Entanglement in many-body systems

Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral

Rev. Mod. Phys. 80, 517 – Published 6 May 2008

Abstract

Recent interest in aspects common to quantum information and condensed matter has prompted a flurry of activity at the border of these disciplines that were far distant until a few years ago. Numerous interesting questions have been addressed so far. Here an important part of this field, the properties of the entanglement in many-body systems, are reviewed. The zero and finite temperature properties of entanglement in interacting spin, fermion, and boson model systems are discussed. Both bipartite and multipartite entanglement will be considered. In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.

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DOI:https://doi.org/10.1103/RevModPhys.80.517

Authors & Affiliations

Luigi Amico

MATIS-CNR-INFM and Dipartimento di Metodologie Fisiche e Chimiche (DMFCI), viale A. Doria 6, 95125 Catania, Italy

Rosario Fazio

International School for Advanced Studies (SISSA) via Beirut 2-4, I-34014 Trieste, Italy and NEST-CNR-INFM and Scuola Normale Superiore, I-56126 Pisa, Italy

Andreas Osterloh

Institut für Theoretische Physik, Leibniz Universität Hannover, 30167 Hannover, Germany

Vlatko Vedral

The School of Physics and Astronomy, University of Leeds, Leeds LS29JT, United Kingdom and Center for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore

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Vol. 80, Iss. 2 — April - June 2008

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Figure 1
Schematic of the nearest-neighbor valence bond state, exact ground state of the spin-1 model in Eq. (27) for $β = 1 ∕ 3$ (AKLT-model). The ground state is constructed regarding every $S = 1$ in the lattice sites as made of a pair of $S = 1 ∕ 2$ , and projecting out the singlet state. The singlets are then formed taking pairs of $S = 1 ∕ 2$ in nearest-neighbor sites.Reuse & Permissions
Figure 2
(Color online) The derivative of the nearest-neighbor concurrence as a function of the reduced coupling strength. The curves correspond to different lattice sizes. On increasing the system size, the minimum gets more pronounced and the position of the minimum tends (see the left-hand side inset) towards the critical point where for an infinite system a logarithmic divergence is present. The right-hand side inset shows the behavior of the concurrence for the infinite system. The maximum is not related to the critical properties of the Ising model. From 317.Reuse & Permissions
Figure 3
Nearest-neighbor concurrence of the $X X Z$ model. From 176.Reuse & Permissions
Figure 4
The rescaled concurrence of the antiferromagnetic LMG model. The first order transition occurs at $h = 0$ . From 447.Reuse & Permissions
Figure 5
(Color online) The rescaled concurrence between two atoms in the Dicke mode. The concurrence is rescaled both for finite $N$ and in the thermodynamic limit. The inset shows the finite size scaling. From 254.Reuse & Permissions
Figure 6
(Color online) The block entropy is evaluated after partitioning the system in regions $A$ and $B$ . For finite range correlations, it is intuitive that the wave function of the system is factorized $∣ Ψ_{A} ⟩ ∣ Ψ_{B} ⟩$ by removing the region at the boundary. Accordingly the reduced entropy would vanish.Reuse & Permissions
Figure 7
The block entropy $S_{ℓ}$ for $L = 200$ in the case of an $X X Z$ Heisenberg chain for a critical value $Δ = 0.0$ (circles) and noncritical value $Δ = 1.8$ (squares). The critical data compared with the conformal field theory prediction (dashed line). Lower inset: Central charge extrapolated by fitting the numerical data $S_{ℓ}$ with Eq. (42) (with a factor $1 ∕ 2$ as done in the numerical calculation with the block at the boundary) for different values of $Δ$ . The data are for $L = 1000$ . Upper inset: Scaling of $c$ extrapolated as a function of $1 ∕ L$ for the worst case $Δ = 0.5$ and compared to a quadratic fit (dashed line). From 115.Reuse & Permissions
Figure 8
The idea of the comb partition suggested by Keating Mezzadri, and Novaes illustrated for three different values of the spacing $p$ . In all cases the subsystem $A$ , denoted by solid circles contains $l$ spins (be careful with the different notation with original paper) while the other (denoted by empty circles) contains the rest of the chain. The case $p = 1$ corresponds to the well-known “block” division. From 238.Reuse & Permissions
Figure 9
(Color online) Entanglement entropy of a subsystem of size $ℓ$ embedded in a closed ring of size $L$ , shown vs $ℓ$ in a log-linear plot. Numerical results obtained by exact diagonalization performed at the $X X$ point. For clean nonrandom systems with $L = 500$ and $L = 2000$ (open circles), $S_{l}$ is in agreement with Eq. (42) $A = 0.8595 + (\ln 2 ∕ 3) \ln ℓ$ , $Y = 0.72602 + \frac{1}{3} \ln [(2000 ∕ π) \sin (π ℓ ∕ 2000)]$ , $Z = 0.72567 + \frac{1}{3} \ln [(500 ∕ π) \sin (π ℓ ∕ 500)]$ . From 251.Reuse & Permissions
Figure 10
(Color online) Logarithmic negativity for harmonic chains. Top panel: Logarithmic negativity for coupling strength $α = 20$ and a bipartition in $n_{1}$ and $n_{2}$ oscillators (total number of oscillators $N = n_{1} + n_{2}$ ). For sufficiently large parts, a plateau is reached. For one part consisting of only a few oscillators, the negativity decreases as a function of the system size. Bottom panel: The logarithmic negativity relative to its plateau value $E_{N} (\infty)$ for $n_{2} = 20$ as a function of $n_{1}$ and varying coupling strength $α$ . For small coupling the plateau limit is reached faster; since the plateau limit is connected to a local quantity (the average energy per oscillator), this can be explained by a longer correlation length that grows with $α$ . From 30.Reuse & Permissions
Figure 11
Logarithmic negativity for harmonic chains. Top panel: The logarithmic negativity of two connected groups of $s$ oscillators as a function of the distance between both groups. It is seen that the range of the negativity is finite but grows with the group size $s$ . For $s = 1$ , its range is zero, meaning that no two single oscillators share pairwise entanglement unless they are neighbors. Bottom panel: Logarithmic negativity as in the top panel but for an open chain. The value of the negativity is roughly half as large as for the periodic chain since the number of connection points of the parts is halved. In contrast to the periodic chain, the negativity of one small part with the rest of the open chain grows with system size From 30.Reuse & Permissions
Figure 12
(Color online) Entanglement entropy for $N = 500$ and $ℓ = 125$ as a function of the parameters $h_{z}$ and $γ$ of the ferromagnetic LMG model. For $γ \neq 1$ , there is an anomaly at the critical point $h_{z} = 1$ , whereas the entropy goes to zero at large $h_{z}$ since the ground state is a fully polarized state in the field direction. In the zero field limit, the entropy saturates when the size of the system increases. For $γ = 1$ , the entropy increases with the size of the system in the region $0 ⩽ h_{z} < 1$ and jumps to zero at $h_{z} = 1$ . From 259.Reuse & Permissions
Figure 13
(Color online) Entropy $S$ between two-level atom and cavity field as a function of the atom-radiation coupling for both $N \to \infty$ and finite $N$ . Inset: The scaling of the value of the entanglement maximum as a function of $\log N$ . From 254.Reuse & Permissions
Figure 14
(Color online) Local entanglement entropy in Hubbard models. Upper panel: The local entropy as a function of the on-site coupling $u$ and nearest-neighbor coupling $v$ . The contour plot below includes the known phase diagram of the model (full solid line). Except the superconducting phase, the phase diagram is nicely reproduced. Lower panel: The local entropy for the standard Hubbard model and different on-site couplings $u$ as a function of the filling (1 corresponds to half filling). Except for $U ∕ t = 0$ , where a maximum appears, this quantity shows a dip at half filling, i.e., at the metal-insulator transition. From 175.Reuse & Permissions
Figure 15
(Color online) Local entanglement entropy in bond-charge Hubbard models. Upper panel: The ground state phase diagram of the Hirsch model at $x = 1$ . Empty, slashed, and solid circles indicate the presence of empty, singly, and doubly occupied sites, respectively. Bottom panel: Except the insulating line IV, the phase diagram is nicely reproduced by $\partial_{u} S_{i}$ . From 22.Reuse & Permissions
Figure 16
Entanglement between two qubits interacting according to the antiferromagnetic Heisenberg model as a function of the external field $h_{z}$ and temperature (multiplied by the Boltzmann’s constant) $T$ with coupling $J = 1$ . The topmost plot shows the variation of nearest-neighbor entanglement with the magnetic field. The middle and bottommost plots show the same for next-nearest and next-to-next-nearest neighbors, respectively, from 27.Reuse & Permissions
Figure 17
(Color online) Temperature effect on the anomalies from the critical divergence of the field derivative $C (R)$ measured by $\partial_{T} [\partial_{a} C (R)]$ . The density plot corresponds to $γ = 1$ and $R = 1$ . $T = T^{*}$ and $T = T_{M}$ are drawn as dashed and thick lines, respectively. Maxima below $T^{*}$ are found at $T_{M} = β T_{cross}$ with $β \sim 0.290 \pm 0.005$ and are independent of $γ$ and $R$ ; the crossover behavior is enclosed between two flexes of $\partial_{T} [\partial_{a} C (R)]$ at $T_{c 1}$ $T_{c 2}$ ; such values are fixed to $T_{c 1} = (0.170 \pm 0.005) T_{cross}$ and $T_{c 2} = (0.442 \pm 0.005) T_{cross}$ and independent of $γ$ and $R$ . For $T ≲ T_{c 1}$ , $\partial_{T} [\partial_{a} C (R)] ≃ 0$ . Scaling properties are inherited in $\partial_{T} [\partial_{a} C (R)]$ from $\partial_{a} C (R)$ ]. From 17.Reuse & Permissions
Figure 18
(Color online) Contour plot of $τ_{2}$ in the $h − T$ plane, for $γ = 0.3$ (i.e., $h_{f} = 0.9539 \dots$ ). The white area indicates where $τ_{2} = 0$ . From 13.Reuse & Permissions
Figure 19
(Color online) Magnetic susceptibility vs temperature $T$ from simulations of the diluted, dipolar-coupled Ising magnet, compared to experimental data (triangles). The solid circles use quantum decimation as well as the correct quantum mechanical form of susceptibility, utilizing the entanglement of the low-lying energy doublet with the excited states. From 163.Reuse & Permissions
Figure 20
(Color online) The temperature dependence of magnetic susceptibility of powder CN (triangles) and a single-crystal CN, measured at low field parallel (open squares) and perpendicular (open circles, crosses, filled circles) to the monoclinic $b$ axis. The data and the figure are from 51. The intersection point of this curve and the experimental one defines the temperature range (left from the intersection point) with entanglement in CN around $\approx 5 K$ . From 68.Reuse & Permissions
Figure 21
(Color online) One-tangle $τ_{1}$ and the sum of squared concurrences $τ_{2}$ as a function of the applied magnetic field in the $z$ direction for the $X Y Z$ model with exchange along $y$ : $J_{y} = 0.25$ (in unit of exhange along $z$ ). Inset: Contributions to the concurrence between $j th$ neighbors; closed symbols stand for $C^{I} (j)$ and open symbols for $C^{II} (j)$ . The dashed line marks the critical field $h_{c}$ . From 357.Reuse & Permissions
Figure 22
(Color online) The geometric entanglement measure Eq. (12) per site versus $γ$ and $h_{z}$ for the $X Y$ model. There are three different phases: the ordered oscillatory phase for $γ^{2} + h_{z}^{2} < 1$ and $γ \neq 0$ , the ordered ferromagnetic phase between $(γ^{2} + h) z^{2} > 1$ and $h < 1$ , and the paramagnetic phase for $h_{z} > 1$ . There is a sharp rise in the entanglement across the quantum phase transition line $λ^{- 1} = h_{z} = 1$ . At $γ = 0$ lies the $X X$ model, which belongs to a different universality class than the anisotropic $X Y$ model. From 460.Reuse & Permissions
Figure 23
(Color online) Entanglement in thermal states of the $X X$ model in a transverse magnetic field. The different regions correspond to different types of multipartite entanglement contained in the equilibrium thermal state. From 181.Reuse & Permissions
Figure 24
(Color online) Purity relative to the observable $S^{z}$ in the ground state of the LMG model. From 393 Reuse & Permissions
Figure 25
(Color online) The purity $P_{u (N)}^{'} = P_{u (N)} - {(1 + γ)}^{- 1}$ of an $X Y$ model in a transverse field as a function of $λ$ for different anisotropies $γ$ . The constant part has been subtracted in order to make the connection to the disorder parameter. From 393.Reuse & Permissions
Figure 26
(Color online) Concurrence between sites $n = - x$ , $m = x$ , symmetrically placed with respect to the state where the singlet was initially created. From their initial position $i = - 1$ and $j = 1$ ; From 16.Reuse & Permissions
Figure 27
(Color online) The nearest-neighbor logarithmic negativity $E_{N}$ as a function of the initial transverse field $h_{z}$ and time $t$ for the anisotropy $γ = 0.5$ . At short times $t ⩽ 1.8$ the dynamical phase transition is a point of reentrance for the entanglement. At later times the state remains separable for values of the field above the critical value. For $t = 0$ , entanglement vanishes, as it should, as $h_{z}$ grows. The transverse magnetization of the evolved state does not show a similar critical behavior as a function of $h_{z}$ . From 385.Reuse & Permissions
Figure 28
(Color online) Evolution of the entropy for different types of quenches in the case of the Ising (top panel) [from 77] and Heisenberg (bottom panel) [from 115] models. The block are of 60 and 6 sites in the Ising and Heisenberg cases, respectively. In the Ising case the quench are obtained by changing the external magnetic field from $h_{0}$ to $h_{1} = 1$ . In the Heisenberg model the anisotropy parameters are used instead, $Δ_{0} = 1.5$ while $Δ_{1} = 0.0$ , 0.2, 0.4, 0.6, and 0.8. The time is expressed in units of the spin wave velocity. The inset of the top panel shows the behavior of the initial slope as a function of the final value of the anisotropy.Reuse & Permissions
Figure 29
Schematic representation of the of block entropy dynamics. Entangled particles are emitted from region $A$ , and they contribute to the block entropy as long as one of the two particles ends in region $B$ . From 77.Reuse & Permissions

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