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arXiv:2306.05464 · Quantum

Bicolor loop models and their long range entanglement

Zhao Zhang

Published Feb 29, 2024• 4 citations
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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.

DOI
10.22331/q-2024-02-29-1268

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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.
Abstract We consider a quantum fully packed loop model on the square lattice with a frustration-free projector Hamiltonian and ring-exchange interactions acting on plaquettes. A boundary Hamiltonian is added to favor domain-wall boundary conditions and link ground state properties to the combinatorics and six-vertex model literature. We discuss how the boundary term fractures the Hilbert space into Krylov subspaces, and we prove that the Hamiltonian is ergodic within each subspace, leading to a series of energy-equidistant exact eigenstates in the lower end of the spectrum. Among them we systematically classify both finitely entangled eigenstates and product eigenstates. Using a recursion relation for enumerating half-plane configurations, we compute numerically the exact entanglement entropy of the ground state, confirming area law scaling. Finally, the spectrum is shown to be gapless in the thermodynamic limit with a trial state constructed by adding a twist to the ground state superposition.
The area law for entanglement entropy captures a fundamental constraint on the complexity of quantum many-body ground states and enables their efficient description. While the area law is rigorously established in one dimension, its status in higher-dimensional local systems remains unresolved, and it does not hold in general for geometrically non-local systems. Here, we establish a generalized area law for gapped ground states of fully connected Hamiltonians. We show that the bipartite entanglement entropy grows at most logarithmically with system size despite the absence of geometric locality. In the ground state, each site is only weakly entangled with the rest, while configurations with extensive local fluctuations are strongly suppressed, effectively restricting the accessible Hilbert space. As a consequence, the ground state admits a matrix product state approximation with polynomial bond dimension with respect to the system size at fixed accuracy. In the permutation-invariant setting, we further prove a constant entanglement bound and demonstrate that the gapped ground state can be computed in time polylogarithmic in the system size. These results show that low entanglement complexity need not rely on geometric locality, broadening the conceptual and computational scope of area laws.
We present a class of exactly solvable 2D models whose ground states violate conventional beliefs about entanglement scaling in quantum matter. These beliefs are (i) that area law entanglement scaling originates from local correlations proximate to the boundary of the entanglement cut, and (ii) that ground state entanglement in 2D Hamiltonians cannot violate area law scaling by more than a multiplicative logarithmic factor. We explicitly present two classes of models defined by local, translation-invariant Hamiltonians, whose ground states can be exactly written as weighted superpositions of framed loop configurations. The first class of models exhibits area-law scaling, but of an intrinsically nonlocal origin so that the topological entanglement entropy scales with subsystem sizes. The second class of models has a rich ground state phase diagram that includes a phase exhibiting volume law entanglement.
The non-factorizing nature of the Hilbert space in Loop Quantum Gravity (LQG) due to gauge invariance requires a generalized definition of entanglement entropy. This work employs the framework of von Neumann algebras to investigate the entanglement entropy in LQG. On a graph, the holonomy and flux operators within a region and on the boundary generate a non-factor type I von Neumann algebra, which is used to define the entanglement entropy for LQG states. This algebraic formalism is applied to ``fixed-area states''--superpositions of spin networks associated with a surface with a definite macroscopic area given by the LQG area spectrum. By maximizing the entropy, we derive a geometrical area law where the entanglement entropy is proportional to the area. In addition, we show that bulk entanglement can renormalize the area-law coefficient and produce logarithmic corrections. The results in this paper closely relate to LQG black hole entropy.
Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically ordered systems such as the toric code, double semion, colour code and quantum double models. As happens for the entanglement entropy, we find that for sufficiently large block sizes the geometric entanglement is, up to possible sub-leading corrections, the sum of two contributions: a bulk contribution obeying a boundary law times the number of blocks and a contribution quantifying the underlying pattern of long-range entanglement of the topologically ordered state. This topological contribution is also present in the case of single-spin blocks in most cases, and constitutes an alternative characterization of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the two-dimensional colour code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments (Bombin et al 2012 New J. Phys. 14 073048). Motivated by these results, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian group symmetry, and which we explicitly use to analyse quantum double models. Furthermore, we also provide an analysis of the robustness of the topological contribution in terms of renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. Some of the results in this paper rely on the ability to disentangle single sites from the quantum state, which is always possible for the systems that we consider. Additionally we relate our results to the behaviour of the relative entropy of entanglement in topologically ordered systems, and discuss a number of numerical approaches based on tensor networks that could be employed to extract this topological contribution for large systems beyond exactly solvable models.
Abstract Graphs are topological spaces that include broader objects than discretized manifolds, making them interesting playgrounds for the study of quantum phases not realized by symmetry breaking. In particular they are known to support anyons of an even richer variety than the two-dimensional space. We explore this possibility by building a class of frustration-free and gapped Hamiltonians based on discrete abelian gauge groups. The resulting models have a ground state degeneracy that can be either a topological invariant, an extensive quantity or a mixture of the two. For two basis of the degenerate ground states which are complementary in quantum theory, the entanglement entropy (EE) is exactly computed. The result for one basis has a constant global term, known as the topological EE, implying long-range entanglement. On the other hand, the topological EE vanishes in the result for the other basis. Comparisons are made with similar occurrences in the toric code. We analyze excitations and identify anyon-like excitations that account for the topological EE. An analogy between the ground states of this system and the θ -vacuum for a U (1) gauge theory on a circle is also drawn.