Dimerization in the SU(4) Heisenberg model on the cubic lattice: iPEPS study

TL;DR

This work addresses the ground-state phase diagram of the SU(4)-symmetric Heisenberg model on a cubic lattice with anisotropic couplings. It employs a suite of advanced tensor-network techniques—2d and 3d iPEPS, MPS+MF, and boundary iPEPS with CTMRG—to access 2d, 1d, and 3d regimes and large unit cells. The key finding is that a dimerized color-ordered phase persists for all finite anisotropy, smoothly connecting the 2d dimerized state, the 1d decoupled-chain limit, and the isotropic 3d point. This demonstrates robust nontrivial SU(4) order in three dimensions and provides a scalable framework for exploring similar SU(N) models on other lattices.

Abstract

We study SU(4)-symmetric Heisenberg model on the cubic lattice with spatially anisotropic magnetic couplings. We utilize several approaches based on the tensor-network representation of the many-body wave functions, which enable accurate analysis of ground-state properties of the model in different regimes of spatial anisotropy including fully isotropic three-dimensional case. Our results point to the persistence of the dimerized color-ordered phase throughout whole range of magnetic couplings excluding only the limit of completely decoupled one-dimensional chains.

Figures (7)

• Figure 1: Idealized representation of the ground-state wave function in SU(4)-symmetric Heisenberg model on cubic lattice with spatially-anisotropic magnetic couplings.
• Figure 2: Illustration of three different tensor network ansatzes employed in this study: (a) 3d iPEPS wave function with tensors $T$ placed on the nodes of cubic lattice and bond matrices $\lambda$; (b) 2d iPEPS wave function with the $2 \times 2$ unit cell (indicated by surrounding dashed lines); (c) MPS + MF ansatz (with the unit cell $2 \times 2 \times 2$), where the wave function is represented as a product of MPSs formed along the $z$ axis. In all cases the different tensor colors correspond to different tensors inside the unit cell. Note that the unit cells are chosen here for illustrative purposes and may differ from the unit cells used in actual computations.
• Figure 3: Three-dimensional double-layer tensor network representing the norm of the 3d iPEPS wave function: (a) From the onsite tensors $T$ we can form the double-layer tensors $A$. Note that additional $\lambda$ matrices on the bonds of 3d iPEPS are assumed to be already absorbed into iPEPS site tensors $T$; (b) 3d tensor network representing the norm of iPEPS wave function (different colors of $A$ tensors show the different sites in the unit cell, which is chosen as $2 \times 2 \times 2$); (c) the boundary iPEPS approximating the leading eigenvector of the matrix formed by the two horizontal layers of $A$ tensors. We introduce the subscripts $u$ and $d$ in the boundary tensors $B$ to distinguish between the boundary iPEPS representing the contraction of top and bottom parts of the tensor network, respectively.
• Figure 4: Contraction of a residual 2d tensor network consisting of a middle layer with the double-layer tensors $A$ (see also Fig. \ref{['fig:3d_contraction']}) sandwiched between the boundary iPEPS, which represent the top and bottom parts of the tensor network. Different colors of tensors show different points in the projected unit cell, which is chosen here as $2 \times 2$. The resulting four-layer tensor network can be mapped into the effective single-layer tensor network with 16 times larger unit cell, as discussed in detail in Refs. lukin2024haghshenas2019single.
• Figure 5: Dependencies of the average energies $E_{x}$ and $E_{y}$ on $x$ and $y$ bonds, respectively, and the average difference of energies between the two consecutive $x$-bonds $|\Delta E_{x}|$ on the strength of interaction on the $y$-bonds $J_{y}$ in the 2d limit of SU(4) Heisenberg model (with $J_{x}=1$). The calculations are performed within the 2d iPEPS with the bond dimensions $D_x=8$ and $D_y=6$ along the $x$ and $y$ bonds, respectively.