In this paper we calculate the entanglement entropy of two coupled gapless systems in general spatial dimension d. The gapless systems can be either conformal field theories (CFT), or Fermi liquids. We assume the two systems are coupled uniformly in a h-dimensional submanifold of the space, with 0 <= h <= d. We will focus on the scaling of the entanglement entropy with the size of the system, and its scaling with the inter-system coupling constant g. Three approaches will be used for our calculation: (1) exact calculation with ground state wave-functional, (2) perturbative calculation with functional path integral, (3) scaling argument.
The entanglement entropy in Kondo impurity systems is studied analytically using conformal field theory. From the impurity contribution to the scaling corrections of the entanglement entropy we extract information about the screening cloud profile for general non-Fermi-liquid fixed points. By also considering the finite-temperature corrections to scaling of the von Neumann entropy we point out a direct connection between the large-distance screening cloud profile and thermodynamic observables such as the specific heat.
The multi-scale entanglement renormalization ansatz (MERA) provides a natural description of the ground state of a quantum critical Hamiltonian on the lattice. From an optimized MERA, one can extract the scaling dimensions of the underlying conformal field theory. However, extracting conformal data from the ascending superoperator derived from MERA seems to have a limited range of accuracy, even after increasing the bond dimension. Here, we propose an alternative ansatz based on an increasing number of disentangling layers. This leads to generalized versions of MERA that improve the extraction…
We consider entanglement through permeable interfaces in the c=1 (1+1)-dimensional conformal field theory. We compute the partition functions with the interfaces inserted. By the replica trick, the entanglement entropy is obtained analytically. The entropy scales logarithmically with respect to the size of the system, similarly to the universal scaling of the ordinary entanglement entropy in (1+1)-dimensional conformal field theory. Its coefficient, however, is not constant but controlled by the permeability, the dependence on which is expressed through the dilogarithm function. The sub-leadin…
We introduce a series of quantities which characterizes a given local operator in conformal field theories from the viewpoint of quantum entanglement. It is defined by the increased amount of (Renyi) entanglement entropy at late time for an excited state defined by acting the local operator on the vacuum. We consider a conformal field theory on an infinite space and take the subsystem in the definition of the entanglement entropy to be its half. We calculate these quantities for a free massless scalar field theory in 2, 4 and 6 dimensions. We find that these results are interpreted in terms of…
We argue that the entanglement of purification for two dimensional holographic CFT can be obtained from conformal blocks with internal twist operators. First, we explain our formula from the view point of tensor network model of holography. Then, we apply it to bipartite mixed states dual to subregion of AdS${}_3$ and the static BTZ blackhole geometries. The formula in CFT agrees with the entanglement wedge cross section in the bulk, which has been recently conjectured to be equivalent to the entanglement of purification.
We show that in 1+1 dimensional conformal field theories, exciting a state with a local operator increases the Renyi entanglement entropies by a constant which is the same for every member of the conformal family. Hence, it is an intrinsic parameter that characterises local operators from the perspective of quantum entanglement. In rational conformal field theories this constant corresponds to the logarithm of the quantum dimension of the primary operator. We provide several detailed examples for the second Renyi entropies and a general derivation.
An exponential deformation of 1D critical Hamiltonians gives rise to ground states whose entanglement entropy satisfies a volume-law. This effect is exemplified in the XX and Heisenberg models. In the XX case we characterize the crossover between the critical and the maximally entangled ground state in terms of the entanglement entropy and the entanglement spectrum.
We study nonchiral wave functions for systems with continuous spins obtained from the conformal field theory (CFT) of a free, massless boson. In contrast to the case of discrete spins, these can be treated as bosonic Gaussian states, which allows us to efficiently compute correlations and entanglement properties both in one (1D) and in two spatial dimensions (2D). In 1D, the computed entanglement entropy and spectra are in agreement with the underlying CFT. Furthermore, we construct a 1D parent Hamiltonian with a low-energy spectrum corresponding to that of a free, massless boson. In 2D, we fi…
Bipartite entanglement entropy of a segment with the length $l$ in $1+1$ dimensional conformal field theories (CFT) follows the formula $S=\frac{c}{3}\ln l+γ$, where $c$ is the central charge of the CFT and $γ$ is a cut-off dependent constant which diverges in the absence of an ultraviolet cutoff. According to this formula, systems with larger central charges have {\it{more}} bipartite entanglement entropy. Using quantum Fisher information (QFI), we argue that systems with bigger central charges not only have larger bipartite entanglement entropy but also have more multipartite entanglement co…
We derive dynamics of the entanglement wedge cross section directly from the two-dimensional holographic CFTs with a local operator quench. This derivation is based on the reflected entropy, a correlation measure for mixed states. We further compare these results with the mutual information and ones for RCFTs. Our results directly suggest the classical correlation also plays an important role in the subregion/subregion duality even for dynamical setup. Besides a local operator quench, we study the reflected entropy in a heavy state and provide improved bulk interpretation. We checked the above…
We review the conformal field theory approach to entanglement entropy. We show how to apply these methods to the calculation of the entanglement entropy of a single interval, and the generalization to different situations such as finite size, systems with boundaries, and the case of several disjoint intervals. We discuss the behaviour away from the critical point and the spectrum of the reduced density matrix. Quantum quenches, as paradigms of non-equilibrium situations, are also considered.
Quantifying entanglement properties of mixed states in quantum field theory via entanglement of purification and reflected entropy is a new and challenging subject. In this work, we study both quantities for two spherical subregions far away from each other in the vacuum of a conformal field theory in any number of dimensions. Using lattice techniques, we find an elementary proof that the decay of both, the entanglement of purification and reflected entropy, is enhanced with respect to the mutual information behaviour by a logarithm of the distance between the subregions. In the case of the Is…
For two dimensional conformal field theories in the ground state, it is known that a conformal interface along the entanglement cut can suppress the entanglement entropy from $S_A\sim c\log L$ to $S_A\sim c_{\text{eff}}\log L$, where $L$ is the length of the subsystem $A$, and $c_{\text{eff}}\in [0, c]$ is the effective central charge which depends on the transmission property of the conformal interface. In this work, by making use of conformal mappings, we show that a conformal interface has the same effect on entanglement evolution in non-equilibrium cases, including global, local and certai…
We study the time evolution of the entanglement negativity after a local quantum quench in (1+1)-dimensional conformal field theories (CFTs), which we introduce by suddenly joining two initially decoupled CFTs at their endpoints. We calculate the negativity evolution for both adjacent intervals and disjoint intervals explicitly. For two adjacent intervals, the entanglement negativity grows logarithmically in time right after the quench. After developing a plateau-like feature, the entanglement negativity drops to the ground-state value. For the case of two spatially separated intervals, a ligh…
We study the entanglement entropies in one-dimensional open critical systems, whose effective description is given by a conformal field theory with boundaries. We show that for pure-state systems formed by the ground state or by the excited states associated to primary fields, the entanglement entropies have a finite-size behavior that depends on the correlation of the underlying field theory. The analytical results are checked numerically, finding excellent agreement for the quantum chains ruled by the theories with central charge $c=1/2$ and $c=1$.
The entanglement entropy of a pure quantum state of a bipartite system $A \cup B$ is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. Critical ground states of local Hamiltonians in one dimension have entanglement that diverges logarithmically in the subsystem size, with a universal coefficient that for conformally invariant critical points is related to the central charge of the conformal field theory. We find the entanglement entropy for a standard class of $z=2$ quantum critical points in two spatial dimensions with scale invari…
We propose a method of computing and studying entanglement quantities in non-Hermitian systems by use of a biorthogonal basis. We find that the entanglement spectrum characterizes the topological properties in terms of the existence of mid-gap states in the non-Hermitian Su-Schrieffer-Heeger (SSH) model with parity and time-reversal symmetry (PT symmetry) and the non-Hermitian Chern insulators. In addition, we find that at a critical point in the PT symmetric SSH model, the entanglement entropy has a logarithmic scaling with corresponding central charge $c=-2$. This critical point then is a fr…
We calculate analytically the Rényi bipartite entanglement entropy $S_α$ of the ground state of $1+1$ dimensional conformal field theories (CFT) after performing a projective measurement in a part of the system. We show that the entanglement entropy in this setup is dependent on the central charge and the operator content of the system. When the measurement region $A$ separates the two parts $B$ and $\bar{B}$, the entanglement entropy between $B$ and $\bar{B}$ decreases like a power-law with respect to the characteristic distance between the two regions with an exponent which is dependent on t…
In this work, we study the time evolution of the entanglement hamiltonian during the process of thermalization in a (1+1)-dimensional conformal field theory (CFT) after a quantum quench from a special class of initial states. In particular, we focus on a subsystem which is a finite interval at the end of a semi-infinite line. Based on conformal mappings, the exact forms of both entanglement hamiltonian and entanglement spectrum of the subsystem can be obtained. Aside from various interesting features, it is found that in the infinite time limit the entanglement hamiltonian and entanglement spe…
We compute the entanglement entropy of the Wilson-Fisher conformal field theory (CFT) in 2+1 dimensions with O($N$) symmetry in the limit of large $N$ for general entanglement geometries. We show that the leading large $N$ result can be obtained from the entanglement entropy of $N$ Gaussian scalar fields with their mass determined by the geometry. For a few geometries, the universal part of the entanglement entropy of the Wilson-Fisher CFT equals that of a CFT of $N$ massless scalar fields. However, in most cases, these CFTs have a distinct universal entanglement entropy even at $N=\infty$. No…
We show that the dynamics resulting from preparing a one-dimensional quantum system in the ground state of two decoupled parts, then joined together and left to evolve unitarily with a translational invariant Hamiltonian (a local quench), can be described by means of quantum field theory. In the case when the corresponding theory is conformal, we study the evolution of the entanglement entropy for different bi-partitions of the line. We also consider the behavior of one- and two-point correlation functions. All our findings may be explained in terms of a picture, that we believe to be valid mo…
We consider the time evolution of mixed state correlation measures in two-dimensional conformal field theories, such as logarithmic negativity, odd entropy, and reflected entropy, after quantum quenches of various kinds. These correlation measures, in the holographic context, are all associated to the entanglement wedge cross section. We contrast various classes of conformal field theories, both rational and irrational (pure) conformal field theories. First, for rational conformal field theories, whose dynamics can be well described by the quasi-particle picture, we find all four quantities fo…
In this paper we apply the formalism of translation invariant (continuous) matrix product states in the thermodynamic limit to $(1+1)$ dimensional critical models. Finite bond dimension bounds the entanglement entropy and introduces an effective finite correlation length, so that the state is perturbed away from criticality. The assumption that the scaling hypothesis holds for this kind of perturbation is known in the literature as finite entanglement scaling. We provide further evidence for the validity of finite entanglement scaling and based on this formulate a scaling algorithm to estimate…
Many quantum information theoretic quantities are similar to and/or inspired by thermodynamic quantities, with entanglement entropy being a well-known example. In this paper, we study a less well-known example, capacity of entanglement, which is the quantum information theoretic counterpart of heat capacity. It can be defined as the second cumulant of the entanglement spectrum and can be loosely thought of as the variance in the entanglement entropy. We review the definition of capacity of entanglement and its relation to various other quantities such as fidelity susceptibility and Fisher info…
The Fermi surface may be usefully viewed as a collection of 1+1 dimensional chiral conformal field theories. This approach permits straightforward calculation of many anomalous ground state properties of the Fermi gas including entanglement entropy and number fluctuations. The 1+1 dimensional picture also generalizes to finite temperature and the presence of interactions. Finally, I argue that the low energy entanglement structure of Fermi liquid theory is universal, depending only on the geometry of the interacting Fermi surface.
By making use of conformal mapping, we construct various time-evolution operators in (1+1) dimensional conformal field theories (CFTs), which take the form $\int dx\, f(x) \mathcal{H}(x)$, where $\mathcal{H}(x)$ is the Hamiltonian density of the CFT, and $f(x)$ is an envelope function. Examples of such deformed evolution operators include the entanglement Hamiltonian, and the so-called sine-square deformation of the CFT. Within our construction, the spectrum and the (finite-size) scaling of the level spacing of the deformed evolution operator are known exactly. Based on our construction, we al…
We study the dynamics of entanglement entropy for weakly excited states in conformal field theories by using the AdS/CFT. This is aimed at a first step to find a counterpart of Einstein equation in the CFT language. In particular, we point out that the entanglement entropy satisfies differential equations which directly correspond to the Einstein equation in several setups of AdS/CFT. We also define a quantity called entanglement density in higher dimensional field theories and study its dynamical property for weakly excited states in conformal field theories.
Quantum phase transitions out of a symmetry-protected topological (SPT) phase in (1+1) dimensions into an adjacent, topologically distinct SPT phase protected by the same symmetry or a trivial gapped phase, are typically described by a conformal field theory (CFT). At the same time, the low-lying entanglement spectrum of a gapped phase close to such a quantum critical point is known(Cho et al., arXiv:1603.04016), very generally, to be universal and described by (gapless) boundary conformal field theory. Using this connection we show that symmetry properties of the boundary conditions in bounda…
Using the density-matrix renormalization-group, we investigate the critical behavior of the anisotropic Heisenberg chains with spins up to $S=9/2$. We show that through the relations arising from the conformal invariance and the DMRG technique it is possible to obtain accurate finite-size estimates of the conformal anomaly $c$, the sound velocity $v_{s}$, the anomalous dimension $x_{bulk}$, and the surface exponent $x_{s}$ of the anisotropic spin-$S$ Heisenberg chains with relatively good accuracy without fitting parameters. Our results indicate that the entanglement entrop $S(L,l_{A},S)$ of t…
I discuss the von Neumann entanglement entropy in two-dimensional quantum Lifshitz criical point, namely in Rokhsar-Kivelson type critical wavefunctions. I follow the approach proposed by B. Hsu et al. [Phys. Rev. B 79, 115421 (2009)], but point out a subtle problem concerning compactification of replica boson fields: although one can define a set of new boson fields by linear combinations of the original fields, the new fields are not compactified independently. In order to systematically study boundary conformal field theory of multicomponent free bosons, I employ a geometric formulation bas…
We calculate the amount of entanglement shared by two intervals in the ground state of a (1+1)-dimensional conformal field theory (CFT), quantified by an entanglement measure $\mathcal{E}$ based on the computable cross norm (CCNR) criterion. Unlike negativity or mutual information, we show that $\mathcal{E}$ has a universal expression even for two disjoint intervals, which depends only on the geometry, the central charge c, and the thermal partition function of the CFT. We prove this universal expression in the replica approach, where the Riemann surface for calculating $\mathcal{E}$ at each o…
The entanglement entropy (EE) of quantum systems is often used as a test of low-energy descriptions by conformal field theory (CFT). Here we point out that this is not a reliable indicator, as the EE often shows the same behavior even when a CFT description is not correct (as long as the system is asymptotically scale-invariant). We use constraints on the scaling dimension given by the CFT with SU(2) symmetry to provide alternative tests with two- and four-point correlation functions, showing examples for quantum spin models in 1+1 dimensions. In the case of a critical amplitude-product state …
The one-dimensional free Fermi gas is a prototype conformally invariant system, whose entanglement properties are well-understood. In this work, the effects of a single impurity on one dimensional free fermion entanglement entropy are studied both analytically and numerically. Such an impurity represents an exactly marginal perturbation to the bulk conformally invariant fixed point. We find that the impurity leads to sub-leading contributions to the entanglement entropy that scale inversely with the subsystem size. The origin of such contributions are identified.
A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from AdS/CFT correspondence. We argue that the entanglement entropy in d+1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS_{d+2}, analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal perfectly reproduces the correct entanglement entropy in 2D CFT when applied to AdS_3. We also compare the entropy computed in AdS_5 \times S^5 with that of the free N=4 super Yang-Mills.
Entanglement evolutions after a global quantum quench and a local quantum quench in 1+1 dimensional conformal field theories (CFTs) show qualitatively different behaviors, and are studied within two different setups. In this work, we bridge global and local quantum quenches in (1+1)-d CFTs in the same setup, by studying the entanglement evolution from a specific inhomogeneous initial state. By utilizing conformal mappings, this inhomogeneous quantum quench is analytically solvable. It is found that the entanglement evolution shows a global quantum quench feature in the short time limit, and a …
We consider the large $N$ interacting vector $O(N)$ model on a sphere in $4-ε$ Euclidean dimensions. The Gaussian theory in the UV is taken to be either conformally or non-conformally coupled. The endpoint of the RG flow corresponds to a conformally coupled scalar field at the Wilson-Fisher fixed point. We take a spherical entangling surface in de Sitter space and compute the entanglement entropy everywhere along the RG trajectory. In $4$ dimensions, a free non-conformal scalar has a universal area term scaling with the logarithm of the UV cutoff. In $4-ε$ dimensions, such a term scales as $1/…
We study the contribution to the entanglement entropy of (2+1)-dimensional conformal field theories coming from a sharp corner in the entangling surface. This contribution is encoded in a function $a(θ)$ of the corner opening angle, and was recently proposed as a measure of the degrees of freedom in the underlying CFT. We show that the ratio $a(θ)/C_T$, where $C_T$ is the central charge in the stress tensor correlator, is an almost universal quantity for a broad class of theories including various higher-curvature holographic models, free scalars and fermions, and Wilson-Fisher fixed points of…
We study the structure of divergences and universal terms of the entanglement and Rényi entropies for singular regions. First, we show that for $(3+1)$-dimensional free conformal field theories (CFTs), entangling regions emanating from vertices give rise to a universal contribution $S_n^{\rm univ}= -\frac{1}{8π}f_b(n) \int_γ k^2 \log^2(R/δ)$, where $γ$ is the curve formed by the intersection of the entangling surface with a unit sphere centered at the vertex, and $k$ the trace of its extrinsic curvature. While for circular and elliptic cones this term reproduces the general-CFT result, it vani…
We study the dynamics of the entanglement in one dimensional critical quantum systems after a local quench in which two independently thermalized semi-infinite halves are joined to form a homogeneous infinite system and left to evolve unitarily. We show that under certain conditions a nonequilibrium steady state (NESS) is reached instantaneously as soon as the entanglement interval is within the light-cone emanating from the contact point. In this steady state, the exact expressions for the entanglement entropy and the logarithmic negativity are in agreement with the steady state density matri…
Given a $generic$ two-dimensional conformal field theory (CFT), we propose an analytically solvable setup to study the Floquet dynamics of the CFT, i.e., the dynamics of a CFT subject to a periodic driving. A complete phase diagram in the parameter space can be analytically obtained within our setup. We find two phases: the heating phase and the non-heating phase. In the heating phase, the entanglement entropy keeps growing linearly in time, indicating that the system keeps absorbing energy; in the non-heating phase, the entanglement entropy oscillates periodically in time, i.e., the system is…
We study the energy and entanglement dynamics of $(1+1)$D conformal field theories (CFTs) under a Floquet drive with the sine-square deformed (SSD) Hamiltonian. Previous work has shown this model supports both a non-heating and a heating phase. Here we analytically establish several robust and `super-universal' features of the heating phase which rely on conformal invariance but not on the details of the CFT involved. First, we show the energy density is concentrated in two peaks in real space, a chiral and anti-chiral peak, which leads to an exponential growth in the total energy. The peak lo…
We carry out a systematic study of entanglement entropy in nonrelativistic conformal field theories via holographic techniques. After a discussion of recent results concerning Galilean conformal field theories, we deduce a novel expression for the entanglement entropy of (1+1)-dimensional Lifshitz field theories --- this is done both at zero and finite temperature. Based on these results, we pose a conjecture for the anomaly coefficient of a Lifshitz field theory dual to new massive gravity. It is found that the Lifshitz entanglement entropy at finite temperature displays a striking similarity…
