Soft Hubbard Gaps in Disordered Itinerant Models with Short-Range Interaction

We study the Anderson-Hubbard model in the Hartree-Fock approximation and the exact diagonalization under the coexistence of short-range interaction and diagonal disorder. We show that there exist unconventional soft gaps, where the single-particle (SP) density of states (DOS) $A$ follows a scaling in energy $E$ as $A(E)\propto \exp [-(-\gamma \log |E-E_F|{)}^d]$ irrespective of electron filling and long-range order. Here, $d$ is the spatial dimension, $E_F$ the Fermi energy and $\gamma$ a nonuniversal constant. We propose a multivalley energy landscape as their origin. Possible experiments to verify the present theory are proposed.

Figure 1

(a) Ground-state phase diagram of 3D Anderson-Hubbard model at half filling for Gaussian distribution of $P_V$. AFI, AF insulator; AFM, AF metal; PI, paramagnetic insulator (Anderson insulator); PM, paramagnetic metal. (b) DOS with system size $8\times 8\times 250$: $A$ ($t=1$, $U=6$, $W=5$), $B$ ($t=1$, $U=4$, $W=30$), $C$ ($t=1$, $U=0$, $W=30$), $D$ ($t=0$, $U=4$, $W=30$). We employ Lorentz broadening with a broadening factor $1.25\times {10}^{-3}$ and $6.25\times {10}^{-4}$ for $A$ and $B$, respectively. The broken lines denote $|E-E_F|={10}^{-2}$ and ${10}^{-1}$. The DOS fits well with $A(E)\propto \exp \mathbf{(}-(-\gamma \log |E-E_F|{)}^3\mathbf{)}$ shown by the fitting lines for ${10}^{-2}<|E-E_F|<{10}^{-1}$ as shown in the lower panel.

Figure 2

(a) DOS by HF in 1D at $t=0.3$, $U=1.0$, $W=2.0$, $E_F=U/2-1$ ($N_s=14$). Fitting of DOS gives $\alpha =0.85\pm 0.07$ (solid line), which is in good agreement with the expected exponent of $b^{\prime }/b=0.79\pm 0.02$ (broken line). (b) Numerical estimates of $P(R)$ and $\Delta (R)$. Fitting by Eqs. (4, 6) gives $b^{\prime }=1.06\pm 0.01$ and $b=1.34\pm 0.01$.

Figure 3

(a) DOS in 1D with ED (open boundary condition): $t=0.1$, $U=1.0$, $W=1.0$, $E_F=U/2$, ($N_s=2$, 3, 4, 5, 6). We average the DOS over $3.2\times {10}^7$ realizations of disorder for $N_s=6$. (b) Scaling plot by $A(ϵ,N_s^{-1})=N_s^{-\beta }f(ϵN_s^{\beta /\alpha })$ ($\alpha =0.075$, $\beta =0.375$).

Figure 4

Schematic illustration of (a) the ground state, (b) a SP excited state, (c) a nearly degenerate state with the ground state and (d) a multiply excited state. (e) Schematic of $V_1$ dependence of excitation energies.