Affine lattice construction of spiral surfaces in frustrated Heisenberg models

Frustration in classical spin models can lead to degenerate ground states without long-range order. In reciprocal space, these degeneracies appear as manifolds of wave vectors, their dimensionality increasing with the degree of frustration and the robustness of the disordered spin-liquid state. Here, we present a recipe to explicitly construct Heisenberg models on Bravais lattices with codimension-one manifolds, i.e., lines in two dimensions and surfaces with different Euler characteristics in three dimensions. Furthermore, we discuss the role of thermal and quantum fluctuations in stabilizing ordered states.

Figure 1

Constructions of the codimension-one ground-state manifold ${\mathcal{M}}_{\mathrm{GS}}$, given by the minima of $J\left(\mathbf{q}\right)$ defined by $f\left(\mathbf{q}\right)=0$ via Eq. (4), on the square lattice $\mathrm{\Lambda }$. (a) Affine lattice construction with ${\mathrm{\Lambda }}^{*}=\mathrm{\Lambda }+{\mathit{\delta }}_1^{*}$. The red arrows connect the origin with the points of set ${\mathrm{\Xi }}_1\subset {\mathrm{\Lambda }}^{*}$ (red dots), dark-green dots are the nearest-neighbor points to the origin (black dot) of $\mathrm{\Lambda }$ with exchange coupling $J_1$, and light-green dots show the second-nearest-neighbor points with $J_2=J_1/2$. (b) Construction when ${\mathrm{\Xi }}_0,{\mathrm{\Xi }}_1\subset \mathrm{\Lambda }$, the ${\mathrm{\Xi }}_0$ has one point, the origin (black dot) and the ${\mathrm{\Xi }}_1$ is the orbit of ${\mathit{\delta }}_1={\mathbf{a}}_1$ (dark-green dots). We get a Heisenberg model with nearest-neighbor exchanges $J_1$ (dark green) and further neighbor exchanges $J_3=J_2/2$ (denoted by lighter colors). (c) ${\mathcal{M}}_{\mathrm{GS}}$ (red square) for ${\mathrm{\Lambda }}^{*}=\mathrm{\Lambda }+{\mathit{\delta }}_1^{*}$, which corresponds to the zeros of the function $f\left(\mathbf{q}\right)$ in Eq. (9), is pinned to the Brillouin zone boundary. (d) ${\mathcal{M}}_{\mathrm{GS}}$ for ${\mathrm{\Lambda }}^{*}=\mathrm{\Lambda }$, given by $f\left(\mathbf{q}\right)=0$ in Eq. (12), are shown as thick colored curves for $J_1/J_2=2$, 0, and $-2$.

Figure 2

Affine lattice constructions for the simple cubic (a) and face-centered-cubic (b) lattices. The shift vector ${\mathit{\delta }}^{*}$ (red arrow) defines the affine lattice ${\mathrm{\Lambda }}^{*}=\mathrm{\Lambda }+{\mathit{\delta }}^{*}$. Red balls with a cage show the set ${\mathrm{\Xi }}_1\subset {\mathrm{\Lambda }}^{*}$. Dark balls are the nearest-neighbor points to the origin (black ball) of $\mathrm{\Lambda }$, with exchange couplings $J_1$. Lighter balls depict the second neighbor points with exchange strengths $J_2=J_1/2$ for both lattices. For the simple cubic lattice a third neighbor exchange $J_3=J_2/2$ is also generated.

Figure 3

Two examples of the ground -tate manifolds of 3D lattices colored according to their free energies for the affine lattice construction with ${\mathit{\delta }}^{*}\ne \mathbf{0}$. Brighter colors correspond to states with smaller values of $\mathcal{A}\left(\mathbf{Q}\right)$ [Eq. (16)], and the minima are selected by thermal fluctuations. (a) Simple cubic lattice: the ground-state manifold is the Brillouin zone boundary, the degenerate minima are the inequivalent points $\left(\pi ,0,0\right), \left(\pi ,\pi ,0\right)$, and $\left(\pi ,\pi ,\pi \right)$ and their symmetry-related partners. (b) Face-centered-cubic lattice: the degenerate minima are the points $\langle \pi ,\pi ,\pi \rangle$. The Brilloin zone boundary is shown as a light wireframe; the enclosing cube is a guide to the eye.