Scalar Boundary Conditions in Hyperscaling Violating Geometry

TL;DR

This paper investigates the boundary conditions of scalar fields in hyperscaling violating geometries with various parameters, revealing how these conditions depend on the hyperscaling violation exponent, scalar mass, and other parameters.

Contribution

It systematically classifies possible boundary conditions for scalar fields in HV geometries across different parameter ranges, clarifying the role of scalar mass and hyperscaling violation.

Findings

Multiple boundary condition types (Neumann, Dirichlet, Robin) are possible depending on parameters.

Scalar mass does not influence boundary conditions for >0.

For <0, scalar mass sign determines available boundary conditions.

Abstract

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 $\theta (\theta\neq0)$. For the case with $\theta>0$, we show that in the parameter range with $1\leq z\leq 2,~-z+d-1<\theta\leq (d-1)(z-1)$ or $z>2,~-z+d-1<\theta\leq d-1$, the boundary conditions have different types, including the Neumann, Dirichlet and Robin conditions, while in the range with $\theta\leq-z+d-1$, 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 $\theta>0$, which has been addressed in Ref. \cite{1201.1905}. Meanwhile, we also do the parallel investigation in the case with $\theta<0$. We find that for $m^2<0$, three types of boundary conditions are available, but for $m^2>0$, only one type is available.