Nonzero-temperature entanglement negativity of quantum spin models: Area law, linked cluster expansions, and sudden death

We show that the bipartite logarithmic entanglement negativity (EN) of quantum spin models obeys an area law at all nonzero temperatures. We develop numerical linked cluster (NLC) expansions for the “area-law” logarithmic entanglement negativity as a function of temperature and other parameters. For one-dimensional models the results of NLC are compared with exact diagonalization on finite systems and are found to agree very well. The NLC results are also obtained for two dimensional $XXZ$ and transverse field Ising models. In all cases, we find a sudden onset (or sudden death) of negativity at a finite temperature above which the negativity is zero. We use perturbation theory to develop a physical picture for this sudden onset (or sudden death). The onset of EN or its magnitude are insensitive to classical finite-temperature phase transitions, supporting the argument for absence of any role of quantum mechanics at such transitions. On approach to a quantum critical point at $T=0$, negativity shows critical scaling in size and temperature.

Figure 1

Sudden death inverse temperate $\beta$ values for the antiferromagnetic $XXZ$ models for different size rings and chains, indicated with circles and crosses respectively. Note that the results converge from above for the chains and from below for the rings.

Figure 2

Comparison of sudden death temperature $T_{\text{SD}}$ for chains of length 4 and 9 with the perturbation theory results.

Figure 3

A second onset in negativity happens for a four-site system below an inverse temperature of $\beta =8$. The inset shows eigenvalues of the partially transposed density matrix as a function of $\beta$. Note that in a small window near these onsets there are multiple eigenvalues which change sign (going from positive to negative and from negative to positive).

Figure 4

Logarithmic negativity for the antiferromagnetic Heisenberg model $\left(\lambda =1\right)$ as a function of the length $L$ from exact diagonalization (ED) of periodic chains at different $\beta$ values, as well as from NLC up to order $L$. The terms in the NLC series are strongly alternating, so a Euler summation is used starting from the fourth term to smoothen the sum. Also shown are the results from the Lanczos based calculation for the ground state negativity. Plots with symbols are from ED while those without symbols are from NLC.

Figure 5

Logarithmic negativity for the transverse field Ising model (TFIM) tuned to its quantum critical point $\left(J=2\right)$ as a function of the length $L$ from exact diagonalization (ED) of periodic chains and also NLC up to order $L$ for different $\beta$ values. Also shown are the results from the Lanczos based calculation for the ground state negativity. Plots with symbols are from ED while those without symbols are from NLC.

Figure 6

Logarithmic negativity of the antiferromagnetic Heisenberg model as a function of $\beta$ calculated by exact diagonalization of rings of size $L$ and by NLC. For the NLC the partial sums of different orders (denoted NLC) and the Euler sums (denoted NLCET) are shown. Linear fits to the Euler transformed NLC results are shown by a dashed line.

Figure 7

Logarithmic negativity for the transverse field Ising model (TFIM) tuned to its quantum critical point $\left(J=2\right)$ as a function of $\beta$ calculated by exact diagonalization of rings of size $L$ and by NLC. For the NLC the partial sums of different orders are shown. At intermediate $\beta$, ED has not yet converged and appears to be increasing towards the NLC value. Linear fits to the NLC results are shown by a dashed line.

Figure 8

Logarithmic negativity for the TFIM on a square lattice as a function of $J$ for varying temperatures from the “low temperature” expansions (see text). The curves with diamonds are for NLC order 8, and solid lines are for order 7. The vertical dotted lines are the corresponding critical $J$ values for this system. Note that the critical $J$ is monotonic in temperature.

Figure 9

Phase diagram for the transverse field Ising model (TFIM) on the square lattice. Also illustrated are the sudden death temperatures found via NLC. Also shown are first order perturbation theory results. Data for the classical phase boundary gathered from [37].