Lanczos transformation for quantum impurity problems in $d$-dimensional lattices: Application to graphene nanoribbons

We present a completely unbiased and controlled numerical method to solve quantum impurity problems in $d$-dimensional lattices. This approach is based on a canonical transformation, of the Lanczos form, where the complete lattice Hamiltonian is exactly mapped onto an equivalent one-dimensional system, in the same spirit as Wilson's numerical renormalization, and Haydock's recursion method. We introduce many-body interactions in the form of a Kondo or Anderson impurity and we solve the low-dimensional problem using the density matrix renormalization group. The technique is particularly suited to study systems that are inhomogeneous, and/or have a boundary. The resulting dimensional reduction translates into a reduction of the scaling of the entanglement entropy by a factor $L^{d-1}$, where $L$ is the linear dimension of the original $d$-dimensional lattice. This allows one to calculate the ground state of a magnetic impurity attached to an $L\times L$ square lattice and an $L\times L\times L$ cubic lattice with $L$ up to 140 sites. We also study the localized edge states in graphene nanoribbons by attaching a magnetic impurity to the edge or the center of the system. For armchair metallic nanoribbons we find a slow decay of the spin correlations as a consequence of the delocalized metallic states. In the case of zigzag ribbons, the decay of the spin correlations depends on the position of the impurity. If the impurity is situated in the bulk of the ribbon, the decay is slow as in the metallic case. On the other hand, if the adatom is attached to the edge, the decay is fast, within few sites of the impurity, as a consequence of the localized edge states, and the short correlation length. The mapping can be combined with ab initio band structure calculations to model the system, and to understand correlation effects in quantum impurity problems starting from first principles.

Figure 1

Mapping of the system into a semichain. Using the Lanczos method, a real lattice can be mapped into a 1D semichain. The coupling between the impurity and the lattice, shown in (a), determines the construction of the Lanczos seed $|{\Psi }_0\rangle$. In this case, the seed consists of just a single orbital, which becomes the first site of the mapped chain, as shown in (b). The dashed lines connect the sites that form the Lanczos orbitals of the resulting 1D system. (c) and (d) Two orthogonal wave functions with the same symmetry $x^2+y^2$. Circles and squares denote different amplitudes, red (blue) color indicates a positive (negative) value.

Figure 2

(a) Hopping parameters for a tilted square lattice (diamond) of size $\sqrt{L}\times \sqrt{L}$ with open boundary conditions. (b) Spin-spin correlations between a Kondo impurity connected to the first site of the chain ($J_K=1$), and the Lanczos orbitals along the chain. We show results for $L=10,20,40$, and a large chain ($L=100$) with “infinite boundary conditions” (see text). The inset is a magnification showing the first 30 sites in more detail.

Figure 3

Choosing the seed. Examples of how the coupling between the impurity and the lattice will determine the construction of the seed. We are using a honeycomb carbon lattice, but the idea can be extended to any geometry. (a) In the simplest case (and the one used for the calculations in the rest of the paper), the impurity is sitting on top of just one atom. (b) The impurity is connected to all six sites. (c) The impurity is on the bond between two C atoms.

Figure 4

Local density of states for CNT and hopping parameters for a single chain mapping. Chain hopping vs $n$ [(a) and (c)] and LDOS as function of $\omega$ [(b) and (d)] are shown for an insulating (top two panels) and a metallic (bottom two panels) CNT. For the LDOS, the exact solution was calculated using Green's functions [(red) dashed line]. The agreement with the mapping calculation [(black) solid line] is excellent.

Figure 5

LDOS for several lattices. The LDOS is plotted as a function of $\omega$ for (a) 2D square lattice with ${800}^2$ sites, (b) rectangular lattice with ${800}^2$ sites, (c) graphene sheet with ${800}^2$ sites, and (d) 3D cubic lattice with ${200}^3$ sites.

Figure 6

Correlations for a 2D square lattice. (a) Spin-spin correlations between a Kondo impurity and the rest of the chain, for different values of $J_K$, keeping $N=100$ sites. We show results for the first 20 sites of the chain. (b) Density-density correlations between the first site of the chain, and the rest. All simulations were done at half-filling. Notice that this system length corresponds to an actual square lattice with a diagonal size $L\sim 140$ sites.

Figure 7

Correlations for a 3D cubic lattice. (a) Spin-spin correlations between a Kondo impurity and the rest of the chain, for different values of $J_K$, keeping $N=100$ sites. As in Fig. 6, we show results for the first 20 sites of the chain. (b) Density-density correlations between the first site of the chain, and the rest. Notice that this system size corresponds to an actual cubic lattice with a diagonal size of 100 sites.

Figure 8

Scaling of correlations in CNTs using DMRG. (a) In order to extrapolate to the thermodynamic limit we first map the band Hamiltonian onto its tridiagonal Lanczos form. We study the convergence as a function of $1/N$, where $N$ is the number of sites kept in the actual simulation. (b) and (c) The spin-spin correlations as function $1/N$ for an adatom at the surface of a CNT. (b) An insulating $\left(0,4\right)$ CNT. In this case, as there are no extended states in the CNT, convergence is fast. (c) A metallic $\left(4,4\right)$ CNT. Due to the gapless metallic extended states, the convergence is slower. A linear extrapolation in $1/N$ is also shown in the figure [(red) dashed curve]. The parameters used are $U=1$, $V_g=-U/2$, and $t^{\prime }=\sqrt{0.05}$.

Figure 9

Representation of a graphene nanoribbon. We show a nanoribbon containing zigzag edges along its length $L$ and armchair edges along its width $W_d$. The two triangular sublattices are represented by squares and open circles. The unit cell in the armchair edge direction, containing $n$ sites, is circumscribed by a dashed box. By definition, a nanoribbon takes its classification (zigzag or armchair) from the geometry of its length edges.

Figure 10

Local density of states for armchair ribbons. LDOS calculated along the $y$ direction of the ribbon (see text for details): (a) metallic, $n=10$, and (b) insulating, $n=12$ nanoribbons.

Figure 11

Local density of states for zigzag ribbons. (a) A color contour plot of the LDOS as a function of $\omega$ (horizontal axis, around the Fermi energy $\omega =0$) and along the $y$ direction of a zigzag ribbon with an $n=24$ unit cell (see Fig. 9 for the $x$ and $y$ axes directions). Details of the LDOS are shown at the center (b) and the edge (c) of the nanoribbon, respectively. In both panels, the total DOS, averaged over all sites within the unit cell, is shown as a (red) dashed line. A resonance at $\omega =0$ for the LDOS at the edge is consequence of the localized state. The absence, or presence, of this resonance, is the main qualitative difference between the (black) solid curves in (b) and (c). We can see, in (a), how this resonance vanishes when we approach, from one of the edges, the center of the nanoribbon.

Figure 12

Spin correlations for an adatom in an armchair nanoribbon. (a) and (b) The spin correlations between an impurity at the center of the nanoribbon and the first site of the chain as a function of the inverse of the system size for an insulating ($n=12$) and a metallic ($n=10$) nanoribbon, respectively. In the insulating case, the correlations saturate at $N\approx 50$, while in the metallic nanoribbon, the correlations scale linearly with $1/N$ [the extrapolated curve is shown as a (red) dashed line]. (c) and (d) The spin correlations between the impurity and the first 100 Lanczos orbitals are shown. In (c) all curves are indistinguishable. Even and odd sites along the chain represent orbitals in different sublattices of the original problem (see text for details).

Figure 13

Spin correlations for an adatom in a zigzag nanoribbon. (a) and (b) The spin correlations between the impurity and the first site of the Lanczos chain as a function of $1/N$. (a) has results for an impurity coupled to an edge site and (b) for a central site. The edge states in zigzag nanoribbons are localized and correlations in (a) tend to saturate. In (b), since the impurity is at the center of the nanoribbon where the bulk states are metallic, no saturation occurs. Rather, one sees a linear dependence with $1/N$, as in the case of a metallic armchair nanoribbon [Fig. 12b]. The correlations between the adatom as a function of the distance along the chain are shown in (c) for the impurity at the edge, and (d) for the impurity at the center (see text for details).