Bicolor loop models and their long range entanglement

Zhao Zhang

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

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Quantum loop models are well studied objects in the context of lattice gauge theories and topological quantum computing. They usually carry long range entanglement that is captured by the topological entanglement entropy. I consider generalization of the toric code model to bicolor loop models and show that the long range entanglement can be reflected in three different ways: a topologically invariant constant, a sub-leading logarithmic correction to the area law, or a modified bond dimension for the area-law term. The Hamiltonians are not exactly solvable for the whole spectra, but admit a tower of area-law exact excited states corresponding to the frustration free superposition of loop configurations with arbitrary pairs of localized vertex defects. The continuity of color along loops imposes kinetic constraints on the model and results in Hilbert space fragmentation, unless plaquette operators involving two neighboring faces are introduced to the Hamiltonian.

Physicists, quantum or statistical, have long been obsessed with binary degrees of freedom. While it is true that a black-and-white pictures capture much of the features of our colorful world, often times things change qualitatively going from 2 to 3, when topologies, dynamics, and entanglement are taken into account. This article offers a peek into the rich consequences of enlarging the local degrees of freedom from qubits to qutrits in various generalizations the toric code and quantum loop models.

► BibTeX data

► References

[1] M B Hastings. ``An area law for one-dimensional quantum systems''. Journal of Statistical Mechanics: Theory and Experiment 2007, P08024 (2007).
https:/​/​doi.org/​10.1088/​1742-5468/​2007/​08/​P08024

[2] Anurag Anshu, Itai Arad, and David Gosset. ``An area law for 2d frustration-free spin systems''. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. Pages 12–18. STOC 2022New York, NY, USA (2022). Association for Computing Machinery.
https:/​/​doi.org/​10.1145/​3519935.3519962

[3] Christoph Holzhey, Finn Larsen, and Frank Wilczek. ``Geometric and renormalized entropy in conformal field theory''. Nuclear Physics B 424, 443–467 (1994).
https:/​/​doi.org/​10.1016/​0550-3213(94)90402-2

[4] Pasquale Calabrese and John Cardy. ``Entanglement entropy and conformal field theory''. Journal of Physics A: Mathematical and Theoretical 42, 504005 (2009).
https:/​/​doi.org/​10.1088/​1751-8113/​42/​50/​504005

[5] Dimitri Gioev and Israel Klich. ``Entanglement Entropy of Fermions in Any Dimension and the Widom Conjecture''. Phys. Rev. Lett. 96, 100503 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.96.100503

[6] G Vitagliano, A Riera, and J I Latorre. ``Volume-law scaling for the entanglement entropy in spin-1/​2 chains''. New Journal of Physics 12, 113049 (2010).
https:/​/​doi.org/​10.1088/​1367-2630/​12/​11/​113049

[7] Giovanni Ramírez, Javier Rodríguez-Laguna, and Germán Sierra. ``From conformal to volume law for the entanglement entropy in exponentially deformed critical spin 1/​2 chains''. Journal of Statistical Mechanics: Theory and Experiment 2014, P10004 (2014).
https:/​/​doi.org/​10.1088/​1742-5468/​2014/​10/​P10004

[8] Zhao Zhang. ``Entanglement blossom in a simplex matryoshka''. Annals of Physics 457, 169395 (2023).
https:/​/​doi.org/​10.1016/​j.aop.2023.169395

[9] Javier Rodríguez-Laguna, Jérôme Dubail, Giovanni Ramírez, Pasquale Calabrese, and Germán Sierra. ``More on the rainbow chain: entanglement, space-time geometry and thermal states''. Journal of Physics A: Mathematical and Theoretical 50, 164001 (2017).
https:/​/​doi.org/​10.1088/​1751-8121/​aa6268

[10] Ian MacCormack, Aike Liu, Masahiro Nozaki, and Shinsei Ryu. ``Holographic duals of inhomogeneous systems: the rainbow chain and the sine-square deformation model''. Journal of Physics A: Mathematical and Theoretical 52, 505401 (2019).
https:/​/​doi.org/​10.1088/​1751-8121/​ab3944

[11] Ramis Movassagh and Peter W. Shor. ``Supercritical entanglement in local systems: Counterexample to the area law for quantum matter''. Proceedings of the National Academy of Sciences 113, 13278–13282 (2016).
https:/​/​doi.org/​10.1073/​pnas.1605716113

[12] Zhao Zhang, Amr Ahmadain, and Israel Klich. ``Novel quantum phase transition from bounded to extensive entanglement''. Proceedings of the National Academy of Sciences 114, 5142–5146 (2017).
https:/​/​doi.org/​10.1073/​pnas.1702029114

[13] L. Dell'Anna, O. Salberger, L. Barbiero, A. Trombettoni, and V. E. Korepin. ``Violation of cluster decomposition and absence of light cones in local integer and half-integer spin chains''. Phys. Rev. B 94, 155140 (2016).
https:/​/​doi.org/​10.1103/​PhysRevB.94.155140

[14] Olof Salberger and Vladimir Korepin. ``Entangled spin chain''. Reviews in Mathematical Physics 29, 1750031 (2017).
https:/​/​doi.org/​10.1142/​S0129055X17500313

[15] Olof Salberger, Takuma Udagawa, Zhao Zhang, Hosho Katsura, Israel Klich, and Vladimir Korepin. ``Deformed fredkin spin chain with extensive entanglement''. Journal of Statistical Mechanics: Theory and Experiment 2017, 063103 (2017).
https:/​/​doi.org/​10.1088/​1742-5468/​aa6b1f

[16] Zhao Zhang and Israel Klich. ``Entropy, gap and a multi-parameter deformation of the fredkin spin chain''. Journal of Physics A: Mathematical and Theoretical 50, 425201 (2017).
https:/​/​doi.org/​10.1088/​1751-8121/​aa866e

[17] Rafael N. Alexander, Amr Ahmadain, Zhao Zhang, and Israel Klich. ``Exact rainbow tensor networks for the colorful motzkin and fredkin spin chains''. Phys. Rev. B 100, 214430 (2019).
https:/​/​doi.org/​10.1103/​PhysRevB.100.214430

[18] Zhao Zhang and Israel Klich. ``Coupled Fredkin and Motzkin chains from quantum six- and nineteen-vertex models''. SciPost Phys. 15, 044 (2023).
https:/​/​doi.org/​10.21468/​SciPostPhys.15.2.044

[19] Zhao Zhang and Israel Klich. ``Quantum colored lozenge tiling and entanglement phase transition'' (2022). arXiv:2210.01098.
arXiv:2210.01098

[20] Alexei Kitaev and John Preskill. ``Topological entanglement entropy''. Phys. Rev. Lett. 96, 110404 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.96.110404

[21] Michael Levin and Xiao-Gang Wen. ``Detecting topological order in a ground state wave function''. Phys. Rev. Lett. 96, 110405 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.96.110405

[22] A. Yu. Kitaev. ``Fault-tolerant quantum computation by anyons''. Annals of Physics 303, 2–30 (2003).
https:/​/​doi.org/​10.1016/​S0003-4916(02)00018-0

[23] Liujun Zou and Jeongwan Haah. ``Spurious long-range entanglement and replica correlation length''. Phys. Rev. B 94, 075151 (2016).
https:/​/​doi.org/​10.1103/​PhysRevB.94.075151

[24] Dominic J. Williamson, Arpit Dua, and Meng Cheng. ``Spurious topological entanglement entropy from subsystem symmetries''. Phys. Rev. Lett. 122, 140506 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.140506

[25] David T. Stephen, Henrik Dreyer, Mohsin Iqbal, and Norbert Schuch. ``Detecting subsystem symmetry protected topological order via entanglement entropy''. Phys. Rev. B 100, 115112 (2019).
https:/​/​doi.org/​10.1103/​PhysRevB.100.115112

[26] Kohtaro Kato and Fernando G. S. L. Brandão. ``Toy model of boundary states with spurious topological entanglement entropy''. Phys. Rev. Res. 2, 032005 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.032005

[27] Isaac H. Kim, Michael Levin, Ting-Chun Lin, Daniel Ranard, and Bowen Shi. ``Universal lower bound on topological entanglement entropy''. Phys. Rev. Lett. 131, 166601 (2023).
https:/​/​doi.org/​10.1103/​PhysRevLett.131.166601

[28] Eduardo Fradkin and Joel E. Moore. ``Entanglement entropy of 2d conformal quantum critical points: Hearing the shape of a quantum drum''. Phys. Rev. Lett. 97, 050404 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.97.050404

[29] H. Casini and M. Huerta. ``Universal terms for the entanglement entropy in 2+1 dimensions''. Nuclear Physics B 764, 183–201 (2007).
https:/​/​doi.org/​10.1016/​j.nuclphysb.2006.12.012

[30] Daniel S. Rokhsar and Steven A. Kivelson. ``Superconductivity and the quantum hard-core dimer gas''. Phys. Rev. Lett. 61, 2376–2379 (1988).
https:/​/​doi.org/​10.1103/​PhysRevLett.61.2376

[31] R. Moessner, S. L. Sondhi, and Eduardo Fradkin. ``Short-ranged resonating valence bond physics, quantum dimer models, and ising gauge theories''. Phys. Rev. B 65, 024504 (2001).
https:/​/​doi.org/​10.1103/​PhysRevB.65.024504

[32] Eddy Ardonne, Paul Fendley, and Eduardo Fradkin. ``Topological order and conformal quantum critical points''. Annals of Physics 310, 493–551 (2004).
https:/​/​doi.org/​10.1016/​j.aop.2004.01.004

[33] Tomoyoshi Hirata and Tadashi Takayanagi. ``Ads/​cft and strong subadditivity of entanglement entropy''. Journal of High Energy Physics 2007, 042 (2007).
https:/​/​doi.org/​10.1088/​1126-6708/​2007/​02/​042

[34] E. M. Stoudenmire, Peter Gustainis, Ravi Johal, Stefan Wessel, and Roger G. Melko. ``Corner contribution to the entanglement entropy of strongly interacting o(2) quantum critical systems in 2+1 dimensions''. Phys. Rev. B 90, 235106 (2014).
https:/​/​doi.org/​10.1103/​PhysRevB.90.235106

[35] Shankar Balasubramanian, Ethan Lake, and Soonwon Choi. ``2d hamiltonians with exotic bipartite and topological entanglement'' (2023). arXiv:2305.07028.
arXiv:2305.07028

[36] Paul Fendley. ``Loop models and their critical points''. Journal of Physics A: Mathematical and General 39, 15445 (2006).
https:/​/​doi.org/​10.1088/​0305-4470/​39/​50/​011

[37] Zhao Zhang and Henrik Schou Røising. ``The frustration-free fully packed loop model''. Journal of Physics A: Mathematical and Theoretical 56, 194001 (2023).
https:/​/​doi.org/​10.1088/​1751-8121/​acc76f

[38] Michael A. Levin and Xiao-Gang Wen. ``String-net condensation: A physical mechanism for topological phases''. Phys. Rev. B 71, 045110 (2005).
https:/​/​doi.org/​10.1103/​PhysRevB.71.045110

[39] H. Bombin and M. A. Martin-Delgado. ``Topological quantum distillation''. Phys. Rev. Lett. 97, 180501 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.97.180501

[40] Jeffrey C. Y. Teo, Abhishek Roy, and Xiao Chen. ``Unconventional fusion and braiding of topological defects in a lattice model''. Phys. Rev. B 90, 115118 (2014).
https:/​/​doi.org/​10.1103/​PhysRevB.90.115118

[41] Zhao Zhang and Giuseppe Mussardo. ``Hidden bethe states in a partially integrable model''. Phys. Rev. B 106, 134420 (2022).
https:/​/​doi.org/​10.1103/​PhysRevB.106.134420

[42] R. Raghavan, Christopher L. Henley, and Scott L. Arouh. ``New two-color dimer models with critical ground states''. Journal of Statistical Physics 86, 517–550 (1997).
https:/​/​doi.org/​10.1007/​BF02199112

[43] B. Normand. ``Multicolored quantum dimer models, resonating valence-bond states, color visons, and the triangular-lattice ${t}_{2g}$ spin-orbital system''. Phys. Rev. B 83, 064413 (2011).
https:/​/​doi.org/​10.1103/​PhysRevB.83.064413

[44] Naoto Shiraishi and Takashi Mori. ``Systematic construction of counterexamples to the eigenstate thermalization hypothesis''. Phys. Rev. Lett. 119, 030601 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.030601

[45] Libor Caha and Daniel Nagaj. ``The pair-flip model: a very entangled translationally invariant spin chain'' (2018). arXiv:1805.07168.
arXiv:1805.07168

[46] Chenjie Wang and Michael Levin. ``Braiding statistics of loop excitations in three dimensions''. Phys. Rev. Lett. 113, 080403 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.080403

[47] Daniel K. Mark, Cheng-Ju Lin, and Olexei I. Motrunich. ``Unified structure for exact towers of scar states in the affleck-kennedy-lieb-tasaki and other models''. Phys. Rev. B 101, 195131 (2020).
https:/​/​doi.org/​10.1103/​PhysRevB.101.195131

[48] Benjamin Doyon. ``Thermalization and pseudolocality in extended quantum systems''. Communications in Mathematical Physics 351, 155–200 (2017).
https:/​/​doi.org/​10.1007/​s00220-017-2836-7

[49] Berislav Buča. ``Unified theory of local quantum many-body dynamics: Eigenoperator thermalization theorems''. Phys. Rev. X 13, 031013 (2023).
https:/​/​doi.org/​10.1103/​PhysRevX.13.031013

[50] Charles Stahl, Rahul Nandkishore, and Oliver Hart. ``Topologically stable ergodicity breaking from emergent higher-form symmetries in generalized quantum loop models'' (2023). arXiv:2304.04792.
arXiv:2304.04792

[51] Alexei Kitaev. ``Anyons in an exactly solved model and beyond''. Annals of Physics 321, 2–111 (2006).
https:/​/​doi.org/​10.1016/​j.aop.2005.10.005

Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2024-04-12 09:56:29). On SAO/NASA ADS no data on citing works was found (last attempt 2024-04-12 09:56:29).