We consider the Renyi entropies S_n in one-dimensional massive integrable models diagonalizable by means of corner transfer matrices (as Heisenberg and Ising spin chains). By means of explicit examples and using the relation of corner transfer matrix with the Virasoro algebra, we show that close to a conformal invariant critical point, when the correlation length xi is finite but large, the corrections to the scaling are of the unusual form xi^(-x/n), with x the dimension of a relevant operator in the conformal theory. This is reminiscent of the results for gapless chains and should be valid for any massive one-dimensional model close to a conformal critical point.
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 determine both analytically and numerically the entanglement between chiral degrees of freedom in the ground state of massive perturbations of 1+1 dimensional conformal field theories quantised on a cylinder. Analytic predictions are obtained from a variational Ansatz for the ground state in terms of smeared conformal boundary states recently proposed by J. Cardy, which is validated by numerical results from the Truncated Conformal Space Approach. We also extend the scope of the Ansatz by resolving ground state degeneracies exploiting the operator product expansion. The chiral entanglement …
We study the entanglement of two disjoint blocks in spin-1/2 chains obtained by merging solvable models, such as XX and quantum Ising models. We focus on the universal quantities that can be extracted from the Rényi entropies S_α. The most important information is encoded in some functions denoted by F_α. We compute F_2 and we show that F_α-1 and F_{v.N.}, corresponding to the von Neumann entropy, can be negative, in contrast to what observed in all models examined so far. An exact relation between the entanglement of disjoint subsystems in the XX model and that in a chain embodying two quantu…
We study the scaling of the traces of the integer powers of the partially transposed reduced density matrix and of the entanglement negativity for two spin blocks as function of their length and separation in the critical Ising chain. For two adjacent blocks, we show that tensor network calculations agree with universal conformal field theory (CFT) predictions. In the case of two disjoint blocks the CFT predictions are recovered only after taking into account the finite size corrections induced by the finite length of the blocks.
We study the scaling of the Renyi and entanglement entropy of two disjoint blocks of critical Ising models, as function of their sizes and separations. We present analytic results based on conformal field theory that are quantitatively checked in numerical simulations of both the quantum spin chain and the classical two dimensional Ising model. Theoretical results match the ones obtained from numerical simulations only after taking properly into account the corrections induced by the finite length of the blocks to their leading scaling behavior.
The Renyi entropies of a massless Dirac fermion on a circle with chemical potential are calculated analytically at nonzero temperature by using the bosonization method. The bosonization of a massive Dirac fermion to the sine-Gordon model lets us obtain the small mass corrections to the entropies. We numerically compute the Renyi entropies by putting a massive fermion on the lattice and find agreement between the analytic and numerical results. In the presence of a mass gap, we show that corrections to Renyi and entanglement entropies in the limit m >> T scale as exp(-m/T). We also show that wh…
We study the Renyi entanglement entropies of two disjoint intervals in XY chains. We exploit the exact solution of the model in terms of free Majorana fermions and we show how to construct the reduced density matrix in the spin variables by taking properly into account the Jordan-Wigner string between the two blocks. From this we can evaluate any Renyi entropy of finite integer order. We study in details critical XX and Ising chains and we show that the asymptotic results for large blocks agree with recent conformal field theory predictions if corrections to the scaling are included in the ana…
We study the continuum limit of the entanglement hamiltonians of a block of consecutive sites in massless harmonic chains. This block is either in the chain on the infinite line or at the beginning of a chain on the semi-infinite line with Dirichlet boundary conditions imposed at its origin. The entanglement hamiltonians of the interval predicted by Conformal Field Theory for the massless scalar field are obtained in the continuum limit. We also study the corresponding entanglement spectra and the numerical results for the ratios of the gaps are compatible with the operator content of the Boun…
Recent studies have shown that logarithmic divergence of entanglement entropy as function of size of a subsystem is a signature of criticality in quantum models. We demonstrate that the ground state entanglement entropy of $ n$ sites for ferromagnetic Heisenberg spin-1/2 chain of the length $L$ in a sector with fixed magnetization $y$ per site grows as ${1/2}\log_{2} \frac{n(L-n)}{L}C(y)$, where $C(y)=2πe({1/4}-y^{2})$
We consider the Renyi alpha-entropies for Luttinger liquids (LL). For large block lengths l these are known to grow like ln l. We show that there are subleading terms that oscillate with frequency 2k_F (the Fermi wave number of the LL) and exhibit a universal power-law decay with l. The new critical exponent is equal to K/(2 alpha), where K is the LL parameter. We present numerical results for the anisotropic XXZ model and the full analytic solution for the free fermion (XX) point.
In this paper we propose an expression for the entanglement entropy of several intervals in a stationary state of a free, translational invariant Hamiltonian in a fermionic chain. We check numerically the accuracy of our proposal and conjecture a new formula for the asymptotic behaviour of principal sub-matrices of a Toeplitz matrix.
The study of the dynamics of entanglement measures after a quench has become a very active area of research in the last two decades, motivated by the development of experimental techniques. However, exact results in this context are available in only very few cases. In this work, we present the proof of the quasiparticle picture for the dynamics of entanglement entropies for two disjoint blocks in the XY chain after a quantum quench. As a byproduct, we also prove the quasiparticle conjecture for the mutual information in that model. Our calculations generalize those presented in [M. Fagotti, P…
We calculate the entanglement entropy of blocks of size x embedded in a larger system of size L, by means of a combination of analytical and numerical techniques. The complete entanglement entropy in this case is a sum of three terms. One is a universal x and L-dependent term, first predicted by Calabrese and Cardy, the second is a nonuniversal term arising from the thermodynamic limit, and the third is a finite size correction. We give an explicit expression for the second, nonuniversal, term for the one-dimensional Hubbard model, and numerically assess the importance of all three contributio…
The analysis of the entanglement entropy of a subsystem of a one-dimensional quantum system is a powerful tool for unravelling its critical nature. For instance, the scaling behaviour of the entanglement entropy determines the central charge of the associated Virasoro algebra. For a free fermion system, the entanglement entropy depends essentially on two sets, namely the set $A$ of sites of the subsystem considered and the set $K$ of excited momentum modes. In this work we make use of a general duality principle establishing the invariance of the entanglement entropy under exchange of the sets…
The block entanglement entropy and fluctuations are investigated in one dimension in finite size correlated electron systems using the Gutzwiller wave function as a prototype correlated electron state. Entanglement entropy shows logarithmic divergence for all values of the correlation projection parameter $g$, as predicted by conformal field theories for critical systems, but the central charge requires finite size corrections. There is an infinite correlation length corresponding to correlation between same kinds of spins, for all values of $g$. A scaling form for the block entropy, as a func…
Starting with the valence bond solid (VBS) ground state of the 1D AKLT Hamiltonian, we make a partition of the system in 2 subsystems $A$ and $B$, where $A$ is a block of $L$ consecutive spins and $B$ is it's complement. In that setting we compute the partial transpose density matrix with respect to $A$, $ρ^{T_A}$. We obtain the spectrum of the transposed density matrix of the VBS pure system. Subsequently we define two disjoint blocks, $A$ and $B$ containing $L_A$ and $L_B$ spins respectively, separated by $L$ sites. Tracing away the spins which do not belong to $A\cup B$, we find an expressi…
We optimize a quantum walk of multiple fermions following a quench in a spin chain to generate near ideal resources for quantum networking. We first prove an useful theorem mapping the correlations evolved from specific quenches to the apparently unrelated problem of quantum state transfer between distinct spins. This mapping is then exploited to optimize the dynamics and produce large amounts of entanglement distributed in very special ways. Two applications are considered: the simultaneous generation of many Bell states between pairs of distant spins (maximal block entropy), or high entangle…
The non-equilibrium evolution of the block entanglement entropy is investigated in the XY chain in a transverse magnetic field after the Hamiltonian parameters are suddenly changed from and to arbitrary values. Using Toeplitz matrix representation and multidimensional phase methods, we provide analytic results for large blocks and for all times, showing explicitly the linear growth in time followed by saturation. The consequences of these analytic results are discussed and the effects of a finite block length is taken into account numerically.
We study entanglement in Valence-Bond-Solid state. It describes the ground state of Affleck, Kennedy, Lieb and Tasaki quantum spin chain. The AKLT model has a gap and open boundary conditions. We calculate an entropy of a subsystem (continuous block of spins). It quantifies the entanglement of this block with the rest of the ground state. We prove that the entanglement approaches a constant value exponentially fast as the size of the subsystem increases. Actually we proved that the density matrix of the continuous block of spins depends only on the length of the block, but not on the total siz…
The evaluation of vacuum expectation values (VEVs) in massive integrable quantum field theory (QFT) is a nontrivial renormalization-group "connection problem" -- relating large and short distance asymptotics -- and is in general unsolved. This is particularly relevant in the context of entanglement entropy, where VEVs of branch-point twist fields give universal saturation predictions. We propose a new method to compute VEVs of twist fields associated to continuous symmetries in QFT. The method is based on a differential equation in the continuous symmetry parameter, and gives VEVs as infinite …
We consider the Rényi entropies $S_n(\ell)$ in the one dimensional spin-1/2 Heisenberg XX chain in a magnetic field. The case n=1 corresponds to the von Neumann ``entanglement'' entropy. Using a combination of methods based on the generalized Fisher-Hartwig conjecture and a recurrence relation connected to the Painlevé VI differential equation we obtain the asymptotic behaviour, accurate to order ${\cal O}(\ell^{-3})$, of the Rényi entropies $S_n(\ell)$ for large block lengths $\ell$. For n=1,2,3,10 this constitutes the 3,6,10,48 leading terms respectively. The o(1) contributions are found to …
We consider the Renyi entanglement entropies in the one-dimensional XX spin-chains with open boundary conditions in the presence of a magnetic field. In the case of a semi-infinite system and a block starting from the boundary, we derive rigorously the asymptotic behavior for large block sizes on the basis of a recent mathematical theorem for the determinant of Toeplitz plus Hankel matrices. We conjecture a generalized Fisher-Hartwig form for the corrections to the asymptotic behavior of this determinant that allows the exact characterization of the corrections to the scaling at order o(1/l) f…
Gaussian matrix product states are obtained as the outputs of projection operations from an ancillary space of M infinitely entangled bonds connecting neighboring sites, applied at each of N sites of an harmonic chain. Replacing the projections by associated Gaussian states, the 'building blocks', we show that the entanglement range in translationally-invariant Gaussian matrix product states depends on how entangled the building blocks are. In particular, infinite entanglement in the building blocks produces fully symmetric Gaussian states with maximum entanglement range. From their peculiar p…
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 Per…
We study a 1-dimensional AKLT spin chain, consisting of spins $S$ in the bulk and $S/2$ at both ends. The unique ground state of this AKLT model is described by the Valence-Bond-Solid (VBS) state. We investigate the density matrix of a contiguous block of bulk spins in this ground state. It is shown that the density matrix is a projector onto a subspace of dimension $(S+1)^{2}$. This subspace is described by non-zero eigenvalues and corresponding eigenvectors of the density matrix. We prove that for large block the von Neumann entropy coincides with Renyi entropy and is equal to $\ln(S+1)^{2}$…
We examine the evaluation of entanglement measures in weakly correlated gaussian states. It is shown that they can be expressed in terms of the singular values of a particular block of the generalized contraction matrix. This result enables to obtain in a simple way asymptotic expressions and related area laws for the entanglement entropy of bipartitions in pure states, as well as for the logarithmic negativity associated with bipartitions and also pairs of arbitrary subsystems. As illustration, we consider different types of contiguous and noncontiguous blocks in two dimensional lattices. Exa…
We introduce a new spin chain which is a deformation of the Fredkin spin chain and has a phase transition between bounded and extensive entanglement entropy scaling. In this chain, spins have a local interaction of three nearest neighbors. The Hamiltonian is frustration-free and its ground state can be described analytically as a weighted superposition of Dyck paths. In the purely spin $1/2$ case, the entanglement entropy obeys an area law: it is bounded from above by a constant, when the size of the block $n$ increases (and $t>1$). When a local color degree of freedom is introduced the entang…
We study the entanglement entropy of a block of contiguous spins in excited states of spin chains. We consider the XY model in a transverse field and the XXZ Heisenberg spin-chain. For the latter, we developed a numerical application of algebraic Bethe Ansatz. We find two main classes of states with logarithmic and extensive behavior in the dimension of the block, characterized by the properties of excitations of the state. This behavior can be related to the locality properties of the Hamiltonian having a given state as ground state. We also provide several details of the finite size scaling.
We study the scaling of the Renyi entanglement entropy of two disjoint blocks of critical lattice models described by conformal field theories with central charge c=1. We provide the analytic conformal field theory result for the second order Renyi entropy for a free boson compactified on an orbifold describing the scaling limit of the Ashkin-Teller (AT) model on the self-dual line. We have checked this prediction in cluster Monte Carlo simulations of the classical two dimensional AT model. We have also performed extensive numerical simulations of the anisotropic Heisenberg quantum spin-chain …
We calculate the reduced density matrix of a block of integer spin-S's in a q-deformed valence-bond-solid (VBS) state. This matrix is diagonalized exactly for an infinitely long block in an infinitely long chain. We construct an effective Hamiltonian with the same spectrum as the logarithm of the density matrix. We also derive analytic expressions for the von Neumann and Rényi entanglement entropies. For blocks of finite length, we calculate the eigenvalues of the reduced density matrix by perturbation theory and numerical diagonalization. These results enable us to describe the effects of fin…
We study the entanglement entropy of blocks of contiguous spins in non-periodic (quasi-periodic or more generally aperiodic) critical Heisenberg, XX and quantum Ising spin chains, e.g. in Fibonacci chains. For marginal and relevant aperiodic modulations, the entanglement entropy is found to be a logarithmic function of the block size with log-periodic oscillations. The effective central charge, c_eff, defined through the constant in front of the logarithm may depend on the ratio of couplings and can even exceed the corresponding value in the homogeneous system. In the strong modulation limit, …
Entanglement spectrum of finite-size correlated electron systems are investigated using the Gutzwiller projection technique. The product of largest eigenvalue and rank of the block reduced density matrix, which is a measure of distance of the state from the maximally entangled state of the corresponding rank, is seen to characterise the insulator to metal crossover in the state. The fraction of distinct eigenvalues exhibits a `chaotic' behaviour in the crossover region, and it shows a `integrable' behaviour at both insulating and metallic ends. The integrated entanglement spectrum obeys confor…
We consider the ground state of the XX chain that is constrained to carry a current of energy. The von Neumann entropy of a block of $L$ neighboring spins, describing entanglement of the block with the rest of the chain, is computed. Recent calculations have revealed that the entropy in the XX model diverges logarithmically with the size of the subsystem. We show that the presence of the energy current increases the prefactor of the logarithmic growth. This result indicates that the emergence of the energy current gives rise to an increase of entanglement.
By means of free fermionic techniques we study the time evolution of the entanglement entropy, S(t), of a block of spins in the random transverse-field Ising chain after a sudden change of the parameters of the Hamiltonian. We consider global quenches, when the parameters are modified uniformly in space, as well as local quenches, when two disconnected blocks are suddenly joined together. For a non-critical final state, the dynamical entanglement entropy is found to approach a finite limiting value for both types of quenches. If the quench is performed to the critical state, the entropy grows …
The characterization of an infinite-order quantum phase transition (QPT) by entanglement measures is analyzed. To this aim, we consider two closely related solvable spin-1/2 chains, namely, the Ashkin-Teller and the staggered XXZ models. These systems display a distinct pattern of eigenstates but exhibit the same thermodynamics, i.e. the same energy spectrum. By performing exact diagonalization, we investigate the behavior of pairwise and block entanglement in the ground state of both models. In contrast with the XXZ chain, we show that pairwise entanglement fails in the characterization of th…
Average block entanglement in the 1D XX-model with uncorrelated random couplings is known to grow as the logarithm of the block size, in similarity to conformal systems. In this work we study random spin chains whose couplings present long range correlations, generated as gaussian fields with a power-law spectral function. Ground states are always planar valence bond states, and their statistical ensembles are characterized in terms of their block entropy and their bond-length distribution, which follow power-laws. We conjecture the existence of a critical value for the spectral exponent, belo…
We study the average entanglement entropy of blocks of contiguous spins in aperiodic XXZ chains which possess an aperiodic singlet phase at least in a certain limit of the coupling ratios. In this phase, where the ground state constructed by a real space renormalization group method, consists (asymptotically) of independent singlet pairs, the average entanglement entropy is found to be a piecewise linear function of the block size. The enveloping curve of this function is growing logarithmically with the block size, with an effective central charge in front of the logarithm which is characteri…
We investigate the scaling of the Rényi $α$-entropies in one-dimensional gapped quantum spin models. We show that the block entropies with $α> 2$ violate the area law monotonicity and exhibit damped oscillations. Depending on the existence of a factorized ground state, the oscillatory behavior occurs either below factorization or it extends indefinitely. The anomalous scaling corresponds to an entanglement-driven order that is independent of ground-state degeneracy and is revealed by a nonlocal order parameter defined as the sum of the single-copy entanglement over all blocks.
We introduce the single-copy entanglement as a quantity to assess quantum correlations in the ground state in quantum many-body systems. We show for a large class of models that already on the level of single specimens of spin chains, criticality is accompanied with the possibility of distilling a maximally entangled state of arbitrary dimension from a sufficiently large block deterministically, with local operations and classical communication. These analytical results -- which refine previous results on the divergence of block entropy as the rate at which EPR pairs can be distilled from many…
We show that block entanglement entropies in one-dimensional systems close to a quantum critical point can in principle be measured in terms of the population of low-lying energy levels following a certain type of local quantum quench.
We consider the ground state of the XY model on an infinite chain at zero temperature. Following Bennett, Bernstein, Popescu, and Schumacher we use entropy of a sub-system as a measure of entanglement. Vidal, Latorre, Rico and Kitaev conjectured that von Neumann entropy of a large block of neighboring spins approaches a constant as the size of the block increases. We evaluated this limiting entropy as a function of anisotropy and transverse magnetic field. We used the methods based on integrable Fredholm operators and Riemann-Hilbert problem. The entropy is singular at phase transitions.
The Affleck-Kennedy-Lieb-Tasaki (AKLT) spin interacting model can be defined on an arbitrary graph. We explain the construction of the AKLT Hamiltonian. Given certain conditions, the ground state is unique and known as the Valence-Bond-Solid (VBS) state. It can be used in measurement-based quantum computation as a resource state instead of the cluster state. We study the VBS ground state on an arbitrary connected graph. The graph is cut into two disconnected parts: the block and the environment. We study the entanglement between these two parts and prove that many eigenvalues of the density ma…
We investigate the structural aspects of genuine multipartite entanglement in Gaussian states of continuous variable systems. Generalizing the results of [Adesso & Illuminati, Phys. Rev. Lett. 99, 150501 (2007)], we analyze whether the entanglement shared by blocks of modes distributes according to a strong monogamy law. This property, once established, allows to quantify genuine N-partite entanglement in terms of the "residual contangle" not encoded into 2,...,K,...,(N-1)-partite quantum correlations. The explicit expression of this entanglement measure is derived, by a recursive formula, for…
We propose an entanglement-based algorithm of the tensor-network strong-disorder renormalization group (tSDRG) method for quantum spin systems with quenched randomness. In contrast to the previous tSDRG algorithm based on the energy spectrum of renormalized block Hamiltonians, we directly utilizes the entanglement structure associated with the blocks to be renormalized. We examine accuracy of the new algorithm for the random antiferromagnetic Heisenberg models on the one-dimensional, triangular, and square lattices. We then find that the entanglement-based tSDRG achieves better accuracy than t…
We simulate the entanglement dynamics in a critical random quantum Ising chain with generic perturbations using the time-evolving block decimation algorithm. Starting from a product state, we observe super-logarithmic growth of entanglement entropy with time. The numerical result is consistent with the analytical prediction of Vosk and Altman using a real-space renormalization group technique.
The entanglement spectrum provides crucial information about correlated quantum systems. We show that the study of the block-like nature of the reduced density matrix in number sectors and the partition dependence of the spectrum in finite systems leads to interesting unexpected insights, which we illustrate for the case of a 1D extended Hubbard model. We show that block symmetry provides an intuitive understanding of the spectral double degeneracy of the Haldane-insulator, revealing as well partial double degeneracy for the Mott-insulator. Moreover, surprisingly, the partition dependence of t…
We investigate entanglement properties in the ground state of the open/periodic SU($n$) generalized valence-bond-solid state consisting of representations of SU($n$). We obtain exact expression for the reduced density matrix of a block of contiguous spins and explicitly evaluate the von Neumann and the Rényi entropies. We discover that the Rényi entropy is independent of the parameter $α$ in the limit of large block sizes and its value $2 \log n$ coincides with that of von Neumann entropy. We also find the direct relation between the reduced density matrix of the subsystem and edge states for …
