Comment on arxiv:1902.06475v1, "Magnetisation plateaus of the quantum pyrochlore Heisenberg antiferromagnet"

TL;DR

This paper comments on previous work about magnetization plateaus in quantum pyrochlore antiferromagnets, highlighting additional plateau states and providing an exact wavefunction for one of them.

Contribution

It identifies and discusses additional magnetization plateaus in pyrochlore magnets not covered in the original paper, including an exact wavefunction for the 5/6 plateau.

Findings

Identification of layered Kagome-like plateaus in pyrochlore magnets.

Derivation of an exact wavefunction for the 5/6 plateau state.

Extension of previous arguments to include new plateau states.

Abstract

This short note documents some of the pyrochlore magnetization plateaus resulting from the arguments of Pal & Lal that were not mentioned in arxiv:1902.06475v1. These are (layered) "Kagome"-like plateaus that are commensurate with the pyrochlore lattice, including an exact wavefunction for the $\frac{m}{m_s} = \frac{5}{6}$ plateau state.

Full text

Recently, the arguments of Refs. Oshikawa et al., 1997, Oshikawa, 2000 were applied to the Kagome Pal et al. (2018) and pyrochlore Pal and Lal (2019) quantum Heisenberg antiferromagnets. The main new ingredient was appropriate twist operatorsLieb et al. (1961) for these highly frustrated magnets that took into account the unit cell geometry. Using “flux-threading” arguments, they arrived at the following magnetization plateaus: for Kagome $\frac{m^{K}}{m_{s}}=\frac{1}{3}$ ($Q_{m}^{K}=3$), $\frac{m^{K}}{m_{s}}=\frac{1}{9},\frac{1}{3},\frac{5}{9},\frac{7}{9}$ ($Q_{m}^{K}=9$). For pyrochlore $\frac{m^{p}}{m_{s}}=0,\frac{1}{2}$ ($Q_{m}^{p}=4$), $\frac{m^{p}}{m_{s}}=0,\frac{1}{8},\frac{1}{4},\frac{3}{8},\frac{1}{2},\frac{5}{8},\frac{7}{8}$ ($Q_{m}^{p}=16$). $m_{s}=\frac{1}{2}$ is the saturation magnetization per site (in units of $\hbar$). $Q_{m}$ indicates the possible enlargement of the unit cell that often corresponds to a spontaneous translational symmetry breaking.Oshikawa (2000)

However, the above pyrochlore list does not include Kagome-like states that can be commensurately accommodated in the pyrochlore lattice. These states may be expected to be present for the pyrochlore lattice because of commensurability. Oshikawa (2000) For Kagome lattice, the $\frac{m^{K}}{m_{s}}=\frac{7}{9},\frac{5}{9},\frac{1}{3}$ plateaus (alternatively in terms of hardcore boson fillings, $\frac{1}{9},\frac{2}{9},\frac{3}{9}$ respectively) can be thought of as crystalline states in a $\sqrt{3}\times\sqrt{3}$ pattern (e.g. see Refs. Nishimoto et al., 2013, Okuma et al., 2019). This leads to the gapped, incompressible physics of the plateau states. For the $m=\frac{7}{9}$ plateau, exact wavefunctions can also be written down Schulenburg et al. (2002); Zhitomirsky and Tsunetsugu (2004); Bergman et al. (2008); Huber and Altman (2010); Changlani et al. (2019) which consist of closely-packed localized modes on a ($Z_{3}$) subset of hexagons consistent with the $\sqrt{3}\times\sqrt{3}$ pattern. As has been argued in some papers, Capponi et al. (2013); Nishimoto et al. (2013) the same idea may lie behind the other plateaus, though exact wavefunctions have not been written down for them.

If we assume the stability of the plateaus coming from such putative closely-packed hexagon modes at various fillings where the sites not on the hexagons are all polarized, then there arises a simple connection between the Kagome and pyrochlore plateaus because the pyrochlore lattice can be geometrically thought of as alternating Kagome and triangular layers connected by tetrahedral bonds. We can accommodate these stable closely-packed hexagon modes on the pyrochlore lattice in a commensurate fashion by populating the Kagome layers with the closely-packed hexagon modes and have the remaining sites as fully polarized including those on the triangular layers. As mentioned before, commensuration gives plausibility to their stability on the pyrochlore lattice. Oshikawa (2000) Then, it remains to work out the fillings.

We first start with an exact statement. The $\frac{m^{K}}{m_{s}}=\frac{7}{9}$ Kagome plateau ($\frac{1}{9}$ boson filling) corresponds to the $\frac{m^{p}}{m_{s}}=\frac{5}{6}$ pyrochlore plateau ($\frac{1}{12}$ bosons). (One has three extra polarized spins from the triangular layers per localized hexagon on the Kagome layer.) This is because the localized Kagome hexagon modes remain localized on the pyrochlore lattice too for the same (quantum interference) reasons, Schulenburg et al. (2002); Zhitomirsky and Tsunetsugu (2004); Bergman et al. (2008); Huber and Altman (2010); Changlani et al. (2019) and we can closely-pack them in the Kagome layers to get the $\frac{m^{p}}{m_{s}}=\frac{5}{6}$plateau. Going further, the $\frac{m^{K}}{m_{s}}=\frac{5}{9},\frac{1}{3},\frac{1}{9}$ plateaus of Kagome will now correspond to $\frac{m^{p}}{m_{s}}=\frac{2}{3},\frac{1}{2},\frac{1}{3}$ pyrochlore plateaus respectively. Summarizing in terms of hardcore bosons, $\frac{1}{9},\frac{2}{9},\frac{3}{9},\frac{4}{9}$ fillings on Kagome correspond to $\frac{1}{12},\frac{2}{12},\frac{3}{12},\frac{4}{12}$ fillings on pyrochlore respectively.

One may ask if this Kagome-pyrochlore connection coming from the layered “Kagome” plateaus get captured by the arguments of Ref. Pal and Lal, 2019, since the corresponding Kagome plateaus were captured by the arguments of Ref. Pal et al., 2018. Some reflection then tells us that these states indeed get captured for $Q_{m}^{p}=12$ (not noted in Ref. Pal and Lal, 2019). This is a tripling of the pyrochlore unit cell, as is to be expected since in Ref. Pal et al., 2018, the corresponding Kagome plateaus got captured by $Q_{m}^{K}=9$ which is again a tripling of the Kagome unit cell. For Kagome, Ref. Pal et al., 2018 argued that the reason behind $Q_{m}^{K}$ = 3 and 9 was that $N_{2}$ (number of unit cells perpendicular to the flux-threading or twist direction) be odd via Eq. 7 of Ref. Pal et al., 2018. For pyrochlore on the other hand, the factor $4N_{2}N_{3}$ in Eq. 6 of Ref. Pal and Lal, 2019 is even owing to even number of sites per pyrochlore unit cell. Thus, $N_{2}$ and $N_{3}$ can be either odd or even without any restrictions. This then a priori does not forbid $Q_{m}^{p}$ = 8 and 12.

For $Q_{m}^{p}=8$, one needs $N_{2}$ or $N_{3}$ to be even for consistency. This leads to the following plateaus via Eq. 6 of Ref. Pal and Lal, 2019: $\frac{m^{p}}{m_{s}}=0,\frac{1}{4},\frac{1}{2},\frac{3}{4}$. This is naturally a subset of $Q_{m}^{p}$=16 plateaus mentioned in Ref. Pal and Lal, 2019. The notable case is $Q_{m}^{p}=12$, where one needs $N_{2}$ or $N_{3}$ to be a multiple of 3. This is where we get the Kagome-pyrochlore connection by having $N_{2}$ as a multiple of 3 which amounts to the tripling of the unit cell in register with the desired Kagome-like layering. Then, we get the following new plateaus via Eq. 6 of Ref. Pal and Lal, 2019: $\frac{m^{p}}{m_{s}}=\frac{1}{6},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{5}{6}$. The last four plateaus were arrived at earlier in the note through heuristic arguments by layering the Kagome plateaus for $Q_{m}^{K}=9$ ($\frac{m^{K}}{m_{s}}=\frac{1}{9},\frac{1}{3},\frac{5}{9},\frac{7}{9}$), inspired by the very last case of $\frac{m^{p}}{m_{s}}=\frac{5}{6}$ and $\frac{m^{K}}{m_{s}}=\frac{7}{9}$ which is exact. Refs. Pal et al., 2018; Pal and Lal, 2019’s arguments building on Refs. Oshikawa et al., 1997; Oshikawa, 2000 now make this heuristic reasoning into a non-perturbative one.

Some final remarks: pyrochlore hosts a new plateau at $\frac{1}{6}$ for a tripled magnetic unit cell which can not be understood as a derivative of Kagome plateaus. What is this state? The $\frac{1}{9}$ Kagome plateau has been numerically argued to have no translation symmetry breaking in Ref. Nishimoto et al., 2013, however taking the tripling of the unit cell for this plateau as argued in Ref. Pal et al., 2018 at face value, we may expect the pyrochlore-Kagome connection for this case as well. Discussions with S. Pal and S. Lal are gratefully acknowledged.