Highly Entangled 2D Ground States: Tensor Network, Order Parameter and Correlation

TL;DR

This work constructs exact 3D tensor-network representations for the first known 2D ground states that transition from area-law to extensive entanglement entropy as a deformation parameter $q$ is varied. By mapping 2D spin configurations to tilings in 3D—via colored 6-vertex and polychromatic lozenge tilings—the authors encode local vertex constraints and Dyck-walk correlations into two bulk tensor families, $H(q)/V(q)$ for the 6-vertex case and prism-based tensors for lozenge tilings. The resulting TNs provide explicit weights $q^{V}$ and 1-to-1 correspondences with GS configurations, revealing how entanglement and correlation functions arise from height-function statistics of random surfaces and their hard-wall constraints. They also derive spin and color correlation behaviors across the phase diagram, including boundary-driven order parameters and scaling laws, illustrating how holographic TNs capture nontrivial entangled phases in higher dimensions and offering a path to generalizations to even higher dimensions.

Abstract

In this article we present analytical results on the exact tensor network representations and correlation functions of the first examples of 2D ground states with quantum phase transitions between area law and extensive entanglement entropy. The tensor networks constructed are one dimension higher than the lattices of the physical systems, allowing entangled physical degrees of freedoms to be paired with one another arbitrarily far away. Contraction rules of the internal legs are specified by a simple translationally invariant set of rules in terms of the tesselation of cubes or prisms in 3D space. The networks directly generalize the previous holographic tensor networks for 1D Fredkin and Motzkin chains. We also analyze the correlation in the spin and color sectors from the scaling of the height function of random surfaces, revealing additional characterizations of the exotic phase transitions.

Figures (29)

• Figure 1: (a) Maximal volume spin configuration present in \ref{['eq: groundStateOfSixVertex']} for the $L = 4$ system. The numbers indicate the height $\phi$ on the dual lattice. The black and gray spins are spins outside our system implicitly fixed by the boundary conditions. The Dyck walks in $y = 1$ and $x = 3$ horizontal and vertical spin chains are shown below and to the right. (b) The 6 different vertex configurations compatible with the ice-rule. Horizontal $S^\mathrm{h}$ and vertical $S^\mathrm{v}$ spins are indicated along with the height function $\phi$ defined at the dual lattice points, marked with the $\sbullet[0.6]$'s.
• Figure 2: (a) Upper panel shows coordinate system and height convention. Lower panel shows the three different lozenges, where the numbers indicate how the height change around a lozenge according to the height convention. (b)-(c) Two color decorated lozenge tilings for a given boundary. The minimal volume tiling is seen in (b), while the maximal volume tiling is seen in (c). The three-dimensional effect is understood by noting that the light is shining onto the lozenge tilings anti-parallel with the $y$-axis, as defined in (a). In both lozenge tilings we have color correlated Dyck walks, in the $xy$-, $xz$- and $yz$-plane.
• Figure 3: (a) The five different tiles $A_{i}(\bm{c})$ for the single Fredkin chain case. The tiles are also defined as rank-4 tensors in terms of Kronecker deltas, where the indices $\bm{k}_{i}$ are defined as for $A(q)$ 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 GS of the single Fredkin chain for $L = 4$. The constituent tensors are defined in the upper panel. The variable $l$ denotes the different levels of the holographic TN.
• Figure 4: (a) Top view of towers of cubical tiles associated to each spin, for a $L = 4$ system. (b) The two sublattices corresponding to horizontal tiles $H$ and vertical tiles $V$. The original lattice vertices are located at $+$'s and height functions are defined on the dual lattice at $\bullet$'s.
• Figure 5: The set of cubical tiles corresponding to nonzero entries of a tensor in the network. The dashed (resp. solid) arrows are due to horizontal (resp. vertical) physical spins on the 2D lattice. Arrows pointing between top and bottom surfaces of a cube are doubled, and share the same color, as they come from the same spin. The labels $\bm{c}_{i}$ denote the colors of the arrows.