Effects of temperature and magnetization on the Mott–Anderson physics in one-dimensional disordered systems

We investigate the Mott–Anderson physics in interacting disordered one-dimensional chains through the average single-site entanglement quantified by the linear entropy, which is obtained via density-functional theory calculations. We show that the minimum disorder strength required to the so-called full Anderson localization—characterized by the real-space localization of pairs—is strongly dependent on the interaction regime. The degree of localization is found to be intrinsically related to the interplay between the correlations and the disorder potential. In magnetized systems, the minimum entanglement characteristic of the full Anderson localization is split into two, one for each of the spin species. We show that although all types of localization eventually disappear with increasing temperature, the full Anderson localization persists for higher temperatures than the Mott-like localization.


Theoretical model
We simulate the disordered interacting lattices via the one-dimensional Hubbard model, with on-site disorder potential V i characterized by a certain concentration C ≡ L V /L of randomly distributed impurities, where L V is the number of impurity sites and L the chain size. The density operator is n iσ =ĉ † iσĉ iσ , the average density is n = N/L = n ↑ + n ↓ and the magnetization is m = n ↑ − n ↓ , where N = N ↑ + N ↓ is the total number of particles and ĉ † iσ ( ĉ iσ ) is the creation (annihilation) operator, with z-spin component σ =↑, ↓ at site i. All the energies are in units of t and we set t = 1.
We consider the average single-site entanglement: a bipartite entanglement between each site with respect to the remaining L − 1 sites averaged over the sites. This ground-state entanglement is quantified via the average linear entropy, where w ↑,i , w ↓,i , w 2,i e w 0,i are the occupation probabilities for the four possible states of site i: single occupation with spin up, single occupation with spin down, double occupation and empty, respectively. At finite temperature, the probabilities are calculated with respect to a thermal state ρ β = n e −βE n |n��n| , where |n� is an eigenstate of the Hamiltonian with energy E n and β = 1/k B T is the inverse temperature. Thus for small chains ( L = 8 ) we calculate Eq. (2) by diagonalizing the full Hamiltonian.
We also explore larger ( L = 100 ) disordered chains at T = 0 via density-functional theory calculations. In this case, instead of Eq. (2), we adopt an approximate density functional 14 for the linear entropy of homogeneous chains, where �(x) is a step function, with �(x) = 0 for x < 0 and �(x) = 1 for x ≥ 0 , and α(U) is given by where J k (x) are Bessel functions of order k. This density functional, Eq. (3), was specially designed to be used in LDA approximations for calculating the linear entropy of inhomogeneous systems via DFT calculations. Thus the entanglement in our large disordered chains are approximately obtained via LDA: where the density profile {n i } is calculated via standard (Kohm-Sham iterative scheme) DFT calculations within LDA for the energy, in which the exact Lieb-Wu 21 energy is used as the homogeneous input.
For each set of parameters (C, V; U, n, m), L is obtained through an average over 100 samples of random disorder samples to ensure that the results are not dependent on specific configurations of impurities. Notice that this huge amount of data would be impracticable via exact methods such as DMRG (for a comparison between our DFT approach and DMRG calculations, see Supplementary Material of Ref. 18 ).

Results and discussion
We start by exploring the Mott-Anderson physics at zero temperature via the entanglement as a function of interaction. In Fig. 1a we consider several concentrations of impurities with a fixed strength V = −20t , thus ranging from strong ( |V | >> U ) to moderate ( |V | ≈ U ) disorder. As disorder becomes more relevant, i.e. for U → 0 , entanglement decreases and saturates for any concentration C > 0 . This saturation characterizes the localization: the disorder potential freezes the electronic degrees of freedom such that L → constant. Figure 1a also shows the non-monotonic behavior of entanglement with C, whose minimum occurs at the critical concentration C C = 100n/2 = 40% for V < 0 (for V > 0 , C C = 100(1 − n/2) ), observed previously in the MIT 18 and in the SIT 19,20 . This minimum entanglement has been associated-in both MIT and SIT cases-to a fully localized state, marked by L → 0 for |V | → ∞ due to real-space localization of pairs (as also confirmed by the average occupation probabilities, see Fig. 2). While for the MIT the full localization was found to appear for |V | ≥ V min ≈ 3t for U = 5t , for the SIT the same minimum V min ≈ 3t was found for any interaction 20 . However we observe now a distinct feature: Fig. 1a reveals that depending on the interaction strength ( U > 10t ) even a strong disorder potential as V = −20t is not enough to fully localize the system, i.e. L = 0 at C C = 40% . In other words, V min in the MIT case is strongly affected by the interaction.
To further analyze this interplay between U and V, in Fig. 1b we focus on the critical concentration C C and vary instead the potential strength V. We find that for |V | U the degree of entanglement is essentially independent on V, suggesting that the system presents the same degree of localization for a given U and that this is a weak localization, since the degree of entanglement is very close to the clean V = 0 case. In contrast, for |V | U the degree of entanglement decreases with U decreasing and is smaller for higher |V|, reaching the full www.nature.com/scientificreports/ localization when |V | → ∞ . As U increases, a stronger V is required for having L → 0 , confirming thus that the full localization in the MIT requires a minimum disorder strength V min which is dependent on the interaction. For the particular case of V = −1t , such that U is always U |V | , we don't find the characteristic decreasing of entanglement when U → 0 , indicating that the full localization does not occur in this case. Next we analyze the impact of the impurities' concentration on the entanglement for several attractive, Fig. 2a, and repulsive, Fig. 2b, disorder strengths. In both cases we see the signature of the full Anderson localization for |V | U : minimum entanglement at the critical concentration C C = 100n/2 for V < 0 and C C = (1 − n/2)100 for V > 0 , with L → 0 for |V | → ∞ . For |V | U the minimum at C C disappears, so the system does not fully localize.
We also see the extra minimum at C * C = 100n for V < 0 (Fig. 2a) and at C C * = (1 − n)100 for V > 0 (Fig. 2b) associated to a Mott-like localization 18 , in which the effective density is equal to 1 either at the impurity sites (for V < 0 ) or at the non-impurity sites (for V > 0 ). For attractive disorder this means that the average double occupancy in the impurity sites ( w V 2 ) tends to zero due to the repulsion U, while the single-particle probability ( w V ↑ ) tends to a maximum, as confirmed by Fig. 2c,d (for repulsive disorder, the same holds for the non-impurity sites: . Notice however that the Mott-like MIT requires a minimum amount of disorder to occur. Thus the two entanglement minima-full Anderson and Mott-like localizations-are intrinsically connected through the interplay between interaction and disorder. In Fig. 2e one can see that if the interaction is too small compared to the disorder strength ( U |V |/2 ) only the minimum related to the full Anderson localization persists, while if U is strong compared to V ( U |V | ) only the minimum related to the Mott-like localization holds, the two minima appearing only for U 10t , |V | U.
In Fig. 3a,b we show the impact of the temperature on both the full Anderson and the Mott-like localization. As the temperature increases the two minima-at C = 100n/2 = 37.5% (full Anderson) and at C = 100n = 75% (Mott-like)-are attenuated. Our results reveal that the full Anderson localization survives for higher temperatures than the Mott-like localization, however for T = 20 there remains no localization in the system, since entanglement is high and maximum for any concentration.
Finally, while all the above calculations were performed with non-magnetized chains, i.e. for n ↑ = n ↓ = n/2 , in Fig. 3c we analyze the impact of the magnetization m = n ↑ − n ↓ � = 0 on the entanglement minimum related to the full Anderson localization. We find that the minimum at C C = 100n/2 = 50% for m = 0 is now split into two minima: one at C C = 100n ↑ and the other at C C = 100n ↓ . Our results thus reveal that the localization occurs separately for each species, thus with two critical densities n C,↑ = L V and n C,↓ = L V . Figure 3d shows however that the magnetized systems never reach the full localization: there remain spin degrees of freedom due to the unpaired majority species such that L saturates finite values.

Conclusion
In summary, we have explored the Mott-Anderson physics by analyzing the entanglement of interacting disordered chains. We find that the interplay between interaction (U) and disorder strength (V) defines the type and the degree of localization. For weak interactions, U |V |/2 , there only appears the full Anderson localization, marked by entanglement approaching zero when |V | → ∞ . In contrast, for weak disorder, |V | U , only the Mott-like localization holds, associated to an effective density equal to 1. The two types of localization, full Anderson and Mott-like, occurring only when both U and V are strong enough: U 10t , |V | U . For sufficiently strong interaction, U |V | , the entanglement is independent on the disorder potential and very close www.nature.com/scientificreports/ to the clean (non-disordered) case, suggesting thus that the localization is weak in this case. Our results also show that the temperature fades the localization phenomena, but that the full Anderson localization minimum survives for higher temperatures ( T ∼ 2 ) than the Mott-like localization. Finally we have shown that that the entanglement minimum related to the full Anderson localization is split into two when there is a magnetization in the system, one for each spin species, but in this case the localization is weaker due to remaining spin degrees of freedom, with entanglement saturating at finite values. www.nature.com/scientificreports/

Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request. www.nature.com/scientificreports/