BMS symmetry, which is the asymptotic symmetry at null infinity of flat spacetime, is an important input for flat holography. In this paper, we give a holographic calculation of entanglement entropy and Rényi entropy in three dimensional Einstein gravity and Topologically Massive Gravity. The geometric picture for the entanglement entropy is the length of a spacelike geodesic which is connected to the interval at null infinity by two null geodesics. The spacelike geodesic is the fixed points of replica symmetry, and the null geodesics are along the modular flow. Our strategy is to first reformulate the Rindler method for calculating entanglement entropy in a general setup, and apply it for BMS invariant field theories, and finally extend the calculation to the bulk.

To understand the new physics and richness of quantum many-body system phenomena is one of the stimuli driving the condensed matter community forward. Importantly, the new insights and solutions for condensed matter theory sometimes come from the developed and developing knowledge of high energy theory, mathematical and particle physics, which is also true the other way around: Condensed matter physics has been providing crucial hints and playgrounds for the fundamental laws of high energy physics. In this thesis, we explore the aspects of symmetry, topology and anomalies in quantum matter with entanglement from both condensed matter and high energy theory viewpoints. The focus of our research is on the gapped many-body quantum systems including symmetry-protected topological states (SPTs) and topologically ordered states (TOs). We first explore the ground state structures of SPTs and TOs: the former can be symmetry twisted and the latter has robust degeneracy. The Berry phases generated by transporting and overlapping ground state sectors potentially provide universal topological invariants that fully characterize the SPTs and TOs. This framework provides us the aspects of symmetry and topology. We establish a field theory representation of SPT invariants in any dimension to uncover group cohomology classification and beyond — the former for SPTs with gapless boundary gauge anomalies, the latter for SPTs with mixed gauge-gravity anomalies. We study topological orders in 3+1 dimensions such as Dijkgraaf-Witten models, which support multi-string braiding statistics; the resulting patterns may be analyzed by the mathematical theory of knots and links. We explore the aspects of surface anomalies of bulk gapped states from the bulk-edge correspondence: The gauge anomalies of SPTs shed light on the construction of bosonic anomalies including Goldstone-Wilczek type, and also guide us to design a non-perturbative lattice model regularizing the low-energy chiral fermion/gauge theory towards the Standard Model while overcoming the Nielsen-Ninomiya fermion-doubling problem without relying on Ginsparg-Wilson fermions. We conclude by utilizing aspects of both quantum mechanical topology and spacetime topology to derive new formulas analogous to Verlinde’s via geometric-topology surgery. This provides new insights for higher dimensional topological states of matter.

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We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS$_3$/CFT$_{2}$. With the guidance from the boundary and bulk modular flows we find a natural slicing of the entanglement wedge with the modular planes, which are co-dimension one bulk surfaces tangent to the modular flow everywhere. This gives an one-to-one correspondence between the points on the boundary interval $\mathcal{A}$ and the points on the Ryu-Takayanagi (RT) surface $\mathcal{E}_{\mathcal{A}}$. In the same sense an arbitrary subinterval $\mathcal{A}_2$ of $\mathcal{A}$ will correspond to a subinterval $\mathcal{E}_2$ of $\mathcal{E}_{\mathcal{A}}$. This fine correspondence indicates that the length of $\mathcal{E}_2$ captures the contribution $s_{\mathcal{A}}(\mathcal{A}_2)$ from $\mathcal{A}_2$ to the entanglement entropy $S_{\mathcal{A}}$, hence gives the contour function for entanglement entropy. Furthermore we propose that $s_{\mathcal{A}}(\mathcal{A}_2)$ in general can be written as a simple linear combination of entanglement entropies of single intervals inside $\mathcal{A}$. This proposal passes several non-trivial tests.

To formulate the universal constraints of quantum statistics data of generic long-range entangled quantum systems, we introduce the geometric-topology surgery theory on spacetime manifolds where quantum systems reside, cutting and gluing the associated quantum amplitudes, specifically in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable of creating anyonic excitations of particles and strings, well-defined in gapped states of matter with intrinsic topological orders. Second, we introduce the braiding statistics data of particles and strings, such as the geometric Berry matrices for particle-string Aharonov-Bohm, 3-string, 4-string, or multi-string adiabatic loop braiding process, encoded by submanifold linkings, in the closed spacetime 3-manifolds and 4-manifolds. Third, we derive new `quantum surgery' formulas and constraints, analogous to Verlinde formula associating fusion and braiding statistics data via spacetime surgery, essential for defining the theory of topological orders, 3d and 4d TQFTs and potentially correlated to bootstrap boundary physics such as gapless modes, extended defects, 2d and 3d conformal field theories or quantum anomalies. This article is meant to be an extended and further detailed elaboration of our previous work [arXiv:1602.05951] and Chapter 6 of [arXiv:1602.05569]. Our theory applies to general quantum theories and quantum mechanical systems, also applicable to, but not necessarily requiring the quantum field theory description.