Entanglement evolution from entangled multipodal states

TL;DR

This work introduces entangled multipodal states as a natural generalization of crosscap states for periodic fermionic chains. Each state |B_N⟩ couples sites at the vertices of an N-sided polygon, producing a two-regime equilibrium entanglement profile with a plateau at $ ext{L}/N$ and an accompanying momentum-space structure that becomes smooth as $N\to\mathcal{L}$. When quenched to a free-fermion Hamiltonian, the dynamics are exactly captured by a quasiparticle multiplet picture, where entanglement evolves via counting functions at multiple length scales and eigenvalues of the correlation matrix take values 0 or ±1/2 for maximally entangled subsystems. A key finding is the emergence of a negative tripartite mutual information, signaling entangled multiplets of quasiparticles and richer correlations than in the crosscap case. Overall, the paper demonstrates tractable solvable dynamics for highly entangled initial states and outlines avenues to probe entanglement structure beyond simple pairs in more complex quantum many-body settings.

Abstract

In a periodic lattice system an entangled antipodal pair state, otherwise known as a crosscap state, is a simple two site product state in which spins at antipodal sites are prepared in Bell pairs. Such states have maximal bipartite entanglement and serve as a useful platform for studying the quench dynamics of systems which have large initial entanglement. In this paper, we study a generalization of these states which we dub entangled mutipodal states. These states, which are defined for fermionic systems, generalize the crosscap states by having correlations among more than two sites, specifically, those which sit at the vertices of regular polygons. By construction, the states are Gaussian and translationally invariant allowing many of their properties to be understood. We study the bipartite entanglement entropy of these states both in and out of equilibrium. In equilibrium, the entanglement profile as a function of subsystem size exhibits two distinct regimes, a volume-law growth followed by a saturation to a constant value, thus generalizing the Page-curve profile of the crosscap state. In the non-equilibrium setting, we study quenches from these initial states to the free-fermion chain, whose ensuing dynamics displays a far richer structure compared to the crosscap case. We interpret our results in terms of the quasiparticle picture, which requires multiplets of quasiparticles to be excited non-locally around the system. This scenario is confirmed by the appearance of a post-quench, negative tripartite information.

Figures (14)

• Figure 1: A depiction of the antipodal entangled or crosscap state. Antipodal sites, which are a distance $L$ apart are prepared in Bell pairs. Also depicted are the subsystem $A$ which is of length $\ell$ and its mirror $B$ which is diametrically opposite $A$. The structure of correlations generates maximal bipartite entanglement in the system as shown on the right hand side where we plot $S[\ket{\mathcal{B}_2}]$ as a function of $\ell$.
• Figure 2: A depiction of the multipodal entangled states. On the left we have the triangle state, $N=3$ and on right we show the $N=4,5$ states. The multipodal entangled states exhibit correlations between sites which sit a distance $L$ apart, indicated by the dotted lines. On the left we mark the subsystems, $A,B,C$ which are all of equal length $\ell$.
• Figure 3: Entanglement entropy as a function of subsystem size for mutlipodal entangled states with $N=2,3,4,5,6,7,8$. We notice that for each case the volume law saturates to a plateau at $\ell_A=L=\mathcal{L}/N$.
• Figure 4: On the left we show the entanglement dynamics from the crosscap state. The solid line is the prediction of the quasiparticle picture while the symbols are exact numerics. On the right we plot the long time average \ref{['eq:crosscap_longtime']}. The dashed line is the initial value for comparison.
• Figure 5: A depiction of the quasiparticle picture for the fermionic crosscap state quench. Entangled counter propagating quasiparticles are created in pairs located at antipodes. The first change in the entanglement occurs when the pair (indicated by the arrows) on the left enters the subsystem. Subsequent drops in entropy occur when the pair emitted from the same point but travelling in the opposite direction, as shown on the right, enter the subsystem.