Structural complexity of snapshots of 2D Fermi-Hubbard systems

In Fig. 2(a) we show the temperature dependence of the structural complexity of the FM Ising model. Our results are in agreement with Ref. [40] and the critical temperature, deduced from the location of the rapid drop in $C_{0}$ or $C_{1}$ or the peak in their derivatives, is in excellent agreement with the exact result. We can gain further understanding about the inner workings of the complexity measure by looking at the dissimilarities as a function of the coarse-graining step $k$ [Fig. 2(b)]. At high temperatures in the disordered phase, $D_{k}$ falls off rapidly with $k$. As the temperature is lowered to near, but still above, the transition, $D_{k}$ decays much more slowly, with non-vanishing values even at large values of $k$, illustrating the existence of correlations between far regions in the model. Finally, in the ordered phase, $D_{k}$ is mostly small and vanishes quickly at all $k$ by decreasing $T$. This makes sense since in a perfect FM, coarse-graining does not lead to a different image.

In contrast to the FM case, where $\mathcal{C}_{0}$ starts at a finite but constant value and then rapidly falls off to zero below the critical temperature, we find that in the AFM Ising model, $\mathcal{C}_{0}$ remains constant at all temperatures, as can be seen in Fig. 2(c). In order to understand this, let us analyze two limiting cases. At infinite temperature, i.e. in the fully disordered phase (where each spin orientation is equally probable), the structural complexity for both models is given by (see Appendix A for a derivation),

On the contrary, in the ground state for the perfect classical FM and AFM orderings it suffices to analyze what happens after the first coarse-graining (c.g.) step, i.e. $\mathcal{C}_{0}=D_{0}$, since it is easy to see that all other $D_{k}$ for $k>1$ vanish:

From these two limiting cases we conclude that while in the FM case $\mathcal{C}_{0}$ must exhibit a drop from $1/2$ to $0$ as one lowers the temperature and the system develops a net magnetization, in the AFM case $\mathcal{C}_{0}$ can remain constant. We note that the variations in $C_{0}$ in Fig. 2(c) are of the order of $10^{-5}$ and not visible in the scale of the plot. Since the derivation at high-temperatures assumes an equal number of spin ups and spin downs, and in the perfect classical AFM ordering this balance is preserved at all temperatures, we infer that $\mathcal{C}_{0}$ is sensitive only to the uniform magnetization of the system. As we will see later, this is also the case for the spin-balanced FHM.

On the other hand, $\mathcal{C}_{1}$ captures the transition temperature in the AFM Ising model with great accuracy, and exhibits a similar behavior to its FM counterpart: the complexity is larger in the disordered phase, but tends to zero in the ground state, illustrating that in the latter state, after a single coarse-graining step, the resulting patterns are no longer complex, and that there is scale invariance.

We observe that the behaviors of the FM and AFM dissimilarities as a function of $k$ are very different. For the AFM case [Fig. 2(d)], in the ordered phase, only $D_{0}$ is nonzero and contributions at higher $k$ come from thermal fluctuations. Interestingly, because $\mathcal{C}_{0}$ is constant at all $T$, we observe a conservation of weights in the dissimilarities. In other words, the same exact weight loss in $D_{k}$ for $k>0$ is gained by $D_{0}$ at every temperature.

Finally, we notice that $\mathcal{D}_{0}$ in the FM Ising model closely tracks, within a prefactor, the temperature behavior of the entropy per spin $S$ (not shown), and $\mathcal{C}_{0}$ correlates with the Boltzmann entropy $S_{B}$, defined as [50]

where $p_{\sigma}=(1\pm m)/2$, and $m$ is the average magnetization of the system. That means $\mathcal{C}_{1}$ follows the trend in the total correlation density $\rho_{c}=S_{B}-S$ [50]. The Boltzmann entropy assumes the lack of internal correlations and corresponds to the isolated spin entropy. Since $\rho_{c}$ only vanishes when no internal correlations occur, i.e. when each spin is independent, it measures how constrained the spin distribution is, and is therefore maximized when the spins are strongly correlated [50].

We now turn our attention to the XY model. In Fig. 2(e) we show the structural complexity of the model as a function of temperature for two system sizes $128\times 128$ (solid markers) and $256\times 256$ (open black markers). The results are averaged over five different initial random seeds. We further apply a moving average with a five-point window over temperature fitted with a local third-order polynomial (the Savitzky-Golay filter) to reduce noise. Similarly to the AFM Ising model, $\mathcal{C}_{0}$ is more or less constant for all temperatures and for both system sizes, indicating the near zero net magnetization of the system. On the contrary, $\mathcal{C}_{1}$ increases as the temperature is lowered, indicating that the low-$T$ phase, where the existence of pairs of vortices with opposite vorticity is expected [47], is the most complex one. We find a strong finite-size dependence in $\mathcal{C}_{1}$, expected to persist in this model for certain properties even with systems an order of magnitude larger in size [51]. The derivative of $\mathcal{C}_{1}$ for the $128\times 128$ system exhibits broad and inconclusive features at $T/J\lesssim 3$. By increasing the system size to $256\times 256$, these features develop into a sharp upturn at the lowest temperatures we could access in our simulations due to inflating autocorrelation times. This feature is consistent with the location of the peak in the specific heat ($T/J\sim 1.04$) and the BKT transition temperature of the model ($T_{c}/J=0.894$) [52, 51]. In order to access lower temperatures and larger system sizes, more sophisticated numerical algorithms are required, which are beyond the scope of this work [51, 53, 54].

We find that in this case, the behavior of the dissimilarities as a function of $k$ is completely different from those for the FM or AFM Ising models as illustrated in Fig. 2(f). At high temperatures $D_{k}$ decays rapidly with $k$. However as the temperature is lowered, we find significant contributions to the complexity from all length scales for $T\lesssim T_{c}$, which we attribute to the existence of a non-trivial topological phase.

Since the structural complexity aggregates information about correlations in the system up to a distance of $\Lambda^{k}$ at step $k$, we define an average for $k$,

and take $\Lambda^{\langle k\rangle}=2^{\langle k\rangle}$ as a relevant length scale of the model. We therefore expect this quantity to reflect the cluster sizes needed to describe the behavior of the system at different temperatures.

We show $2^{\langle k\rangle}$ vs $T/J$ in Fig. 3 for the classical models considered here. In the case of FM Ising model, it grows rapidly near $T_{c}$ as the temperature is lowered and then rapidly falls in the ordered phase. Its behavior is reminiscent of the critical behavior of the magnetic susceptibility of the model. The location of the peak in this quantity (black vertical line in Fig. 3) provides a great estimate for $T_{c}$. Interestingly, we find that the behavior of $2^{\langle k\rangle}$ for the AFM Ising model resembles the shape of the energy, and its derivative (red dashed line in Fig. 3) predicts an accurate estimate for $T_{c}$ too (its peak location is denoted by the red vertical line). Finally, for the XY model, $2^{\langle k\rangle}$ grows as the temperature is lowered, indicating a growing length scale for relevant correlations, consistent with the non-trivial topological order the system exhibits.

In Fig. 5(a) we show the structural complexity of single-species snapshots of the half-filled FHM as a function of temperature. We observe that $\mathcal{C}_{0}$ is constant for all temperatures, similar to what was observed in the AFM Ising model, indicating a spin-balanced mixture. On the contrary, $\mathcal{C}_{1}$ decays monotonically as the temperature is lowered. Its derivative (dashed red line) appears to diverge as $T\to 0$ in agreement with the fact that for the 2D FHM, the Néel transition occurs at $T_{\mathrm{N}}=0$.

Results for the corresponding dissimilarities as a function of the coarse-graining step at different temperatures exhibit a similar behavior as for the AFM Ising model [see Fig. 5(b)]. Namely, the weight lost by $D_{0}$ as the temperature increases is gained by $D_{k>0}$. There is however a fundamental difference here corresponding to the non-vanishing $D_{1}$ at all temperatures, which signals the presence of quantum fluctuations in the model. $\mathcal{C}_{0}$ is constant despite that.

For both the complexities and the dissimilarities, results based on snapshots that are obtained experimentally using a quantum gas microscope [37] (denoted by open markers) are in excellent agreement with the theory curves. Furthermore, this agreement between the numerical and experimental values of the dissimilarities holds for every $D_{k}$ at all temperatures considered, as evidenced in Fig. 6 where we show $D_{k}$ vs $T/t$ for the half-filled case and the 18% doped FHM. Such excellent agreement demonstrates the applicability of the method to current experiments involving ultracold atoms in optical lattices.

More interestingly, we find that $\mathcal{C}_{1}$ vs temperature or doping exhibits the same trends as the entropy per site $S/N$ in the model (see Fig. 7 for the doping dependence at different temperatures). We observe that below $T/t<1$, the structural complexity develops a peak around 20% doping. Although the peak in the entropy occurs at a slightly lower doping (see the inset of Fig. 7), the shape and trends are fairly similar. Additionally, we also observe good agreement with the experimental data at all temperatures considered, which further supports the validity of our results. Because of the correlation between $\mathcal{C}_{1}$ and $S/N$, this suggests the structural complexity of Fermi-Hubbard snapshots can be used as a quick and rapid proxy for the entropy of the system. This can be very useful in optical lattice experiments, where after loading the lattice, the calculation of the entropy is not straightforward [56].

It is worth mentioning that previous studies in one-dimensional systems: bit-strings [41] and the extended Hubbard model [43] have demonstrated that the structural complexity draws a close analog to the entanglement entropy since both capture the existence of nonlocal correlations in the models. In the present study we do not have access to the entanglement entropy and therefore only focus on the thermodynamic entropy. Although the entanglement entropy and the thermodynamic entropy are not equivalent to each other, they are associated with the development of correlations in the system and with the freezing of degrees of freedom, respectively, which generally occur together as we vary a tuning parameter. The establishment of connections between the structural complexity and the entanglement entropy in doped Fermi Hubbard models at finite-$T$ is an open question that requires further exploration.

Another important observation is the fact that at the temperatures considered, the region where the structural complexity is maximal corresponds to the strange metal region. The broad peak in $\mathcal{C}_{1}$ spanning $\delta\in[10\%,30\%]$ is consistent with the doping value $\delta_{c}\sim 30\%$ that corresponds to the crossover to the Fermi liquid phase [57]. Moreover, the trends in the structural complexity vs doping illustrates that features related to phases that emerge in this doping range at lower temperatures (such as fluctuating stripes [31, 33, 32]) are more complex than the AFM parent state at half-filling, and that this metallic region is also more complex than the conventional Fermi liquid found at larger dopings. As optical lattice experiments evolve and stripe physics becomes available, we expect that signatures of the presence and periodicity of stripe orderings should be detectable by analyzing the contributions of the dissimilarities at different scales and $\mathcal{C}_{1}$.

Before quantum gas microscopy had access to images of both fermionic species of the same system at the same time, density snapshots of the FHM could not be obtained; only local moments could be detected separately. Nowadays, with spin-resolved imaging, local moment as well as total density images can be created [58, 28, 29]. In this section we study their structural complexities and dissimilarities.

In Fig. 8(a) we show the structural complexities of the local moments (open markers and dashed lines) and the total density (solid markers and solid lines) of the half-filled FHM as a function of temperature. In contrast to what we observed for the spin-resolved snapshots, $\mathcal{C}_{0}$ is not constant but exhibits a non-monotonic behavior as a function of $T/t$. This is because unlike for single-species snapshots, as the temperature changes, the number of doublons and therefore the balance between filled and empty pixels in the snapshots changes too. Such non-monotonic behavior persists for $C_{1}$, albeit with smaller magnitude changes.

We find that for $T/t\leq 2$, $\mathcal{C}_{0}$ for the density snapshots is larger than for the local moment snapshots, but the opposite is true for $\mathcal{C}_{1}$. We can understand this through the following argument: Below the characteristic AFM temperature, where the upturn in double occupancy starts as we lower $T$, for density snapshots, the doublon-hole fluctuations take place on nearest-neighbor sites, and so the first coarse-graining step already renders mostly uniform images. On the contrary, for local moment snapshots, because both holes and doublons show up as empty pixels, one still has significant structure for later coarse-graining steps.

To complement these results, in Fig. 8(b) we show the dissimilarities of the half-filled FHM as a function of coarse-graining steps at different temperatures. $D_{k}$ for both the local moments and total density exhibit an exponential decay with $k$ at fixed $T/t$.

In Fig. 9, we show the temperature dependence of the dissimilarities for the density snapshots at $\delta=0.18$. Here we observe that (i) $D_{k}$ decays rapidly with increasing $k$, (ii) $D_{0}$ exhibits a non monotonic behavior with $T/t$, akin to the double occupancy, $\mathcal{D}=\langle n_{i\uparrow}n_{i\downarrow}\rangle$, and (iii) it closely follows the function $\mathcal{D}^{\prime}=(3/4)[\mathcal{D}+0.5\delta(1-\delta)]$. This expression can be derived analytically under the assumption that the uncorrelated parts of the correlation functions dominate (see Appendix B), and it holds for other dopings too. The agreement at high-$T$, where the assumption is valid, illustrates that $D_{0}$ for the density snapshots is in fact closely related to the double occupancy fraction in the model. More important are the discrepancies at $T/t\leq 1$, which signal the presence of non-trivial 2-point nearest- and next-nearest-neighbor spin-resolved density-density correlations in the model. A similar approximation for $D_{0}$ based on local moment snapshots proves challenging as 3-point and 4-point density correlators enter the analysis (see Appendix B).

Fig. 10(a) shows $\mathcal{C}_{0}$ as a function of $\delta$ for the density snapshots along with a shifted $\mathcal{D}$ from the numerical linked-cluster expansion (NLCE) [59, 60, 55] for each value of the doping. $\mathcal{C}_{0}$ exhibits an interesting behavior: (i) it is non-monotonic with temperature for all values of $\delta$, (ii) is largest at $\delta=0.5$, except at very high temperatures, and remarkably (iii) fully matches the shifted double occupancies at all temperatures and all dopings considered. Previously, we discussed that $D_{0}$ is closely correlated to the double occupancy fraction in the model. When higher order $D_{k}$’s are added, they capture the remaining features of the double occupancy, and therefore, $\mathcal{C}_{0}$ captures its temperature dependence extremely well.

In contrast, the structural complexities of the local moment snapshots [Fig. 10(b)] are more complicated to interpret. Although the behavior of the curves for all dopings except $\delta=0.5$ look fairly similar to their counterparts based on density, we were unable to derive a simple analytic expression based on thermodynamic properties in this case due to the presence of higher order correlations even at the first coarse graining step. The most interesting region for this complexity occurs at $\delta=0.5$ where for $T/t\leq 1$, $\mathcal{C}_{0}=0.125$, the same constant value of the structural complexity as for the single-species snapshots at half-filling. This region corresponds to the CDW phase observed at quarter-filling in the FHM [26, 37], where one sublattice is occupied and other sublattice is empty, and therefore, $\mathcal{C}_{0}$ is merely capturing the classical complexity associated with having the largest entropy in the images with an equal number of empty and filled pixels.

Contrary to what occurred with the single-species snapshots, here we do not find excellent agreement between experiment and theory. Figure 10(b) displays experimental data at different dopings as open markers. For $\delta=0.35$ the open diamond indeed matches with the simulations at the same doping. However, $\mathcal{C}_{0}$ at $\delta=0.6$ has the same value. On the other hand, for data at $\delta=0.18$, the open left triangles match fairly well with the simulations at a slightly larger doping of $\delta=0.23$, which might suggest the experimental data was taken at such higher doping. Finally, $\mathcal{C}_{0}$ based on experimental data at half-filling does not show good agreement with that calculated for the simulated images, and therefore could signal issues with the imaging procedure. Hence, the structural complexity of local moments can be used to further diagnose imaging issues.

Fig. 11 presents $2^{\langle k\rangle}$ vs doping at different temperatures for the three different types of snapshots analyzed. At the highest temperature considered, $T/t=4$, $2^{\langle k\rangle}$ increases slowly with $\delta$ for the $n_{\sigma}$ and $n$ snapshots, but decreases for the $m_{z}$ snapshots. This trend persists down to all temperatures considered for the $m_{z}$ snapshots, becoming sharper as the temperature is lowered, but surprisingly exhibiting a crossing at $\delta\sim 18\%$, which points to a possible temperature-independent length scale in $m_{z}$ at this doping. This coincides with the region where structural complexity is maximal and corresponds to the strange metal region. Since this occurs at easily experimentally-accessible regimes, it is a phenomenon that can be explored immediately in the experiment.

$2^{\langle k\rangle}$ vs temperature for $n_{\sigma}$ and $n$ snapshots also displays interesting behaviors at the same doping. As seen in Fig. 11(a) for $n_{\sigma}$, the curves develop a plateau that starts around $\delta\sim 18\%$ and extends up to $\delta\sim 40\%$, while for $n$ in Fig. 11(b) the curves develop a broad peak around $\delta\sim 18\%$.