The system of inverted Y-type four-level atoms interacting with multi-mode light fields is discussed. Through the mechanism of quantum interference, the relative dielectric permittivity and the relative magnetic permeability will changed remarkably. Using appropriately chosen parameters, the real parts of both the relative dielectric permittivity and the relative magnetic permeability will be negative, and then the left handedness effect happens and the left-handed material is realized. The properties of left-handed materials such as the range of frequencies of the negative refractive index can be manipulated via changing certain parameters of the system.