Matrix product state approach for a two-lead multilevel Anderson impurity model

We exploit the common mathematical structure of the numerical renormalization group and the density-matrix renormalization group, namely, matrix product states, to implement an efficient numerical treatment of a two-lead multilevel Anderson impurity model. By adopting a starlike geometry, where each species (spin and lead) of conduction electrons is described by its own Wilson chain, instead of using a single Wilson chain for all species together, we achieve a very significant reduction in the numerical resources required to obtain reliable results. We illustrate the power of this approach by calculating ground-state properties of a four-level quantum dot coupled to two leads. The success of this proof-of-principle calculation suggests that the star geometry constitutes a promising strategy for future calculations the ground-state properties of multiband multilevel quantum impurity models. Moreover, we show that it is possible to find an “optimal” chain basis, obtained via a unitary transformation (acting only on the index distinguishing different Wilson chains), in which degrees of freedom on different Wilson chains become effectively decoupled from each other further out on the Wilson chains. This basis turns out to also diagonalize the model’s chain-to-chain scattering matrix. We demonstrate this for a spinless two-lead model, presenting DMRG results for the mutual information between two sites located far apart on different Wilson chains, and NRG results with respect to the scattering matrix.

Figure 1

Quantum dot coupled to two leads.

Figure 2

Iterative generation of matrix product states for a chain.

Figure 3

(a) Single chain geometry: a single Wilson chain of local dimension $2^{2N_{\text{l}}}$ coupled to one dot site of local dimension $2^{2m}$. (b) Star geometry: $2N_{\text{l}}$ Wilson chains (here $N_{\text{l}}=2$ and $\alpha =l,r$), each with local dimension 2, coupled to two dot sites of local dimension $2^m$.

Figure 4

Graphical representation of Eq. (10a).

Figure 5

MPS representation for a quantum dot coupled to two spinful leads. The lead chains are combined to big boxes for clarity. The indices of the dot matrices are labeled explicitly.

Figure 6

Procedure for moving the actual site from $k$ to $k-1$. The matrices that are not orthonormalized in any direction are printed with gray background. The gray lines within the boxes indicate whether the row or column vectors are orthonormal (with the local level associated with row or column, respectively).

Figure 7

(Color online) The solid line shows the dimension $D_k$ needed at bond $k$ of the spin up chain to satisfy $w_{\text{min}}={10}^{-6}$ for the reduced density matrix at each bond (negative $k$ correspond to the left chain). The dashed line displays the exponentiated bond entropy $e^{S_k}$ multiplied by 4.5 to visually match the $D_{k,\text{min}}$ curve for large $k$. Here $k=0$ corresponds to the “vertical” bond between the two spin subsystems. The two insets show spectra of reduced density matrices at different bonds $k$ indicated by the vertical dashed lines of the main plot. The data shown in this figure has been obtained from the ground state of the four-level model shown in Fig. 10 with $ϵ=-1.7U$. In general, the maximum dimension needed depends strongly on the model parameters.

Figure 8

(Color online) Sweeping sequence. For clarity we place the spin up and spin down parts on top of each other to emphasize the starlike structure. The solid blue line depicts the standard sweeping sequence.

Figure 9

(Color online) Dot level occupation for a spinless two-level system, with ${ϵ}_{1,2}=ϵ\pm \Delta /2$, level spacing $\Delta =0.1U$ and couplings ${\Gamma }_{1l}={\Gamma }_{1r}=0.005U$, ${\Gamma }_{2l}={\Gamma }_{2r}=30{\Gamma }_{1l}$. This parameter set was used in Sindel et al. (Ref. 20) $N=\frac{1}{2}\left(n_1+n_2\right)$ is half the total dot occupation. Note that the sign in ${\Gamma }_{i\alpha }$ just serves as an indication of the sign of the related hopping matrix element $V_{i\alpha }$ in the Hamiltonian.

Figure 10

(Color online) Dot level occupation for a spinful four-level system. We parametrize the dot level energies as ${ϵ}_{is}=ϵ+{ϵ}_i\pm B/2$ for $s=\uparrow ,\downarrow$, where $B$ represents the applied magnetic field with $B=0.2U$ and $ϵ$ a gate voltage, with ${ϵ}_i=\left(-0.1,-0.03,0.07,0.1\right)U$. The coupling of the dot levels are chosen asymmetrically ${\Gamma }_{ir}=s_i{\Gamma }_{il}$ with $s_i=\left(1,-1,-1,1\right)$ and ${\Gamma }_{il}=\left(0.5,0.02,1,0.7\right)0.2U$.

Figure 11

(Color online) (a) Mutual information $I_{\rho }^{l,r}$ between two sites situated in different leads but at equal distances $k$ from the dot, for a spinless four-level, two-lead model with dot levels ${ϵ}_i/U=\left(-0.1,-0.03,0.07,0.1\right)+ϵ$, $ϵ=-2U$ fixed, couplings ${\Gamma }_{ir}=\left(0.3,-0.02,-1,0.2\right)$ and ${\Gamma }_{il}=\left(0.5,0.08,1,0.7\right)$ and $\Lambda =3$. The dashed line shows $I_{\rho }^{l,r}$ for the system with the leads in the original basis of Eq. (1), whereas the solid line shows $I_{\rho }^{l,r}$ after the leads have been rotated by the (fixed $k$ independent) optimal angle ${\theta }_{\text{opt}}$ obtained from Fig. 12a. (b) Exponentiated bond entropy $e^{S_k}$ along the right chain of the system both prior (dashed line) and after (solid line) the rotation with ${\theta }_{\text{opt}}$, indicating an effective reduction in the required matrix dimension $D_k$ close to the impurity for the rotated system by about $\frac{1}{2}$ for the same numerical accuracy.

Figure 12

(Color online) Optimal basis for the leads of a spinless four-level, two-lead system (same parameters as for Fig. 11, but with varying $ϵ$). (a) Optimal angle of rotation ${\theta }_{\text{opt}}$ for the leads obtained by diagonalizing ${\rho }_{\text{red}}^{l,r}\left(k=L\right)$ for the DMRG calculation (red symbols) in comparison with angle that diagonalizes the scattering matrix calculated with NRG (blue line). ${\theta }_{\text{opt}}$ is defined $\text{mod}\pi /2$. (b) Dot level occupation. $N=\frac{1}{4}{\sum }_{i=1}^4{n}_i$ is the rescaled total dot occupation. Rapid changes in the angle ${\theta }_{\text{opt}}$ coincide with rapid shifting of dot-level occupations. (c) Truncation error (accumulated discarded density-matrix eigenvalues) of the DMRG calculation considering two neighboring sites at a time for a rotated and nonrotated system. We typically used 20 sweeps for the DMRG calculations. The truncation error is significantly reduced for the rotated system except for the points where ${\theta }_{\text{opt}}$ actually shows rather rapid transitions through ${\theta }_{\text{opt}}=0$ itself. At these points the leads are already decoupled from the outset.