Disorder and interactions in systems out of equilibrium: The exact independent-particle picture from density functional theory

Density functional theory (DFT) exploits an independent-particle-system construction to replicate the densities and current of an interacting system. This construction is used here to access the exact effective potential and bias of non-equilibrium systems with disorder and interactions. Our results show that interactions smoothen the effective disorder landscape, but do not necessarily increase the current, due to the competition of disorder screening and effective bias. This puts forward DFT as a diagnostic tool to understand disorder screening in a wide class of interacting disordered systems.


I. INTRODUCTION
How disorder and electron correlations shape material properties is a major question of current condensed matter research 1 . The interest in this problem is many decades old [2][3][4][5][6][7][8] , and significant progress has been made in important directions, e.g. in describing the correlation-induced Mott-Hubbard 9 and the disorderinduced Anderson 10 metal-insulator transition. Yet, a complete general understanding of the joint effect of interactions and disorder remains elusive to this day.
Advances in ultracold-atoms experiments 11,12 have boosted interest in scenarios where disorder and interactions are simultaneously important and new implications emerge from their interplay. An example of recently observed phenomena 13 is many-body localization (MBL) 14,15 , a new experimental and theoretical paradigm where several notions of many-body physics blend coherently 16 . In fact, MBL is part of a broad palette of situations. For example, disorder or interactions alone can produce insulating behavior but, between these limits, how they simultaneously affect conductance is not fully settled [17][18][19][20][21][22][23] . In equilibrium, interactions can increase or decrease conductivity in a disordered system 21,24,25 . Out of equilibrium, results for quantum rings 26 and quantum transport setups 27 suggest that at fixed disorder strength the current depends non-monotonically on interactions.
To facilitate the description of disordered and interacting systems, it would be useful to have a simple picture. A recent example in this direction was to look at a reduced quantity, the one-body density matrix, to establish a link between MBL, Anderson localization and Fermiliquid-type features in closed systems 28,29 . Another possible reduced description would be in terms of an independent-particle Hamiltonian. In a traditional meanfield-intuitive description of disorder vs interactions 22 , the low-energy pockets of the rugged potential landscape attract high particle density, but this is opposed by interparticle repulsion, resulting in a flatter effective potential landscape, i.e. disorder is screened by interactions. It is not unambiguous how to define such potential, and different conclusions are reached in the literature 26,[30][31][32][33][34] . The question is even more delicate for open systems, where typical localization signatures are unavailable 35 . As such, a simple, rigorous picture valid also in the presence of reservoirs would be of utmost importance.
Motivated by these arguments, we introduce here a picture of disorder and interactions based on the Kohn-Sham (KS) independent-particle scheme 36 of density functional theory (DFT) 37,38 . In DFT, the exact density of the interacting system can be obtained from a KS system subjected to an effective potential v eff (Fig. 1). For the density, v eff is the best effective potential in an independent-particle picture. We propose that v eff can be identified as the independent-particle effective energy landscape in a disordered and interacting system, which unambiguously defines disorder screening. To assess disorder screening for conductance and currents, we consider out-of-equilibrium systems. In extending DFT to non-equilibrium, we also have to include the notion of an effective bias [39][40][41] .
Our main findings are: i) interactions smoothen the effective landscape seen by the electrons (we interpret this as disorder screening); ii) a non-monotonic dependence of the current on the interaction strength cannot be explained by disorder screening alone; an "effective bias" (corresponding to a screening of the applied bias due to electron correlations) has to be taken into account; iii) The picture from i) and ii) applies to both isolated and open systems and to different dimensionality; iv) More in general, our works paves the way to a rigorous understanding of the notion of effective disorder 42 in a variety of situations, including open systems in-and out-of equilibrium, a topic which is the object of recent and fast-growing interest 35,[43][44][45] .
Systems considered.-In this work, we focus on the transition from the weakly to the strongly correlated regime, and consider a single disorder strength. This specific choice is enough to display how the competition of disorder and interaction is captured within a DFT picture. We study quantum rings pierced by magnetic fields and electrically biased quantum-transport setups (Fig. 1). Both situations show the aforementioned current crossover as function of the interaction strength. The rings are solved numerically exactly, while for quantum transport we use the Non-Equilibrium Green's Function (NEGF) formalism within many-body perturbation theory 46-51 to obtain steady-state currents and densities. The effective potentials and biases were found via a numerical reverseengineering algorithm 41 within non-equilibrium lattice DFT 52,53 .

II. QUANTUM RINGS
We study disordered Hubbard rings with L = 10 sites, N electrons, and spin-compensated, i.e. N ↑ = N ↓ = N/2. Currents are set by a magnetic field threading the rings, via the so-called Peierls substitution 54,55 . The Hamiltonian iŝ whereĉ † mσ creates an electron with spin projection σ = ±1 at site m.n mσ =ĉ † mσĉmσ is the density operator, and ... denotes nearest-neighbor sites. φ is the Peierls phase and x mn = ±1 depending on the direction of the hop from m to n. U is the onsite interaction. We consider onsite energies with box disorder of strength W with v m ∈ [−W/2, W/2]. In passing, we note that Peierls phases can be realized experimentally in cold atoms by artificial gauge fields 56 . We study currents in rings regimes via exact diagonalization, obtaining the manybody ground-state wavefunction |ψ(φ) , and the corresponding density matrix ρ mn = ψ(φ)|ĉ † nσĉmσ |ψ(φ) . This gives the density at site m as n m = 2ρ mm and the bond current as I m+1,m = −4T Im e iφ/L ρ m,m+1 . As we are in a steady-state scenario, all nearest-neighbor bond currents are equal, and the current I ≡ I m+1,m for any lattice site m. The corresponding effective KS Hamiltonian is 57,58 L xmn for nearest neighbors and 0 otherwise, and (v KS ) mn = δ mn v KS m . Imposing that the KS density n m = 2 N/2 ν=1 |ϕ ν (m)| 2 and KS bond cur- No physical meaning should be a priori given to the KS orbitals ϕ ν or the KS eigenvalues ν ; they pertain to an auxiliary system giving the exact density and current but not necessarily other quantities.
The KS potential, referred to as v eff hereafter, is our proposed measure of disorder screening. It can be split into external (disorder) and Hartree-exchangecorrelation parts: v eff = v + v Hxc (similarly, φ eff ≡ φ KS = φ + φ xc ). Thus, in DFT, the screening of disorder by interactions (i.e. when |v eff | < |v|) comes from v Hxc . This is an improvement over standard mean-field descriptions, in which the effective potential does not include correlations and the applied phase is unscreened.
Both v eff and φ eff are obtained by mapping the exact many-body ring system into a DFT-KS one. In lattice models, existence and uniqueness issues for such a DFTbased map can occur 52,53,57,59-62 . Of relevance here, φ and φ + 2πkL (k integer) give the same current (uniqueness issue); this periodicity also implies that the magnitude of the current has an upper bound (existence issue). Further, a non-interacting (or described within KS-DFT) homogeneous ring has energy degeneracy for even N σ (the degeneracy is lifted by many-body interactions).
To circumvent these occurrences, we choose N σ odd to avoid degeneracies. Furthermore, we fix −π/L < φ eff ≤ π/L in the reverse engineering scheme. However, even with this restriction, two different phases can yield the same current. Practically, we consistently choose the region for φ eff that smoothly connects to φ for small U . Finally, in practice the "maximal current" existence issue is largely mitigated since the target current comes from a physical many-body system.
In the numerical reverse-engineering implementation of the DFT map, φ eff and v eff are recursively updated until the interacting MB system and the KS system have the same current and density. Using exact diagonalization, we obtain the exact many-body density n MB and current I MB . These quantities are then used as input to obtain φ eff and v eff according to the protocol 41 where (k) denotes the kth iteration, and α 1 , α 2 < 1 are convergence parameters.
For reference, we start our discussion with homogeneous rings, (i.e. v i = 0, which gives a constant density n i = N/L and a constant v eff ). In Fig. 2, we show HF and NQF currents and the corresponding φ eff :s as function of the interaction U , for fixed external phase φ = −0.5. Both HF and NQF currents I decrease monotonically with U , but tend to zero and nonzero values, respectively. This is consistent with Mott insulator behavior at HF and metallic behavior otherwise for the infinite (L → ∞) one-dimensional Hubbard model 63 . The homogeneity singles out the importance of the effective phase. The KS orbitals are plane waves for any value of U , and as such the current is determined solely by φ eff . This shows the importance of the effective phase in our Hamiltonian picture, and highlights that standard meanfield descriptions, which yields φ eff = φ, cannot capture the correct physics.
We now address the effect of disorder in rings. We use M = 150 box-disorder configurations. For a given configuration, the spread ∆X of a quantity X over the L = 10 sites is measured by Results are presented for i) histograms collecting data from each disorder configuration and ii) arithmetic averages over all M configurations. We examine the dependence on the interactions U of the current I, φ eff , ∆n,and ∆v eff . The latter is a measure of disorder screening (in the homogeneous case, ∆v eff = 0 for all U ). With disorder (W = 2), for both NQF and HF the current I is hindered by disorder at low U and by interactions at large U , with a maximum in between (Fig. 3). As for W = 0 (Fig. 2), for HF I vanishes at very large U . The non-monotonic behavior of I results from competing disorder and interactions 21,26,27 . Conversely, the density spread ∆n decreases monotonically as a function of U at both NQF and HF, i.e. interactions favor a more homogeneous density. For NQF, ∆n seems to tend to a finite value for large U , while for the HF case, ∆n → 0, i.e. a fully homogeneous density. In the KS system, ∆v eff also decreases monotonically as function of U , tending to a finite value for NQF and to zero for HF. This means that the exact v eff for a strongly correlated system is smoother than for a weakly correlated system, and similarly for the density. Thus, we cannot simply look at the spread of the density to predict the current through the system: Including the effective phase is crucial. The competition of disorder and interactions thus translates into a competition of a decreasing effective potential spread (favoring the current) and a decreasing effective phase (reducing the current). Mean-field 26 or DFT-LDA treatments 64 fail to explain the current drop since they only take the effective potential into account: With an effective potential and no effective phase, the current can only increase with interactions. This ends our discussion on exact treatments of quantum rings.

III. OPEN SYSTEMS
We study short clusters connected to semi-infinite leads, with Hamiltonian where C, l, and Cl label the cluster, leads, and clusterleads coupling parts, respectively. With the same notation as for rings, As in the case of the quantum rings, we consider box disorder, v m ∈ [−W/2, W/2]. Depending on the cluster dimensionality, the leads are either 1D (chain) or 2D (strip) semi-infinite tight-binding structures. The latter case is shown in Fig. 1. The lead Hamiltonian isĤ l = αĤ α , and α = r(l) refers to the right (left) contact: Here, b α (t) is the (site-independent) bias in lead α, and N α = m∈α,σn mσ the number operator in lead α. Finally, the lead-cluster couplingĤ Cl connects the edges of the central region to the leads ( Fig. 1) with tunneling parameter −T . In the following, we put T = 1, which defines the energy unit. We focus on the steady-state scenario with b r (t) = 0, and b l ≡ b l (t → ∞) = 1, beyond the linear regime. Our 1D and 2D clusters have L = 10 sites, but are large enough to illustrate the relevant physics and the scope of a DFT perspective. Also, we put n ↑ = n ↓ = n (non-magnetic case) and the temperature to zero.

A. Steady-state Green's functions
Both our many-body (MB) and KS treatments of open systems are based on NEGF in its steady-state formalism. Thus we keep our presentation general, and later specialize to MB or KS. To describe the steady-state regime, we use retarded G R (ω) and lesser G < (ω) Green's functions: Here, boldface quantities denote L×L matrices in site indices of the cluster region. G A = (G R ) † is the advanced Green's function, (T ) mn = T mn is the kinetic term of Eq. (7), and v is not specified yet. The self-energy Σ contains many-body (MB) and embedding (emb) parts: emb . All correlation effects are contained in the many-body self-energy Σ R/< MB , whilst the embedding term accounts in an exact way for the left (l) and right (r) leads 65 : Σ . Thus, information of the actual structure of the leads, the bias b α and the chemical potential µ enters via f α and Σ R emb . Explicit expressions of Σ R emb exist for 1D and 2D 66 semi-infinite leads, since they are determined by the uncontacted-lead case.
For our system, the steady-state particle density and current are spin-independent. In each spin channel 67 , with I l the left lead current, and the spin-labels omitted, where the spectral function is 2πA = i(G R − G A ). Both the interacting MB system as well as the KS system are described by Eqs. (9)(10)(11)(12). We now specialize the discussion to the separate cases.

The interacting MB system
Here (v) ij = δ ij v i are the disordered onsite energies and b α is the applied bias. While the NEGF formalism provides a formally exact description for open systems, in practice the MB self-energies Σ R/< MB need to be approximated. We consider the self-consistent Σ 27,65,68,69 , keeping all diagrams up to second order. While the numerical details depend on the chosen approximation, our conclusions do not, as discussed in more detail below. We solve the equations self-consistently, with the convergence rate improved with the Pulay scheme 69,70 . Fully self-consistent NEGF calculations guarantee the satisfaction of general conservation laws 71,72 , and in particular the continuity equation 73 . In the context of steady-state transport, the continuity equation leads to the condition that I l = −I r ≡ I.

The independent-particle KS system
Being an independent-particle system, Σ R/< MB = 0. Thus, the KS system is described exactly by steady-state NEGF. Further, (v) ij ≡ δ ij v i,eff and b α ≡ b eff,α are found iteratively to make the KS and MB density and current the same 41 . The same iteration protocol as for the quantum rings was used, Eq. (3) and Eq. (4), replacing φ eff with b eff . The embedding self-energies in the KS and MB systems differ only by the bias (effective in KS, applied in MB). We restrict b eff,r = 0 and define b eff = b eff,l . The KS independent-particle scheme permits to write the Meir-Wingreen formula, Eq. (12), in a Landauer-Büttiker form I = µ+b eff µ dω 2π T KS (ω), with T KS = Tr Γ l G R Γ r G A the KS transmission function. Although recast in a Landauer-Büttiker form used for independent-particle systems, we stress that the KS current still equals the true current of the original interacting system.

B. Open systems: results
In the following, we put µ α = 0 (half-filled leads). To address the behavior of v eff in quantum transport setups, we find it useful to start with one disorder configuration and two interaction values for a biased 10-site onedimensional chain (Fig. 4). At U = 0, the density n k is non-uniform, since v eff = v and b eff = b. For U = 6, both n k and v eff (now incorporating correlations) become smoother: interactions thus provide a smoother energy landscape also for open systems. However, I U =6 < I U =0 , even if the effective energy landscape is smoother. This is due to b eff , that at U = 6 is much smaller than b 40,41,74 .
To corroborate this analysis, we consider the 2D open system of Fig. 1. Results from 150 disorder configurations for ∆n, ∆v eff (defined as for rings, Eq. (5)), I, and b eff are shown in Fig. 5 75 . The current through the system is a non-monotonic function of U , while ∆n, ∆v eff decrease monotonically. At low U , b eff almost equals b, and I increases since ∆v eff decreases. At larger U , however, the drop in b eff grows, and I is smaller. This is why I shows a crossover. Thus, the competition between disorder and interactions in open systems transfers to a competition between the smoothness of the energy landscape favoring current flow and screening of the effective bias hindering such flow. We have performed the same analysis for one-dimensional linear chains, with the same qualitative results (not shown).
While the 2nd Born approximation is quite accurate at low interaction strengths 68,76,77 , one can of course question the quantitative agreement for higher interaction strengths. We find no reason to question the qualitative results of the approximation, however, since the behavior is similar for the quantum rings, which were treated exactly, and also other calculations suggest similar conclusions 21,24,25 . In order to further confirm the aforementioned qualitative behavior, we also performed calculations for selected disorder configurations (not shown) using the T-matrix approximation 68,78-80 , which takes higher-order processes into account in the self-energy. We found the same qualitative behavior as for the 2nd Born approximation, also for the higher interaction strengths considered.

IV. CONCLUSIONS
We introduced an exact independent-particle characterization of coexisting disorder and interaction effects, based on density-functional theory (DFT). Its scope as a diagnostic for disorder screening was shown for open-sample geometries and small quantum rings. The many-body treatment of the quantum rings was exact, which allowed us to unambiguously characterize disorder screening. For open systems, where no exact solutions are available, we used non-equilibrium Green's functions (NEGF), with biased reservoirs treated exactly and electronic correlations treated via the 2nd Born approximation. We stress that the use of an approximation was simply an expedient way to provide an input to our reverse engineering algorithm; more sophisticated methods can of course be used for the same purpose.
Our DFT-based analysis consistently shows that interactions smoothen the energy landscape in disordered systems out of equilibrium, for both closed quantum rings and open one-and two-dimensional quantum transport systems. In line with earlier qualitative pictures from the literature, it is tempting to think that the spread in the effective potential or the density can be taken as a measure of the conductance of a system. This is not the case, as this picture is not accurate enough to explain the nonmonotonic behavior for the current when changing the interaction strength. To make the picture complete, the effective bias (phase) has to be taken into account. The fact that the quantum rings, the one-dimensional and the two-dimensional quantum transport systems yield the same behavior reinforces our conclusions that the independent-particle picture is general and can be applied to a wide range of systems.
Based on this interpretation, we can provide a simple explanation why mean-field theories can predict a too high current in disordered systems. These methods neglects the correlation screening of the disordered potential, and fully neglects the screening of the applied bias. To improve the picture, correlation effects need to be added.
To conclude, within our Hamiltonian independentparticle picture, strong correlation effects are behind the appearance of the effective potential and bias, and this is the essence of disorder screening. As possible extensions of our approach, we mention applications to real materials and the generalization to finite temperatures 81 to describe, for example, the many-body localized regime.
These are deferred to future work.

ACKNOWLEDGMENTS
We acknowledge Fabian Heidrich-Meisner for useful discussions. D.K. would like to thank the Academy of Finland for support under project no. 267839.