Edge theory approach to topological entanglement entropy, mutual information and entanglement negativity in Chern-Simons theories

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 $g$, 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 $R$-symbols, monodromy and topological spins of quasiparticles. Furthermore, by using our method, we calculate other entanglement measures such as the mutual information and the entanglement negativity. In particular, it is found that the entanglement negativity of two adjacent non-contractible regions on a torus provides a simple way to distinguish Abelian and non-Abelian topological orders.Comment: 30 pages, 8 figures; Reference and discussions on double torus are adde

Quantum dynamics in sine-square deformed conformal field theory: Quench from uniform to non-uniform CFTs

In this work, motivated by the sine-square deformation (SSD) for (1+1)-dimensional quantum critical systems, we study the non-equilibrium quantum dynamics of a conformal field theory (CFT) with SSD, which was recently proposed to have continuous energy spectrum and continuous Virasoro algebra. In particular, we study the time evolution of entanglement entropy after a quantum quench from a uniform CFT, which is defined on a finite space of length $L$, to a sine-square deformed CFT. We find there is a crossover time $t^{\ast}$ that divides the entanglement evolution into two interesting regions. For $t\ll t^{\ast}$, the entanglement entropy does not evolve in time; for $t\gg t^{\ast}$, the entanglement entropy grows as $S_A(t)\simeq \frac{c}{3}\log t$, which is independent of the lengths of the subsystem and the total system. This $\log t$ growth with no revival indicates that a sine-square deformed CFT effectively has an infinite length, in agreement with previous studies based on the energy spectrum analysis. Furthermore, we study the quench dynamics for a CFT with M$\ddot{\text{o}}$bius deformation, which interpolates between a uniform CFT and a sine-square deformed CFT. The entanglement entropy oscillates in time with period $L_{\text{eff}}=L\cosh(2\theta)$, with $\theta=0$ corresponding to the uniform case and $\theta\to \infty$ corresponding to the SSD limit. Our field theory calculation is confirmed by a numerical study on a (1+1)-d critical fermion chain.Comment: are welcome; 10 pages, 4 figures; v2: refs added; v3: refs added; A physical interpretation of t* is added; v4: published version (selected as Editors' Suggestion

Entanglement entropy of (3+1)D topological orders with excitations

Excitations in (3+1)D topologically ordered phases have very rich structures. (3+1)D topological phases support both point-like and string-like excitations, and in particular the loop (closed string) excitations may admit knotted and linked structures. In this work, we ask the question how 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 orders? We are mainly interested in (3+1)D topological orders that can be realized in Dijkgraaf-Witten gauge theories, which are labeled by a finite group $G$ and its group 4-cocycle $\omega\in\mathcal{H}^4[G;U(1)]$ 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 $(G,\,\omega)$. The entanglement entropy of the excitations of the linked/unlinked topology can capture different information of the DW theory $(G,\,\omega)$. In particular, the entanglement entropy introduced by Hopf-link loop excitations can distinguish certain group 4-cocycles $\omega$ from the others.Comment: 12 pages, 4 figures; v2: minor changes, published versio