  We study a ground-state ansatz for the single-hole doped $t$-$J$ model in two
dimensions via a variational Monte Carlo (VMC) method. Such a single-hole wave
function possesses finite angular momenta generated by hidden spin currents,
which give rise to a novel ground state degeneracy in agreement with recent
exact diagonalization (ED) and density matrix renormalization group (DMGR)
results. We further show that the wave function can be decomposed into a
quasiparticle component and an incoherent momentum distribution in excellent
agreement with the DMRG results up to an $8\times 8 $ lattice. Such a
two-component structure indicates the breakdown of Landau's one-to-one
correspondence principle, and in particular, the quasiparticle spectral weight
vanishes by a power law in the large sample-size limit. By contrast, turning
off the phase string induced by the hole hopping in the so-called $\sigma\cdot
t\text{-}J$ model, a conventional Bloch-wave wave function with a finite
quasiparticle spectral weight can be recovered, also in agreement with the ED
and DMRG results. The present study shows that a singular effect already takes
place in the single-hole-doped Mott insulator, by which the bare hole is turned
into a non-Landau quasiparticle with translational symmetry breaking.
Generalizations to pairing and finite doping are briefly discussed.
