We study three different measures of quantum correlations -- entanglement spectrum, entanglement entropy, and logarithmic negativity -- for (1+1)-dimensional massive scalar field in flat spacetime. The entanglement spectrum for the discretized scalar field in the ground state indicates a cross-over in the zero-mode regime, which is further substantiated by an analytical treatment of both entanglement entropy and logarithmic negativity. The exact nature of this cross-over depends on the boundary conditions used -- the leading order term switches from a $\log$ to $\log-\log$ behavior for the Periodic and Neumann boundary conditions. In contrast, for Dirichlet, it is the parameters within the leading $\log-\log$ term that are switched. We show that this cross-over manifests as a change in the behavior of the leading order divergent term for entanglement entropy and logarithmic negativity close to the zero-mode limit. We thus show that the two regimes have fundamentally different information content. Furthermore, an analysis of the ground state fidelity shows us that the region between critical point $Λ=0$ and the crossover point is dominated by zero-mode effects, featuring an explicit dependence on the IR cutoff of the system. For the reduced state of a single oscillator, we show that this cross-over occurs in the region $Nam_f\sim \mathscr{O}(1)$.
We investigate the behavior of entanglement entropy in the holographic QCD model proposed by Gubser et al. By choosing suitable parameters of the scalar self-interaction potential, this model can exhibit various types of phase structures: crossover, first order and second order phase transitions. We use entanglement entropy to probe the crossover/phase transition, and find that it drops quickly/suddenly when the temperature approaches the critical point which can be seen as a signal of confinement. Moreover, the critical behavior of the entanglement entropy suggests that we may use it to chara…
For quantum matter, eigenstate entanglement entropies obey an area law or log-area law at low energies and small subsystem sizes and cross over to volume laws for high energies and large subsystems. This transition is captured by crossover functions, which assume a universal scaling form in quantum critical regimes. We demonstrate this for the harmonic lattice model, which describes quantized lattice vibrations and is a regularization for free scalar field theories, modeling, e.g., spin-0 bosonic particles. In one dimension, the groundstate entanglement obeys a log-area law. For dimensions $d\…
We postulate the existence of universal crossover functions connecting the universal parts of the entanglement entropy to the low temperature thermal entropy in gapless quantum many-body systems. These scaling functions encode the intuition that the same low energy degrees of freedom which control low temperature thermal physics are also responsible for the long range entanglement in the quantum ground state. We demonstrate the correctness of the proposed scaling form and determine the scaling function for certain classes of gapless systems whose low energy physics is described by a conformal …
A new numerical approach to entanglement entropies of the Renyi type is proposed for one-dimensional quantum field theories. The method extends the truncated conformal spectrum approach and we will demonstrate that it is especially suited to study the crossover from massless to massive behavior when the subsystem size is comparable to the correlation length. We apply it to different deformations of massless free fermions, corresponding to the scaling limit of the Ising model in transverse and longitudinal fields. For massive free fermions the exactly known crossover function is reproduced alre…
We compute the crossover exponents of all quadratic and cubic deformations of critical field theories with permutation symmetry $S_q$ in $d=6-ε$ (Landau-Potts field theories) and $d=4-ε$ (hypertetrahedral models) up to three loops.We use our results to determine the $ε$-expansion of the fractal dimension of critical clusters in the most interesting cases, which include spanning trees and forests ($q\to0$), and bond percolations ($q\to1$). We also explicitly verify several expected degeneracies in the spectrum of relevant operators for natural values of $q$ upon analytic continuation, which are…
We study the entanglement entropy of a region of length 2L with the remainder of an infinite one dimensional gapless quantum system in the case where the region is centered on a quantum impurity. The coupling to this impurity is not scale invariant, and the physics involves a crossover between weak and strong coupling regimes. While the impurity contribution to the entanglement has been computed numerically in the past, little is known analytically about it, since in particular the methods of conformal invariance cannot be applied because of the presence of a crossover length. We show in this …
We study the time evolution of the entanglement entropy in bosonic systems with time-independent, or time-periodic, Hamiltonians. In the first part, we focus on quadratic Hamiltonians and Gaussian initial states. We show that all quadratic Hamiltonians can be decomposed into three parts: (a) unstable, (b) stable, and (c) metastable. If present, each part contributes in a characteristic way to the time-dependence of the entanglement entropy: (a) linear production, (b) bounded oscillations, and (c) logarithmic production. In the second part, we use numerical calculations to go beyond Gaussian st…
We study the Von Neumann and Rényi entanglement entropy of long-range harmonic oscillators (LRHO) by both theoretical and numerical means. We show that the entanglement entropy in massless harmonic oscillators increases logarithmically with the sub-system size as $S=\frac{c_{eff}}{3}\log l$. Although the entanglement entropy of LRHO's shares some similarities with the entanglement entropy at conformal critical points we show that the Rényi entanglement entropy presents some deviations from the expected conformal behavior. In the massive case we demonstrate that the behavior of the entanglement…
In this work we study the flow of holographic entanglement entropy in dimensions $d \geq 3$ in the gauge/gravity duality set up. We observe that a generalized entanglement temperature $T_g$ can be defined which gives the Hawking temperature $T_H$ in the infrared region and leads to a generalized thermodynamics like law $E= \left(\frac{d-1}{d}\right)T_g~S_{REE}$, which becomes an exact relation in the entire region of the subsystem size $l$, including both the infrared ($l\rightarrow\infty$) as well as the ultraviolet ($l\rightarrow 0$) regions. Furthermore, in the IR limit, $T_g$ produces the …
We calculate the thermal correction to the entanglement spectrum for separating a single interval of two dimensional conformal field theories. Our derivation is a direct extension of the thermal correction to the Rényi entropy. Within a low-temperature expansion by including only the first excited state in the thermal density matrix, we approach analytical results of the thermal correction to the entanglement spectrum at both of the small and large interval limit. We find the temperature correction reduces the large eigenvalues in the entanglement spectrum while increases the small eigenvalues…
The holographic complexity of a `black' non-susy $D3$-brane is computed. The difference in the holographic complexity between this geometry in the Fefferman-Graham coordinates and that of the $AdS_5$ geometry is obtained for a strip type subsystem. This is then related to the changes in the energy and the entanglement entropy of the system. We next take the high temperature limit of the change in complexity and observe that it scales with the temperature in the same way as the holographic entanglement entropy. The crossover of the holographic complexity to its corresponding thermal counterpart…
The entanglement entropy of free fermions with a Fermi surface is known to obey a logarithmic scaling and violate the area law in all dimensions. Here, we would like to see how temperature affects the logarithmic scaling behavior. To this end, we compute the entanglement negativity of free fermions using the fermionic partial transpose developed in our earlier paper [Phys. Rev. B 95, 165101 (2017)]. In one dimension, we analytically derive the leading order term in the finite-temperature entanglement negativity and show how the entanglement negativity indicates a crossover from a quantum entan…
Using Rindler method we derive the logarithmic correction to the entanglement entropy of a two dimensional BMS-invariant field theory (BMSFT). In particular, we present a general formula for extraction of the logarithmic corrections to both the thermal and the entanglement entropies. We also present a CFT formula related to the logarithmic correction of the BTZ inner horizon entropy which results in our formula after taking appropriate limit.
We study the time dependence of the entanglement between two quantum wires after suddenly connecting them via tunneling through an impurity. The result at large times is given by the well known formula $S(t) \approx {1\over 3}\ln {t}$. We show that the intermediate time regime can be described by a universal cross-over formula $S=F(tT_K)$, where $T_K$ is the crossover (Kondo) temperature: the function $F$ describes the dynamical "healing" of the system at large times. We discuss how to obtain analytic information about $F$ in the case of an integrable quantum impurity problem using the massles…
In recent years, various aspects of theoretical models with long range interactions have attracted attention, ranging from out-of-time-ordered correlators to entanglement. In the present paper, entanglement properties of a simple non-local model with long-range interactions in the form of a fractional Laplacian is investigated in both static and a quantum quench scenario. Logarithmic negativity, which is a measure for entanglement in mixed states is calculated numerically. In the static case, it is shown that the presence of long-range interaction ensures that logarithmic negativity decays muc…
We compute the entanglement entropy of minimally coupled scalar fields on subtracted geometry black hole backgrounds, focusing on the logarithmic corrections. We notice that matching between the entanglement entropy of original black holes and their subtracted counterparts is only at the order of the area term. The logarithmic correction term is not only different but also, in general, changes sign in the subtracted case. We apply Harrison transformations to the original black holes and find out the choice of the Harrison parameters for which the logarithmic corrections vanish.
We study holographic entanglement entropy for certain logarithmic conformal field theories by making use of their gravity descriptions. The corresponding gravity descriptions are provided by higher derivative gravity at critical points where the equations of motion degenerate leading to a log gravity. When the central charge of the dual theory is zero, the entanglement entropy has a new divergent term whose coefficient is given by the new anomaly of the logarithmic conformal field theory.
We compute the entanglement entropy of a wide class of exactly solvable models which may be characterized as describing matter coupled to gauge fields. Our principle result is an entanglement sum rule which states that entropy of the full system is the sum of the entropies of the two components. In the context of the exactly solvable models we consider, this result applies to the full entropy, but more generally it is a statement about the additivity of universal terms in the entropy. We also prove that the Renyi entropy is exactly additive and hence that the entanglement spectrum factorizes. …
We discuss and compute entanglement entropy (EE) in (1+1)-dimensional free Lifshitz scalar field theories with arbitrary dynamical exponents. We consider both the subinterval and periodic sublattices in the discretized theory as subsystems. In both cases, we are able to analytically demonstrate that the EE grows linearly as a function of the dynamical exponent. Furthermore, for the subinterval case, we determine that as the dynamical exponent increases, there is a crossover from an area law to a volume law. Lastly, we deform Lifshitz field theories with certain relevant operators and show that…
Logarithmic negativity typically equals exact entanglement cost
SJTU Paris Elite Institute of Technology, Shanghai Jiao Tong University（上海交通大学巴黎卓越工程师学院）；Martin Fisher School of Physics, Brandeis University（布兰代斯大学马丁费舍尔物理系）；School of Natural Sciences, Institute for Advanced Study（高等研究院自然科学学院）
The entanglement entropy correlates two quantum sub-systems which are the part of the larger system. A logarithmic divergence term present in the entanglement entropy is universal in nature and directly proportional to the conformal anomaly. We study this logarithmic divergence term of entropy for massive scalar field in $(2+1)$ dimension by applying numerical techniques to entanglement entropy approach. This (2+1) dimensional massive theory can be obtained from (3+1) dimensional massless scalar field via dimensional reduction. We also calculated mass corrections to entanglement entropy for sc…
The entanglement entropy for smooth regions $\cal A$ has a logarithmic divergent contribution with a shape dependent coefficient and that for regions with conical singularities an additional $\log ^2$ term. Comparing the coefficient of this extra term, obtained by direct holographic calculation for an infinite cone, with the corresponding limiting case for the shape dependent coefficient for a regularised cone, a mismatch by a factor two has been observed in the literature. We discuss several aspects of this issue. In particular a regularisation of $\cal A$, intrinsically delivered by the holo…
The logarithmic negativity of a bipartite quantum state is a widely employed entanglement measure in quantum information theory, due to the fact that it is easy to compute and serves as an upper bound on distillable entanglement. More recently, the $κ$-entanglement of a bipartite state was shown to be the first entanglement measure that is both easily computable and has a precise information-theoretic meaning, being equal to the exact entanglement cost of a bipartite quantum state when the free operations are those that completely preserve the positivity of the partial transpose [Wang and Wild…
In a $d$-dimensional conformal field theory, it has been known that a relevant deformation operator with the conformal dimension, $Δ=\frac{d+2}{2}$, generates a logarithmic correction to the entanglement entropy. In the large 't Hooft coupling limit, we can investigate such a logarithmic correction holographically by deforming an AdS space with a massive scalar field dual to the operator with $Δ=\frac{d+2}{2}$. There are two sources generating the logarithmic correction. One is the metric deformation and the other is the minimal surface deformation. In this work, we investigate the change of t…
The entropy of entanglement between a three-dimensional slab of thickness l and its complement is studied numerically for four-dimensional SU(2) lattice gauge theory. We find a signature of a nonanalytic behavior of the entanglement entropy, which was predicted recently for large N_c confining gauge theories in the framework of AdS/CFT correspondence. The derivative of the entanglement entropy over l is likely to have a discontinuity at some l = l_c. It is argued that such behavior persists even at finite temperatures, probably turning into a sort of crossover for temperatures larger than the …
We present a direct lattice gauge theory computation that, without using dualities, demonstrates that the entanglement entropy of Yang-Mills theories with arbitrary gauge group $G$ contains a generic logarithmic term at sufficiently weak coupling $e$. In two spatial dimensions, for a region of linear size $r$, this term equals $\frac{1}{2} \dim(G) \log\left(e^2 r\right)$ and it dominates the universal part of the entanglement entropy. Such logarithmic terms arise from the entanglement of the softest mode in the entangling region with the environment. For Maxwell theory in two spatial dimension…
We compute the logarithmic coefficient of the entanglement entropy on a sphere for a Maxwell field in $d=4$ dimensions. In spherical coordinates the problem decomposes into one dimensional ones along the radial coordinate for each angular momentum. We show the entanglement entropy of a Maxwell field is equivalent to the one of two identical massless scalars from which the mode of $l=0$ has been removed. This shows the relation $c^M_{\log}=2 (c^S_{\log}-c^{S_{l=0}}_{\log})$ between the logarithmic coefficient in the entropy for a Maxwell field $c^M_{\log}$, the one for a $d=4$ massless scalar $…
We study corner contributions to holographic entanglement entropy in non-conformal backgrounds: a kink for D2-branes as well as a cone and two different types of crease for D4-branes. Unlike 2+1-dimensional CFTs, the corner contribution to the holographic entanglement entropy of D2-branes exhibits a power law behaviour rather than a logarithmic term. However, the logarithmic term emerges in the holographic entanglement entropy of D4-branes. We identify the logarithmic term for a cone in D4-brane background as the universal contribution under appropriate limits and compare it with other physica…
It was shown in Dey and Roy (2012) that the near horizon limit of the 1/4 BPS threshold F-D$p$ (for $0\leq p \leq 5$, $p \neq 4$) bound state solutions of type II string theories give rise to space-time metrics endowed with Lifshitz scaling along with hyperscaling violation. Here we compute the holographic entanglement entropy of this system for all $p \neq 4$ (for $p=4$ the space-time has AdS$_2$ structure). For $p=3,5$, we get the expected area law behavior of the entanglement entropy. For $p=0,1$, the entanglement entropy has new area law violations and has the behavior which is in between …
