Discrete lattice symmetry breaking in a two-dimensional frustrated spin-1 Heisenberg model

Spontaneous discrete symmetry breaking can be described in the framework of projected entangled pair states (PEPS) by linearly superposing local tensors belonging to two (or more) symmetry classes of tensors. This is illustrated in the case of a frustrated spin-1 Heisenberg model on the square lattice, which hosts a nematic spin liquid spontaneously breaking lattice $\pi /2$-rotation symmetry. A superposition of SU(2)-symmetric PEPS tensors belonging to two irreducible representations of the lattice point group is shown to capture accurately the properties of the nematic phase, as shown from a comparison to exact diagonalizations and density matrix renormalization group results.

Figure 1

Low-energy spectra of the Heisenberg Hamiltonian (1) vs $K_1$ obtained by Lanczos ED of periodic (a) $4\times 4$ and (b) $\sqrt{20}\times \sqrt{20}$ clusters. The different symbols (colors) correspond to different momenta (IRREP of the lattice point group). Open (filled) symbols correspond to singlet (triplet) states. The shaded/white regions are characterized by the singlet/triplet nature of the first excitation above the GS, suggesting possible nematic/magnetic phases (see text).

Figure 2

DMRG results for spin-spin correlations (3) at distance $(\mathrm{\Delta }x,\mathrm{\Delta }y)$ obtained on a $16\times 8$ cylinder for various $K_1$ indicated on the plot. Data are averaged in the middle $8\times 8$ region to reduce finite-size effects.

Figure 3

DMRG results vs $K_1$ obtained on various $2W\times W$ cylinders (with open boundary conditions in the long direction, see text): (a) Square of the staggered magnetization (4); (b) nematic order parameter (2); (c) square of the $\mathbf{q}=(\pi ,0)$ magnetic structure factor (4). Extrapolation leads to a phase diagram with three successive phases: Néel, nematic, and magnetic stripe phase, respectively, when increasing $K_1$ parameter.

Figure 4

CTM RG for nematic PEPS with one-site unit cell. The $\pi /2$ lattice rotation symmetry breaking is indicated by red/blue line in horizontal/vertical direction. (a) Nematic PEPS tensor $\mathcal{A}$. (b) Double tensor $E$ obtained by tracing out physical indices $E={\sum }_s{\overline{\mathcal{A}}}^s{\mathcal{A}}^s$, where ${\overline{\mathcal{A}}}^s$ is complex conjugate of ${\mathcal{A}}^s$. (c) CTM RG environment for $2\times 2$ cluster, constructed with corner matrix $C$, boundary tensor $T_{x,y}$ in horizontal/vertical direction, as shown separately in (d)–(f). The environment bond dimension is chosen to be $\chi =k{D}^2(k\in {\mathbb{N}}_{+})$. Energy density is calculated by inserting either identity operator or the local Hamiltonian operator in the red shaded $2\times 2$ cluster. (g), (h) is effective transfer matrix ${\mathcal{T}}_{\mathrm{eff}}^{x,y}$ in horizontal/vertical direction, constructed with $T_{x,y}$ and $E$ tensor. The maximal correlation length in horizontal/vertical direction can be obtained from the largest two eigenvalues of ${\mathcal{T}}_{\mathrm{eff}}^{x,y}$.

Figure 5

iPEPS variational energies per site (extrapolated in the limit $\chi \to \infty )$ vs $1/D$. A comparison to DMRG data obtained on $2W\times W$ cylinders (see text) and plotted vs $1/W$ is shown. In (a), data for $W\times W$ tori [9] are also shown.

Figure 6

iPEPS nematic order parameter (2) for bond dimension $D$ from 3 to 6, plotted as a function of $D^2/\chi$. The extrapolated value $\left(\chi \to \infty \right)$ is also shown, and the error bars are smaller than symbol sizes.

Figure 7

iPEPS maximal correlation length ${\xi }_{\max }$ in the (a) symmetric PEPS and (b)–(d) nematic PEPS vs $\chi /D^2$, for different values of $K_1$. The data for different values of $D$ ranging from 3 to 6 are shown with different symbols according to the legends. (a) and (b) $K_1=0$; (c) $K_1=0.05$; (d) $K_1=0.1$. In (b)–(d), open/filled symbols with same color correspond to the weak/strong bond directions of the nematic state.

Figure 8

Extrapolations performed for DMRG simulations on cylinder $2W\times W$ at fixed $K_1=0.05$. (a) The ground-state energy per site $e_0$ is extrapolated with respect to the discarded weight. (b) The nematic order parameter measured in the bulk of the system is extrapolated with respect to $1/m$.

Figure 9

Bond energies obtained by DMRG on a $16\times 8$ cylinder keeping up to $m=5000$ states. (a) $K_1=0$. (b) $K_1=0.05$. (c) $K_1=0.1$.

Figure 10

Finite-size scaling of DMRG data obtained on cylinders $2W\times W$ vs $1/W$ for three typical values of $K_1=-0.2$, 0.05, and 0.2 corresponding to the Néel, nematic, and stripe phases, respectively.

Figure 11

iPEPS energy density for various bond dimensions $D$, plotted as a function of $D^2/\chi$. (a) $K_1=0$. (b) $K_1=0.05$. (c) $K_1=0.1$.

Figure 12

iPEPS correlation length ${\xi }_{\max }$ vs $\chi /D^2$. Each panel corresponds to a given value of the virtual dimension $D$. Open squares/crosses with same color correspond to the weak/strong bond directions of the nematic ${\mathcal{A}}_1+{\mathcal{B}}_1$ PEPS. Open circles correspond to the symmetric ${\mathcal{A}}_1$ PEPS.