Entanglement in interacting Majorana chains and transitions of von Neumann algebras

TL;DR

This work presents a controlled, large-$N$ solvable Majorana lattice model with general two-site interactions, enabling exact two-point function solutions and a detailed study of entanglement through a constrained one-sided modular Hamiltonian. It reveals a quantum phase transition in a four-site chain controlled by hopping parameters, accompanied by a discontinuity in the entanglement entropy and a sequence of transitions between von Neumann algebra types (II$_1$, III, I$_\infty$) across the phase diagram. In the strong-coupling limit with exponential potentials, the II$_1$ to I$_\infty$ transition aligns with a ground-state factorization, mirroring holographic Hawking–Page-type behavior in a many-body setting. The authors extend the analysis to periodic chains and arbitrary potentials, illustrating the generality of algebraic transitions and their connection to entanglement structure in interacting quantum matter.

Abstract

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 II$_1\leftrightarrow\,$III$\,\,\leftrightarrow\,\,$I$_\infty$ that reduce to II$_1\leftrightarrow\,\,$I$_\infty$ in the strongly interacting limit, where they connect non-factorized and factorized ground states. Our results provide novel realizations of such transitions in a controlled many-body model.

Figures (6)

• Figure 1: Free energy of the four-site chain with potential $h_{x,x+1}(\xi)=\mu_x \left(1-e^{J\xi}\right)/(2J)$ for different interaction strengths $J$. We observe a non-analyticity 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 \ref{['eq:G_aba']}, whose limiting cases for $r\to 0$ and $r\to \infty$ are shown in the two embedded diagrams.
• Figure 2: Entanglement entropy \ref{['eq:entropy_from_DM']} of subregion $A$ on solutions to \ref{['eq:self_cons_eq_aba']} as function of $r \equiv \mu_a/\mu_b$ for different coupling $J$. Solid lines denote all SC solutions, while a sample of physical solutions minimizing $F$ in Fig. \ref{['fig:free_energy_aba']} 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.
• Figure 3: Interaction vertices for terms in \ref{['SMeq:hhhh']} for orders $p=1,2,3$. Each dot indicates the non-local bilinear operator $\psi_x^j\psi_y^j$ and the dotted lines indicate the synchronicity of the interacting operators
• Figure 4: 1PI diagram constructed from the diagrams in Fig. \ref{['SMfig:feynman1']} through their combination and subsequent amputation of external legs.
• Figure 5: Solutions to the self-consistency equation \ref{['eq:Self_Cons_Eq']} for an exponential potential with different interaction strengths, as function of the hopping ratio $r \equiv \mu_a/\mu_b$. Above the critical value $J=J_c=\sqrt{2}$, the equation starts having three possible solutions. Solid lines denote all SC solutions, while the dots represent a subset of all the thermodynamically dominant solutions, which are the ones minimizing the free energy \ref{['SMeq:free_energy_aba']} (derived in the next section) and shown in Fig. 1 of the main text. For $J>J_c$ the SC solutions which are thermodynamically dominant exhibit a discontinuous jump at $r=1/2$, indicating that the systems undergoes a phase transition. The two phases are characterized by their correlation structure, as shown schematically in the insets of Fig. 1 of the main text.