Variational Monte Carlo method for fermionic models combined with tensor networks and applications to the hole-doped two-dimensional Hubbard model

The conventional tensor-network states employ real-space product states as reference wave functions. Here, we propose a many-variable variational Monte Carlo (mVMC) method combined with tensor networks by taking advantages of both to study fermionic models. The variational wave function is composed of a pair product wave function operated by real-space correlation factors and tensor networks. Moreover, we can apply quantum number projections, such as spin, momentum, and lattice symmetry projections, to recover the symmetry of the wave function to further improve the accuracy. We benchmark our method for one- and two-dimensional Hubbard models, which show significant improvement over the results obtained individually either by mVMC or by tensor network. We have applied the present method to a hole-doped Hubbard model on the square lattice, which indicates the stripe charge/spin order coexisting with a weak $d$-wave superconducting order in the ground state for the doping concentration of less than 0.3, where the stripe oscillation period gets longer with increasing hole concentration. The charge homogeneous and highly superconducting state also exists as a metastable excited state for the doping concentration less than 0.25.

Figure 1

Example of an FTTN for a $4\times 4$ lattice. Red solid circles represent the lattice sites and blue diamonds represent the tensors in the FTTN. $t_{1,i}$ is the leaf tensor which contains four physical lattice points bridged by tensor indices represented by green bonds in the left dashed circle, and $t_{2,i}$, $t_{3,i}$, $t_{4,i}$, $\cdots$ are internal-node tensors which are connected only by virtual indices represented as black bonds. Every two neighboring leaf tensors share two common physical indices. Note that the two sites [such as the two right bottom solid red circles (sites)] at the border of a block (such as the dashed red circle) are connected also through a leaf tensor in a nearby block (such as the block below the red circle). Therefore, these two sites (red circles) are shared and connected by green-bond tensor indices (not shown) with the block below them. The right-most bottom site is shared by the block to the right of the red dashed circle as well.

Figure 2

Relative errors of the ground state energy $\left|E-E_{\mathrm{exact}}\right|/\left|E\right|$ as a function of $D$ for the 1D Hubbard model with $L=16$, $U=10.0$, and the number of electrons $N_e=10$. The red line is the conventional MPS result, and the blue line is obtained by applying mVMC on the variational wave function of Eq. (8) together with the MPS for the tensor network part $\mathcal{M}$. In this calculation, the full sublattice is employed with the spin projection.

Figure 3

Relative errors of the ground-state energy as a function of $D$ in $4\times 4$ two-dimensional (2D) Hubbard model with $U=10.0$ and $N_e=10$. The red and blue lines are obtained by the combination of the mVMC with the TTN and FTTN, respectively. In this calculation, the full sublattice is employed with the spin and space group projections.

Figure 4

Variance dependence of energies for $D=0,2,4,8,16$ FTTN in $4\times 4$ 2D Hubbard model with $U=10.0$, and $N_e=10$. The red broken line represents the linear fitting of energies, and the error bar on the $y$ axis is the fitting prediction bounds. The blue line is the exact ground-state energy.

Figure 5

Lattice size dependence of the ground-state energies for 2D Hubbard model with $U=4.0$ at half filling. The blue line is the mVMC result without tensor network. The red line represents the result of combination of the mVMC and FTTN with $D=16$. The black line is the result of the first step Lanczos. The magenta line is obtained from variance extrapolation. The green dotted line represents the QMC result [43]. The inset shows a magnified view of a portion of the main figure. In this calculation, the full sublattice is employed with the spin and space group projections.

Figure 6

Hole-doping concentration ($\delta$) dependence of the ground-state energy per site for $16\times 16$ square lattice of 2D square lattice Hubbard model at $U=10$ with the periodic-antiperiodic boundary condition. The blue line is the mVMC result without tensor network. The red line represents the result of combination of the mVMC and FTTN with $D=16$. The black line is the result of the first step Lanczos, and the magenta line is obtained from variance extrapolation. In this calculation, we have performed the optimization from the initial wave function with the optimized period of the stripe order coexisting with the $d$-wave superconductivity and employ the $16\times 2$ sublattice for $f_{ij}$, which allows various charge and spin orders. For the doping smaller than $15\%$, the ground-state spin stripe period is 16 and charge stripe period is 8, while for the doping larger than $15\%$, the spin stripe period is 8 and the charge stripe period is 4, irrespective of the methods. In addition to the ground states, we show metastable excited states obtained from the optimization performed from homogeneous superconducting initial wave function: The green crosses are the mVMC results without tensor network, the orange pentagrams represent the results of the combination of the mVMC and FTTN with $D=16$, both of which preserve the charge homogeneity even after the optimization, the purple stars are the results of the first step Lanczos, and the cyan squares are obtained from variance extrapolation estimated in the way shown in the lower inset (see below). The upper inset shows the energy per site of the mVMC results with periodic-periodic boundary condition (red point) and periodic-antiperiodic boundary condition (blue line). The lower inset shows the variance dependence of energies for the mVMC, combination of the mVMC and FTTN with $D=16$, and the first step Lanczos with stripe and homogeneous initial states at $\delta \sim 0.11$, in which the broken lines represent the linear fitting of energies, and the energy extrapolation to zero variance is represented as the magenta diamond and cyan square for stripe and homogeneous initial states respectively.

Figure 7

Hole-doping concentration ($\delta$) dependence of the ground-state energy per site for the Hubbard model on the $16\times 16$ square lattice at $U=10$. The data are the same as Fig. 6, but here, a linear function $f\left(\delta \right)=-1.7297\delta -0.4270$ has been subtracted from the energy to enhance the visibility. Notations are the same as Fig. 6. The black dashed line indicates the range of the phase separation near half filling ($0<\delta \lesssim 0.11$) for the example of VMC+FTTN+Lanczos.

Figure 8

Distance ($r$) dependence of superconducting correlation for $\delta \sim 0.11$ on $16\times 16$ lattice with $U=10$. The blue line is the mVMC result, and the red line is the result of the combination of the mVMC and $D=16$ FTTN for the ground states, which are obtained by starting from the stripe order coexisting with the $d$-wave superconductivity initial wave function. We also plot the correlation for the excited states, where the green line is the mVMC result, and the orange line is the result of combination of the mVMC and $D=16$ FTTN, obtained by starting from the homogeneous $d$-wave superconducting initial wave function, which results in the metastable excited states after the optimization as we see in Figs. 6 and 7. Here, for a given distance, the maximum value of the correlation is plotted.

Figure 9

Distance ($r$) dependence of superconducting correlation along $x$ (blue circles and black +signs) and $y$ (red crosses and magenta diamonds) directions for $\delta \sim 0.11$ on $16\times 16$ lattice with $U=10$. The black and magenta lines are the mVMC result without tensor network. The red and blue lines represent the result of the combination of the mVMC and the $D=16$ FTTN. The stripe direction is along the $y$ direction.

Figure 10

Distance ($r$) dependence of superconducting correlation along $x$ (blue circles and black + signs) and $y$ (red crosses and magenta diamonds) directions for $\delta \sim 0.11$ on $16\times 16$ lattice with $U=10$ plotted by taking the origin of the correlation at the minimum (blue circles and red crosses) and maximum (black + signs and magenta diamonds) columns of the hole density. Here only the results calculated by the combination of mVMC and FTTN with $D=16$ are shown. The stripe direction is along the $y$ direction.

Figure 11

Spin (upper panel) and charge (lower panel) structure factors at $\delta \sim 0.11$ on $16\times 16$ lattice with $U=10$.

Figure 12

Spin structure factor at $k_y=0$ (upper panel) and spin configuration (lower panel) at $\delta \sim 0.11$ on $16\times 16$ lattice with $U=10$. In the upper panel, the blue line is the mVMC result, the red line is the result by combining the mVMC and the FTTN with $D=16$, and the black line is with the first Lanczos step. $S_s\left(\mathbf{k}\right)$ has a peak at $\mathbf{k}=\left(\frac{7}{8}\pi ,\pi \right)$.

Figure 13

Charge structure factor (upper panel) and charge configuration (lower panel) at $\delta \sim 0.11$ on $16\times 16$ lattice with $U=10$. In the upper panel, the blue line is the mVMC result, the red line is the result of combining the mVMC and the FTTN with $D=16$, and the black line is with the first Lanczos step. $S_c\left(\mathbf{k}\right)$ has a peak at $\mathbf{k}=\left(\frac{1}{4}\pi ,0\right)$.

Figure 14

Spin (upper panel) and charge (lower panel) structure factors at $\delta \sim 0.22$ on $16\times 16$ lattice with $U=10$.

Figure 15

Spin structure factor (upper panel) and spin configuration (lower panel) at $\delta \sim 0.22$ on $16\times 16$ lattice with $U=10$. In the upper panel, the blue line is the mVMC result, the red line is the result of combining the mVMC and the FTTN with $D=16$, and the black line is with the first Lanczos step. $S_s\left(\mathbf{k}\right)$ has a peak at $\mathbf{k}=\left(\frac{3}{4}\pi ,\pi \right)$.

Figure 16

Charge structure factor (upper panel) and charge configuration (lower panel) at $\delta \sim 0.22$ on $16\times 16$ lattice with $U=10$. In the upper panel, the blue line is the mVMC result, the red line is the result of combining the mVMC and the FTTN with $D=16$, and the black line is with the first Lanczos step. $S_c\left(\mathbf{k}\right)$ has a peak at $\mathbf{k}=\left(\frac{1}{2}\pi ,0\right)$.

Figure 17

Spin and charge orders at $\delta \sim 0.11$ (upper panel) and $\delta \sim 0.22$ (lower panel) on $16\times 16$ lattice with $U=10$. The radius of every circle is proportional to the hole density $1-n$. The length of every arrow is proportional to the spin density along $z$ direction $\left|S^z\right|$, and up and down arrows represent positive and negative $S^z$, respectively. The values of $\left|S^z\right|$ and $1-n$ are shown above and below the order plots, respectively.

Figure 18

Spin (upper panel) and charge (lower panel) structure factors for the charge homogeneous superconducting state at $\delta \sim 0.11$ on $16\times 16$ lattice with $U=10$. No prominent structure is visible in the charge correlations.

Figure 19

Example of a TTN for a $8\times 8$ lattice. The red solid circles represent the lattice sites, and the blue diamonds represent the tensors in the TTN. $t_{1,i}$ is the leaf tensor which contains two physical indices, and $t_{2,i},t_{3,i},...,t_{6,1}$ are internal-node tensors which only contain virtual indices.