The Motzkin Spaghetto

TL;DR

This work addresses realizing highly entangled ground states in two dimensions with strictly local, frustration-free Hamiltonians. It constructs a square-lattice model by twisting a Motzkin chain into the plane, yielding a ground state that exhibits three entanglement regimes controlled by a deformation parameter $q$: a highly entangled phase with $S \sim L^2$, a critical regime with $S \sim L$, and a low-entanglement phase with $S \sim O(1)$. It develops both a tensor-network description with lower-rank tensors and a tiling-based TN for the 2D GS, and analyzes the induced antiferromagnetic order in the radial direction, with a correlation length $\xi \propto 1/\log q$. It also shows that spectral-gap scaling can differ from entanglement scaling and provides upper bounds that map from the 1D Motzkin chain by replacing the 1D length with $N=L(L+2)$. The results offer a simpler route to volume-law entanglement in 2D and highlight the nuanced relationship between EE and spectral gaps, with implications for higher-dimensional generalizations and holographic tensor networks.

Abstract

While highly entangled ground states of gapless local Hamiltonians have been known to exist in one dimension, their two-dimensional counterparts were only recently found, with rather sophisticated interactions involving at least four neighboring degrees of freedom. Here, we show that similar bipartite entanglement properties can be realized on a square lattice with anisotropic interactions in four different quadrants. The interaction to generate such entanglement is much simpler than the previous constructions by coupling orthogonal arrays of highly entangled chains. The new construction exhibits an entanglement phase transition with different scalings of entanglement entropy at the critical point and in the lowly entangled phase, and faster decay of the spectral gap in the highly entangled phase. The tensor network representation of the new ground state consists of tensors with lower rank, while preserving a global geometry similar to that of the original networks.

Figures (6)

• Figure 1: (a) A square lattice with $L(L+1)$ spins on the vertices. (b) An $L\times L$ square lattice with spins on the edges of the square lattice that are covered by a Motzkin chain of length $2N=L(L+2)$, both shown here for $L=8$. Four symmetric bipartition cuts are shown in different colors. (c) The two subsystems resulting from the yellow cut in (b). The northeast subsystem is represented with the thickened black lines, whereas the southwest is marked by thin gray lines. This is asymptotically equivalent to the yellow bipartition in (a), separating the upper half plane from the lower. The configuration depicted corresponds to the probabilistically dominant Motzkin path in the GS superposition for the $q>1$ phase, to illustrate the entanglement across subsystems. Typical height configuration at the critical point is depicted in blue. (d) The uncoupled arrays of orthogonal chains with different EE and gap scaling for comparison in \ref{['tab:table1']} and \ref{['tab:table2']}.
• Figure 2: Scaling of the bipartite EE $S$ computed from \ref{['eq:S_scaling_eq_MPS']} and the MPS representation of the GS for various linear system sizes $L$ and for the different colored cuts in \ref{['fig:snake']} (a). In (a) $q = 0.99$, in (b) $q = 1$ and in (c) $q = 1.01$.
• Figure 3: Order parameter $\langle S^z\rangle$ for (a) $q > 1$, and (b) $q=1$. Warm color and cold colors denote respective spin up and down, and darker colors correspond to larger magnitude. The neutral muted shade represents 2D lattice sites not occupied by degrees of freedom of the 1D chain.
• Figure 4: (a) TN representation of the 2D GS of the twisted Motzkin chain, where the contractions in the vertical direction are represented by placing the tensor cubes adjacent to each other, and in the horizontal planes, contraction only happens between neighbors along the twisted chain direction. (b) TN representation of the 2D GS of the coupled chain model, where contractions in each horizontal plane happen between neighboring tensor cubes in both directions.
• Figure 5: (a) The five different tiles $B_{i}(\bm{c})$ for the new Fredkin tensor network. The tiles are also defined as rank-4 tensors in terms of Kronecker deltas, where the indices $\bm{k}_{i}$ are defined as for $B(q)$ in panel (d). We have $\bm{c} = (1, 0)$ for red arrow, $\bm{c} = (0, 1)$ for blue arrow. No arrow corresponds to $\bm{0}$. (b)-(c) Valid tilings corresponding to the maximal and minimal height Dyck walk for the $L = 4$ system. (d) TN representation of the ground state of the single Fredkin chain for $L = 4$. $l$ denotes the different levels of the holographic TN. (e) The constituent tensors of the TN, where the white circle projects out the index value $\bm{0}$ corresponding to no arrow in the tilings.