Phase Diagram for a Hiden Competing Order in the Mixed State of YBa$_2$Cu$_4$O$_8$ CRYSTALS UP TO 150 kG

Torque measurements on YBa$_2$Cu$_4$O$_8$ crystals show multiple peaks as a function of $\theta_{ca}$ between the c axis and the magnetic field $H$ in the c-a plane. We propose that the first peaks at $\theta_{ca} \simeq 80$ and 100 degrees arise from a possible long-range spiral spin density wave through the matching pinning between vortices and spins. We construct a contour map for a competing order parameter in YBa$_2$Cu$_4$O$_8$ which starts at a finite $H$ and is located far from Hc2 line. The regime of the proposed spin density wave goes to higher temperatures when field increases. The first peak does not appear when the field direction is scanned in the c-b plane. The NMR line width of the c-axis aligned YBa$_2$Cu$_4 $O$_8$ powders in H perpendicular c shows broadening at temperatures below 30 K. These features might be intrinsic in the cleanest underdoped cuprate.