Entanglement in Interacting Majorana Chains and Transitions of von Neumann Algebras

We consider Majorana lattices with two-site interactions consisting of a general function of the fermion bilinear. The models are exactly solvable in the limit of a large number of on-site fermions. The four-site chain exhibits a quantum phase transition controlled by the hopping parameters and manifests itself in a discontinuous entanglement entropy, obtained by constraining the one-sided modular Hamiltonian. Inspired by recent work within the AdS/CFT correspondence, we identify transitions between types of von Neumann operator algebras throughout the phase diagram. We find transitions of the form ${\mathrm{II}}_1\leftrightarrow \mathrm{III}\leftrightarrow {\mathrm{I}}_{\infty }$ that reduce to ${\mathrm{II}}_1\leftrightarrow {\mathrm{I}}_{\infty }$ in the strongly interacting limit, where they connect nonfactorized and factorized ground states. Our results provide novel realizations of such transitions in a controlled many-body model.

Figure 1

Free energy of the four-site chain with potential $h_{x,x+1}(\xi )={\mu }_x(1-e^{J\xi })/(2J)$ for different interaction strengths $J$. We observe a nonanalyticity at $r\equiv {\mu }_a/{\mu }_b=1/2$ for $J$ above the critical value $J_c=\sqrt{2}$, signaling a phase transition. The two phases of the system are characterized by the correlation structure given by (10), whose limiting cases for $r\to 0$ and $r\to \infty$ are shown in the two embedded diagrams.

Figure 2

Entanglement entropy (9) of subregion $A$ on solutions to (11) as a function of $r\equiv {\mu }_a/{\mu }_b$ for different couplings $J$. Solid lines denote all SC solutions, while a sample of physical solutions minimizing $F$ in Fig. 1 is marked with dots. The phase transition is signaled by the discontinuity for $J>J_c$. Inset: von Neumann type of the algebra ${\mathcal{A}}_A$ in different regimes of the phase diagram. Each type is denoted by a different color, and the black dot (line) represents a phase transition of second (first) order.