the spectra for an elementary excitation. Our method is artificial
broadening free and is adaptive for various quasi-particle pictures. Using the
triplet spectrum, the application to $\infty$-by-$N$ antiferromagnetic
spin-$\frac{1}{2}$ ladders for $N=2, 4, 6, 8$, and $10$ confirms a previous
report that there is a quantum dimensional transition, namely, the lattice
transits from quasi-1D to 2D at a finite critical value $N_c=10$. The massless
triplet dispersion at $\left( \pi, \pi \right)$ sees a vanishing gap. Our
results detect the anomaly at $\left(\pi,0\right)$ in the triplet spectrum,
agreeing well with the inelastic neutron scattering measurement of a
macroscopic sample. Surprisingly, our results also reveal a gapless and massive
1D singlet dispersion channel that is much lower than the triplet excitation.
We note, however, the dimensional transition is determined by the massless
triplet dispersion.
