Stacked tree construction for free-fermion projected entangled pair states

The tensor network representation of a state in higher dimensions, say a projected entangled-pair state (PEPS), is typically obtained indirectly through variational optimization or imaginary-time Hamiltonian evolution. Here, we propose a divide-and-conquer approach to directly construct a PEPS representation for free-fermion states admitting descriptions in terms of filling exponentially localized Wannier functions. Our approach relies on first obtaining a tree tensor network description of the state in local subregions. Next, a stacking procedure is used to combine the local trees into a PEPS. Lastly, the local tensors are compressed to obtain a more efficient description. We demonstrate our construction for states in one and two dimensions, including the ground state of an obstructed atomic insulator on the square lattice.

Figure 1

The overall procedure of the “tree, stack and compress”. (a) Exponentially decaying WFs centered at the centers of plaquettes of the square lattice, while atomic sites are the vertices. (b) The tree decomposition of a single WF, where the center of the WF is indicated by a blue dot. The green circle encloses the truncated region for the WF. The dashed lines indicate the square lattice, while the solid lines are legs of the tensors. The legs pointing out-of-page are the physical legs, whereas those in-plane are virtual legs/bonds of local tensors. Blue legs connect the diamond in the center to the neighboring sites, and black legs connect among the square tensors. (c) PEPS obtained by stacking the trees over the whole lattice using translation symmetry. Note that both blue (between square and diamond tensors) and black (among square tensors) are present along the diagonal direction. (d) The local tensors all over the lattice after the compression.

Figure 2

Compression procedure. The thickness of the legs indicates the size of the bond space. Red dashed lines indicate the successive Schmidt decomposition performed along the red arrow. The hollow arrow points to the final result of each compression step, which is the middle piece among all the decomposed local tensors. The ordering of compression direction is first along (a) horizontal direction, then (b) vertical direction and lastly along the two diagonal directions (c) and (d).

Figure 3

Correlation function $\langle {\widehat{c}}_{x,l}{\widehat{c}}_{0,O_1}^{†}\rangle$ for a 100-site SSH model. Only data inside the range $(-20,20)$ is shown as the values are practically zero outside of this range. The blue and red colors represent the two orbitals $O_1$ and $O_2$, respectively. The dashed lines represent exact values while the markers represent the compressed ones obtained from the constructed TN. The inset is the log scale plot of the same range while the vertical axis being ${log}_{10}\left|\langle {\widehat{c}}_{x,l}{\widehat{c}}_{0,O_1}^{†}\rangle \right|$.

Figure 4

Real part of correlation function $\mathfrak{R}\left(\langle {\widehat{c}}_{\vec{r},l}{\widehat{c}}_{{\vec{r}}_0,s}^{†}\rangle \right)$ on a $100\times 100$ square lattice, where ${\vec{r}}_0=(0,0)$. Data for $\vec{r}$ along direction $\left[11\right]$ within the range $(-10,10)$ and $(10,10)$ is demonstrated. $\mathfrak{R}\left(\langle {\widehat{c}}_{\vec{r},l}{\widehat{c}}_{{\vec{r}}_0,s}^{†}\rangle \right)$ is practically 0 for $\vec{r}$ outside the region. The red, green, and blue colors represent the $s$, $d_{x^2-y^2}$ and $p_{+}=p_x+i{p}_y$ orbitals, respectively. The dashed lines are for exact values and the markers show the reconstructed results obtained from our TN. The inset is the log scale plot within the same range while the vertical axis is for ${log}_{10}\left|\mathfrak{R}\left(\langle {\widehat{c}}_{\vec{r},l}{\widehat{c}}_{{\vec{r}}_0,s}^{†}\rangle \right)\right|$.