  The spin Nernst effect describes a transverse spin current induced by the
longitudinal thermal gradient in a system with the spin-orbit coupling. Here we
study the spin Nernst effect in a mesoscopic four-terminal cross-bar Weyl
semimetal device under a perpendicular magnetic field. By using the
tight-binding Hamiltonian combining with the nonequilibrium Green's function
method, the three elements of the spin current in the transverse leads and then
spin Nernst coefficients are obtained. The results show that the spin Nernst
effect in the Weyl semimetal has the essential difference with the traditional
one: The z direction spin currents is zero without the magnetic field while it
appears under the magnetic field, and the x and y direction spin currents in
the two transverse leads flows out or flows in together, in contrary to the
traditional spin Nernst effect, in which the spin current is induced by the
spin-orbit coupling and flows out from one lead and flows in on the other. So
we call it the anomalous spin Nernst effect. In addition, we show that the Weyl
semimetals have the center-reversal-type symmetry, the mirror-reversal-type
symmetry and the electron-hole-type symmetry, which lead to the spin Nernst
coefficients being odd function or even function of the Fermi energy, the
magnetic field and the transverse terminals. Moreover, the spin Nernst effect
in the Weyl semimetals are strongly anisotropic and its coefficients are
strongly dependent on both the direction of thermal gradient and the direction
of the transverse lead connection. Three non-equivalent connection modes (x-z,
z-x and x-y modes) are studied in detail, and the spin Nernst coefficients for
three different modes exhibit very different behaviors. These strongly
anisotropic behaviors of the spin Nernst effect can be used as the
characterization of magnetic Weyl semimetals.
