Hybrid-space density matrix renormalization group study of the doped two-dimensional Hubbard model

The performance of the density matrix renormalization group (DMRG) is strongly influenced by the choice of the local basis of the underlying physical lattice. We demonstrate that, for the two-dimensional Hubbard model, the hybrid–real-momentum-space formulation of the DMRG is computationally more efficient than the standard real-space formulation. In particular, we show that the computational cost for fixed bond dimension of the hybrid-space DMRG is approximately independent of the width of the lattice, in contrast to the real-space DMRG, for which it is proportional to the width squared. We apply the hybrid-space algorithm to calculate the ground state of the doped two-dimensional Hubbard model on cylinders of width four and six sites; at $n=0.875$ filling, the ground state exhibits a striped charge-density distribution with a wavelength of eight sites for both $U/t=4.0$ and $8.0$. We find that the strength of the charge ordering depends on $U/t$ and on the boundary conditions. Furthermore, we investigate the magnetic ordering as well as the decay of the static spin, charge, and pair-field correlation functions.

Figure 1

Mapping of the one-dimensional DMRG chain (black solid line) onto a $10\times 4$ square lattice. The bold gray lines depict nearest neighbors on the square lattice, and the dashed lines A and B depict two possible cuts through the DMRG chain and the corresponding bipartitions of the system.

Figure 2

Performance comparison between real-space and hybrid-space DMRG, calculated for $U/t=4.0$ and $n=0.875$; (a) wall time per DMRG sweep and (b) peak memory consumption for a $16\times 6$ cylinder as a function of the MPS bond dimension $m$. The dashed gray lines depict the $m^3$ and $m^2$ scaling expected in the $m\to \infty$ limit. (c) Speedup and (d) memory savings of the hybrid-space DMRG compared to real-space DMRG as a function of $m$ for $16\times 4,16\times 6$, and $16\times 8$ cylinders.

Figure 3

(a) CPU time per sweep and (b) peak memory consumption of hybrid-space DMRG calculations for Hubbard cylinders with length 16 and width $L_y$ as a function of the MPS bond dimension $m$. The calculations were carried out for $n=0.875$ and $U/t=4.0$.

Figure 4

Von Neumann entropy $S\left(i\right)$ as a function of the MPS bond index $i$ for (a) $8\times 6$ cylinders at $U/t=4.0$ andhalf-filling and (b) $16\times 6$ cylinders at $U/t=8.0$ and $n=0.875$. The sites of the real-space and hybrid-space lattices are mapped to the MPS sites in an $x$-direction-first ordering; accordingly, the gray vertical lines indicate cuts between neighboring rings of the cylinders.

Figure 5

Ground-state energy obtained from hybrid-space DMRG and real-space DMRG as a function of the discarded weight per site $\mathrm{\Delta }\xi$ for (a) $8\times 6$ cylinders at $U/t=4.0$ and half-filling and (b) $16\times 6$ cylinders at $U/t=8.0$ and $n=0.875$. The MPS bond dimension is increased every other sweep and is written alongside the corresponding data points. The solid lines are linear fits through the last five data points of each series and indicate the zero-truncation-error extrapolations of the ground-state energies.

Figure 6

Charge-density distribution $n\left(x\right)$ for length-32 Hubbard cylinders at $U/t=4.0$ and $n=0.875$. Panels (a)–(d) show width-4 and width-6 cylinders with periodic (PBC) and antiperiodic (ABC) transverse boundary conditions, as indicated. The black lines show the zero-truncation-error extrapolated densities, and the gray lines show the nonextrapolated values. The gray dashed lines indicate the average filling.

Figure 7

Charge-density distribution $n\left(x\right)$ for length-32 Hubbard cylinders at $U/t=8.0$ and $n=0.875$. (a)–(d) show cylinder with different width, boundary conditions, and wavelength $\lambda$ of the charge-density stripes, as indicated. The black lines show the zero-truncation-error extrapolated densities, and the gray lines show the nonextrapolated values. The gray dashed lines indicate the average filling.

Figure 8

Discarded weight extrapolation of the energy for $\lambda =5.3$ and 8.0 on $16\times 6$ cylinders at $U/t=8.0$ and $n=0.875$. The energy contribution of the pinning field has been subtracted, and the MPS bond dimension $m$ is indicated by the labels alongside selected data points.

Figure 9

Amplitude of the charge-density stripes $\mathrm{\Delta }n$ for width-4 and width-6 cylinders with periodic (PBC) and antiperiodic (ABC) transverse boundary conditions at $n=0.875$ with (a) $U/t=4.0$ and (b) $U/t=8.0$ as a function of the inverse cylinder length, $1/L_x$.

Figure 10

Spin structure factor $S_S\left(\mathbf{k}\right)$ for (a) and (b) $32\times 4$ and (c) and (d) $32\times 6$ Hubbard cylinders with periodic [(a) and (c)] and antiperiodic [(b) and (d)] boundary conditions at $n=0.875$ and $U/t=4.0$.

Figure 11

Spin structure factor $S_S\left(\mathbf{k}\right)$ at $n=0.875$ and $U/t=8.0$ for (a) $L_y=4$, PBC, $\lambda =8.0$, (b) $L_y=4$, ABC, $\lambda =8.0$, (c) $L_y=6$, PBC, $\lambda =8.0$, and (d) $L_y=6$, PBC, $\lambda =5.3$.

Figure 12

Equal-time pair-field $D_{\mathrm{y}\mathrm{y}}\left(l_x\right)$, spin $S\left(l_x\right)$, and charge $C\left(l_x\right)$ correlations as a function of the longitudinal distance $l_x$ for $64\times 4$ Hubbard cylinders at $n=0.875$ and $U/t=4.0$ for (a) periodic and (b) antiperiodic boundary conditions.

Figure 13

Equal-time pair-field correlation functions for $64\times 4$ Hubbard cylinders at $n=0.875$ for different $U/t$ for (a) periodic and (b) antiperiodic boundary conditions.

Figure 14

Ground-state energies for doped width-4 and width-6 Hubbard cylinders with periodic (PBC) and antiperiodic (ABC) transverse boundary conditions at $n=0.875$ and (a) $U/t=4.0$ and (b) $8.0$ as a function of the inverse cylinder length. In (b), the wavelength $\lambda$ of the charge-density stripe is indicated. The energies are extrapolated to zero discarded weight, with the estimated error from the extrapolation indicated by the error bars. The dashed and the solid lines show the extrapolation to infinite cylinder length for width-4 and width-6 cylinders, respectively.