Numerical results

Now we take the model in Eq. (5.5) and, for each range of interactions α and transverse eld h, obtain a ground state via DMRG and look for non-local correlations as explained in Section 5.3.1. In Figure 5.1

we show the maximal violation obtained with the two-body PIBI in Eq. (5.1), as well as the half-chain von Neumann Entanglement En- tropy (EE). Once observes that for values α → 0 the violation of the two-bodyPIBIis higher, as expected due to the nature of the two-body

PIBIunder consideration. Then, as the range of interactions decrease, the violation of Eq. (5.1) also decreases until eventually there is not detection of nonlocality for α . 3, which can be explained by the almost absence of spin squeezing at the nearest neighbour QCP [Liu et al., 2013; Frérot and Roscilde, 2018]. Notice that the detection of non-local correlations appears in the vicinity of the QCP.

Fig. 5.1 Numerical results through the phase diagram for the tunable range TFIM described in Eq. (5.5) corresponding to an n = 40 chain. The stars correspond to extrapolations n → ∞ of the maximal half-systemEE, which marks the QCP between the FM phase and the PM phase. Left: Maxi- mal violation of the two-body PIBI in Eq. (5.1), denoted as Wmin, given a

ground state. Negative values Wmin < 0 violate the inequality exhibiting the presence of non-local correlations. Right: Half-chainsEEof the ground state.

EEand the maximal violation of Eq. (5.1) happen at the same transverse- eld value when extrapolating to the thermodynamic limit. Therefore, since in the asymptotic limit the half-chainEEexhibits a maximum at the QCP independently of α, the coincidence of maximums manifests the quantum-critical origin of the violation for Eq. (5.1).

The ts considered in Section 5.3.2 to extrapolate the nite-size computations follow the expression hc(N ) = hc(∞) + aN−2/3. Finite-

size scaling theory predicts hc(n) − hc(∞) ∼ L−1/ν, where ν denotes

the exponent of the correlation length. For innite-range interactions, L is replaced by n1/dc where d

c = 3 is the upper critical dimension

of the quantum Ising model [Botet et al., 1982]. Hence, with ν = 1/2 (expected for innite-range interactions), hc(n)−hc(∞) ∼ n−2/3. Since

LSW theory predicts that for α < 1 the model is equivalent to the α = 0 limit, it is natural to expect that the same scaling law holds up to α = 1. On the other hand, for α > 1 there is no reason to

expect the same exponent. However, strong nite-size eects do not allow us to observe signicant deviations from the mean-eld behaviour for α < 2.2 (taking into account the system sizes accessible with our

DMRG computations). Furthermore, in Section 5.3.2 we also show some k-producible bounds from the DIWEDs previously constructed in Chapter 4 for Eq. (5.1). One we have observed some non-local correlations, the certication of entanglement depth needs no other procedure than to check which k-producible bound is violated. In this way, it can serve as a certication tool to provide insight on how the quantum correlations spread in the vicinity of QCPs, and to certify whether entanglement depth diverges at the critical point.

In Figure 5.3 we show the maximal violation of Eq. (5.1) obtained at the nite-size precursor of the QCPs, as a function of the system size n and the power-law exponent α. We call the value h at which the maximal violation is found the nite-size precursor of theQCPdenoted hc(n, α). With LSW theory applied on the witness in Eq. (5.3), that

we shall consider in Section5.4, we observe that for α < d and n → ∞ (where d is the lattice dimension), the maximal violation tends to hW i_min → −1/4. On the other hand, we also observe a quantiable error on the extrapolation for n ≤ 170 compared to the prediction, which we attribute it to be due to strong nite-size eects. As one increases α, the violation of Eq. (5.1) weakens up to α & 3 where the detection of nonlocality with Eq. (5.1) vanishes. To carry out the t in Figure5.3 for the extrapolation we have used the analytical results for α = 0 in [Dusuel and Vidal, 2004], which predict hJzi → n/2

and h(Jy)2i ∼ n2/3 and, therefore, Wmin(n) = Wmin(∞) + an−1/3 from

Eq. (5.4). Since for α ≤ 2.2 and the system sizes accessible via our

DMRG we did not observe deviations from the α = 0 behaviour, we used the same tting function in the α ≤ 2.2 cases.

Fig. 5.2 Numerical results for an n = 170 chainTFIMdescribed in Eq. (5.5) with α = 1.2 and as a function of the transverse eld h. In (a) we show the half-chainEE. In (b) we show the maximal violation of Eq. (5.1), where the dashed lines correspond to the k-producible bound for the DIWEDs built upon Eq. (5.1) in Chapter 4. An entanglement depth of k ≥ 30 is certied at the maximal violation. Finally, the inset in (a) shows the maximum of the half-chain EE and the maximal violation of Eq. (5.1) for several dis- tinct system sizes n, from which the critical point is extrapolated in the thermodynamic limit. The extrapolations follow hc(n) = hc(∞) + an−2/3.

Remarkably, in the asymptotic limit both the half-chainEEand the maximal violation of Eq. (5.1) occur at the same transverse eld hc.

Fig. 5.3 Violation of the two-bodyPIBI in Eq. (5.1) obtained at the QCP of the 1-dimensional TFIM in Eq. (5.5), as a function of α. The dots correspond to the nite-size DMRG computations (from top to bottom, n ∈ {150, 160, 170} respectively). The diamonds correspond to the ex- trapolations n → ∞ which have been obtained by using the results for n ∈ {30, 40, . . . , 170}. The stars correspond to the LSW theory results for n = 105. The inset shows the extrapolation for α = 0.2 as an example, where the t follows Wmin(n) = Wmin(∞) + an−1/3.

5.4 Analytical exploration: Linear spin-wave