We calculate the topological entanglement entropy (TEE) for a three-dimensional hyperhoneycomb lattice generalization of Kitaev's honeycomb lattice spin model. We find that for this model TEE is not directly determined by the total quantum dimension of the system. This is in contrast to general two-dimensional systems and many three-dimensional models, where TEE is related to the total quantum dimension. Our calculation also provides TEE for a three-dimensional toric-code-type Hamiltonian that emerges as the effective low-energy theory for the Kitaev model in a particular limit.
We discuss entanglement entropy of gapped ground states in different dimensions, obtained on partitioning space into two regions. For trivial phases without topological order, we argue that the entanglement entropy may be obtained by integrating an ``entropy density'' over the partition boundary that admits a gradient expansion in the curvature of the boundary. This constrains the expansion of entanglement entropy as a function of system size and points to an even-odd dependence on dimensionality. For example, in contrast to the familiar result in two dimensions, a size-independent constant contribution to the entanglement entropy can appear for trivial phases in any odd spatial dimension. We then discuss phases with topological entanglement entropy (TEE) that cannot be obtained by adding local contributions. We find that in three dimensions there is just one type of TEE, as in two dimensions, that depends linearly on the number of connected components of the boundary (the ``zeroth Betti number''). In
D>3 dimensions, new types of TEE appear which depend on the higher Betti numbers of the boundary manifold. We construct generalized toric code models that exhibit these TEEs and discuss ways to extract TEE in $D\ensuremath{\ge}3$.
Abstract The topological entanglement entropy is used to measure long-range quantum correlations in the ground space of topological phases. Here we obtain closed form expressions for the topological entropy of (2+1)- and (3+1)-dimensional loop gas models, both in the bulk and at their boundaries, in terms of the data of their input fusion categories and algebra objects. Central to the formulation of our results are generalized
S S -matrices. We conjecture a general property of these
S S -matrices, with proofs provided in many special cases. This includes constructive proofs for categories up to rank 5.
Entanglement entropies of two-dimensional gapped ground states are expected to satisfy an area law, with a constant correction term known as the topological entanglement entropy (TEE). In many models, the TEE takes a universal value that characterizes the underlying topological phase. However, the TEE is not truly universal: it can differ even for two states related by constant-depth circuits, which are necessarily in the same phase. The difference between the TEE and the value predicted by the anyon theory is often called the "spurious" topological entanglement entropy. We show that this spurious contribution is always non-negative, thus the value predicted by the anyon theory provides a universal lower bound. This observation also leads to a definition of TEE that is invariant under constant-depth quantum circuits.
It is well known that the topological entanglement entropy (
Stopo) of a topologically ordered ground state in two spatial dimensions can be captured efficiently by measuring the tripartite quantum information (
I3) of a specific annular arrangement of three subsystems. However, the nature of the general
N-partite information (
IN) and correlation of a topologically ordered ground state remains unknown. In this work, we study such
IN measure and its nontrivial dependence on the arrangement of
N subsystems. For the collection of subsystems (CSS) forming a closed annular structure, the
IN measure ($N\ensuremath{\ge}3$) is a topological invariant equal to the product of
Stopo and the Euler characteristic of the CSS embedded on a planar manifold, $|{I}^{N}|=\ensuremath{\chi}{S}_{\mathrm{topo}}$. Importantly, we establish that
IN is robust against several deformations of the annular CSS, such as the addition of holes within individual subsystems and handles between nearest-neighbor subsystems. While the addition of a handle between further neighbor subsystems causes
IN to vanish, the multipartite information measures of the two smaller annular CSS emergent from this deformation again yield the same topological invariant. For a general CSS with multiple holes (
nh>1), we find that the sum of the distinct, multipartite information measured on the annular CSS around those holes is given by the product of ${S}_{\mathrm{topo}}, \ensuremath{\chi}$ and ${n}_{h}, {\ensuremath{\sum}}_{{\ensuremath{\mu}}_{i}=1}^{{n}_{h}}|{I}_{{\ensuremath{\mu}}_{i}}^{{N}_{{\ensuremath{\mu}}_{i}}}|={n}_{h}\ensuremath{\chi}{S}_{\mathrm{topo}}$. This constrains the concomitant measurement of several multipartite information on any complicated CSS. The
Nth order irreducible correlations for an annular CSS of
N subsystems is also found to be bounded from above by
∣IN∣, which shows the presence of correlations among subsystems arranged in the form of closed loops of all sizes. Thus, our results offer important insight into the nature of the many-particle entanglement and correlations within a topologically ordered state of matter.
Topological order is a new paradigm for quantum phases of matter developed to explain phase transitions which do not fit the symmetry breaking scheme for classifying phases of matter. They are characterized by patterns of entanglement that lead to topologically depended ground state degeneracy and anyonic excitations. One common approach for studying such phases in two-dimensional systems is through exactly solvable lattice Hamiltonian models such as quantum double models and String-Net models. The former can be understood as the Hamiltonian formulation of lattice gauge theories and, as such, it is defined by a finite gauge group. However, not much is known about topological phases in tridimensional systems. Motivated by this we develop a new class of three-dimensional exactly solvable models which go beyond quantum double models by using finite crossed modules instead of gauge groups. This approach relies on a lattice implementation of 2-gauge theory to obtain models with a richer topological structure. We construct the Hamiltonian model explicitly and provide a rigorous proof that the ground state degeneracy is a topological invariant and that the ground states can only be characterized with nonlocal order parameters.