  Reconfigurable intelligent surface (RIS) has gained much traction due to its
potential to manipulate the propagation environment via nearly-passive
reconfigurable elements. In our previous work, we have analyzed and proposed a
beyond diagonal RIS (BD-RIS) model, which is not limited to traditional
diagonal phase shift matrices, to unify different RIS modes/architectures. In
this paper, we create a new branch of BD-RIS supporting a multi-sector mode. A
multi-sector BD-RIS is modeled as multiple antennas connected to a multi-port
group-connected reconfigurable impedance network. More specifically, antennas
are divided into $L$ ($L \ge 2$) sectors and arranged as a polygon prism with
each sector covering $1/L$ space. Different from the recently introduced
concept of intelligent omni-surface (or simultaneously transmitting and
reflecting RIS), the multi-sector BD-RIS not only achieves a full-space
coverage, but also has significant performance gains thanks to the highly
directional beam of each sector.We derive the constraint of the multi-sector
BD-RIS and the corresponding channel model taking into account the relationship
between antenna beamwidth and gain. With the proposed model, we first derive
the scaling law of the received signal power for a multi-sector BD-RIS-assisted
single-user system. We then propose efficient beamforming design algorithms to
maximize the sum-rate of the multi-sector BD-RIS-assisted multiuser system.
Simulation results verify the effectiveness of the proposed design and
demonstrate the performance enhancement of the proposed multi-sector BD-RIS.
