Hamiltonian renormalization. VIII. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:msub> <mml:mo stretchy="false

In previous works in this series we focused on Hamiltonian renormalization of free field theories in all spacetime dimensions. In this paper, we address the Hamiltonian renormalization of the self-interacting scalar field in two spacetime dimensions with polynomial potential, called <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>P</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi mathvariant="normal">Φ</a:mi> <a:msub> <a:mo stretchy="false">)</a:mo> <a:mn>2</a:mn> </a:msub> </a:math> . We consider only the finite volume case. The <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>P</f:mi> <f:mo stretchy="false">(</f:mo> <f:mi mathvariant="normal">Φ</f:mi> <f:msub> <f:mo stretchy="false">)</f:mo> <f:mn>2</f:mn> </f:msub> </f:math> theory is one of the few interacting quantum field theories that can be rigorously constructed nonperturbatively. We find that our Hamiltonian renormalization flow finds this theory indeed as a fixed point.