Projective symmetry group classifications of quantum spin liquids on the simple cubic, body centered cubic, and face centered cubic lattices

We perform extensive classifications of ${\mathbb{Z}}_2$ quantum spin liquids on the simple cubic, body centered cubic, and face centered cubic (fcc) lattices using a spin-rotation-invariant fermionic projective symmetry group approach. Taking into account that all three lattices share the same point group $O_h$, we apply an efficient gauge where the classification for the simple cubic lattice can be partially carried over to the other two lattices. We identify hundreds of projective representations for each of the three lattices, however, when constructing short-range mean-field models for the fermionic partons (spinons) these phases collapse to only very few relevant cases. We self-consistently calculate the corresponding mean-field parameters for frustrated Heisenberg models on all three lattices with up to third-neighbor spin interactions and discuss the spinon dispersions, ground-state energies, and dynamical spin structure factors. Our results indicate that phases with nonuniform spinon hopping or pairing amplitudes are energetically favored. An unusual situation is identified for the fcc lattice where the spinon dispersion minimizing the mean-field energy features a network of symmetry-protected linelike zero modes in reciprocal space. We further discuss characteristic fingerprints of these phases in the dynamical spin structure factor which may help to identify and distinguish them in future numerical or experimental studies.

Figure 1

Illustration of the bcc lattice where blue (red) points denote sublattice $A$ ($B$). The bold black lines in the upper right part of the figure highlight a cubic unit cell where the dark blue and dark red points are considered to lie inside this unit cell. The eight red points are the first neighbors of the dark blue site in the center.

Figure 2

Nearest-neighbor model SC $1_1$. (a) Spinon dispersion of Eq. (30) along the high-symmetry path through the Brillouin zone where $\mathrm{\Gamma }=(0,0,0)$, $X=(\pi ,0,0)$, $M=(\pi ,\pi ,0)$, and $R=(\pi ,\pi ,\pi )$ (and symmetry-related wave vectors). The Fermi surface is depicted in (b). (c) Dynamical spin structure factor plotted along the high-symmetry path in reciprocal space.

Figure 3

Quantum phase diagram of the $S=\frac{1}{2}$ antiferromagnetic $J_1\text{−}{J}_2\text{−}{J}_3$ Heisenberg model on the sc lattice. Gray regions denote the classical phases with the corresponding ordering wave vectors indicated. Spin configurations are illustrated for all classical orders. Thick black lines are the classical phase boundaries. The red area is the regime where Ref. [21] identifies a quantum paramagnetic phase. Red points mark the sets of Heisenberg couplings considered here.

Figure 4

SC $1_{3a}$ model with mean-field amplitudes up to third neighbors. (a) Self-consistent spinon dispersion for the first-neighbor terms in Eq. (30), second- and third-neighbor terms in Eqs. (32) [fixed to their self-consistently determined values given in Eq. (34)] along the high-symmetry path through the first Brillouin zone. (b) Corresponding dynamical spin structure factor along the same path in reciprocal space.

Figure 5

Nearest-neighbor model SC $2_1$. (a) Spinon dispersion of Eq. (35) along the high-symmetry path through the Brillouin zone. (b) Dynamical spin structure factor plotted along the high-symmetry path in reciprocal space.

Figure 6

Nearest-neighbor model BCC $1_1$. (a) Spinon dispersion of Eq. (38) along a path through the first Brillouin zone where $H=(0,0,2\pi )$, $N=(0,\pi ,\pi )$, and $P=(\pi ,\pi ,\pi )$ (and symmetry-related wave vectors). The Fermi surface is depicted in (b) where the green region indicates the first Brillouin zone. (c) Dynamical spin structure factor along a path in reciprocal space.

Figure 7

Phase diagram of the classical antiferromagnetic $J_1\text{−}{J}_2\text{−}{J}_3$ Heisenberg model on the bcc lattice. Gray regions denote the classical phases with the corresponding ordering wave vector indicated. Thick black lines are the classical phase boundaries. The red area is the regime where Ref. [24] identifies a nonmagnetic phase. Red points mark the sets of Heisenberg couplings considered here. On the left and right sides of the phase diagram we depict the states with ordering wave vectors $\mathbf{q}=(2\pi ,0,0)$ and $\mathbf{q}=(\pi ,\pi ,\pi )$. Note that in the $\mathbf{q}=(2\pi ,0,0)$ state the two sublattices have opposite spin orientations. For the $\mathbf{q}=(\pi ,\pi ,\pi )$ order, the $B$ sublattice has the same spin configuration as the $A$ sublattice, but globally rotated by an angle $\pi /2$.

Figure 8

BCC $1_2$ model with mean-field amplitudes up to second neighbors. (a) Self-consistent spinon dispersion for the first-neighbor terms in Eq. (38) and second-neighbor terms in Eq. (40) along a path through the first Brillouin zone. (b) Corresponding dynamical spin structure factor along the same path in reciprocal space.

Figure 9

Nearest-neighbor model BCC $2_1$. (a) Spinon dispersion of Eq. (43) along a path through the first Brillouin zone. The Fermi surface is depicted in (b) where the green region indicates the first Brillouin zone. (c) Dynamical spin structure factor along a path in reciprocal space.

Figure 10

BCC $2_3$ model with mean-field amplitudes up to third neighbors. (a) Self-consistent spinon dispersion for the first-neighbor terms in Eq. (43) and third-neighbor terms in Eq. (45) along a path through the first Brillouin zone. (b) Corresponding dynamical spin structure factor along the same path in reciprocal space.

Figure 11

Nearest-neighbor model FCC $1_1$. (a) Spinon dispersion of Eq. (47) along a path through the first Brillouin zone where $X=(0,2\pi ,0)$, $W=(\pi ,2\pi ,0)$, $L=(\pi ,\pi ,\pi )$, and $K=(\frac{3}{2}\pi ,\frac{3}{2}\pi ,0)$ (and symmetry-related wave vectors). The Fermi surface is depicted in (b) where the green region indicates the first Brillouin zone. (c) Dynamical spin structure factor along a path in reciprocal space.

Figure 12

Relevant magnetic orders of an antiferromagnetic classical $J_1\text{−}{J}_2$ Heisenberg model on the fcc lattice: At $J_2/J_1=0.5$ the $\mathbf{q}=(2\pi ,\pi ,0)$ magnetic order (top) shows a phase transition into $\mathbf{q}=(\pi ,\pi ,\pi )$ magnetic order (bottom).

Figure 13

FCC $1_2$ model with mean-field amplitudes up to second neighbors. (a) Self-consistent spinon dispersion for the first-neighbor terms in Eq. (47) and second-neighbor terms in Eq. (49) along a path through the first Brillouin zone. (b) Corresponding dynamical spin structure factor along the same path in reciprocal space.

Figure 14

Nearest-neighbor model FCC $2_1$. (a) Spinon dispersion of Eq. (52) along a path through the first Brillouin zone. Note the symmetry-protected zero-energy modes along the line $\overline{\mathrm{\Gamma }L}$. (b) Fermi lines emanating from the $\mathrm{\Gamma }$ point and forming a cubelike pattern. The green region indicates the first Brillouin zone. (c) Dynamical spin structure factor along a path in reciprocal space.