  HfTe3 single crystal undergoes a charge-density-wave (CDW) transition at TCDW
= 93 K without the appearance of superconductivity (SC) down to 50 mK at
ambient pressure. Here, we determined its CDW vector q = 0.91(1) a* + 0.27(1)
c* via low-temperature transimission electron microscope and then performed
comprehensive high-pressure transport measurements along three major
crystallographic axes. Our results indicate that the superconducting pairing
starts to occur within the quasi-one-dimensional (Q1D) -Te2-Te3- chain at 4-5 K
but the phase coherence between the superconducting chains cannot be realized
along either the b- or c-axis down to at least 1.4 K, giving rise to an
extremely anisotropic SC rarely seen in real materials. We have discussed the
prominent Q1D SC in pressurized HfTe3 in terms of the anisotropic Fermi
surfaces arising from the unidirectional Te-5px electronic states and the local
pairs formed along the -Te2-Te3- chains based on the first-principles
electronic structure calculations.
