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

This paper presents analytical results on exact tensor network representations and correlation functions for the first examples of 2D ground states exhibiting quantum phase transitions between area law and extensive entanglement entropy, utilizing 3D tessellation-based networks that generalize 1D holographic models to reveal exotic phase characteristics through random surface scaling.

Original authors: Olai B. Mykland, Zhao Zhang

Published 2026-08-25

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Olai B. Mykland, Zhao Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, particles are often linked in ways that defy our everyday experience, a phenomenon known as entanglement. When a group of particles is entangled, the state of one cannot be described without the others, no matter how far apart they are. For decades, physicists have understood that in most stable, low-energy states of matter, this entanglement is surprisingly limited. It tends to scale with the surface area of a region rather than its volume, a rule known as the "area law." This makes sense intuitively: particles usually only interact with their immediate neighbors, so the connection between two distant parts of a system is weak. However, there are special, exotic states of matter where this rule breaks down. In these rare cases, the entanglement grows with the volume of the system, meaning that particles are deeply connected across vast distances. These states are not just theoretical curiosities; they represent a frontier in our understanding of quantum matter, offering a glimpse into how complex systems can organize themselves in ways that challenge our classical intuition.

The challenge for scientists has been to find concrete examples of these highly entangled states in two-dimensional systems, which are more complex than the one-dimensional chains often studied in textbooks. While one-dimensional models have been solved, extending these solutions to two dimensions has proven difficult because the particles are entangled in multiple directions simultaneously. In a new study, researchers have successfully constructed the first exact mathematical descriptions, known as tensor networks, for the ground states of two specific two-dimensional quantum models. These models feature a dramatic shift in behavior: by adjusting a single parameter, the system can switch from a state where entanglement is limited to the surface area, to one where it fills the entire volume. The researchers did not just simulate these states; they built a precise, analytical framework that maps the quantum particles to a three-dimensional structure of geometric tiles, revealing exactly how the entanglement is organized.

The team focused on two distinct models. The first involves a grid of particles arranged in a square pattern, where the particles interact according to rules similar to those governing the flow of ice crystals. The second model uses a triangular grid, where the particles are arranged like tiles that fit together in three different orientations. In both cases, the researchers introduced a "deformation parameter," a dial that controls the weight given to different configurations of the system. When this parameter is small, the system behaves normally, with entanglement confined to the boundaries. But when the parameter is large, the system enters a new phase where the entanglement becomes extensive, spreading throughout the entire volume. The breakthrough was finding a way to represent these complex, two-dimensional quantum states using a three-dimensional network of connections.

To achieve this, the researchers imagined the quantum system not as a flat sheet, but as the surface of a three-dimensional structure. They constructed a network where each physical particle on the surface corresponds to a column of geometric blocks rising into a third dimension. For the square grid model, these blocks are cubes; for the triangular grid, they are prisms. The rules for how these blocks stack and connect are strict and simple, ensuring that every valid arrangement of blocks corresponds to exactly one possible state of the quantum system. This method allows the researchers to see how information flows through the system. In the highly entangled phase, the connections between particles are not just local; they can reach across the entire system, linking particles that are far apart. The three-dimensional nature of the network is crucial here, as it provides the extra room needed to connect these distant pairs without breaking the rules of the quantum system.

The study also explored how the system behaves at the transition point between the two phases. By analyzing the correlations between particles, the researchers found that the nature of the connection changes dramatically. In the low-entanglement phase, the influence of one particle on another fades away quickly, much like a whisper that dies out after a few feet. In the high-entanglement phase, however, the connection persists over long distances, and the system exhibits a unique type of order that is not seen in ordinary materials. The researchers showed that this long-range connection is driven by the way the particles are colored and paired. In these models, particles come in different "colors," and the rules of the system require that particles of the same color must pair up in a specific way. When the system is in the highly entangled phase, these color pairings create a complex web of connections that spans the entire system, leading to the extensive entanglement.

One of the most significant findings is that the researchers could prove these results mathematically, rather than relying on computer simulations. They demonstrated that their three-dimensional tile construction is a perfect match for the quantum states, with no gaps or errors. This exact solution provides a rare window into the mechanics of highly entangled matter. The researchers also calculated how the system responds to changes in size and shape, confirming that the entanglement scales with the volume of the system in the new phase. This is a stark contrast to the area law, where entanglement would only scale with the surface. The ability to describe these states with such precision opens the door to understanding other complex quantum systems that were previously too difficult to analyze.

The work also sheds light on the nature of phase transitions in quantum systems. The transition from the low-entanglement state to the high-entanglement state is not a smooth change but a sharp shift, similar to how water suddenly turns to ice. The researchers showed that this transition is characterized by a sudden change in how the system's correlations decay. In the low-entanglement phase, correlations drop off exponentially, meaning they become negligible very quickly. In the high-entanglement phase, the decay is much slower, following a power law that allows connections to persist over long distances. This behavior is a hallmark of critical systems, where the rules of the system change fundamentally. The researchers' ability to map these transitions using their three-dimensional tile models provides a new tool for studying critical phenomena in two dimensions.

The implications of this work extend beyond just these specific models. The method used to construct the tensor networks—mapping a two-dimensional system to a three-dimensional structure—could be applied to other problems in quantum physics. The researchers suggest that their approach might help in understanding other exotic states of matter, such as those found in high-temperature superconductors or topological insulators. By providing a clear, analytical picture of how entanglement works in two dimensions, this study offers a foundation for future research. It shows that even in complex, multi-directional systems, there are underlying patterns that can be uncovered with the right mathematical tools. The study does not claim to have solved all the mysteries of quantum entanglement, but it has provided a concrete, verified example of how these phenomena can manifest in two-dimensional systems, bridging the gap between theory and the complex reality of quantum matter.

In the end, the paper presents a clear and detailed picture of a quantum world where entanglement is not just a local phenomenon but a global one. By building a bridge between the flat world of the quantum particles and the three-dimensional world of geometric tiles, the researchers have shown how these systems can organize themselves into states of extreme complexity. The findings confirm that the transition from limited to extensive entanglement is real and can be described with mathematical precision. This work stands as a testament to the power of analytical methods in physics, demonstrating that even the most intricate quantum behaviors can be understood when viewed from the right perspective. The researchers have not only found the first examples of these states in two dimensions but have also provided the tools to study them in depth, paving the way for a deeper understanding of the quantum universe.

Technical Summary: Highly Entangled 2D Ground States: Tensor Networks and Correlation Functions

Problem Statement
While the area law of entanglement entropy (EE) is well-established for gapped ground states (GSs) of locally interacting systems, finding analytically solvable models in two dimensions (2D) that exhibit extensive (volume-law) EE remains a significant challenge. Previous work has successfully constructed 1D models, such as the Fredkin and Motzkin spin chains, which display a quantum phase transition from area-law to extensive EE via a deformation parameter $q$ . However, generalizing these 1D "rainbow" entanglement structures to 2D presents fundamental difficulties. Specifically, constructing exact tensor network (TN) representations for 2D systems where spins are entangled in multiple directions requires a bulk geometry that is one dimension higher than the physical system. Existing prescriptions, such as the Multiscale Entanglement Renormalization Ansatz (MERA), generalize to 2D but do not naturally capture the specific coupling of independent yet interconnected entanglement flows required for these highly entangled 2D ground states.

Methodology
The authors construct exact analytical tensor network representations for two distinct 2D generalizations of the Fredkin spin chain, both of which exhibit a phase transition between area-law and extensive EE depending on the deformation parameter $q$ .

Model Construction:
- 6-Vertex Coupled Chains: Two orthogonal arrays of Fredkin chains on a square lattice are coupled at intersections via the "ice rule" (six-vertex model). The ground state is a superposition of spin configurations satisfying the ice rule and color-correlated Dyck walks in all horizontal and vertical chains.
- Lozenge Tiling Model: Three arrays of Fredkin chains are coupled on a triangular lattice via lozenge tiling rules (equivalent to dimer coverings on a honeycomb lattice). The ground state involves color-correlated Dyck walks in three orthogonal planes ($xy$, $xz$, $yz$).
Tensor Network Construction via Tiling:
- The authors employ a tiling method where valid 3D tilings of specific geometric shapes correspond one-to-one with the physical spin configurations in the ground state superposition.
- 6-Vertex Model: The physical spins are associated with towers of cubic tiles. The construction utilizes two sublattices (horizontal and vertical) with distinct 6-leg tensors ( $H$ and $V$ ). The tiles incorporate arrow continuity to enforce both the Dyck walk rules (in $xz$ and $yz$ planes) and the ice rule (in the $xy$ plane). The authors demonstrate a bijection between valid 3D cube tilings and the 2D spin configurations.
- Lozenge Tiling Model: The physical degrees of freedom (triangles of the lattice) are associated with towers of prismatic tiles. The construction uses two sublattices (right-pointing and left-pointing) with 5-leg tensors ( $R$ and $L$ ). The tiles utilize three types of arrows (solid, dashed, dotted) to represent Dyck walks in the three orthogonal planes. The authors prove a bijection between valid 3D prism tilings and the 2D lozenge tilings.
Weighting and Deformation:
- The tensors are weighted by powers of the deformation parameter $q$ to reproduce the volume-weighted ground state superposition found in the Hamiltonian formulations. The weights are derived by analyzing the volume of the spin configurations (or height of the random surfaces) and ensuring the contraction of the TN yields the correct statistical weight $q^V$ .
Correlation Analysis:
- The paper analyzes spin and color correlation functions in the scaling limit.
- Spin Correlations: The spin order parameter is derived from the scaling of the height function $\phi$ . The authors argue that for $q < 1$ , the system is gapped, leading to exponential decay of spin correlations. For $q > 1$ , the system exhibits extensive entanglement, but the spin-spin correlations in the spin sector remain characterized by exponential decay due to the finite spectral gap.
- Color Correlations: The color correlation function is analyzed by evaluating the probability of nearest up-down spin pairs forming at a specific height. In the $q < 1$ phase, this decays exponentially. At the critical point $q=1$ , the authors derive a power-law decay ( $r^{-3/2}$ in 1D, $r^{-\eta}$ in 2D) based on the statistics of random surfaces and Dyck paths. In the $q > 1$ phase, color correlations grow with distance, exhibiting exponential growth.

Key Contributions and Results

First Exact 2D TN Representations: The paper presents the first exact tensor network representations for 2D ground states with extensive entanglement entropy. These networks are 3D bulk structures (pyramids of cubes or prisms) where the physical degrees of freedom reside on the boundary.
Geometric Enforcement of Rules: The construction demonstrates how complex coupling rules (ice rule and multi-plane Dyck walks) can be intrinsically enforced in the TN bulk through the continuity of arrows in the 3D tilings, without requiring explicit external constraints on the tensor indices.
Phase Transition Characterization: The authors provide analytical characterizations of the quantum phase transition at $q=1$ $q = 1$ :
- Spin Sector: The transition is identified as an order-disorder transition. For $q < 1$ , the system is gapped with exponential decay of correlations. For $q > 1$ , the system exhibits extensive EE, yet the spin correlations continue to decay exponentially due to the gapped nature of the spin sector.
- Color Sector: The transition is marked by a change in the scaling of color correlations from exponential decay ( $q < 1$ ) to power-law decay ( $q = 1$ ) and eventually growing (exponential) correlations ( $q > 1$ ).
Scaling Laws: The paper tabulates the scaling behaviors of the height function, spin order parameters, spectral gaps, and correlation functions for both 1D and 2D cases across the three phases ( $q < 1, q = 1, q > 1$ ).

Significance and Claims
The authors claim that their work provides the first explicit classical description (via 3D tensor networks) of extensively entangled 2D quantum states. By establishing a bijection between 3D tilings and 2D physical configurations, the paper offers a new method for generalizing 1D solvable models to higher dimensions.

The significance lies in:

Holographic Realization: The TNs serve as discrete realizations of the AdS/CFT correspondence, where the 3D bulk geometry naturally encodes the extensive entanglement of the 2D boundary.
Analytical Insight: The exact TN representation allows for the derivation of correlation functions and scaling laws that are typically inaccessible in 2D strongly correlated systems.
Generalization Framework: The construction reveals how entanglement in multiple directions can be managed in a single tensor network structure, providing a blueprint for future generalizations to higher dimensions.

The paper concludes modestly, noting that while the TNs provide exact representations, future work could utilize these structures to numerically confirm or algebraically derive the correlation functions and explore further generalizations. The authors do not propose experimental realizations but emphasize the theoretical utility of these models for understanding entanglement in 2D systems.

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