Quantum lozenge tiling and entanglement phase transition

Zhao Zhang; Israel Klich

doi:10.22331/q-2024-10-10-1497

Quantum lozenge tiling and entanglement phase transition

¹Department of Physics, University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo, Norway
²SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136, Trieste, Italy
³Department of Physics, University of Virginia, Charlottesville, VA, USA

Published:	2024-10-10, volume 8, page 1497
Eprint:	arXiv:2210.01098v3
Doi:	https://doi.org/10.22331/q-2024-10-10-1497
Citation:	Quantum 8, 1497 (2024).

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Abstract

While volume violation of area law has been exhibited in several quantum spin chains, the construction of a corresponding ground state in higher dimensions, entangled in more than one direction, has been an open problem. Here we construct a 2D frustration-free Hamiltonian with maximal violation of the area law. We do so by building a quantum model of random surfaces with color degree of freedom that can be viewed as a collection of colored Dyck paths. The Hamiltonian may be viewed as a 2D generalization of the Fredkin spin chain. It relates all the colored random surface configurations subject to a Dirichlet boundary condition and hard wall constraint from below to one another, and the ground state is therefore a superposition of all such classical states and non-degenerate. Its entanglement entropy between subsystems undergoes a quantum phase transition as the deformation parameter is tuned. The area- and volume-law phases are similar to the one-dimensional model, while the critical point scales with the linear size of the system $L$ as $L\log L$. Further it is conjectured that similar models with entanglement phase transitions can be built in higher dimensions with even softer area law violations at the critical point.

Featured image: Left: An enriched version of the Rokhsar-Kivelson move, or dimer flipping, that encodes both correlated swapping of an internal colour degree of freedom and conditioning on next-nearest-neighbour spin configurations. Right: A coloured lozenge tiling configuration viewed as a pile of unit cubes, coloured lines within each three cross-sectional planes are entangled in the same way as Fredkin spin chains, an example of which is thickened to be highlighted as a coloured Dyck path.

Popular summary

Can entanglement be accumulated in two, three and higher dimensions as much as they can in one-dimensional quantum many-body systems? The answer is, somewhat surprisingly, yes, when the configuration of neighbouring spins is subject to some constraint such as the tessellation of lozenges, which allows an emergent gauge degree of freedom. That can be viewed as the height variable in an extra dimension perpendicular to the space where physical spins live. The spin variables, being $\pm 1$, are then the gradients of the height field, exactly as electric fields are gradients of their scalar potential. The height field specifies a classical random surface that describes the continuous limit of a discrete tiling configuration, the superposition of many of which gives a quantum state. A carefully designed local interaction among a small number of neighbouring spins guarantees that all the random surfaces that meet certain conditions are included in the superposition of the ground state. They can further be weighted in a way that favours higher or lower height configurations, corresponding to extensive or area-law entanglement entropy. The competition between surface tension and weighting by volume determines the critical scaling behaviour, where an entanglement phase transition takes place.

► BibTeX data

@article{Zhang2024quantumlozenge,
  doi = {10.22331/q-2024-10-10-1497},
  url = {https://doi.org/10.22331/q-2024-10-10-1497},
  title = {Quantum lozenge tiling and entanglement phase transition},
  author = {Zhang, Zhao and Klich, Israel},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1497},
  month = oct,
  year = {2024}
}

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Cited by

[1] Wucheng Zhang, "Sequential Generation of Two-Dimensional Super-Area-Law States with Local Parent Hamiltonian", PRX Quantum 7 1, 010311 (2026).

[2] Olai B. Mykland and Zhao Zhang, "Highly entangled 2D ground states: Tensor network, order parameter and correlation", SciPost Physics 20 5, 126 (2026).

[3] Olai B. Mykland and Zhao Zhang, "Highly Entangled 2D Ground States: Tensor Network, Order Parameter and Correlation", arXiv:2502.20192, (2025).

[4] Olai B. Mykland and Zhao Zhang, "Exact critical exponents of the Motzkin and Fredkin Chains", arXiv:2507.14656, (2025).

[5] Zhao Zhang and Olai B. Mykland, "Highly Entangled Quantum Spin Chains on Fermat's Spiral", arXiv:2506.02103, (2025).

[6] Henrik Schou Røising and Zhao Zhang, "Ergodic Archimedean dimers", SciPost Physics Core 6 3, 054 (2023).

[7] Zhao Zhang and Henrik Schou Røising, "The frustration-free fully packed loop model", Journal of Physics A Mathematical General 56 19, 194001 (2023).

[8] Shankar Balasubramanian, Ethan Lake, and Soonwon Choi, "2D Hamiltonians with exotic bipartite and topological entanglement", arXiv:2305.07028, (2023).

[9] Zhao Zhang and Israel Klich, "Coupled Fredkin and Motzkin chains from quantum six- and nineteen-vertex models", SciPost Physics 15 2, 044 (2023).

[10] Zhao Zhang, "Entanglement blossom in a simplex matryoshka", Annals of Physics 457, 169395 (2023).

[11] Zhao Zhang and Giuseppe Mussardo, "Hidden Bethe states in a partially integrable model", Physical Review B 106 13, 134420 (2022).

[12] Zhao Zhang, "Bicolor loop models and their long range entanglement", Quantum 8, 1268 (2024).

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