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Carrying on with the idea of introducing staggered laser fields to the system, in this section, we will demonstrate that a critical point has been identified from numerical simulation. The system, with the staggered laser field, is depicted in Fig. 3.1(b). Here, to make it clear, the sublattice fugacity parameter ξodd (ξeven), is

related to the laser parameters, {Ωodd, ∆odd} ({Ωeven, ∆even}) through Eqs. (3.5).

For the upcoming analysis, it is convenient to define the difference ξd= ξodd− ξeven

and the sum ξs = ξodd+ ξeven of the two sublattice fugacity parameters.

To investigate the effect of the staggered laser fields, let us now specifically study the expectation value of the Rydberg density on the odd sublattice |ni_odd= P

k=oddnk/L. Using the ground state (3.4) and following the transfer matrix

method one finds,

|ni_odd = 1 2 " 1 + ξd− ξ −1 s p(1 + ξ−2 s )(1 + ξd2) # . (3.7)

To verify the analytical results, we numerically diagonalise Hamiltonian (2.40) to obtain the sublattice density, hni_odd, and compare it to Eq. (3.7) with fixed ξs=

5 and ξs = 20. The numerical diagonalisation used here is very similar to the one

that we introduced in Sec. 5.6. In short, we write HRyd in a matrix representation

that is calculated from the sum of the tensor products in this Hamiltonian. e.g. P2

Figure 3.4: We plot the analytical result obtained from Eq. (3.7) in black and compare it to the numerically obtained hni_odd from diagonalising Hamiltonian (2.40) within the ξk-manifold given in Eq. (2.42), with ξs = 5 < L in (a) and with

ξs= 20 > L in (b). In (a), we see an excellent agreement and a clear convergence

of the numerical results toward the analytical one with increasing lattice size. However, in (b), the results disagree due to the fact that the correlation length given by ηc = ξs/2 is beyond the lattice size and a large ξs and relatively small

L leads to non-negligible contribution from the 2nd eigenvalues of the transfer matrix [c.f. Sec. 2.3.2] for more details)

express them in terms of ξk in the ξk-manifold introduced in Eqs. (3.5). One can

then obtain classical observables such as hni_odd with the numerical ground state obtained from the diagonalisation.

The results are compared in Fig. 3.4. In (a), we see the analytical result is in good agreement with the numerics. As one increases the lattice size in the numerical calculation, a convergence of the results toward the analytical results is also found. However, in plot (b) where ξs = 20 the numerics and analytic are

no longer in agreement. We notice that the major difference between the two cases is the choice of ξs in respect with the value of the lattice size, L. With

further investigation of different combination of ξs and L, we find that as long as

ξs < L, the results are in good agreement. To physically explain this, one can

show that the correlation length ηc = ξs/2 at ξd = 0 meaning that for ξs > L,

the correlation length of the system is beyond the lattice size. Another reason is due to the finite size where the approximation used in the transfer matrix method is not well justified. For instance, as demonstrated in Sec. 2.3.2, when calculating the partition function of the hard-dimer gases, one of the eigenvalues in the transfer matrix has be neglected in the thermodynamic limit. Nonetheless, for small system size and large ξs, one notices that the two eigenvalues are on the

same orders of magnitudes. Therefore, in the following calculations, we restrict ourselves to relatively large L with ξs always smaller than L.

Figure 3.5: (a) Mean density of the odd sublattice as a function of ξd. Here,

with ξs = 5, we have plotted the analytical result given in Eq. (3.7) in black and

the numerical result obtained from diagonalising Hamiltonian (3.1) in red (with circles) with L = 14. The latter shows a significantly steeper switching of the sublattice populations at ξd = 0. (b) Susceptibility χodd(ξs = 5, ξd) for different

lattice sizes: L = 10 (red circles), L = 12 (blue triangles), and L = 14 (black squares). The data suggests a divergence of the susceptibility at ξd = 0 in the

limit of large lattice sizes L.

Having justified the analytical results obtained in Eq. (3.7), now let us discuss the features in Fig. 3.4(a) in more details. For small ξ_s−1 Eq. (3.7) predicts a transition between two states in which Rydberg atoms predominantly occupy the odd/even sublattice which takes place when the difference between the sublattice fugacity parameters vanishes ξd = 0 [see Fig. 3.5(a)]. This is expected since

for ξ−1_s = 0 and ξd = 0 both |↑↓↑↓↑ ...i and |↓↑↓↑↓ ...i are ground states and

any non-zero value of ξd will favour one over the other. Note, that according to

Eq. (3.7), which is plotted in Fig. 3.5 in black, this switching between the two anti-ferromagnetic ground states is smooth as the susceptibility, which can be calculated as,

χodd(ξs, ξd) = ∂hnoddi/∂ξd, (3.8)

saturates at a value 1/2 at the “transition point” {ξd= 0, ξs−1 = 0} [see Eq. (3.7)].

Since a tiny perturbation to the system would lead to a symmetry breaking, e.g. an impurity, one would naturally expect the transition occurring at ξd = 0 to be sharp

rather than a crossover. Therefore, although the frustration-free Hamiltonian (2.40) excellently describes the Rydberg gas along the curve parameterised by Eqs. (2.42) as shown in Ref. [31], it is very questionable whether this Hamiltonian and Eq. (3.7) faithfully describe the actual sublattice occupation of the ground state of the Rydberg gas Hamiltonian (3.1) in one-dimension at the “transition point”. The suspicion is confirmed by numerically calculating hnoddi in the ground

Figure 3.6: The scaling behaviour of the variance at ξdclose to critical value ξd = 0.

The dashed line indicates the boundary of the scaling region for the L = 18 curve. From the left to the right, we have corresponding lattice size L = 18, 16, 14, 12. The gradient seems to not depend on the lattice size.

clearly displays a significantly sharper transition. Moreover, as shown in Fig. 3.5(b), one can anticipate a diverging behaviour of the susceptibility χodd(ξs, ξd)

with increasing lattice sizes.

This strongly suggests that {ξd= 0, ξs−1 = 0} is a critical point of the Rydberg

gas Hamiltonian (3.1) in one-dimension which is not captured by the ground state of the frustration-free approximation (2.40). To investigate the nature of this point, we will perform a scaling analysis in the next section by using the results shown in Fig. 3.5(b).