  Mutually coupled spin Hall nano-oscillators (SHNO) can exhibit binarized
phase state, offering pathways to realize Ising machines and efficient
neuromorphic hardware. Conventionally, phase binarization is achieved in
coupled SHNOs via injecting an external microwave at twice of the oscillator
frequency in presence of a biasing magnetic field. However, this technology
poses potential challenges of higher energy consumption and complex circuit
design. Moreover, fabrication-induced mismatch in SHNO dimensions may hinder
mutual synchronization. Addressing these challenges, we demonstrate purely DC
current-driven mutual synchronization and phase binarization of two
non-identical nanoconstriction SHNOs without biasing magnetic field and
microwave injection. We thoroughly investigate these phenomena and underlying
mechanisms using micromagnetic simulation. We further demonstrate the bias
field-free synchronized SHNO pair efficiently performing a reservoir computing
benchmark learning task: sin and square wave classification, utilizing current
tunable phase binarization. Our results showcase promising magnetization
dynamics of coupled bias field-free SHNOs for future computing applications.
