m-15 IG.2:(col of Nott-n 12 0.06 with the inte te at ze d lines at U=1 and field becomes ins gat electric fieldof vith thee pr r the electric-field is inc the perat that ence (b)Spectral functi and distribution functio ight rapidly disa near the MITdriven b with qualitative agreement beyond the linear response limit. onotonic cold-hot-cold temperature evo a two-step resonant process which can be viewed as acon r the the lowe o limit in th h27 e of interaction.T he in at the Fom then deviates this behavior outside the nar row linear regime,as discussed below. Hub The scattering rate can be directly related to the elec 1 c() tatCeariowhichtheCoiombicitoracion interacting limit.the linear ductivity can be written the dominant scattering process and is rapidly modified as 0o.DC =22/(VT2+4)[15.nFIG.2(b),w ormula with he ing in Fig.2(a)that the ttering rate from the Coulomb interaction, =-m(w=0), a wide range of the E-field,well beyond the linear regime for different sets of the I collapse onto a scaling e as a function of (E proximate as brium retarded self-en eparture from the linear (83).This estimate is valid away from and the metal-insulator limit,and agrees well hile nega eriodic structures due to the Bloch the dashed lines (U=0)in Fig.1.the NDR here come from strong non-linear scattering enhanced by the Joule of weak dissipation and str 1 comes more dramatic.With the effective temperature Eq.(8),having a singular limit as →0，the ectro dashed lines in Fig.2(a of th DOS at=0.The robust agreement in the self-energies enormalized coherent energy scales causes the system leaves no doubt that the electron scattering is dominated to immediately deviate from the linear response regime, by the Joule heating with T given with Eq.(8)in the preventing itself from overheating.This mechanism,in 3 0 0.005 0.01 0.015 0.02 (E/K) 2 0 0.001 0.002 0.003 0.004 0.005 o U -1 = - Im Y U r(0) U/W = 1.5 U/W = 1 (a) 0 0.05 0.1 0.15 0.2 0.25 electric-field, E/W 0 0.01 0.02 0.03 current, J U = 1 U = 1.5 K/W = 0.025 (b) FIG. 2: (color online) (a) Interacting scattering rate, τ −1 U = −ImΣr U (ω = 0), plotted against (E/Γ)2 . Different colors de￾note different damping Γ = 0.0125, · · · , 0.06 with the interval of 0.0025. For small (E/Γ), the numerical results on the 1-d chain collapse on well-defined lines at U = 1 and 1.5. The dashed lines are predictions based on the equilibrium self￾energy with the temperature replaced by the non-interacting effective temperature Teff given in Eq. (8). The remarkable agreement proves that Joule heating controls the scattering in the small field limit. (b) Comparison of the current and the Drude formula estimate with the total scattering rate Γ+τ −1 U , with qualitative agreement beyond the linear response limit. a two-step resonant process which can be viewed as a con￾sequence of the energy overlap between the lower/upper Hubbard bands of the left/right neighboring sites with the in-gap states present at the Fermi level [27]. The current peak at E = U is due to the direct overlap of the Hubbard bands on neighboring sites [18, 27]. The immediate departure from the linear conductivity at very small fields can be well understood with a Joule heating scenario in which the Coulombic interaction is the dominant scattering process and is rapidly modified by an increasing effective temperature as the field is in￾creased. We first demonstrate this effective temperature effect by showing in Fig. 2(a) that the scattering rates from the Coulomb interaction, τ −1 U = −ImΣr U (ω = 0), for different sets of the damping Γ collapse onto a scaling curve as a function of (E/Γ)2 for small E. This scal￾ing is clearly evocative of the well known T 2 behavior of equilibrium retarded self-energies. In the non-interacting 1-d chain with Tb = 0, the ef￾fective temperature has been obtained in the small field limit as [15, 17] Teff = √ 6 π γ E Γ . (8) Inserting this Teff into the equilibrium perturbative self￾energy [30], we obtain in the weak-U limit τ −1 U = −ImΣr eq(ω = 0, Teff) ≈ π 3 2 A0(0)3 U 2 T 2 eff, (9) which is represented by the dashed lines in Fig. 2(a). Here A0(0) = (π p Γ2 + 4γ 2) −1 is the non-interacting DOS at ω = 0. The robust agreement in the self-energies leaves no doubt that the electron scattering is dominated by the Joule heating with Teff given with Eq. (8) in the 0 0.0002 0.0004 0.0006 0 0.001 0.002 0.003 0.004 0.005 current, J electric-field, E/W U = 1.225 (a) increasing E decreasing E metal-insulator coexistence 0 1 2 -1 -0.5 0 0.5 1 spectral function, A(ω) frequency, ω/W (b) E = 0.0 0.0017 0.0050 -0.02 0 0.02 0 0.5 1 ω/W floc(ω) FIG. 3: (color online) (a) Electric-field driven metal-to￾insulator transition (MIT) in the vicinity of a Mott-insulator at U = 1.225, Γ = 0.00167 and Tb = 0.0025 in a 3-dimensional cubic lattice with electric field in x-direction. The metallic state at zero field becomes insulating at electric field of magni￾tude orders of magnitude smaller than bare energy scales. De￾pending on whether the electric-field is increased or decreased, metal-insulator hysteresis occurs with a window for phase￾coexistence. (b) Spectral function and distribution function floc(ω) with increasing electric-field. The quasi-particle (QP) spectral weight rapidly disappears near the MIT driven by the electric-field, opening an insulating gap. The non-equilibrium energy distribution function indicates that the system under￾goes a highly non-monotonic cold-hot-cold temperature evo￾lution near the MIT. linear response limit in the presence of interaction. Teff then deviates strongly from this behavior outside the nar￾row linear regime, as discussed below. The scattering rate can be directly related to the elec￾tric current via the Drude conductivity J(E) = σDC(E)E with the non-linear DC conductivity σDC(E). In the non￾interacting limit, the linear conductivity can be written as σ0,DC = 2γ 2/(πΓ p Γ2 + 4γ 2) [15]. In FIG. 2(b), we plot the Drude formula with the scattering rate Γ re￾placed by the total scattering Γtot = Γ +τ −1 U . The quali￾tative agreement with the numerical results extends over a wide range of the E-field, well beyond the linear regime. Using Eq. (9), the current at small field can be ap￾proximated as J = σ0,DC E/(1 + E2/E2 lin) with the departure from the linear behavior occuring around (from the condition Γ ≈ τ −1 U at E = Elin), Elin ≈ (8π 2/3)1/2γ 1/2Γ 3/2/U. This estimate is valid away from U = 0 and the metal-insulator limit, and agrees well with FIG. 2(b) [31]. We emphasize that, while negative￾differential-resistance (NDR) behaviors occur typically in periodic structures due to the Bloch oscillations [32] as the dashed lines (U = 0) in Fig. 1, the NDR here comes from strong non-linear scattering enhanced by the Joule heating. In the presence of weak dissipation and strong elec￾tronic interactions, the non-equilibrium evolution be￾comes more dramatic. With the effective temperature, Eq. (8), having a singular limit as Γ → 0, the electron temperature tends to rise very sharply as the field is ap￾plied. This effect, together with a small value of the renormalized coherent energy scales, causes the system to immediately deviate from the linear response regime, preventing itself from overheating. This mechanism, in