Effective dimension reduction with mode transformations: Simulating two-dimensional fermionic condensed matter systems with matrix-product states

Tensor network methods have progressed from variational techniques based on matrix-product states able to compute properties of one-dimensional condensed-matter lattice models into methods rooted in more elaborate states, such as projected entangled pair states aimed at simulating the physics of two-dimensional models. In this work, we advocate the paradigm that for two-dimensional fermionic models, matrix-product states are still applicable to significantly higher accuracy levels than direct embeddings into one-dimensional systems allow for. To do so, we exploit schemes of fermionic mode transformations and overcome the prejudice that one-dimensional embeddings need to be local. This approach takes the insight seriously that the suitable exploitation of both the manifold of matrix-product states and the unitary manifold of mode transformations can more accurately capture the natural correlation structure. By demonstrating the residual low levels of entanglement in emerging modes, we show that matrix-product states can describe ground states strikingly well. The power of the approach is exemplified by investigating a phase transition of spinless fermions for lattice sizes up to $10\times 10$.

Figure 1

(Left panel) convergence of the ground-state energy for the half-filled $6\times 6$ spinless fermion model for $V=1$ as a function of mode transformation iterations for fixed bond dimension of $D_{\mathrm{opt}}=64$ and 256 is shown by red and black curves, respectively. In the inset, the scaling of the ground-state energy with inverse bond dimension obtained in the real-space basis and in the optimized basis for $D_{\mathrm{opt}}=64$ and 256 are shown, respectively. (Right panel) Charge density wave order parameter for the $N\times N$ half-filled spinless fermion model as a function of $V$ for various values of $D$ obtained in the real-space basis. Black crosses indicate extrapolated data to the $N\to \infty$ limit obtained in the optimized basis and using finite-size scaling data shown for various interaction strengths in the inset (curves correspond from bottom to top to $V=0.25$, 0.5, 1, 2, 4, and 8). The solid black line is a spline fit to the extrapolated data. The error in the extrapolated data is indicated by the symbol sizes.

Figure 2

(Left panel) block entropy for the $6\times 6$ half-filled spinless fermion model for $V=1$ for some selected mode transformation iterations with $D_{\mathrm{opt}}=256$, i.e., for the $0\mathrm{th}, 1\mathrm{st}, 2\mathrm{nd}, 3\mathrm{rd}, 5\mathrm{th}, 10\mathrm{th}, 20\mathrm{th}$, and $40\mathrm{th}$ iterations. (Right panel) maximum of the block entropy as a function of $V$ for various $D$ values and for the real-space basis (solid lines) and for the optimized basis (dashed lines). In the inset, this is shown for various systems sizes obtained with the optimized basis and DMRG with $D_{\min }=1024$ and ${\varepsilon }_{\mathrm{tr}}={10}^{-6}$. Here, a spline is fitted as guide for the eye trough the data points.

Figure 3

Error norm of the one-particle reduced density matrix with respect to the reference obtained with the real-space basis with $D=8192$. Red and black symbols show result for $D=64,256,512,1024,2048$ but using the optimized basis for $D_{\mathrm{opt}}=64$ and 256, respectively. For both quantities reference data obtained with the real-space basis for various $D$ values up to 8192 are shown with dashed lines and labeled as $\mathrm{rs}\left(D\right)$.

Figure 4

Convergence of the ground-state energy for the half-filled $8\times 8$ spinless fermion model as a function of mode transformation iterations for fixed bond dimension of $D_{\mathrm{opt}}=64$, 256, and 512 is shown by blue, red and black curves, respectively, for $V=1$ (left panel) and for $V=8$ (right panel). Reference data obtained in the real-space basis for $D$ up to 8192 are shown with dashed lines. In the inset, the scaling of $E\left(D\right)$ and $\tilde{E}(256,D)$ with the inverse bond dimension is shown.

Figure 5

Half chain block entropy obtained for the $6\times 6$ spinless fermion model as a function of inverse bond dimension for various interaction strengths, including a fit to the data. The solid line is a fit defined by Eq. (A2).

Figure 6

Similar to Fig. 5, but for the $8\times 8$ spinless fermion model.

Figure 7

Convergence of the ground-state energy for the half-filled $8\times 8$ spinless fermion model as a function of mode transformation iterations with fixed bond dimension of $D_{\mathrm{opt}}=256$ for various $V$ values if we rotate to the natural orbitals after the seventh sweep of each iteration instead of using the local updates and perform another seven sweeps to obtain a converged ground state in the current rotated basis to determine the optimal ordering for the next iteration. Therefore, each iteration based on natural orbitals corresponds to every second iteration based on fermionic mode transformation.

Figure 8

Site entropy profiles $\left\{s_i\right\}$, sorted values of the natural orbital occupation numbers $\left\{{\lambda }_i\right\}$, occupation numbers $\left\{\langle n_i\rangle \right\}$ and mutual informations $\left\{I_{i,j}\right\}$ for the real-space basis (first row), and for the $2\mathrm{nd}$ and $40\mathrm{th}$ mode transformation iterations for the half-filled $8\times 8$ spinless fermion model for $V=1$ and $D_{\mathrm{opt}}=256$. The ground-state energy, the sum of the site entropy $I_{\mathrm{tot}}$, and the entanglement distance $I_{\mathrm{dist}}={\sum }_{i,j}{I}_{i,j}{|i-j|}^2$, are printed below the corresponding panels.