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 based on compactification lattices. The result from the boundary conformal field theory agrees exactly with alternative calculations by J.-M. Stephan et al. [Phys. Rev. B 80, 184421 (2009)], confirming its universality as argued originally by B. Hsu et al.
We compute universal finite corrections to entanglement entropy for generalised quantum Lifshitz models in arbitrary odd spacetime dimensions. These are generalised free field theories with Lifshitz scaling symmetry, where the dynamical critical exponent $z$ equals the number of spatial dimensions $d$, and which generalise the 2+1-dimensional quantum Lifshitz model to higher dimensions. We analyse two cases: one where the spatial manifold is a $d$-dimensional sphere and the entanglement entropy is evaluated for a hemisphere, and another where a $d$-dimensional flat torus is divided into two cy…
We study different aspects of quantum entanglement and its measures, including entanglement entropy in the vacuum state of a certain Lifshitz scalar theory. We present simple intuitive arguments based on "non-local" effects of this theory that the scaling of entanglement entropy depends on the dynamical exponent as a characteristic parameter of the theory. The scaling is such that in the massless theory for small entangling regions it leads to area law in the Lorentzian limit and volume law in the $z\to\infty$ limit. We present strong numerical evidences in (1+1) and (2+1)-dimensions in suppor…
We study propagation of entanglement after a mass quench in free scalar Lifshitz theories. We show that entanglement entropy goes across three distinct growth regimes before relaxing to a generalized Gibbs ensemble, namely 'initial rapid growth', 'main linear growth' and 'tortoise saturation'. We show that although a wide spectrum of quasi-particles are responsible for entanglement propagation, as long as the occupation number of the zero mode is not divergent, the linear main growth regime is dominated by the fastest quasi-particle propagating on the edges of a widen light-cone. We present st…
In this work we calculate the entanglement entropy of certain excited states of the quantum Lifshitz model. The quantum Lifshitz model is a 2 + 1-dimensional bosonic quantum field theory with an anisotropic scaling symmetry between space and time that belongs to the universality class of the quantum dimer model and its generalizations. The states we consider are constructed by exciting the eigenmodes of the Laplace-Beltrami operator on the spatial manifold of the model. We perform a replica calculation and find that, whenever a simple assumption is satisfied, the bipartite entanglement entropy…
The behaviour of the 3D axial next-nearest neighbour Ising (ANNNI) model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are $α=0.18(2)$, $β=0.238(5)$ and $γ=1.36(3)$. Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling {\em at} the Lifsh…
Extending the background metric optimization procedure for Euclidean path integrals of two-dimensional conformal field theories, introduced by Caputa et al. (Phys. Rev. Lett. 119, 071602 (2017)), to a $z=2$ anisotropically scale-invariant $(2+1)$-dimensional Lifshitz field theory of a free massless scalar field, we find optimal geometries for static and dynamic correlation functions. For the static correlation functions, the optimal background metric is equivalent to an AdS metric on a Poincare patch, while for dynamical correlation functions, we find Lifshitz like metric. This results suggest…
We investigate the entanglement entropy (EE) of circular entangling cuts in the 2+1-dimensional quantum Lifshitz model, whose ground state wave function is a spatially conformal invariant state of the Rokhsar-Kivelson type, whose weight is the Gibbs weight of 2D Euclidean free boson. We show that the finite subleading corrections of EE to the area-law term as well as the mutual information are conformal invariants and calculate them for cylinder, disk-like and spherical manifolds with various spatial cuts. The subtlety due to the boson compactification in the replica trick is carefully address…
We study the growth of entanglement entropy(EE) of local operator excitation in the quantum Lifshitz model which has dynamic exponent z = 2. Specifically, we act a local vertex operator on the groundstate at a distance $l$ to the entanglement cut and calculate the EE as a function of time for the state's subsequent time evolution. We find that the excess EE compared with the groundstate is a monotonically increasing function which is vanishingly small before the onset at $t \sim l^2$ and eventually saturates to a constant proportional to the scaling dimension of the vertex operator. The quasi-…
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…
The entanglement entropy of a pure quantum state of a bipartite system 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 an entanglement entropy that diverges logarithmically in the subsystem size, with a universal coefficient that is is related to the central charge of the associated conformal field theory. Here I will discuss recent extensions of these ideas to a class of quantum critical points with dynamic critical exponent $z=2$ in two space dimensions and to…
We investigate the entanglement properties of an infinite class of excited states in the quantum Lifshitz model (QLM). The presence of a conformal quantum critical point in the QLM makes it unusually tractable for a model above one spatial dimension, enabling the ground state entanglement entropy for an arbitrary domain to be expressed in terms of geometrical and topological quantities. Here we extend this result to excited states and find that the entanglement can be naturally written in terms of quantities which we dub "entanglement propagator amplitudes" (EPAs). EPAs are geometrical probabi…
We study the time evolution of the entanglement entropy after quantum quenches in Lifshitz free scalar theories, with the dynamical exponent $z>1$, by using the correlator method. For quantum quenches we consider two types of time-dependent mass functions: end-critical-protocol (ECP) and cis-critical-protocol (CCP). In both cases, at early times the entanglement entropy is independent of the subsystem size. After a critical time ($t_c$), the entanglement entropy starts depending on the subsystem size significantly. This critical time $t_c$ for $z = 1$ in the fast ECP and CCP has been explained…
We consider Lifshitz criticalities with dynamical exponent $z=2$ that emerge in a class of topological chains. There, such a criticality plays a fundamental role in describing transitions between symmetry-enriched conformal field theories (CFTs). We report that, at such critical points in one spatial dimension, the finite-size correction to the energy scales with system size, $L$, as $\sim L^{-2}$, with universal and anomalously large coefficient. The behavior originates from the specific dispersion around the Fermi surface, $ε\propto \pm k^2$. We also show that the entanglement entropy exhibi…
We investigate quantum entanglement in a non-relativistic critical system by calculating the logarithmic negativity of a class of mixed states in the quantum Lifshitz model in one and two spatial dimensions. In 1+1 dimensions we employ a correlator approach to obtain analytic results for both open and periodic biharmonic chains. In 2+1 dimensions we use a replica method and consider spherical and toroidal spatial manifolds. In all cases, the universal finite part of the logarithmic negativity vanishes for mixed states defined on two disjoint components. For mixed states defined on adjacent com…
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 calculate the ground state entanglement entropy between two heterogeneous parts of a free fermion chain. The two parts could be XX chains with different parameters or an XX half chain connected with a quantum Ising half chain. It is shown that logarithmic behavior holds if the two parts are conformally critical. In other cases, area law holds with abundant convergence behaviors. In particular, when XX chain at Lifshitz point is connected with a conformally or Lifshitz critical part, entanglement entropy converges algebraically with a fractional subleading index.
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.
We explore the dynamical behavior at and near a special class of two-dimensional quantum critical points. Each is a conformal quantum critical point (CQCP), where in the scaling limit the equal-time correlators are those of a two-dimensional conformal field theory. The critical theories include the square-lattice quantum dimer model, the quantum Lifshitz theory, and a deformed toric code model. We show that under generic perturbation the latter flows toward the ordinary Lorentz-invariant (2+1) dimensional Ising critical point, illustrating that CQCPs are generically unstable. We exploit a corr…
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.
We study the entanglement entropies of an interval on the infinite line in the free fermionic spinless Schrödinger field theory at finite density and zero temperature, which is a non-relativistic model with Lifshitz exponent $z=2$. We prove that the entanglement entropies are finite functions of one dimensionless parameter proportional to the area of a rectangular region in the phase space determined by the Fermi momentum and the length of the interval. The entanglement entropy is a monotonically increasing function. By employing the properties of the prolate spheroidal wave functions of order…
Quantum many-body systems have a rich structure in the presence of boundaries. We study the groundstates of conformal field theories (CFTs) and Lifshitz field theories in the presence of a boundary through the lens of the entanglement entropy. For a family of theories in general dimensions, we relate the universal terms in the entanglement entropy of the bulk theory with the corresponding terms for the theory with a boundary. This relation imposes a condition on certain boundary central charges. For example, in $2+1$ dimensions, we show that the corner-induced logarithmic terms of free CFTs an…
We consider the entanglement entropy for the 2D Ising model at the conformal fixed point in the presence of interfaces. More precisely, we investigate the situation where the two subsystems are separated by a defect line that preserves conformal invariance. Using the replica trick, we compute the entanglement entropy between the two subsystems. We observe that the entropy, just like in the case without defects, shows a logarithmic scaling behavior with respect to the size of the system. Here, the prefactor of the logarithm depends on the strength of the defect encoded in the transmission coeff…
We enumerate the cases in 2d conformal field theory where the logarithm of the reduced density matrix (the entanglement or modular hamiltonian) may be written as an integral over the energy-momentum tensor times a local weight. These include known examples and new ones corresponding to the time-dependent scenarios of a global and local quench. In these latter cases the entanglement hamiltonian depends on the momentum density as well as the energy density. In all cases the entanglement spectrum is that of the appropriate boundary CFT. We emphasize the role of boundary conditions at the entangli…
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 $c=1$.
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…
We reconsider the computation of the entanglement entropy of two disjoint intervals in a (1+1) dimensional conformal field theory by conformal block expansion of the 4-point correlation function of twist fields. We show that accurate results may be obtained by taking into account several terms in the operator product expansion (OPE) of twist fields and by iterating the Zamolodchikov recursion formula for each conformal block. We perform a detailed analysis for the Ising conformal field theory and for the free compactified boson. Each term in the conformal block expansion can be easily analytic…
We investigate the effect of a global degeneracy in the distribution of entanglement spectrum in conformal field theories in one spatial dimension. We relate the recently found universal expression for the entanglement hamiltonian to the distribution of the entanglement spectrum. The main tool to establish this connection is the Cardy formula. It turns out that the Affleck-Ludwig non-integer degeneracy, appearing because of the boundary conditions induced at the entangling surface, can be directly read from the entanglement spectrum distribution. We also clarify the effect of the non-integer d…
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…
We explore a conformal field theoretic interpretation of the holographic entanglement of purification, which is defined as the minimal area of entanglement wedge cross section. We argue that in AdS3/CFT2, the holographic entanglement of purification agrees with the entanglement entropy for a purified state, obtained from a special Weyl transformation, called path-integral optimizations. By definition, this special purified state has the minimal path-integral complexity. We confirm this claim in several examples.
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 revisit conformal quantum mechanics (CQM) from the perspective of sine-square deformation (SSD) and the entanglement Hamiltonian. The operators that correspond to SSD and the entanglement Hamiltonian are identified. Thus, the nature of SSD and entanglement can be discussed in a much simpler CQM setting than higher-dimensional field theories.
For a generic conformal field theory (CFT) in four dimensions, the scale anomaly dictates that the universal part of entanglement entropy across a sphere ($\mathcal{C}_{\text{univ}}(S^{2})$) is positive. Based on this fact, we explore the consequences of assuming positive sign for $\mathcal{C}_{\text{univ}}(S^{2})$ in a four dimensional scale invariant theory (SFT). In absence of a dimension two scalar operator $\mathcal{O}_{2}$ in the spectrum of a SFT, we show that this assumption suggests that SFT is a CFT. In presence of $\mathcal{O}_{2}$, we show that this assumption can fix the coefficie…
We provide numerical evidence that the low-lying part of the entanglement spectrum of a real-space block (i.e. a single interval) of a one-dimensional quantum many body system at a conformal critical point corresponds to the energy spectrum of a boundary conformal field theory (CFT). This correspondence allows to uncover a subset of the operator content of a conformal field theory by inspection of the entanglement spectrum of a single wave function, thus providing important information on a CFT beyond its central charge. As a practical application we show that for many systems described by a c…
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 derive exact formulas for bipartite von Neumann entanglement entropy after partial projective local measurement in $1+1$ dimensional conformal field theories with periodic and open boundary conditions. After defining the set up we will check numerically the validity of our results in the case of Klein-Gordon field theory (coupled harmonic oscillators) and spin-$1/2$ XX chain in a magnetic field. The agreement between analytical results and the numerical calculations is very good. We also find a lower bound for localizable entanglement in coupled harmonic oscillators.
We discuss the Renyi entanglement entropies of descendant states in critical one-dimensional systems with boundaries, that map to boundary conformal field theories in the scaling limit. We unify the previous conformal-field-theory approaches to describe primary and descendant states in systems with both open and closed boundaries. We provide universal expressions for the first two descendants in the identity family. We apply our technique to critical systems belonging to different universality classes with non-trivial boundary conditions that preserve conformal invariance, and find excellent a…
We study the scaling behavior of the entanglement entropy of two dimensional conformal quantum critical systems, i.e. systems with scale invariant wave functions. They include two-dimensional generalized quantum dimer models on bipartite lattices and quantum loop models, as well as the quantum Lifshitz model and related gauge theories. We show that, under quite general conditions, the entanglement entropy of a large and simply connected sub-system of an infinite system with a smooth boundary has a universal finite contribution, as well as scale-invariant terms for special geometries. The unive…
We study the entanglement entropies of an interval adjacent to the boundary of the half line for the free fermionic spinless Schrödinger field theory at finite density and zero temperature, with either Neumann or Dirichlet boundary conditions. They are finite functions of the dimensionless parameter given by the product of the Fermi momentum and the length of the interval. The entanglement entropy displays an oscillatory behaviour, differently from the case of the interval on the whole line. This behaviour is related to the Friedel oscillations of the mean particle density on the half line at …
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.
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…
We study the entanglement of two disjoint intervals in the conformal field theory of the Luttinger liquid (free compactified boson). Trρ_A^n for any integer n is calculated as the four-point function of a particular type of twist fields and the final result is expressed in a compact form in terms of the Riemann-Siegel theta functions. In the decompactification limit we provide the analytic continuation valid for all model parameters and from this we extract the entanglement entropy. These predictions are checked against existing numerical data.
We use the conformal invariance and the holographic correspondence to fully specify the dependence of entanglement entropy on the extrinsic geometry of the 2d surface $Σ$ that separates two subsystems of quantum strongly coupled ${\mathcal{N}}=4$ SU(N) superconformal gauge theory. We extend this result and calculate entanglement entropy of a generic 4d conformal field theory. As a byproduct, we obtain a closed-form expression for the entanglement entropy in flat space-time when $Σ$ is sphere $S_2$ and when $Σ$ is two-dimensional cylinder. The contribution of the type A conformal anomaly to ent…
We present a new method of deriving the geometry of entanglement wedges in holography directly from conformal field theories (CFTs). We analyze an information metric called the Bures metric of reduced density matrices for locally excited states. This measures distinguishability of states with different points excited. For a subsystem given by an interval, we precisely reproduce the expected entanglement wedge for two dimensional holographic CFTs from the Bures metric, which turns out to be proportional to the AdS metric on a time slice. On the other hand, for free scalar CFTs, we do not find a…
We consider the Aharonov-Bohm effect on entanglement entropy for one interval in (1+1) dimensional conformal field theory on a one dimensional ring. The magnetic field is confined inside the ring, i.e. there is a Wilson loop on the ring. The Aharonov-Bohm phase factor which is proportional to the Wilson loop is represented as insertion of twist operators. We compute exactly the Renyi entropy from a four point function of twist operators in a free charged scalar field.
Entanglement or modular Hamiltonians play a crucial role in the investigation of correlations in quantum field theories. In particular, in 1+1 space-time dimensions, the spectra of entanglement Hamiltonians of conformal field theories (CFTs) for certain geometries are related to the spectra of the physical Hamiltonians of corresponding boundary CFTs. As a result, conformal invariance allows exact computation of the spectra of the entanglement Hamiltonians for these models. In this work, we perform this computation of the spectrum of the entanglement Hamiltonian for the free compactified boson …
We investigate how entanglement spreads in time-dependent states of a 1+1 dimensional conformal field theory (CFT). The results depend qualitatively on the value of the central charge. In rational CFTs, which have central charge below a critical value, entanglement entropy behaves as if correlations were carried by free quasiparticles. This leads to long-term memory effects, such as spikes in the mutual information of widely separated regions at late times. When the central charge is above the critical value, the quasiparticle picture fails. Assuming no extended symmetry algebra, any theory wi…
We continue the study of the entanglement entropy of two disjoint intervals in conformal field theories that we started in J. Stat. Mech. (2009) P11001. We compute Trρ_A^n for any integer n for the Ising universality class and the final result is expressed as a sum of Riemann-Siegel theta functions. These predictions are checked against existing numerical data. We provide a systematic method that gives the full asymptotic expansion of the scaling function for small four-point ratio (i.e. short intervals). These formulas are compared with the direct expansion of the full results for free compac…
