Scalar Boundary Conditions in Hyperscaling Violating Geometry

We study the possible boundary conditions of scalar field modes in a hyperscaling violation(HV) geometry with Lifshitz dynamical exponent $z (z\geqslant1)$ and hyperscaling violation exponent $θ(θ\neq0)$. For the case with $θ>0$, we show that in the parameter range with $1\leq z\leq 2,~-z+d-1<θ\leq (d-1)(z-1)$ or $z>2,~-z+d-1<θ\leq d-1$, the boundary conditions have different types, including the Neumann, Dirichlet and Robin conditions, while in the range with $θ\leq-z+d-1$, only Dirichlet type condition can be set. In particular, we further confirm that the mass of the scalar field does not play any role in determining the possible boundary conditions for $θ>0$, which has been addressed in Ref. \cite{1201.1905}. Meanwhile, we also do the parallel investigation in the case with $θ<0$. We find that for $m^2<0$, three types of boundary conditions are available, but for $m^2>0$, only one type is available.

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