  There is controversy whether dusty starbursts selected at submillimeter
wavelengths can trace galaxy overdensities. We perform the first systematic
search for protoclusters around a homogeneously selected sample of 12
spectroscopically confirmed submillimeter galaxies (SMGs) at $z\sim1.2-5.3$ in
the GOODS-N field. We applied the Poisson Probability Method (PPM) to search
for Mpc scale overdensities around these SMGs using three photometric redshift
catalogs. We detect galaxy overdensities for 11 out of the 12 SMGs
($92\%\pm8$\%), distributed over eight protoclusters. We confirm three
previously discovered protoclusters, and we detect five new ones around the
SMGs SMMJ123634 ($z=1.225$), ID.19 ($z=2.047$), SMMJ123607 ($z=2.487$),
SMMJ123606 ($z=2.505$), and GN10 ($z=5.303$). A wavelet-based analysis shows
that the SMGs live in protocluster cores with a complex morphology (compact,
filamentary, or clumpy) and an average size of $\sim(0.4-1)$Mpc. By comparing
the PPM results obtained using independently the three redshift catalogs, we
possibly witness a transitioning phase at $z\gtrsim4$ for the galaxy
populations. While $z\lesssim4$ protoclusters appear to be populated by dusty
galaxies, those at highest redshifts $z\sim5$ are detected as overdensities of
Lyman$\alpha$ emitters or Lyman break galaxies. We also find a good correlation
between the molecular (H$_2$) gas mass of the SMG and the overdensity
significance. To explain the overall phenomenology, we suggest that galaxy
interactions in dense environments likely triggered the starburst and gas-rich
phase of the SMGs. Altogether, we support the scenario that SMGs are excellent
tracers of distant protoclusters. Those presented in this work are excellent
targets for the {\it James Webb Space Telescope.} Surveys with forthcoming
facilities (e.g., {\it Euclid}, LSST) can be tuned to detect even larger
samples of distant protoclusters.
