A single-layer framework of variational tensor network states

TL;DR

The paper tackles the bottleneck of variational tensor-network methods for two-dimensional quantum lattice models by introducing a single-layer framework built on nested tensor networks (NTN) and automatic differentiation. This approach reduces memory from about $D^8$ to $D^6$ and computation from about $D^{12}$ to $D^9$, enabling larger bond dimensions $D$ without GPU acceleration or symmetry exploitation. The method is validated on the infinite square-lattice Heisenberg model and the Shastry-Sutherland model, achieving $D=9$ with energies and order parameters aligning with prior results and providing evidence for an empty-plaquette valence-bond solid in the intermediate regime. The work demonstrates the viability of large-scale 2D tensor-network calculations and outlines pathways to extensions, including symmetry implementation and applications to superconductivity, excited states, and dynamics.

Abstract

We propose a single-layer tensor network framework for the variational determination of ground states in two-dimensional quantum lattice models. By combining the nested tensor network method [Phys. Rev. B 96, 045128 (2017)] with the automatic differentiation technique, our approach can reduce the computational cost by three orders of magnitude in bond dimension, and therefore enables highly efficient variational ground-state calculations. We demonstrate the capability of this framework through two quantum spin models: the antiferromagnetic Heisenberg model on a square lattice and the frustrated Shastry-Sutherland model. Even without GPU acceleration or symmetry implimention, we have achieved the bond dimension of 9 and obtained accurate ground-state energy and consistent order parameters compared to prior studies. In particular, we confirm the existence of an intermediate empty-plaquette valence bond solid ground state in the Shastry-Sutherland model. We have further discussed the convergence of the algorithm and its potential improvements. Our work provides a promising route for large-scale tensor network calculations of two-dimensional quantum systems.

Figures (12)

• Figure 1: An illustration of the infinite PEPS ansatz with a $2\times 2$ sublattice, as expressed in Eq. (\ref{['Eq:PEPS']}). (a) A $2\times 2$ unit cell. (b) Infinite PEPS wave function with translational invariance. Local tensors defined on the dots with the same color are identical. Dashed lines separate different unit cells.
• Figure 2: An illustration of the NTN method. (a) The unit cell of the reduced network $\langle\Psi|\Psi\rangle$, corresponding to the unit cell of $|\Psi\rangle$ in Fig. \ref{['Fig:PEPS']}. The double-layer structure of $T$ is also shown, corresponding to Eq. (\ref{['Eq:Tred']}). (b) The actual structure of (a), where $\langle\Psi|$ and $|\Psi\rangle$ are colored by blue and red, respectively. The slashed black lines denote physical indices $\{\sigma\}$. (c) The nested representation of (a) and (b) used in this work. Two additional tensors $X$ and $Y$ are introduced to form a compact single-layer tensor network. (d) The definition of $X$ and $Y$, as expressed in Eq. (\ref{['Eq:XY']}).
• Figure 3: A sketch of the CTMRG algorithm used in this work, for a tensor network with a $4\times 4$ unit cell. The superscripts $i$ and $j$ should be taken modulo 4. (a) The environment of the local tensor located at the $i$-th row and $j$-th column in the unit cell. (b) The recursive relation between the environment of $T^{(i,j)}$ and $T^{(i+1,j)}$, in a left-move. (c) The $4\times 4$ cluster used to determine the transformation matrices used in (b). Here $P^{(i,j)}$ and $Q^{(i+1,j)}$ will be determined, as expressed in Eqs. (\ref{['Eq:PQ']}-\ref{['Eq:QRSVD']}), and be inserted in the bonds denoted by $a_5a_6$. The resulting framework constitutes an optimal approximation of the bond density matrix $\rho$, which has subscripts $(a_1a_2,a_3a_4)$.
• Figure 4: The PEPS ansatz used in this work to study the square lattice Heisenberg model. Dashed lines separate the $2\times 2$ unit cells.
• Figure 5: The obtained ground-state energy $E_g$ of the square lattice Heisenberg model. (a) $E_g$ as a function of bond dimension $D$, with $D$ ranging from 2 to 9. (b) A power-law fitting of data $E_g(1/D)$. The extrapolation to the large-$D$ limit gives $E_g = -0.66941$. A quantum Monte Carlo estimation from Ref. HM-MC, $E_g=-0.6694421$, is also shown as red solid.