We study the entanglement Hamiltonian for finite intervals in infinite quantum chains for two different free-particle systems: coupled harmonic oscillators and fermionic hopping models with dimerization. Working in the ground state, the entanglement Hamiltonian describes again free bosons or fermions and is obtained from the correlation functions via high-precision numerics for up to several hundred sites. Far away from criticality, the dominant on-site and nearest-neighbour terms have triangular profiles that can be understood from the analytical results for a half-infinite interval. Near criticality, the longer-range couplings, although small, lead to a more complex picture. A comparison between the exact spectra and entanglement entropies and those resulting from the dominant terms in the Hamiltonian is also reported.
In this Letter we study the effect of topological zero modes on entanglement Hamiltonians and entropy of free chiral fermion systems in (1+1)d. We show how Riemann-Hilbert solutions combined with finite rank perturbation theory allow us to obtain explicit expressions for entanglement Hamiltonians. We consider both chiral Majorana and Dirac fermions, and explore the effects of boundary conditions (periodic/anti-periodic for Majorana and generic for Dirac) and associated zero modes on entanglement. In the periodic sector, we derive explicitly the non-local contribution to the entanglement Hamilt…
We study the temporal evolution of the entanglement hamiltonian of an interval after a global quantum quenchin free lattice models in one spatial dimension. In a harmonic chain we explore a quench of the frequency parameter. In a chain of free fermions at half filling we consider the evolution of the ground state of a fully dimerised chain through the homogeneous hamiltonian. We focus on critical evolution hamiltonians. The temporal evolutions of the gaps in the entanglement spectrum are analysed. The entanglement hamiltonians in these models are characterised by matrices that provide also con…
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 consider the entanglement Hamiltonian for an interval in a chain of free fermions in its ground state and show that the lattice expression goes over into the conformal one if one includes the hopping to distant neighbours in the continuum limit. For an infinite chain, this can be done analytically for arbitrary fillings and is shown to be the consequence of the particular structure of the entanglement Hamiltonian, while for finite rings or temperatures the result is based on numerical calculations.
In past work we introduced a method which allows for exact computations of entanglement Hamiltonians. The method relies on computing the resolvent for the projected (on the entangling region) Green's function using a solution to the Riemann-Hilbert problem combined with finite rank perturbation theory. Here we analyze in detail several examples involving excited states of chiral fermions (Dirac and Majorana) on a spatial circle. We compute the exact entanglement Hamiltonians and an exact formula for the change in entanglement entropy due to the introduction of a particle above the Dirac sea. F…
We study the continuum limit of the entanglement Hamiltonian of a sphere for the massless scalar field in its ground state by employing the lattice model defined through the discretisation of the radial direction. In two and three spatial dimensions and for small values of the total angular momentum, we find numerical results in agreement with the corresponding ones derived from the entanglement Hamiltonian predicted by conformal field theory. When the mass parameter in the lattice model is large enough, the dominant contributions come from the on-site and the nearest-neighbour terms, whose we…
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…
We introduce the concept of entanglement features of unitary gates, as a collection of exponentiated entanglement entropies over all bipartitions of input and output channels. We obtained the general formula for time-dependent $n$th-Renyi entanglement features for unitary gates generated by random Hamiltonian. In particular, we propose an Ising formulation for the 2nd-Renyi entanglement features of random Hamiltonian dynamics, which admits a holographic tensor network interpretation. As a general description of entanglement properties, we show that the entanglement features can be applied to s…
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 study the ground-state entanglement Hamiltonian of several disjoint intervals for the massless Dirac fermion on the half-line. Its structure consists of a local part and a bi-local term that couples each point to another one in each other interval. The bi-local operator can be either diagonal or mixed in the fermionic chiralities and it is sensitive to the boundary conditions. The knowledge of such entanglement Hamiltonian is the starting point to evaluate the negativity Hamiltonian, i.e. the logarithm of the partially transposed reduced density matrix, which is an operatorial characterisat…
We show that a proportionality between the entanglement Hamiltonian and the Hamiltonian of a subsystem exists near the limit of maximal entanglement under certain conditions. Away from that limit, solvable models show that the coupling range differs in both quantities and allow to investigate the effect.
Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where t…
In the context of ground states of quantum many-body systems, the locality of entanglement between connected regions of space is directly tied to the locality of the corresponding entanglement Hamiltonian: the latter is dominated by local, few-body terms. In this work, we introduce the negativity Hamiltonian as the (non hermitian) effective Hamiltonian operator describing the logarithm of the partial transpose of a many-body system. This allows us to address the connection between entanglement and operator locality beyond the paradigm of bipartite pure systems. As a first step in this directio…
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 calculate exactly the entanglement content of magnon excited states in the integrable spin-1/2 XXX and XXZ chains in the scaling limit. In particular, we show that as far as the number of excited magnons with respect to the size of the system is small one can decompose the entanglement content, in the scaling limit, to the sum of the entanglement of particular excited states of free fermionic or bosonic theories. In addition we conjecture that the entanglement content of the generic translational invariant free fermionic and bosonic Hamiltonians can be also classified, in the scaling limit,…
The entanglement entropy for the ground state of a XY spin chain is related to the corner transfer matrices of the triangular Ising model and expressed in closed form.
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.
The time evolution of entanglement tracks how information propagates in interacting quantum systems. We study entanglement entropy in CFT$_2$ with a time-dependent Hamiltonian. We perturb by operators with time-dependent source functions and use the replica trick to calculate higher order corrections to entanglement entropy. At first order, we compute the correction due to a metric perturbation in AdS$_3$/CFT$_2$ and find agreement on both sides of the duality. Past first order, we find evidence of a universal structure of entanglement propagation to all orders. The central feature is that int…
Entanglement entropy obeys area law scaling for typical physical quantum systems. This may naively be argued to follow from locality of interactions. We show that this is not the case by constructing an explicit simple spin chain Hamiltonian with nearest neighbor interactions that presents an entanglement volume scaling law. This non-translational model is contrived to have couplings that force the accumulation of singlet bonds across the half chain. Our result is complementary to the known relation between non-translational invariant, nearest neighbor interacting Hamiltonians and QMA complete…
We consider a local quench where two free-fermion half-chains are coupled via a defect. We show that the logarithmic increase of the entanglement entropy is governed by the same effective central charge which appears in the ground-state properties and which is known exactly. For unequal initial filling of the half-chains, we determine the linear increase of the entanglement entropy.
We consider fermionic and bosonic quantum chains where a defect separates two subsystems and compare the corresponding entanglement spectra. With these, we calculate their Rényi entanglement entropies and obtain analytical formulae for the continuously varying coefficient of the leading logarithmic term. For the bosonic case we also present numerical results.
We review results about entanglement (or modular) Hamiltonians of quantum many-body systems in field theory and statistical mechanics models, as well as recent applications in the context of quantum information and quantum simulation.
In this paper, we review how to obtain the central charge of a critical entanglement Hamiltonian through the nested entanglement entropy which was first introduced in [J. Lou et al. PRB 84, 245128 (2011)]. The critical phenomena of the entanglement Hamiltonian can be identified by the central charge obtained by the nested entanglement entropy. We review our previous studies[J. Lou et al. PRB 84, 245128 (2011), S. Tanaka et al. PRA 87, 214401 (2013)] in which we investigated certain entanglement nature of two-dimensional valence-bond-solid (VBS) state and quantum hard-square models on square an…
We study solvable spin chains where either fields or couplings vary linearly in space and create a sandwich-like structure of the ground state. We find that the entanglement entropy between two halves of a chain varies logarithmically with the interface width. After quenching to a homogeneous critical system, the entropy grows logarithmically in time in the XX model, but quadratically in the transverse Ising chain. We explain this behaviour and indicate generalizations to other power laws.
We study the ground-state entanglement Hamiltonian for an interval of $N$ sites in a free-fermion chain with arbitrary filling. By relating it to a commuting operator, we find explicit expressions for its matrix elements in the large-$N$ limit. The results agree with numerical calculations and show that deviations from the conformal prediction persist even for large systems.
We study the entanglement properties of anisotropic open spin one-half Heisenberg chains with a modified central bond. The entanglement entropy between the two half-chains is calculated with the density-matrix renormalization method (DMRG).We find a logarithmic behaviour with an effective central charge c' varying with the length of the system. It flows to one in the ferromagnetic region and to zero in the antiferromagnetic region of the model. In the XX case it has a non-universal limit and we recover previous results.
We study the entanglement Hamiltonian for fermionic hopping models on rings and open chains and determine single-particle spectra, eigenfunctions and the form in real space. For the chain, we find a commuting operator as for the ring and compare with its properties in both cases. In particular, a scaling relation between the eigenvalues is found for large systems. We also show how the commutation property carries over to the critical transverse Ising model.
A powerful perspective in understanding non-equilibrium quantum dynamics is through the time evolution of its entanglement content. Yet apart from a few guiding principles for the entanglement entropy, to date, not much else is known about the refined characters of entanglement propagation. Here, we unveil signatures of the entanglement evolving and information propagation out-of-equilibrium, from the view of entanglement Hamiltonian. As a prototypical example, we study quantum quench dynamics of a one-dimensional Bose-Hubbard model by means of time-dependent density-matrix renormalization gro…
We present the results of magnetic properties and entanglement of the distorted diamond chain model for azurite using pure quantum exchange interactions. The magnetic properties and concurrence as a measure of pairwise thermal entanglement have been studied by means of variational mean-field like treatment based on Gibbs-Bogoliubov inequality. Such a system can be considered as an approximation of the natural material azurite, Cu3(CO3)2(OH)2. For values of exchange parameters, which are taken from experimental results, we study the thermodynamic properties, such as azurite specific heat and ma…
We study zero-temperature XX chains and transverse Ising chains and join an initially separate finite piece on one or on both sides to an infinite remainder. In both critical and non-critical systems we find a typical increase of the entanglement entropy after the quench, followed by a slow decay towards the value of the homogeneous chain. In the critical case, the predictions of conformal field theory are verified for the first phase of the evolution, while at late times a step structure can be observed.
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 investigate the presence of multipartite entanglement in macroscopic spin chains. We discuss the Heisenberg and the XY model and derive bounds on the internal energy for systems without multipartite entanglement. Based on this we show that in thermal equilibrium the above mentioned spin systems contain genuine multipartite entanglement, even at finite modest temperatures.
We carry out a systematic study of the exact block entanglement in XXZ spin-chain at Delta=-1/2. We present, the first analytic expressions for reduced density matrices of n spins in a chain of length L (for n<=6 and arbitrary but odd L) of a truly interacting model. The entanglement entropy, the moments of the reduced density matrix, and its spectrum are then easily derived. We explicitely construct the "entanglement Hamiltonian" as the logarithm of this matrix. Exploiting the degeneracy of the ground-state, we find the scaling behavior of entanglement of the zero-temperature mixed state.
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 eigenstate entanglement entropy has been recently shown to be a powerful tool to distinguish integrable from generic quantum-chaotic models. In integrable models, a unique feature of the average eigenstate entanglement entropy (over all Hamiltonian eigenstates) is that the volume-law coefficient depends on the subsystem fraction. Hence, it deviates from the maximal (subsystem fraction independent) value encountered in quantum-chaotic models. Using random matrix theory for quadratic Hamiltonians, we obtain a closed-form expression for the average eigenstate entanglement entropy as a functio…
We study the unitary time evolution of the entanglement Hamiltonian of a free Fermi lattice gas in one dimension initially prepared in a domain wall configuration. To this aim, we exploit the recent development of quantum fluctuating hydrodynamics. Our findings for the entanglement Hamiltonian are based on the effective field theory description of the domain wall melting and are expected to exactly describe the Euler scaling limit of the lattice gas. However, such field theoretical results can be recovered from high-precision numerical lattice calculations only when summing appropriately over …
The thermal entanglement of a two-qubit anisotropic Heisenberg $XYZ$ chain under an inhomogeneous magnetic field b is studied. It is shown that when inhomogeneity is increased to certain value, the entanglement can exhibit a larger revival than that of less values of b. The property is both true for zero temperature and a finite temperature. The results also show that the entanglement and critical temperature can be increased by increasing inhomogeneous exteral magnetic field.
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…
Entanglement is central to our understanding of many-body quantum matter. In particular, the entanglement spectrum, as eigenvalues of the reduced density matrix of a subsystem, provides a unique footprint of properties of strongly correlated quantum matter from detection of topological order to characterisation of quantum critical systems. However, direct experimental measurement of the entanglement spectrum has so far remained elusive due to lack of direct experimental probes. Here we show that the entanglement spectrum of the ground state of a broad class of Hamiltonians becomes directly acc…
Recent numerical advances in the field of strongly correlated electron systems allow the calculation of the entanglement spectrum and entropies for interacting fermionic systems. An explicit determination of the entanglement (modular) Hamiltonian has proven to be a considerably more difficult problem, and only a few results are available. We introduce a technique to directly determine the entanglement Hamiltonian of interacting fermionic models by means of auxiliary field quantum Monte Carlo simulations. We implement our method on the one-dimensional Hubbard chain partitioned into two segments…
We consider two prototypical quantum models, the spin-1/2 XY chain and the quantum Ising chain and study their entanglement entropy, S(l,L), of blocks of l spins in homogeneous or inhomogeneous systems of length L. By using two different approaches, free-fermion techniques and perturbational expansion, an exact relationship between the entropies is revealed. Using this relation we translate known results between the two models and obtain, among others, the additive constant of the entropy of the critical homogeneous quantum Ising chain and the effective central charge of the random XY chain.
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.
In a recent Letter [Phys. Rev. Lett. 125, 180604 (2020)], we introduced a closed-form analytic expression for the average bipartite von Neumann entanglement entropy of many-body eigenstates of random quadratic Hamiltonians. Namely, of Hamiltonians whose single-particle eigenstates have random coefficients in the position basis. A paradigmatic Hamiltonian for which the expression is valid is the quadratic Sachdev-Ye-Kitaev (SYK2) model in its Dirac fermion formulation. Here we show that the applicability of our result is much broader. Most prominently, it is also relevant for local Hamiltonians…
We examine the entanglement of cyclic spin 1/2 chains with anisotropic XY Z Heisenberg couplings of arbitrary range at transverse factorizing magnetic fields. At these fields the system exhibits a degenerate symmetry-breaking separable ground state (GS). It is shown, however, that the side limits of the GS pairwise entanglement at these fields are actually non-zero in finite chains, corresponding such fields to a GS spin-parity transition. These limits exhibit universal properties like being independent of the pair separation and interaction range, and are directly related to the magnetization…
We investigate the entanglement entropy (EE) of gapped S=1 and $S=1/2$ spin chains with dimerization. We find that the effective boundary degrees of freedom as edge states contribute significantly to the EE. For the $S=1/2$ dimerized Heisenberg chain, the EE of the sufficiently long chain is essentially explained by the localized $S=1/2$ effective spins on the boundaries. As for S=1, the effective spins are also $S=1/2$ causing a Kennedy triplet that yields a lower bound for the EE. In this case, the residual entanglement reduces substantially by a continuous deformation of the Heisenberg mode…
