Casimir Effect for Quantum Field theory in Networks

TL;DR

The article develops a quantum field theory on networks by deriving a node junction condition that enforces current conservation, enabling controlled Casimir effects on network edges. Using a massless scalar in 1+1D and Green-function methods, it shows that Dirichlet boundaries yield always-attractive forces, while Neumann boundaries allow the Casimir force on an edge to switch sign by tuning other edge lengths, with a clear conformal relation $W=\sum_i F^i L_i$. The work extends to general networks and higher dimensions, providing explicit spectra and energy formulas for tree and loop networks, and general expressions for $W$ and $P^i$ in arbitrary $d$, always preserving the conformal relation. These results offer a practical route to modulate Casimir forces in networked nanostructures and broaden the understanding of boundary-geometry effects in quantum field theories on graphs.

Abstract

This paper studies quantum field theories defined in networks, which are the multi-branch generalizations of interface conformal field theory (ICFT). We propose a novel junction condition on the node and show that it is consistent with energy conservation in the sense that the total energy flow into the node is zero. As an application, we explore the Casimir effect on networks. Remarkably, the Casimir force on one edge can be changed from attractive to repulsive by adjusting the lengths of the other edges, providing a straightforward way to control the Casimir effect. We begin by discussing the Casimir effect for $(1+1)$-dimensional free massless scalars on a simple network. We then extend this discussion to various types of networks and higher dimensions. Finally, we offer brief comments on some open questions.

Figures (8)

• Figure 1: The Casimir force at the left edge can switch from attractive to repulsive by adjusting the lengths of the right edges.
• Figure 2: A network with $n$ edges $E_i$ linked by one node $N$ (redpoint).
• Figure 3: Simplest network: three edges linked by one node. We impose the junction condition (\ref{['sect2: junction condition']}) on the node (red point), while Dirichlet and Neumann boundary conditions on the blue and yellow endpoints of the edges in the left and right figures, respectively. Here $x_i$ is the space coordinate on edge $E_i$, which vanishes on the node (red point) $N: x_i=0$. The length of edge $E_i$ is $L_i$.
• Figure 4: Casimir force on edge $E_1$ varies with the lengths of other edges. The left figure is for $L_1=1, L_2=L_3$, and the right figure is for $L_1=L_2=1$. For DBC (blue curve), the amplitude of force $F_1$ decreases with $L_3$ but remains the same signs. While for NBC (orange curve), the force $F_1$ flips signs as $L_3$ increases. For both DBC and NBC, $F_1$ approaches to a constant value in the large $L_3$ limit.
• Figure 5: Tree (left) and loop (right) networks with two nodes. Note that the arrows label the directions of the coordinates $0\le x_i\le L_i$. It does not mean it is a directed network. We impose the junction condition (\ref{['sect2: junction condition']}) on the nodes (red points), while either DBC (\ref{['sect3: DBC']}) or NBC (\ref{['sect3: NBC']}) on the outside edge endpoints (black points).