Quantum magnetism on the Cairo pentagonal lattice

We present an extensive analytical and numerical study of the antiferromagnetic Heisenberg model on the Cairo pentagonal lattice, the dual of the Shastry-Sutherland lattice with a close realization in the $S=5/2$ compound Bi${}_2$Fe${}_4$O${}_9$. We consider a model with two exchange couplings suggested by the symmetry of the lattice, and we investigate the nature of the ground state as a function of their ratio $x$ and the spin $S$. After establishing the classical phase diagram, we switch on quantum mechanics in a gradual way that highlights the different role of quantum fluctuations on the two inequivalent sites of the lattice. The most important findings for $S=1/2$ include (i) a surprising interplay between a collinear and a four-sublattice orthogonal phase due to an underlying order-by-disorder mechanism at small $x$ (related to an emergent $J_1$-$J_2$ effective model with $J_2\gg J_1$), and (ii) a nonmagnetic and possibly spin-nematic phase with $d$-wave symmetry at intermediate $x$.

Figure 1

The Heisenberg model on the Cairo pentagonal lattice considered in this paper. Solid thick and thin bonds stand for the $J_{33}$ and $J_{43}$ exchange couplings. The (green) rectangle in the middle denotes the six-site unit cell of the Cairo lattice. The fourfold-coordinated sites form a tilted square lattice (denoted by thin dashed lines) that is further divided into two square sublattices denoted by large open (red) and filled (blue) circles.

Figure 2

The Heisenberg model on the Cairo pentagonal lattice. Left panel: Evolution of the phase diagram as we go from the classical (top) to the quantum limit (bottom). Quantum fluctuations are included gradually in a way that highlights the different role of the two inequivalent sites of the lattice. The regions indicated by “SF” in the second and fourth lines are those with strong semiclassical fluctuations. Right panel: The main magnetic phases appearing in the phase diagram. The mixed phase (not shown) interpolates between the orthogonal (a) and the 1/3-ferrimagnetic phase (c) as described in the text. In (a) and (b), we also show our labeling scheme for the corresponding bosonic operators (for each of the six sites of the unit cell enclosed by the green solid line) that appear in LSW theory. Note that in the collinear phase (b), the spins in half of the $J_{33}$ bonds (e.g., the ones labeled by $c_i$ and $e_i$) do not feel any exchange field from the neighboring fourfold sites.

Figure 3

Classical GS energy per site (measured from the energy of the orthogonal phase) as a function of $x=J_{43}/J_{33}$. The lines stand for the analytical expressions given in Eqs. (7) and (8), and the symbols are numerical data from CMC simulations at $\beta J_{33}={10}^3$.

Figure 4

Upper panel: Classical ($E_{\text{cl}}$) and semiclassical ($E_{\text{cl}}+\delta E$) GS energy per site from LSW theory around the orthogonal (black), the mixed (blue), and the ferrimagnetic state (red), as a function of $x=J_{43}/J_{33}$. For comparison, we also show (with symbols) the GS energy per site obtained from a fully quantum-mechanical numerical calculation (ED) on two clusters with 12 and 24 sites with periodic boundary conditions. Lower panel: Correction to the local spin length of the fourfold- (circles) and the threefold- (squares) coordinated sites. The (green) diamonds show the results from LSW theory around the orthogonal phase in the effective model (see Sec. 7).

Figure 5

Comparison between the numerical minimization of the variational ansatz described in the text (symbols) and the analytical prediction (solid lines) from the problem of an AFM dimer in the presence of a staggered field $h_s=\sqrt{2}Sx$ (see Appendix ). Circles show the squared overlap between the optimal quantum-mechanical state $|\Phi \rangle$ of a dimer and the full singlet state, and squares show the local spin lengths on the $J_{33}$ bonds.

Figure 6

Bosonic operators and local quantization axes (for the fourfold sites and the first site of each dimer) in the orthogonal state, where all spins lie on the $xz$ plane.

Figure 7

Main results from the spin $+$ bond-wave calculation, which captures the quadratic fluctuations around the variational GS. Left panel: The eight branches of hybridized spin $+$ bond-wave excitations at $x=0.5$ (see text). Middle panel: GS energy per site as a function of $x$. Comparison between linear spin-wave theories around the orthogonal and the mixed state (dashed lines), the spin $+$ bond-wave theory, and ED data from the 12- and 24-site clusters. Right panel: Various GS expectation values as a function of $x$. $\langle S_a^z\rangle$ stands for the local spin length of the fourfold sites. ${\langle S_1^z\rangle }_0$ and $\langle S_1^z\rangle$ stand for the staggered polarization on the $J_{33}$ dimers in the variational GS and when we include quadratic fluctuations, respectively, and similarly for ${\langle {\mathbf{S}}_1·{\mathbf{S}}_2\rangle }_0$ and $\langle {\mathbf{S}}_1·{\mathbf{S}}_2\rangle$. The symbols are ED results from the 12- (triangles) and 24- (circles) site clusters.

Figure 8

Left: Lowest-order processes in the effective model. The fourfold sites form a tilted square lattice (dashed lines). In second order in $x$, $J_1$ vanishes because of quantum interference between two different paths (shown by arrows) with opposite amplitude. In contrast, $J_2$ is finite since there is a single path. In fourth order in $x$, a four-spin term appears that invokes the four spins around a square plaquette. Right: Dependence of $J_1$, $J_2$, and $K$ on $x$, as given by Eqs. (28)--(30).

Figure 9

Left panel: Self-consistent solution for the mean-field pairing fields that appear in the interacting spin-wave theory around the collinear phase. Middle panel: The energy [in units of $J_2\left(x\right)$] of the second excitation mode at $\mathbf{k}=\mathbf{0}$ and at $\mathbf{k}=(\pi ,\pi )$. Inset: The BZ of the effective model is enclosed by the dashed (blue) lines and is set by the reciprocal vectors ${\mathbf{G}}_x$ and ${\mathbf{G}}_y$. At small enough $x$, the effective model reduces to the $J_1$-$J_2$ model on the square lattice whose unit cell contains one site only. Its BZ is enclosed by the solid (red) lines and is set by the vectors ${\mathbf{G}}_1$ and ${\mathbf{G}}_2$. Right panel: Spin-wave dispersions from NLSWT around the collinear phase for two representative values of $x$.

Figure 10

The unit cell of the Cairo pentagonal lattice along with a clarification of the space-group symmetries. The vectors ${\mathbf{t}}_x$ and ${\mathbf{t}}_y$ denote the primitive translations of the Bravais square lattice. In addition to the $C_4$ rotations around the fourfold-coordinated sites, we also have the four nonsymorphic operations $\left({\sigma }_i\right|\tau )$, which stand for reflections ${\sigma }_i$ ($i=x$, $y$, $d$, and $d^{\prime }$), followed by the nonprimitive translation $\tau$. In the absence of the threefold sites, this point group reduces to $C_{4v}$, which is the point group of the square lattice.

Figure 11

Left panel: Low-energy spectrum in the 24-site Cairo pentagonal cluster as a function of $(x-1)/(x+1)$ (bottom $x$ axis), where $x=J_{43}/J_{33}$ (top $x$ axis). The energy is measured from the respective GS and is scaled with the function $f\left(x\right)=J_{33}{x}^2$ for $x<1$ and $f\left(x\right)=J_{43}$ for $x>1$. Open symbols stand for $S_z=0$ states while crosses denote $S_z>0$ states. The legends specify the different symmetry sectors for this cluster. The last portion of the symmetry label accounts for the parity under spin inversion, with “Sze” and “Szo” specifying, respectively, even and odd sectors. The arrow denotes the 1/3-ferrimagnetic state (here with total spin $S=4$) which becomes the GS for $x\gtrsim 1.96$. Right panel: Low-energy spectrum of the 24-site Cairo pentagonal cluster vs the total spin $S$ for $x=0.8$. Different symbols correspond to different symmetry sectors as shown by the legends. The parentheses next to the legends denote the degeneracy of each sector (apart from the Zeeman degeneracy).

Figure 12

Left panel: The effective $J_1$-$J_2$-$K$ model description of the 24-site Cairo pentagonal cluster. There are two interpenetrating square plaquettes that consist of the sites (1, 0, 3, 5) and (2, 4, 7, 6). The $J_1$ and $J_2$ interactions are denoted by thin dotted and thick solid lines, respectively, while the orientation of the $K$ term in each $J_1$ plaquette is denoted by a thick segment (that coincides with the underlying $J_{33}$ dimer of the Cairo lattice) at the middle. At small $x$, the two square plaquettes are decoupled from each other and only the $J_2$ coupling survives (here $J\equiv 2J_2$). The total energy spectrum (last panel) results from a direct sum of the energy spectra of the two plaquettes (second and third panels). In addition to the spin quantum numbers, we also specify the angular momentum $l$ with respect to $C_4$ rotations around the site numbered 5. All degenerate levels are split by hand for clarity.

Figure 13

Left: Ground-state spin-spin correlation profiles $\langle {\mathbf{S}}_6·{\mathbf{S}}_i\rangle$ at $x=0.5$ for the 24-site Cairo pentagonal cluster. Filled (blue) circles denote positive correlations while open (red) circles denote negative correlations. Right: Correlations between the reference site (6) and the set of all symmetry-inequivalent sites of the cluster as a function of $x$, up to $x=1$. Note that the limiting correlation values at $x=0$ are consistent with the exactly known GS wave function of the eight-site effective cluster (see text).

Figure 14

Ground-state bond nematic $\langle ({\mathbf{S}}_1·{\mathbf{S}}_2)({\mathbf{S}}_k·{\mathbf{S}}_l)\rangle$ and vector-chiral $\langle {({\mathbf{S}}_1\times {\mathbf{S}}_2)}_z{({\mathbf{S}}_k\times {\mathbf{S}}_l)}_z\rangle$ correlations at $x=0.1$ for the 24-site Cairo pentagonal cluster. The thick bond $(1,2)$ is the reference bond. Solid (blue) lines denote positive correlations and dashed (red) lines denote negative correlations. The width of each line is proportional to the corresponding expectation value.

Figure 15

Low-$E$ spectra as a function of total spin $S$ for three different cases. The left is the spectra of the 24-site cluster for $x=1.667$, i.e., deep inside the intermediate phase. The middle panel shows the spectrum of the eight-site unconstrained $J_1$-$J_2$-$K$ model in the large $K/J_{1,2}$ regime. The third panel shows the spectrum of the eight-site effective model, but this time we have kept only the $K_x$ term among the three plaquette terms of Eq. (26). The pairs of numbers inside the parentheses indicate the total spins $S_A$ and $S_B$ of the two sublattices, which are separately conserved in this $\text{SU}\left(2\right)\times \text{SU}\left(2\right)$ model. In all three cases, the ovals indicate the sequence of GS's visited by the system in a field, while the different symbols follow the same convention as in the right panel of Fig. 11.

Figure 16

First two panels: Low-$E$ spectra as a function of total $S_z$ for the 16- and 32-site clusters using only the effective $K_x$ plaquette term. Here all energies are measured for the GS energy $E_0\left(S_z\right)$. The ovals indicate the sequence of GS's visited by the system in a field while the pairs of numbers inside the parentheses indicate the total spins $S_A$ and $S_B$ of the two sublattices in these states. Note that for the 32-site case, the 1/2-magnetization states with $(S_A,S_B)=(0,8)$ and $(8,0)$ are not the GS's. Right panel: Magnetization process in a field for the clusters with 8, 16, and 32 sites.

Figure 17

GS nematic correlations $\langle \left(S_i^{+}{S}_j^{+}\right)\left(S_k^{-}{S}_l^{-}\right)\rangle$ for the ${\mathcal{K}}_x$ model in the 32-site cluster. The cluster is denoted by the (green) rectangle. The think (black) line is the reference bond $\left(ij\right)$. The solid (blue) lines denote the bonds $\left(kl\right)$ with positive correlations while the dashed (red) lines denote negative correlations. The width of the bonds is proportional to the actual strength of the correlations, the largest of which is equal to $0.0228$. Note that there are no correlations between the reference bond and bonds belonging to a different sublattice.

Figure 18

The plaquettized version of the square lattice. Thick diagonal bonds denote the strong ${\mathcal{K}}_x$ terms, while dashed ones are treated with first-order perturbation theory.

Figure 19

Classical collinear and orthogonal states.