Von Neumann Entanglement Entropy Research Articles

One-dimensional Bose-Hubbard model with local three-body interactions

We employ the (dynamical) density-matrix renormalization-group technique to investigate the ground-state properties of the Bose-Hubbard model with nearest-neighbor transfer amplitudes $t$ and local two-body and three-body repulsion of strength $U$ and $W$, respectively. We determine the phase boundaries between the Mott-insulating and superfluid phases for the lowest two Mott lobes from the chemical potentials. We calculate the tips of the Mott lobes from the Tomonaga-Luttinger liquid parameter and confirm the positions of the Kosterlitz-Thouless points from the von Neumann entanglement entropy. We find that physical quantities in the second Mott lobe such as the gap and the dynamical structure factor scale almost perfectly in $t/(U+W)$, even close to the Mott transition. Strong-coupling perturbation theory shows that there is no true scaling but deviations from it are quantitatively small in the strong-coupling limit. This observation should remain true in higher dimensions and for not too large attractive three-body interactions.

Fisher–Hartwig expansion for Toeplitz determinants and the spectrum of a single-particle reduced density matrix for one-dimensional free fermions

We study the spectrum of the Toeplitz matrix with a sine kernel, which corresponds to the single-particle reduced density matrix for free fermions on the one-dimensional lattice. For the spectral determinant of this matrix, a Fisher–Hartwig expansion in the inverse matrix size has been recently conjectured. This expansion can be verified order by order, away from the line of accumulation of zeros, using the recurrence relation known from the theory of discrete Painlevé equations. We perform such a verification to the tenth order and calculate the corresponding coefficients in the Fisher–Hartwig expansion. Under the assumption of the validity of the Fisher–Hartwig expansion in the whole range of the spectral parameter, we further derive expansions for an equation on the eigenvalues of this matrix and for the von Neumann entanglement entropy in the corresponding fermion problem. These analytical results are supported by a numerical example.

Incommensurability effects in odd lengthJ1-J2quantum spin chains: On-site magnetization and entanglement

For the antiferromagnetic ${J}_{1}$-${J}_{2}$ quantum spin chain with an even number of sites, the point ${J}_{2}^{d}={J}_{1}/2$ is a disorder point. It marks the onset of incommensurate real space correlations for ${J}_{2}>{J}_{2}^{d}$. At a distinct larger value of ${J}_{2}^{L}=0.520\phantom{\rule{0.16em}{0ex}}36(6){J}_{1}$, the Lifshitz point, the peak in the static structure factor begins to move away from $k=\ensuremath{\pi}$. Here, we focus on chains with an odd number of sites. In this case, the disorder point is also at ${J}_{2}^{d}={J}_{1}/2$ but the behavior close to the Lifshitz point, ${J}_{2}^{L}\ensuremath{\simeq}0.538{J}_{1}$, is quite different: starting at ${J}_{2}^{L}$, the ground state goes through a sequence of level crossings as its momentum changes away from $k=\ensuremath{\pi}/2$. An even length chain, on the other hand, is gapped for any ${J}_{2}>0.24{J}_{1}$ and has the ground-state momentum $k=0$. This gradual change in the ground-state wave function for chains with an odd number of sites is reflected in a dramatic manner directly in the ground-state on-site magnetization as well as in the bipartite von Neumann entanglement entropy. Our results are based on DMRG calculations and variational calculations performed in a restricted Hilbert space defined in the valence bond picture. In the vicinity of the point ${J}_{2}={J}_{1}/2$, we expect the variational results to be very precise.

Equilibrium and nonequilibrium entanglement properties of two- and three-dimensional Fermi gases

We investigate the entanglement properties of the equilibrium and nonequilibrium quantum dynamics of two-dimensional (2D) and three-dimensional (3D) Fermi gases, by computing entanglement entropies of extended space regions, which generally show multiplicative logarithmic corrections to the leading power-law behaviors, corresponding to the logarithmic corrections to the area law. We consider 2D and 3D Fermi gases of $N$ particles constrained within a limited space region, for example, by a hard-wall trap, at equilibrium at $T=0$, i.e., in their ground state, and compute the first few terms of the asymptotic large-$N$ behaviors of entanglement entropies and particle fluctuations of subsystems with some convenient geometries, which allow us to significantly extend their computation. Then we consider their nonequilibrium dynamics after instantaneously dropping the hard-wall trap, which allows the gas to expand freely. We compute the time dependence of the von Neumann entanglement entropy of space regions around the original trap. We show that at small time it is characterized by the relation $S\ensuremath{\approx}{\ensuremath{\pi}}^{2}V/3$ with the particle variance, and multiplicative logarithmic corrections to the leading power law, i.e., $S\ensuremath{\sim}{t}^{1\ensuremath{-}d}\mathrm{ln}(1/t)$.

Weak and strong typicality in quantum systems

We study the properties of mixed states obtained from eigenstates of many-body lattice Hamiltonians after tracing out part of the lattice. Two scenarios emerge for generic systems: (i) The diagonal entropy becomes equivalent to the thermodynamic entropy when a few sites are traced out (weak typicality); and (ii) the von Neumann (entanglement) entropy becomes equivalent to the thermodynamic entropy when a large fraction of the lattice is traced out (strong typicality). Remarkably, the results for few-body observables obtained with the reduced, diagonal, and canonical density matrices are very similar to each other, no matter which fraction of the lattice is traced out. Hence, for all physical quantities studied here, the results in the diagonal ensemble match the thermal predictions.

Quantum dynamics and entanglement in one-dimensional Fermi gases released from a trap

We investigate the entanglement properties of the nonequilibrium dynamics of one-dimensional noninteracting Fermi gases released from a trap. The gas of N particles is initially in the ground state within hard-wall or harmonic traps, then it expands after dropping the trap. We compute the time dependence of the von Neumann and Renyi entanglement entropies and the particle fluctuations of spatial intervals around the original trap, in the limit of a large number N of particles. The results for these observables apply to one-dimensional gases of impenetrable bosons as well. We identify different dynamical regimes at small and large times, depending also on the initial condition, whether it is that of a hard-wall or harmonic trap. In particular, we analytically show that the expansion from hard-wall traps is characterized by the asymptotic small-time behavior $S \approx (1/3)\ln(1/t)$ of the von Neumann entanglement entropy, and the relation $S\approx \pi^2 V/3$ where V is the particle variance, which are analogous to the equilibrium behaviors whose leading logarithms are essentially determined by the corresponding conformal field theory with central charge $c=1$. The time dependence of the entanglement entropy of extended regions during the expansion from harmonic traps shows the remarkable property that it can be expressed as a global time-dependent rescaling of the space dependence of the initial equilibrium entanglement entropy.

Unruh effect without trans-horizon entanglement

We estimate the transition rates of a uniformly accelerated Unruh-DeWitt detector coupled to a quantum field with reflecting conditions on a boundary plane (a ``mirror''). We find that these are essentially indistinguishable from the usual Unruh rates, viz. that the Unruh effect persists in the presence of the mirror. This shows that the Unruh effect (thermality of detector rates) is not merely a consequence of the entanglement between left and right Rindler quanta in the Minkowski vacuum. Since in this setup the state of the field in the Rindler wedge is pure, we argue furthermore that the relevant entropy in the Unruh effect cannot be the von Neumann entanglement entropy. We suggest, an alternative, that it is the Shannon entropy associated with Heisenberg uncertainty.

Entanglement and particle correlations of Fermi gases in harmonic traps

We investigate quantum correlations in the ground state of noninteracting Fermi gases of N particles trapped by an external space-dependent harmonic potential, in any dimension. For this purpose, we compute one-particle correlations, particle fluctuations and bipartite entanglement entropies of extended space regions, and study their large-N scaling behaviors. The half-space von Neumann entanglement entropy is computed for any dimension, obtaining S_HS = c_l N^(d-1)/d ln N, analogously to homogenous systems, with c_l=1/6, 1/(6\sqrt{2}), 1/(6\sqrt{6}) in one, two and three dimensions respectively. We show that the asymptotic large-N relation S_A\approx \pi^2 V_A/3, between the von Neumann entanglement entropy S_A and particle variance V_A of an extended space region A, holds for any subsystem A and in any dimension, analogously to homogeneous noninteracting Fermi gases.

Characterization of Mott-insulating and superfluid phases in the one-dimensional Bose-Hubbard model

We use strong-coupling perturbation theory, the variational cluster approach (VCA), and the dynamical density-matrix renormalization group (DDMRG) method to investigate static and dynamical properties of the one-dimensional Bose--Hubbard model in both the Mott-insulating and superfluid phases. From the von Neumann entanglement entropy we determine the central charge and the transition points for the first two Mott lobes. Our DMRG results for the ground-state energy, momentum distribution function, boson correlation function decay, Mott gap, and single particle-spectral function are reproduced very well by the strong-coupling expansion to fifth order, and by VCA with clusters up to 12 sites as long as the ratio between the hopping amplitude and on-site repulsion, t/U, is smaller than 0.15 and 0.25, respectively. In addition, in the superfluid phase VCA captures well the ground-state energy and the sound velocity of the linear phonon modes. This comparison provides an authoritative estimate for the range of applicability of these methods. In strong-coupling theory for the Mott phase, the dynamical structure factor is obtained from the solution of an effective single-particle problem with an attractive potential. The resulting resonances show up as double-peak structure close to the Brillouin zone boundary. These high-energy features also appear in the superfluid phase which is characterized by a pronounced phonon mode at small momenta and energies, as predicted by Bogoliubov and field theory. In one dimension, there are no traces of an amplitude mode in the dynamical single-particle and two-particle correlation functions.

Detecting Quantum Critical Points Using Bipartite Fluctuations

We show that the concept of bipartite fluctuations F provides a very efficient tool to detect quantum phase transitions in strongly correlated systems. Using state-of-the-art numerical techniques complemented with analytical arguments, we investigate paradigmatic examples for both quantum spins and bosons. As compared to the von Neumann entanglement entropy, we observe that F allows us to find quantum critical points with much better accuracy in one dimension. We further demonstrate that F can be successfully applied to the detection of quantum criticality in higher dimensions with no prior knowledge of the universality class of the transition. Promising approaches to experimentally access fluctuations are discussed for quantum antiferromagnets and cold gases.

Tomonaga-Luttinger-liquid criticality: Numerical entanglement entropy approach

The von Neumann entanglement entropy is studied with the density-matrix renormalization-group technique. We propose a simple approach to calculate the central charge using the entanglement entropy for a one-dimensional (1D) quantum system. This approach is applied to a couple of quantum systems: (i) a 1D frustrated spin model and (ii) a 1D half-filled spinless fermions with nearest-neighbor repulsion; it is confirmed that the central charge is estimated very accurately for the both systems. Also, a new method to determine the critical point between Tomonaga-Luttinger-liquid and gapped (or ordered) phases from the proposed approach is suggested. Furthermore, we mention that the Tomonaga-Luttinger parameter can be obtained in a like manner as the central charge, using the charge-density fluctuation of a part of the 1D system.

Von Neumann entropy and bipartite number fluctuation in quantum phase transitions

In order to characterize quantum phase transitions, we consider von Neumann entanglement entropy of bipartition systems. We propose a bipartition such as even-odd in a one-dimensional system or checkerboard in a two-dimensional case. The entanglement entropy successfully indicates the critical point in the spinless fermion system where the metal-insulator transition is known to exist. Furthermore, we calculate the bipartite number fluctuation which shows a shoulder near the transition.

Entanglement entropy between two coupled Tomonaga-Luttinger liquids

We consider a system of two coupled Tomonaga-Luttinger liquids (TLL) on parallel chains and study the Renyi entanglement entropy $S_n$ between the two chains. Here the entanglement cut is introduced between the chains, not along the perpendicular direction as used in previous studies of one-dimensional systems. The limit $n\to1$ corresponds to the von Neumann entanglement entropy. The system is effectively described by two-component bosonic field theory with different TLL parameters in the symmetric/antisymmetric channels as far as the coupled system remains in a gapless phase. We argue that in this system, $S_n$ is a linear function of the length of the chains (boundary law) followed by a universal subleading constant $\gamma_n$ determined by the ratio of the two TLL parameters. The formulae of $\gamma_n$ for integer $n\ge 2$ are derived using (a) ground-state wave functionals of TLLs and (b) boundary conformal field theory, which lead to the same result. These predictions are checked in a numerical diagonalization analysis of a hard-core bosonic model on a ladder. Although the analytic continuation of $\gamma_n$ to $n\to 1$ turns out to be a difficult problem, our numerical result suggests that the subleading constant in the von Neumann entropy is also universal. Our results may provide useful characterization of inherently anisotropic quantum phases such as the sliding Luttinger liquid phase via qualitatively different behaviors of the entanglement entropy with the entanglement partitions along different directions.

Valence-bond entanglement entropy of frustrated spin chains

We extend the definition of the recently introduced valence-bond entanglement entropy to arbitrary SU(2) wave functions of $S=1/2$ spin systems. Thanks to a reformulation of this entanglement measure in terms of a projection, we are able to compute it with various numerical techniques for frustrated spin models. We provide extensive numerical data for the one-dimensional ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ spin chain where we are able to locate the quantum phase transition by using the scaling of this entropy with the block size. We also systematically compare with the scaling of the von Neumann entanglement entropy. We finally underline that the valence-bond entropy definition does depend on the choice of bipartition so that, for frustrated models, a ``good'' bipartition should be chosen, for instance, according to the Marshall sign.

Quantum phase transitions in an interacting atom-molecule boson model

We study the quantum phase transitions of a model that describes the interconversion of interacting bosonic atoms and molecules. Using a classical analysis, we identify a threshold coupling line separating a molecular phase and a mixed phase. Through studies of the energy gap, von Neumann entanglement entropy, and fidelity, we give evidence that this line is associated to a boundary line in the ground-state phase diagram of the quantum system.

Exact results for the bipartite entanglement entropy of the AKLT spin-1 chain

We study the bipartite entanglement between a subsystem of size l and the rest of the system of total size L as it occurs in a spin-1 Affleck–Kennedy–Lieb–Tasaki (AKLT) chain subject to open boundary conditions. In this case, the ground-state manifold is four-fold degenerate and there is strong dependence on the parity of the number of spins, L. We present exact analytical results for the von Neumann entanglement entropy, as a function of both the size of the subsystem, l, and the total system size, L, for all four degenerate ground states for both odd and even L. In the large l, L limits the entanglement entropy approaches ln (2) for the SzT = ±1 while it approaches twice that value, 2ln (2), for the SzT = 0 states. In all cases, it is found that this constant is approached exponentially fast defining a length scale ξ = 1/ln (3) equal to the known bulk correlation length.

Valence Bond and von Neumann Entanglement Entropy in Heisenberg Ladders

We present a direct comparison of the recently proposed valence bond entanglement entropy and the von Neumann entanglement entropy on spin-1/2 Heisenberg systems using quantum Monte Carlo and density-matrix renormalization group simulations. For one-dimensional chains we show that the valence bond entropy can be either less or greater than the von Neumann entropy; hence, it cannot provide a bound on the latter. On ladder geometries, simulations with up to seven legs are sufficient to indicate that the von Neumann entropy in two dimensions obeys an area law, even though the valence bond entanglement entropy has a multiplicative logarithmic correction.

Entanglement entropy and multifractality at localization transitions

The von Neumann entanglement entropy is a useful measure to characterize a quantum phase transition. We investigate the nonanalyticity of this entropy at disorder-dominated quantum phase transitions in noninteracting electronic systems. At these critical points, the von Neumann entropy is determined by the single particle wave function intensity, which exhibits complex scale invariant fluctuations. We find that the concept of multifractality is naturally suited for studying von Neumann entropy of the critical wave functions. Our numerical simulations of the three dimensional Anderson localization transition and the integer quantum Hall plateau transition show that the entanglement at these transitions is well described using multifractal analysis.

Valence Bond Entanglement Entropy

We introduce for SU(2) quantum spin systems the valence bond entanglement entropy as a counting of valence bond spin singlets shared by two subsystems. For a large class of antiferromagnetic systems, it can be calculated in all dimensions with quantum Monte Carlo simulations in the valence bond basis. We show numerically that this quantity displays all features of the von Neumann entanglement entropy for several one-dimensional systems. For two-dimensional Heisenberg models, we find a strict area law for a valence bond solid state and multiplicative logarithmic corrections for the Néel phase.

On maximal entanglement between two pairs in four-qubit pure states

We show that the state with the highest known average two-particle von Neumann entanglement entropy proposed by Sudbery and one of the authors gives a local maximum of this entropy. We also show that this is not the case for an alternative highly entangled state proposed by Brown et al.