We extend previous work on the perturbative expansion of the Renyi entropy, $S_q$, around $q=1$ for a spherical entangling surface in a general CFT. Applied to conformal scalar fields in various spacetime dimensions, the results appear to conflict with the known conformal scalar Renyi entropies. On the other hand, the perturbative results agree with known Renyi entropies in a variety of other theories, including theories of free fermions and vector fields and theories with Einstein gravity duals. We propose a resolution stemming from a careful consideration of boundary conditions near the entangling surface. This is equivalent to a proper treatment of total-derivative terms in the definition of the modular Hamiltonian. As a corollary, we are able to resolve an outstanding puzzle in the literature regarding the Renyi entropy of ${\cal N}=4$ super-Yang-Mills near $q=1$. A related puzzle regards the question of stationarity of the renormalized entanglement entropy (REE) across a circle for a (2+1)-dimensional massive scalar field. We point out that the boundary contributions to the modular Hamiltonian shed light on the previously-observed non-stationarity. Moreover, IR divergences appear in perturbation theory about the massless fixed point that inhibit our ability to reliably calculate the REE at small non-zero mass.
We introduce and study generalized Rényi entropies defined through the traces of products of ${\rm Tr}_B (|Ψ_i\rangle\langle Ψ_j|)$ where $|Ψ_i\rangle$ are eigenstates of a two-dimensional conformal field theory (CFT). When $|Ψ_i\rangle=|Ψ_j\rangle$ these objects reduce to the standard Rényi entropies of the eigenstates of the CFT. Exploiting the path integral formalism, we show that the second generalized Rényi entropies are equivalent to four-point correlators. We then focus on a free bosonic theory for which the mode expansion of the fields allows us to develop an efficient strategy to comp…
We consider the ground state of a theory composed by $M$ species of massless complex bosons in one dimension coupled together via a conformal interface. We compute both the Rényi entropy and the negativity of a generic partition of wires, generalizing the approach developed in a recent work for free fermions. These entanglement measures show a logarithmic growth with the system size, and the universal prefactor depends both on the details of the interface and the bipartition. We test our analytical predictions against exact numerical results for the harmonic chain.
We investigate the ground state of a (1+1)-dimensional conformal field theory built with $M$ species of massless free Dirac fermions coupled at one boundary point via a conformal junction/interface. Each CFT represents a wire of finite length $L$. We develop a systematic strategy to compute the Rényi entropies for a generic bipartition between the wires and the entanglement negativity between two non-complementary sets of wires. Both these entanglement measures turn out to grow logarithmically with $L$ with an exactly calculated universal prefactor depending on the details of the junction and …
Renyi熵和中心荷，$C_T$，计算一个共精确的p-形式上的一个偶数球，特别是参考共形不变的情况下。它示出，例如，纠缠熵是减去标准的共形异常，没有“移位”被要求。当使用双曲技术时，共形p-形式所需的移位在数值基础上被预测为（减去）共形（p-1）形式的纠缠熵。中心荷在数值上与Buchel { \it et al}的一般公式一致。
The effect of a co--dimension--2 spherical monodromy defect on the free--field theory of a conformal scalar is investigated. The conformal anomaly is computed (in two ways) and thence the Rényi and entanglement entropies. The $C_T$ central charge is found to be negative for a range of monodromy flux.The calculations are pursued for any even dimensional sphere. A Possible conflict with existing results is raised.
Renyi熵S_q是量子纠缠的有用度量;它们可以从提升到q次幂的约化密度矩阵的迹计算，其中q>=0。对于（d+1）维共形场论，S^{d-1}上的Renyi熵可以从这些理论在（d+1）维de Sitter空间或R x H^d上的热配分函数中提取，其中H^d是d维双曲空间。这些热配分函数又可以分别表示为（d+1）维球面和S ^1 x H^d的分支覆盖上的路径积分。我们计算了d=2时自由无质量标量和费米子的Renyi熵，并展示了如何利用ζ函数正则化，发现S^3的分支覆盖和S^1 × H^2的分支覆盖的计算结果是一致的。大量的自由场的类似计算提供单调插值函数之间的Renyi熵在高斯和平凡的不动点。最后，我们讨论了类似的Renyi熵计算d>2。
We study the excited state Rényi entropy and subsystem Schatten distance in the two-dimensional free massless non-compact bosonic field theory, which is a conformal field theory. The discretization of the free non-compact bosonic theory gives the harmonic chain with local couplings. We consider the field theory excited states that correspond to the harmonic chain states with excitations of more than one quasiparticle, which we call multi-particle states. This extends the previous work by the same authors to more general excited states. In the field theory we obtain the exact Rényi entropy and …
We calculate the $N$ interval replicated negativity for the Ising and free compact boson conformal field theory using correlation functions of branch point twist fields. For some subset $P\subset N$, this is a calculation of $\rm{Tr}(ρ_A^{T_P})^n$ where $n$ is the replica index. This can be reformulated as a calculation of partition functions over superelliptic Riemann surfaces, and for the models in question, this partition function can be expressed in terms of the period matrix of this surface. We detail how to construct the period matrices for these surfaces, giving an analytic expression f…
A universal approach to Renyi entropy of multiple disjoint intervals
Beihang University（北京航空航天大学）；Center for Gravitational Physics, Department of Space Science, Beihang University（北京航空航天大学空间科学学院引力物理中心）；Peng Huanwu Collaborative Center for Research and Education, Beihang University（北京航空航天大学彭桓武院）
We study the properties of pseudo entropy, a new generalization of entanglement entropy, in free Maxwell field theory in $d = 4$ dimension. We prepare excited states by the different components of the field strengths located at different Euclidean times acting on the vacuum. We compute the difference between the pseudo Rényi entropy and the Rényi entropy of the ground state and observe that the difference changes significantly near the boundary of the subsystems and vanishes far away from the boundary. Near the boundary of the subsystems, the difference between pseudo Rényi entropy and Rényi e…
We study the Rényi entropy and subsystem distances on one interval for the finite size and thermal states in the critical XY chains, focusing on the critical Ising chain and XX chain with zero transverse field. We construct numerically the reduced density matrices and calculate the von Neumann entropy, Rényi entropy, subsystem trace distance, Schatten two-distance, and relative entropy. As the continuum limit of the critical Ising chain and XX chain with zero field are, respectively, the two-dimensional free massless Majorana and Dirac fermion theories, which are conformal field theories, we c…
