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

We simulate the disordered networks in interaction via the one-dimensional Hubbard model,

$$begin{aligned} H = -tsum _{langle ij rangle sigma }({hat{c}}^{dagger }_{isigma }{hat{c}}_ {jsigma }+ Hc) + Usum _i{hat{n}}_{iuparrow }{hat{n}}_{idownarrow } + sum _{isigma }V_i{ hat{n}}_{isigma }, end{aligned}$$

(1)

with potential for on-site disorder (V_i) characterized by a certain concentration (Cequiv L_V/L) randomly distributed impurities, where (L_V) is the number of impurity sites and *L* the chain size. The density operator is ({hat{n}}_{isigma } = {hat{c}}^{dagger }_{isigma }{hat{c}}_{isigma })the average density is (n=N/L=n_uparrow +n_downarrow ) and the magnetization is (m=n_uparrow -n_downarrow )or (N = N_{uparrow } + N_{downarrow }) is the total number of particles and ({hat{c}}^{dagger }_{isigma }) (({hat{c}}_{isigma })) is the creation (annihilation) operator, with *z*– rotational component (sigma = uparrow ,downarrow ) on the site *I*. All energies are in units of *you* and we fix (t=1).

We consider the average single-site entanglement: a bipartite entanglement between each site with respect to the rest (L-1) sites on average over sites. This ground state entanglement is quantified via mean linear entropy,

$$begin{aligned} {mathcal {L}}=frac{1}{L}sum _i mathcal L_i=1-frac{1}{L}sum _ileft( text {w }_{uparrow ,i}^2+text {w}_{downarrow ,i}^2+text {w}_{2,i}^2+text {w}_{0,i }^2right) , end{aligned}$$

(2)

or (w_{uparrow ,i}), (w_{downarrow ,i}), (w_{2,i}) and (w_{0,i}) are the occupancy probabilities for the four possible states of the site *I*: Single occupancy with spin up, single occupancy with spin down, double occupancy and empty, respectively. At finite temperature, the probabilities are calculated with respect to a thermal state (rho _beta = sum _n e^{-beta E_n} {|{n}rangle } {langle {n}|})or ({|{n}rangle }) is a proper state of the energy Hamiltonian (In) and (beta = 1/k_B T) is the reciprocal of the temperature. So for small strings ((L=8)) we calculate the Eq. (2) by diagonalizing the complete Hamiltonian.

We are also exploring larger ((L=100)) unordered strings to (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,

$$begin{aligned} {mathcal L^{hom}(n,U>0)approx 2n-frac{3n^2}{2}+[(4n-2)alpha (U)-4alpha (U)^2]times theta [n-alpha (U)-1/2],} end{aligned}$$

(3)

or (The tax)) is a step function, with (Theta (x)=0) for (x and (Theta (x)=1) for (xge 0)and (alpha (U)) is given by

$$begin{aligned} {alpha (U)=2int _0^infty frac{J_0(x)J_1(x)exp left[ Ux/2right] }{left( 1+exp [Ux/2]right) ^2},} end{aligned}$$

(4)

or (J_k(x)) are Bessel order functions *k*. This functional density, Eq. (3), was specifically designed for use in LDA approximations to calculate the linear entropy of inhomogeneous systems via DFT calculations. Thus the entanglement in our large disordered chains is approximately obtained via LDA:

$$begin{aligned} {{mathcal {L}}approx {mathcal {L}}^{LDA}equiv frac{1}{L}sum _i{mathcal {L}}^{ hom}(n,U>0)|_{nrightarrow n_i},} end{aligned}$$

(5)

where the density profile ({neither}) is calculated via standard DFT calculations (Kohm-Sham iterative scheme) in LDA for energy, in which the exact Lieb–Wu^{21} energy is used as a homogeneous input.

For each set of parameters (*VS*, *V*; *you*, *not*, *m*), ({mathcal {L}}) is obtained through an average of more than 100 samples of random disorder samples to ensure that the results are not dependent on specific impurity configurations. Note that this huge amount of data would be unfeasible via exact methods such as DMRG (for a comparison between our DFT approach and DMRG calculations, see supplementary material in ref.^{18}).