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arXiv:1710.11168 · Physical review. B./Physical review. B

Entanglement entropy for (3+1)-dimensional topological order with excitations

Xueda Wen, Huan He, Apoorv Tiwari, Yunqin Zheng, Peng Ye

Published Feb 26, 2018• 35 citations
arXivHTMLPDF

Abstract

Excitations in (3+1)-dimensional [(3+1)D] topologically ordered phases have very rich structures. (3+1)D topological phases support both pointlike and stringlike excitations, and in particular the loop (closed string) excitations may admit knotted and linked structures. In this work, we ask the following question: How do different types of topological excitations contribute to the entanglement entropy or, alternatively, can we use the entanglement entropy to detect the structure of excitations, and further obtain the information of the underlying topological order? We are mainly interested in (3+1)D topological order that can be realized in Dijkgraaf-Witten (DW) gauge theories, which are labeled by a finite group $G$ and its group 4-cocycle $\ensuremath{\omega}\ensuremath{\in}{\mathcal{H}}^{4}[G;\text{U}(1)]$ up to group automorphisms. We find that each topological excitation contributes a universal constant $ln{d}_{i}$ to the entanglement entropy, where ${d}_{i}$ is the quantum dimension that depends on both the structure of the excitation and the data $(G,\phantom{\rule{0.16em}{0ex}}\ensuremath{\omega})$. The entanglement entropy of the excitations of the linked/unlinked topology can capture different information of the DW theory $(G,\phantom{\rule{0.16em}{0ex}}\ensuremath{\omega})$. In particular, the entanglement entropy introduced by Hopf-link loop excitations can distinguish certain group 4-cocycles $\ensuremath{\omega}$ from the others.

DOI
10.1103/physrevb.97.085147

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Excitations in (3+1)-dimensional [(3+1)D] topologically ordered phases have very rich structures. (3+1)D topological phases support both pointlike and stringlike excitations, and in particular the loop (closed string) excitations may admit knotted and linked structures. In this work, we ask the following question: How do different types of topological excitations contribute to the entanglement entropy or, alternatively, can we use the entanglement entropy to detect the structure of excitations, and further obtain the information of the underlying topological order? We are mainly interested in (3+1)D topological order that can be realized in Dijkgraaf-Witten (DW) gauge theories, which are labeled by a finite group GGG and its group 4-cocycle \ensuremathω\ensuremath∈H4[G;U(1)]\ensuremath{\omega}\ensuremath{\in}{\mathcal{H}}^{4}[G;\text{U}(1)]\ensuremathω\ensuremath∈H4[G;U(1)] up to group automorphisms. We find that each topological excitation contributes a universal constant lndiln{d}_{i}lndi​ to the entanglement entropy, where di{d}_{i}di​ is the quantum dimension that depends on both the structure of the excitation and the data (G,\ensuremathω)(G,\phantom{\rule{0.16em}{0ex}}\ensuremath{\omega})(G,\ensuremathω). The entanglement entropy of the excitations of the linked/unlinked topology can capture different information of the DW theory (G,\ensuremathω)(G,\phantom{\rule{0.16em}{0ex}}\ensuremath{\omega})(G,\ensuremathω). In particular, the entanglement entropy introduced by Hopf-link loop excitations can distinguish certain group 4-cocycles \ensuremathω\ensuremath{\omega}\ensuremathω from the others.
Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group GGG and its group 4-cocycle ω4∈H4[G;U(1)]ω_4 \in \mathcal H^4[G;U(1)]ω4​∈H4[G;U(1)] up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.
Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems, there is no nontrivial topological order, while in 2+1D bosonic systems, the topological orders are classified by the following pair: a modular tensor category and a chiral central charge. In this paper, following a new line of thinking, we find that in 3+1D the classification is much simpler than it was thought to be; we propose a partial classification of topological orders for 3+1D bosonic systems: If all the pointlike excitations are bosons, then such topological orders are classified by a simpler pair (G,ω_{4}): a finite group G and its group 4-cocycle ω_{4}∈H^{4}[G;U(1)] (up to group automorphisms). Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.
(3+1)(3+1)(3+1)-dimensional topological phases can support loop-like excitations in addition to point-like ones, allowing for nontrivial loop-loop and point-loop braiding statistics not permitted to point-like excitations alone. Furthermore, these loop-like excitations can be linked together, changing their properties. In particular, this can lead to distinct three-loop braiding, involving two loops undergoing an exchange process while linked to a third loop. In this work, we investigate the loop-like excitations in a (3+1)(3+1)(3+1)-dimensional Hamiltonian realization of Dijkgraaf-Witten theory through direct construction of their membrane operators, for a general finite Abelian group and 4-cocycle twist. Using these membrane operators, we find the braiding relations and fusion rules for the loop-like excitations, including those linked to another loop-like excitation. Furthermore, we use these membrane operators to construct projection operators that measure the topological charge and show that the number of distinct topological charges measured by the 2-torus matches the ground-state degeneracy of the model on the 3-torus, explicitly confirming a general expectation for topological phases. This direct construction of the membrane operators sheds significant light on the key properties of the loop-like excitations in (3+1)(3+1)(3+1)-dimensional topological phases.
Abstract We compute the topological entanglement entropy for a large set of lattice models in d-dimensions. It is well known that many such quantum systems can be constructed out of lattice gauge models. For dimensionality higher than two, there are generalizations going beyond gauge theories, which are called higher gauge theories and rely on higher-order generalizations of groups. Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories. In this paper, we derive a general formula for the bipartition entanglement entropy for this class of models, and from it we extract both the area law and the sub-leading terms, which explicitly depend on the topology of the entangling surface. We show that the entanglement entropy SA in a sub-region A is proportional to log (GSDA~) \left({GSD}_{\tilde{A}}\right) (GSDA~​) GSD A ˜ , where GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ is the ground state degeneracy of a particular restriction of the full model to A. The quantity GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ can be further divided into a contribution that scales with the size of the boundary ∂A and a term which depends on the topology of ∂A. There is also a topological contribution coming from A itself, that may be non-zero when A has a non-trivial homology. We present some examples and discuss how the topology of A affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension d. The result is in agreement with entanglement calculations for known topological models.
We develop an approach based on edge theories to calculate the entanglement entropy and related quantities in (2+1)-dimensional topologically ordered phases. Our approach is complementary to, e.g., the existing methods using replica trick and Witten's method of surgery, and applies to a generic spatial manifold of genus ggg, which can be bipartitioned in an arbitrary way. The effects of fusion and braiding of Wilson lines can be also straightforwardly studied within our framework. By considering a generic superposition of states with different Wilson line configurations, through an interference effect, we can detect, by the entanglement entropy, the topological data of Chern-Simons theories, e.g., the RRR symbols, monodromy, and topological spins of quasiparticles. Furthermore, by using our method, we calculate other entanglement/correlation measures such as the mutual information and the entanglement negativity. In particular, it is found that the entanglement negativity of two adjacent noncontractible regions on a torus provides a simple way to distinguish Abelian and non-Abelian topological orders.
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>3D>3D>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 $D\ensuremath{\ge}3$.