Anomalies of kagome antiferromagnets on magnetization plateaus

Let us begin by clarifying an assumption about the effect of boundary conditions that we rely on in this paper. The assumption is that if a system has the unique gapped ground state in the periodic boundary condition under certain symmetries, it also does in any other closed boundary conditions that respect the symmetries. This assumption is natural since the closed boundary condition is just an artificial condition of theories to minimize the boundary effect. The bulk properties should be independent of a specific choice of symmetric boundary conditions.

Taking the contraposition of the assumption, we can state that if the unique gapped ground state is forbidden in one symmetric closed boundary condition, it is also forbidden in the periodic boundary condition. This statement motivates us to search for an appropriate symmetric closed boundary condition that clarifies a universal constraint forbidding the unique gapped ground state. Among such symmetric closed boundary conditions is the tilted boundary condition Yao and Oshikawa (2020) that we consider in this section.

Before defining the tilted boundary condition, we define the model and its symmetries that we use in this paper. We discuss spin-$1/2$ kagome antiferromagnets with translation symmetries and the $\mathrm{U(1)}$ spin-rotation symmetry. We assume translation symmetries of a unit cell by one unit in the $\bm{e}_{1}$ and the $\bm{e}_{2}$ directions of Fig. 1, where the smallest upward triangle is considered as the unit cell. All the results derived in this section also hold for spin-$S$ models with $S>1/2$.

Let us specify the location of the unit cell by a two-dimensional vector $\bm{R}=n_{1}\bm{e}_{1}+n_{2}\bm{e}_{2}$ with $\bm{e}_{1}=(1,0)$, $\bm{e}_{2}=(-\frac{1}{2},\frac{\sqrt{3}}{2})$ (Fig. 1), and $n_{1},n_{2}\in\mathbb{Z}$. We employed a unit of the lattice spacing $a=1$. Accordingly, the spin operator can be denoted as $\bm{S}_{\mu}(n_{1},n_{2})$. In the figures, the indices $\mu=1,2$, and $3$ are distinguished visually by red, white, and blue circles, respectively. Then the Hamiltonian of the spin-$S$ Heisenberg antiferromagnet is represented as

	$\displaystyle\mathcal{H}$	$\displaystyle=J\sum_{n_{1},n_{2}}[\bm{S}_{1}(n_{1},n_{2})\cdot\bm{S}_{2}(n_{1},n_{2})$
		$\displaystyle+\bm{S}_{2}(n_{1},n_{2})\cdot\bm{S}_{3}(n_{1},n_{2})+\bm{S}_{3}(n_{1},n_{2})\cdot\bm{S}_{1}(n_{1},n_{2})]$
		$\displaystyle+J\sum_{n_{1},n_{2}}[\bm{S}_{3}(n_{1},n_{2})\cdot\bm{S}_{2}(n_{1}+1,n_{2})$
		$\displaystyle+\bm{S}_{1}(n_{1},n_{2})\cdot\{\bm{S}_{3}(n_{1},n_{2}+1)+\bm{S}_{2}(n_{1}+1,n_{2}+1)\}]$
		$\displaystyle-h\sum_{n_{1},n_{2},\mu}S_{\mu}^{z}(n_{1},n_{2}),$		(1)

with $J>0$ and $h\geq 0$. This uniform kagome Heisenberg antiferromagnet is merely a specific example that satisfies the $\mathrm{U(1)}$ and the $T_{1}$ and $T_{2}$ translation symmetries, where $T_{n}$ represents the translation by the one unit in the $\bm{e}_{n}$ direction:

	$\displaystyle T_{1}\bm{S}_{\mu}(n_{1},n_{2})T_{1}^{-1}$	$\displaystyle=\bm{S}_{\mu}(n_{1}+1,n_{2}),$		(2)
	$\displaystyle T_{2}\bm{S}_{\mu}(n_{1},n_{2})T_{2}^{-1}$	$\displaystyle=\bm{S}_{\mu}(n_{1},n_{2}+1).$		(3)

We can add any symmetric interactions to the Hamiltonian whenever we want.

To make the translation symmetries well-defined, we need to specify the boundary condition. The Hamiltonian (1) possesses the $\mathrm{U}(1)$ spin-rotation symmetry and translation symmetries if the periodic boundary condition is imposed on the $x$ and the $y$ directions (Fig. 1). When we adopt the flux insertion argument Oshikawa (2000) to this system with the periodic boundary condition, we face the previously mentioned problem of the ambiguous thermodynamic limit Pal et al. (2020). In the case of the $d$-dimensional hyper cubic lattice, this problem can be resolved in Ref. Yao and Oshikawa (2020) by another closed boundary condition that respects the symmetries, the tilted boundary condition.

We define the tilted boundary condition on the kagome lattice. The kagome lattice is a non-Bravais lattice whose unit cell consists of three sites in the periodic boundary condition. Likewise, the unit cell contains three sites in the tilted boundary condition. The one-unit translation operator $T_{1}$ in the $\bm{e}_{1}$ direction acts on the spin operator $\bm{S}_{\mu}(n_{1},n_{2})$ ($\mu=1,2,3)$ as Eq. (2). To define the tilted boundary condition, we first consider a finite-size cluster of the kagome lattice of the rhombic shape (Figs. 2 and 3) and next take the thermodynamic limit by making the system size infinite. The rhombic cluster breaks some of the symmetries that the infinite-size kagome lattice possesses, for example, a $C_{6}$ rotation symmetry whose rotation axis pierces the center of a hexagon. The rhombic cluster is chosen because this paper is focused on an anomaly between the $\mathrm{U(1)}$ spin-rotation symmetry and the translation symmetry. The rhombic shape excludes the effects of the $C_{6}$ and other symmetries on the ground-state degeneracy.

Let us define the origin $\bm{R}=0$ as the left bottom corner of the rhombic finite-size cluster (the center of a triangle with a label 1 in Fig. 2). Suppose $0\leq n_{1}<N_{1}$ and $0\leq n_{2}<N_{2}$ for positive integers $N_{1},N_{2}$. On the rhombic finite-size cluster, the tilted boundary condition is defined as

$\displaystyle T_{1}\bm{S}_{\mu}\bigl{(}N_{1}-1,\,n_{2}\bigr{)}T_{1}^{-1}$

$\displaystyle=\bm{S}_{\mu}\bigl{(}0,\,n_{2}+1\bigr{)},$

(4)

for $n_{2}\in[0,N_{2})$ and

$\displaystyle T_{1}\bm{S}_{\mu}\bigl{(}N_{1}-1,N_{2}-1\bigr{)}T_{1}^{-1}$

$\displaystyle=\bm{S}_{\mu}\bigl{(}0,0\bigr{)}.$

(5)

When we reach the right seam of the system, we reenter the system from the left seam with the dislocation by the unit vector $\bm{e}_{2}$ (Fig. 3). It immediately follows from Eqs. (4) and (5) that the $T_{2}$ translation symmetry, that is, the translation symmetry in the $\bm{e}_{2}$ direction, depends on the $T_{1}$ one in the tilted boundary condition for a relation $T_{2}=(T_{1})^{N_{1}}$. The aspect ratio $N_{2}/N_{1}$ of the rhombus can be arbitrary. The total number of sites is given by

$\displaystyle V$

$\displaystyle=3N_{1}N_{2}$

(6)

We can sweep all the upward triangles on the kagome lattice one dimensionally by applying $T_{1}$ of Eqs. (2), (4), and (5) repeatedly. This path allows us to relabel the spin $\bm{S}_{\mu}(n_{1},n_{2})$ with a one-dimensional coordinate along that path as

$\displaystyle\bm{S}_{\mu}(n_{1},n_{2})=\bm{S}_{r_{1},\mu},$

(7)

where the one-dimensional coordinate $r_{1}\in[1,V/3]$ of the unit cell is related to $\bm{R}=n_{1}\bm{e}_{1}+n_{2}\bm{e}_{2}$ through

$\displaystyle r_{1}$

$\displaystyle=1+n_{1}+n_{2}N_{1}$

(8)

The Hamiltonian (1) in the tilted boundary condition is $T_{1}$-symmetric, that is, $[\mathcal{H},T_{1}]=0$.

We insert the flux adiabatically into the Hamiltonian (1) by replacing transverse exchange interactions,

	$\displaystyle S_{r_{1},\mu}^{+}S_{r^{\prime}_{1},\mu^{\prime}}^{-}$	$\displaystyle+\mathrm{H.c.}$
		$\displaystyle\to e^{i(r_{1}-r^{\prime}_{1})\phi/V}S_{r_{1},\mu}^{+}S_{r^{\prime}_{1},\mu^{\prime}}^{-}+\mathrm{H.c.},$		(9)

and increase the flux amount $\phi\in\mathbb{R}$ slowly from zero to the unit amount, $2\pi$. The $2\pi$ flux can be absorbed by a $\mathrm{U(1)}$ large gauge transformation,

$\displaystyle U$

$\displaystyle=\exp\biggl{(}i\frac{2\pi}{V/3}\sum_{r_{1}=1}^{V/3}r_{1}n(r_{1})\biggr{)},$

(10)

where $n(r_{1})$ is a number density of magnons at the unit cell $r_{1}$

$\displaystyle n(r_{1})=3S-S_{r_{1}}^{z},$

(11)

with $S_{r_{1}}^{z}=\sum_{\mu=1}^{3}S_{r_{1},\mu}^{z}$. The Hamiltonian with the flux, whatever the amount of the inserted flux is, keeps the $T_{1}$ symmetry and the global $\mathrm{U(1)}$ spin-rotation symmetry at the same time. On the other hand, the translation $T_{1}$ and the $\mathrm{U(1)}$ large gauge transformation $U$ satisfy a relation,

$\displaystyle T_{1}UT_{1}^{-1}$

$\displaystyle=Ue^{i\theta},$

(12)

with a nontrivial angle,

$\displaystyle\frac{\theta}{2\pi}=3(S-m),$

(13)

with the magnetization density per a site, $m=\sum_{r_{1}}S_{r_{1}}^{z}/(V/3)$. When $\theta$ is trivial (i.e. $\theta=0\mod 2\pi$), the two operators $T_{1}$ and $U$ are commutative with each other. Then nothing prevents the ground state from being unique and gapped. When the system has nontrivial $\theta\not=0\mod 2\pi$, it is forbidden to have any unique gapped ground state as follows Oshikawa (2000). Let us denote a ground state of the kagome antiferromagnet without the flux as $\ket{\Psi_{0}}$. If the adiabatic flux insertion deforms smoothly $\ket{\Psi_{0}}$ to $\ket{\Psi^{\prime}_{0}}$, the latter is a ground state of the kagome antiferromagnet with the unit flux. The large-gauge-transformed $U\ket{\Psi^{\prime}_{0}}$ is a ground state of the kagome antiferromagnet without the flux. Note that both $\ket{\Psi_{0}}$ and $\ket{\Psi^{\prime}_{0}}$ have the same eigenvalue of $T_{1}$ because the adiabatic flux insertion is compatible with the translation symmetry. If $\theta$ satisfies $\theta\not=0\mod 2\pi$, $U\ket{\Psi^{\prime}_{0}}$ is orthogonal to $\ket{\Psi_{0}}$ thanks to Eqs. (12) and (13), in other words, $U\ket{\Psi^{\prime}_{0}}$ is a degenerate ground state or a gapless excited state of the kagome antiferromagnet without the flux.

Because the angle (13) depends on neither $N_{1}$ nor $N_{2}$, we can take the thermodynamic limit, $V\to+\infty$, without any ambiguity. This well-defined thermodynamic limit is a great advantage of the tilted boundary condition over the periodic boundary condition Oshikawa (2000); Pal et al. (2020).

In the absence of the magnetic field, the model (1) possesses the time-reversal symmetry that imposes $m=0$ unless the spontaneous ferromagnetic order is generated, which is unlikely. Therefore, $\theta/2\pi=3S$ follows at zero magnetic field. When $S\in\mathbb{Z}+1/2$, the ground state of the spin-$S$ kagome Heisenberg antiferromagnet has either the gapless ground state or (at least) doubly degenerate gapped ground states. The former is consistent with the $\mathrm{U}(1)$ Dirac spin liquid scenario Ran et al. (2007); Iqbal et al. (2011); He et al. (2017) and the latter is consistent with the gapped $\mathbb{Z}_{2}$ spin liquid Yan et al. (2011); Depenbrock et al. (2012); Jiang et al. (2012). The degeneracy predicted by the relation (12) refers only to that by the intrinsic anomaly between the $\mathrm{U}(1)$ spin-rotation symmetry and the translation symmetry for a fixed filling. The ground state can, in principle, be more degenerate than Eq. (12) tells. Therefore, the ground-state degeneracy predicted by Eq. (12), which we denote as $d_{\rm m}$, gives the minimum possible value of the actual ground-state degeneracy.

In the presence of the magnetic field, the spin-$1/2$ Heisenberg antiferromagnet on the kagome lattice (1) is believed to have $1/9$, $1/3$, $5/9$, and $7/9$ magnetization plateaus. The angles of Eq. (13) for those fractions of the magnetization are listed in Table 1. When $\theta=2\pi p/q$ with coprime integers $p$ and $q$, the following $q$ states, $\ket{\Psi_{0}}$, $U\ket{\Psi^{\prime}_{0}}$, $U^{2}\ket{\Psi^{\prime}_{0}},\cdots$, and $U^{q-1}\ket{\Psi^{\prime}_{0}}$ have the same eigenenergy in the thermodynamic limit but have different eigenvalues of $T_{1}$. If the ground state is gapped, these $q$ states are $q$-fold degenerate gapped ground states. Given this possible minimum ground-state degeneracy, of particular interest is the $1/3$ plateau where the unique gapped ground state is allowed. In fact, Ref. Parameswaran et al. (2013) constructed the unique gapped ground state of a model explicitly on the $1/3$ plateau without breaking any symmetry of the kagome lattice. This is consistent with the relation (12). However, it remains obscure what allows for the unique gapped ground state on the $1/3$ plateau because the condition (12) with the angle (13) only tells that the minimum number of the ground-state degeneracy allowed by the translation and the $\mathrm{U}(1)$ spin-rotation symmetries is $1$. In the subsequent section, we propose one interpretation of the unique gapped ground state’s appearance possible on the $1/3$ plateau from the viewpoint of an anomaly in a quasi-one-dimensional limit.

The most significant advantage of the tilted boundary condition is that it reduces the number of spins in the cross-section $M_{d-1}$ down to $O(1)$. In the periodic boundary condition, it is $O(V^{(d-1)/d})$. The system with $O(1)$ spins on each cross-section seems like a one-dimensional system. In fact, we can view the kagome antiferromagnet in the tilted boundary condition as a one-dimensional quantum spin system in the periodic boundary condition where the upward triangle is one-dimensionally aligned. The flux insertion argument is independent of whether the system is viewed as a $d$-dimensional one or a one-dimensional one. Since the flux insertion argument picks up the anomaly between the $\mathrm{U(1)}$ symmetry and the translation symmetry, the anomaly is also independent of the viewpoint. This observation about dimensionality motivates us to describe the anomaly of kagome quantum antiferromagnets by using one-dimensional theoretical tools.

In general, relating $d$-dimensional quantum many-body systems to one-dimensional ones is a useful idea (e.g. the coupled wire construction of topological phases Kane et al. (2002); Lu and Vishwanath (2012)). This is partly because the latter is usually much better equipped with theoretical tools than the former Giamarchi (2004). The anomaly is no exception Furuya and Oshikawa (2017); Cho et al. (2017); Yao et al. (2019); Tanizaki and Sulejmanpasic (2018). In our case, however, the effective one-dimensional system inevitably contains long-range interactions that are extremely inconvenient for the anomaly argument, in particular, for the anomaly matching Hooft (1980). For example, a nearest-neighbor exchange interaction $\bm{S}_{1}(n_{1},n_{2})\cdot\bm{S}_{2}(n_{1},n_{2}+1)$ of the Hamiltonian (1) can be seen as an exchange interaction, $\bm{S}_{r_{1},1}\cdot\bm{S}_{r_{1}+N_{1},2}$ over the long distance $N_{1}$. This is a disadvantage of the tilted boundary condition.

To relate anomalies of kagome antiferromagnets to one-dimensional physics with avoiding this inconvenience, we notice the insensitivity of anomalies to spatial anisotropies implied by the flux-insertion argument with the tilted boundary condition. The large-gauge transformation operator (10) depends explicitly on the number of spins inside the unit cell and the lattice structure through $r_{1}$ but is independent of the strength of coupling constants. In general, ’t Hooft anomalies are robust against continuous variation of the Hamiltonian as long as the symmetries in question are preserved since an anomaly is identified with a surface term of a topological action of a symmetry-protected topological phase, which is manifestly topologically invariant Cheng et al. (2016); Jian et al. (2018); Thorngren and Else (2018). We can weaken or strengthen interactions in a specific spatial direction without interfering with the flux insertion argument. More generally, if an anomaly involves translation symmetries and on-site symmetries and is unrelated to spatial rotation symmetries, we can weaken interaction strengths in the directions perpendicular to the $\bm{e}_{1}$ direction with keeping the symmetries. The insensitivity to the spatial anisotropy opens a way to access the accumulated knowledge about anomalies of (1+1)-dimensional quantum field theories.

This section gives particular attention to the $1/3$ plateau where the unique gapped ground state is allowed in the flux-insertion argument and is indeed constructed in a specific model Parameswaran et al. (2013). We discuss that kagome antiferromagnets on the $1/3$ plateau, such as that of Ref. Parameswaran et al. (2013), can have the unique gapped ground state as a result of an explicit breaking of a symmetry that involves an anomaly. This section first discusses an anomaly of a one-dimensional quantum spin system, an $S=1/2$ three-leg spin tube. Next, using this knowledge of one-dimensional physics, we give an anomaly interpretation of the unique gapped ground state of the kagome antiferromagnet on the $1/3$ plateau.

In order to bridge physics in the one dimension to that on the kagome lattice, we consider a deformed kagome antiferromagnet with the following Hamiltonian:

	$\displaystyle\mathcal{H}_{\delta}$	$\displaystyle=\mathcal{H}_{\rm 1d}+\delta\mathcal{H}^{\prime},$		(14)
	$\displaystyle\mathcal{H}_{\rm 1d}$	$\displaystyle=J_{1}\sum_{n_{1},n_{2}}[\bm{S}_{1}(n_{1},n_{2})\cdot\bm{S}_{2}(n_{1},n_{2})+\bm{S}_{2}(n_{1},n_{2})\cdot\bm{S}_{3}(n_{1},n_{2})+\bm{S}_{3}(n_{1},n_{2})\cdot\bm{S}_{1}(n_{1},n_{2})]$
		$\displaystyle\quad+J_{3}\sum_{n_{1},n_{2},\mu}\bm{S}_{\mu}(n_{1},n_{2})\cdot\bm{S}_{\mu}(n_{1}+1,n_{2})-h\sum_{n_{1},n_{2},\mu}S_{\mu}^{z}(n_{1},n_{2}),$		(15)
	$\displaystyle\mathcal{H}^{\prime}$	$\displaystyle=J_{1}\sum_{n_{1},n_{2}}[\bm{S}_{1}(n_{1},n_{2})\cdot\bm{S}_{2}(n_{1}+1,n_{2}+1)+\bm{S}_{2}(n_{1}+1,n_{2})\cdot\bm{S}_{3}(n_{1},n_{2})+\bm{S}_{3}(n_{1},n_{2}+1)\cdot\bm{S}_{1}(n_{1},n_{2})]$
		$\displaystyle\quad+J_{3}\sum_{n_{1},n_{2}}\sum_{\mu=1,2,3}[\bm{S}_{\mu}(n_{1},n_{2})\cdot\{\bm{S}_{\mu}(n_{1},n_{2}+1)+\bm{S}_{\mu}(n_{1}+1,n_{2}+1)\}],$		(16)

where the parameter $\delta\in[0,1]$ controls the spatial anisotropy. The coupling constants $J_{1}$ and $J_{3}$ represent the nearest-neighbor and the third-neighbor exchange interactions when we view the model (14) as a two-dimensional system.

Let us impose the periodic boundary condition in the $\bm{e}_{1}$ and $\bm{e}_{2}$ directions. For $\delta=1$, all the nearest-neighbor bonds have the same strength of the exchange interaction and so do the third-neighbor ones. For $\delta=0$, the model (14) is a set of mutually independent three-leg spin tubes (Fig. 4). Note that the smooth change of $\delta$ keeps the $T_{1}$ and $T_{2}$ translation symmetries and the $\mathrm{U}(1)$ spin-rotation symmetry. We use the periodic boundary condition to investigate the model (14).

In this section, we set $\delta=0$ for a while and discuss its ’t Hooft anomaly on the $1/3$ plateau. Later in Sec. III.5, we will resurrect $\delta$ and discuss the anomaly in the quasi-one-dimensional limit, $0<\delta\ll 1$.

It is now a good occasion to review the original derivation of the OYA condition in the specific case of the three-leg spin tube. Let us consider a situation where three spin chains are weakly coupled to each other.

Each spin chain is equivalent to an interacting spinless fermion chain thanks to the Jordan-Wigner transformation Giamarchi (2004). Let us denote an annihilation operator of the spinless fermion at a location $x$ as $\psi_{\mu}(x)$, where $\mu=1,2,3$ are degrees of freedom that specify the leg. In the continuum limit, $\mu$ represent internal degrees of freedom, which we call “color” in this paper: $\mu=1,2,3$ for red, white, and blue circles, respectively in Figs. 2 and 4. At low energies, the spinless fermion operator can further be split into two species:

$\displaystyle\psi_{\mu}(x)$

$\displaystyle\approx e^{ik_{F}x}\psi_{R,\mu}(x)+e^{-ik_{F}x}\psi_{L,\mu}(x),$

(17)

where $k_{F}>0$ is the Fermi wave number and $\psi_{R,\mu}(x)$ and $\psi_{L,\mu}(x)$ are annihilation operators of the right-moving and left-moving spinless fermions, respectively.

The magnetization process of the three-leg spin tube is described by a $\mathrm{U(1)}$ conformal field theory (CFT) of compactified bosons. The spinless fermion operators $\psi_{R,\mu}(x)$ and $\psi_{L,\mu}(x)$ can be bosonized as $e^{i\phi_{\mu}/R}\propto\psi_{L,\mu}^{\dagger}\psi_{R,\mu}$ and $e^{2\pi iR\theta_{\mu}}\propto\psi_{L,\mu}\psi_{R,\mu}$. Here, $R>0$ is the compactification radius of those bosons. The $\mathrm{U(1)}$ CFT is written in two compactified $\mathrm{U(1)}$ bosons $\phi=\sum_{\mu=1}^{3}\phi_{\mu}$ and $\theta=\sum_{\mu=1}^{3}\theta_{\mu}$.

The Fermi wave number $k_{F}$ is related to the magnetization per site $m$ as

$\displaystyle k_{F}=\frac{\pi}{a}\biggl{(}\frac{1}{2}-m\biggr{)},$

(18)

where $a$ is the lattice spacing of the spin chain. The $T_{1}$ translation $x\to x+a$ along the leg effectively turns into an on-site symmetry, $\psi_{R,\mu}(x)\to e^{ik_{F}a}\psi_{R,\mu}(x)$ and $\psi_{L,\mu}(x)\to e^{-ik_{F}a}\psi_{L,\mu}(x)$ in the continuum limit $a\to 0$. In the boson language, the $T_{1}$ translation $x\to x+a$ affects only $\phi_{\mu}$:

	$\displaystyle\phi_{\mu}$	$\displaystyle\to\phi_{\mu}+2ak_{F}R,$		(19)
	$\displaystyle\phi$	$\displaystyle\to\phi+6ak_{F}R.$		(20)

When $6ak_{F}\in 2\pi\mathbb{Z}$, a perturbation $\cos(\phi/R)$ of the $\mathrm{U(1)}$ CFT is permitted by the translation symmetry. Note that cosines and sines of $\theta$ are forbidden by the $\mathrm{U(1)}$ spin-rotation symmetry. If $\cos(\phi/R)$ is relevant in the sense of the renormalization group, this cosine interaction opens the gap without any spontaneous symmetry breaking. If it is irrelevant, the ground state is gapless because it is the most relevant operator allowed by the symmetries. The condition $6ak_{F}\in 2\pi\mathbb{Z}$ is rephrased as

$\displaystyle 3(S-m)\in\mathbb{Z},$

(21)

with $S=1/2$. The relation (21) is the OYA condition in the case of the three-leg spin tube. It is straightforward to derive the same condition (21) for higher spin quantum numbers $S>1/2$.

When $6ak_{F}\in 2\pi(\mathbb{Z}+p/q)$ with coprime positive integers $p$ and $q$, the most relevant symmetric interaction in the $\mathrm{U(1)}$ CFT is $\cos(q\phi/R)$. If the ground state is gapped, the ground state is at least $q$-fold degenerate by breaking the translation symmetry $\phi\to\phi+2\pi pR/q\mod 2\pi R$ spontaneously. We thus reached the same conclusion as that derived from Eqs. (12) and (13). However, this is not the end of the story.

$S=1/2$ three-leg spin tubes possess the $1/3$ plateau Okunishi et al. (2012); Plat et al. (2012); Yonaga and Shibata (2015). According to the OYA condition (21), the $S=1/2$ three-leg spin tube can, in principle, have the unique gapped ground state on the $1/3$ magnetization plateau since $3(S-m)=1\in\mathbb{Z}$. On the other hand, to the best of our knowledge, the unique gapped ground state has not yet been reported for the simple three-leg spin tube with the Hamiltonian equivalent to Eq. (15) on the $1/3$ magnetization plateau Okunishi et al. (2012); Yonaga and Shibata (2015). This is not a coincidence. The unique gapped ground state is indeed forbidden, as we show below.

We reuse the spinless fermion picture. When three spinless fermion chains are gapless interacting Dirac ones decoupled from each other, the system has the $\mathrm{U(3)}$ symmetry. Interchain interactions in the spin tube reduce this symmetry. The charge of the spinless fermion corresponds to the magnetization of the spin tube. On the $1/3$ magnetization plateau where the total $S^{z}$ is frozen, magnetic excitations can be gapped without breaking any symmetry, as the OYA condition (21) implies. The remaining degrees of freedom, which are the color, are effectively described by a perturbed $\mathrm{SU(3)}$ WZW theory Affleck and Haldane (1987). At the fixed point, the $\mathrm{SU(3)}$ WZW theory has a global $\frac{\mathrm{SU(3)}_{R}\times\mathrm{SU(3)}_{L}}{\mathbb{Z}_{3}}$ symmetry Ohmori et al. (2019), where $V_{R/L}\in\mathrm{SU(3)}_{R/L}$ act on the field $g\in\mathrm{SU(3)}$ of the WZW theory as $g\to V_{L}gV_{R}^{\dagger}$. The symbols $R$ and $L$ represent the right-moving and the left-moving parts of particles of the WZW theory. In terms of spinless fermions $\psi_{R/L,\mu}$, a $(\mu,\nu)$ component, $g_{\mu\nu}$, of the $\mathrm{SU(3)}$ matrix $g$ can be represented as $g_{\mu\nu}\propto\psi_{L,\mu}^{\dagger}\psi_{R,\nu}$ Affleck (1988). Generally, the $R$ and $L$ degrees of freedom are coupled to each other by intrachain and interchain interactions when the field theory deviates from the fixed point. $\mathrm{SU(3)}_{R}\times\mathrm{SU(3)}_{L}$ is reduced to a single $\mathrm{SU(3)}$ with $V_{R}=V_{L}$, away from the fixed point. Then, the global symmetry is reduced to $\mathrm{PSU(3)}$.

We did not include this $\mathrm{SU(3)}$ WZW theory in the previous subsection. The $\mathrm{SU(3)}$ WZW theory is actually crucial on the $1/3$ magnetization plateau because of the following reason. The Fermi wave number $k_{F}$ is given by $k_{F}=\pi/3a$ on the $1/3$ plateau. The field $g_{\mu\nu}\propto\psi_{L,\mu}^{\dagger}\psi_{R,\nu}$ transforms under the $T_{1}$ translation as

$\displaystyle g\to e^{2\pi i/3}g.$

(22)

The $T_{1}$ translation symmetry is turned into the on-site $\mathbb{Z}_{3}$ symmetry (22). The $\mathrm{SU(3)}$ WZW theory has an anomaly between the $\mathbb{Z}_{3}$ symmetry (22) and the $\mathrm{PSU(3)}$ symmetry Yao et al. (2019). According to Ref. Yao et al. (2019), the $\mathbb{Z}_{3}$ symmetry (22) and the $\mathrm{PSU(3)}$ symmetry give an LSM index $\mathcal{I}_{3}$ that equals the number of the Young-tableau box per unit cell. The LSM index is a quantity related to the ground-state degeneracy. If and only if $\mathcal{I}_{3}=0\mod 3$, the ground state can be unique and gapped. Otherwise, the ground state is either gapless or gapped with at least $q$-fold degeneracy. Here, $q$ is given by

$\displaystyle q=\frac{3}{\operatorname{gcd}(\mathcal{I}_{3},3)},$

(23)

where $\operatorname{gcd}(n_{1},n_{2})$ is the greatest common divisor of two integers $n_{1}$ and $n_{2}$. The $3\times 3$ matrix $g$ belongs to either the fundamental or the conjugate representation of $\mathrm{SU(3)}$. The number of the Young tableau of the fundamental (conjugate) representations contains one box (two boxes, respectively). Hence, the three-leg spin tube on the $1/3$ plateau has the LSM index $\mathcal{I}_{3}=1$ or $\mathcal{I}_{3}=2$, both of which lead to $q=3$. The spin tube’s ground state on the $1/3$ plateau is either gapless or gapped with at least three-fold degeneracy. The gapless ground state is indeed possible on the $1/3$ magnetization plateau, which is a liquid state of chirality degrees of freedom Okunishi et al. (2012); Plat et al. (2012).

We saw that the translation and the $\mathrm{PSU(3)}$ color symmetries forbid the unique gapped ground state on the $1/3$ plateau. However, the $\mathrm{PSU(3)}$ color symmetry is too large to be naturally realized in three-leg spin tubes. In fact, in general, spin tubes have a $\mathbb{Z}_{3}$ color-rotation symmetry instead of $\mathrm{PSU(3)}$. It is desirable to reduce the symmetries that forbid the unique gapped ground state as much as possible.

We can reduce the color symmetry relevant to the ground-state degeneracy by referring to another quantum field theory, a (1+1)-dimensional $\mathrm{SU(3)/U(1)^{2}}$ nonlinear sigma model Lajkó et al. (2017); Tanizaki and Sulejmanpasic (2018). This nonlinear sigma model, as well as the $\mathrm{SU(3)}$ WZW theory, is also an effective field theory of $\mathrm{SU(3)}$ spin chains Lajkó et al. (2017); Tanizaki and Sulejmanpasic (2018); Yao et al. (2019). Though the $\mathrm{SU(3)/U(1)^{2}}$ model is not directly related to our spin tube (15), we consider this nonlinear sigma model for its relation to the $\mathrm{SU(3)}$ WZW theory; the low-energy limit of the (1+1)-dimensional $\mathrm{SU(3)/U(1)^{2}}$ nonlinear sigma model perturbed by local interactions is described by the $\mathrm{SU(3)}$ WZW theory Ohmori et al. (2019). According to the anomaly matching argument Hooft (1980), the $\mathrm{SU(3)/U(1)^{2}}$ nonlinear sigma model and the $\mathrm{SU(3)}$ WZW theory share the mixed ’t Hooft anomaly in common. In fact, Ref. Tanizaki and Sulejmanpasic (2018) shows that the $\mathrm{SU(3)/U(1)^{2}}$ nonlinear sigma model has an anomaly between the $\mathrm{PSU(3)}$ symmetry and the one-unit translation symmetry. This is consistent with the result of Ref. Yao et al. (2019). Reference Tanizaki and Sulejmanpasic (2018) further argued that the nonlinear sigma model has an anomaly between a finite subgroup $\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ of $\mathrm{PSU(3)}$ and the translation symmetry. Note that $\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ acts on the sigma fields projectively Tanizaki and Sulejmanpasic (2018).

The $\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ group is related to the following matrices,

$\displaystyle M_{1}$

$\displaystyle=\begin{pmatrix}1&0&0\\ 0&\omega&0\\ 0&0&\omega^{2}\end{pmatrix},\qquad M_{2}=\begin{pmatrix}0&1&0\\ 0&0&1\\ 1&0&0\end{pmatrix}.$

(24)

An isomorhpism $H:\mathrm{SU(3)}\to\mathrm{PSU(3)}$ maps $M_{n}$ for $n=1,2$ to $H(M_{n})$ that generate the $\mathbb{Z}_{3}\times\mathbb{Z}_{3}$. Note that $M_{1}$ is a gauge symmetry of the $\mathrm{SU(3)/U(1)^{2}}$ nonlinear sigma model, which does not forbid any interaction that perturbs the nonlinear sigma model. By contrast, $M_{2}$ forbids some interactions. It acts on the $g$ field as $M_{2}gM_{2}^{\dagger}$. In other words, $M_{2}$ represents the $\mathbb{Z}_{3}$ color-rotation symmetry, which is essential in what follows.

The $\mathbb{Z}_{3}$ color-rotation symmetry forbids interactions that treat the legs of the spin tube unequally. By imposing the $\mathbb{Z}_{3}$ color-rotation symmetry, the $\mathrm{U(1)}$ spin-rotation symmetry, and the translation symmetry on the spin tube, we can forbid the system (15) from having the unique gapped ground state on the $1/3$ plateau. On the other hand, this anomaly is absent in generic kagome antiferromagnets such as Eq. (1), as we show below.

While the anomaly argument in Sec. III.4 is already nontrivial as quantum physics in one dimension, it also gives interesting feedback to the original two-dimensional problem. Let us look back on the kagome antiferromagnet (15). As we mentioned, the anomaly in the two-dimensional system is inherited by the quasi-one-dimensional system where, in our case, three-leg spin tubes are weakly coupled with short-range interactions. Recall that the periodic boundary conditions are imposed on the $\bm{e}_{1}$ and the $\bm{e}_{2}$ directions. Hence, the anomaly argument in the quasi-one-dimensional limit reversely gives us a glimpse of the anomaly in the isotropic two-dimensional case. Let us suppose that these intertube interactions respect the $\mathbb{Z}_{3}$ color-rotation symmetry, the $\mathrm{U}(1)$ spin-rotation symmetry, and the $T_{1}$ and $T_{2}$ translation symmetries.

The unique gapped ground state is forbidden under those symmetries of intertube interactions. When the single spin tube has degenerate gapped ground states with a spontaneous symmetry breaking, the weak enough symmetric intertube interactions keep the gap open and do not lift the degeneracy.

On the other hand, when the single spin tube has the gapless ground state on the $1/3$ plateau, that is, the chirality liquid state Okunishi et al. (2012), a weak symmetric intertube interaction can induce a quantum phase transition. This phase transition occurs even if the coupling constant of the intertube interaction is infinitesimal Schulz (1996); Furuya et al. (2016). This is because the chirality liquid state is a Tomonaga-Luttinger liquid. At zero temperature, the Tomonaga-Luttinger liquid state has a divergent susceptibility of the order parameter Giamarchi (2004). This quantum phase transition drives the chirality liquid state into an ordered phase where the $\mathbb{Z}_{3}$ symmetry is spontaneously broken. After all, the unique gapped ground state is forbidden in both cases.

Now we go back to the $J_{1}-J_{3}$ model (14) of the kagome antiferromagnet. We showed that the model with $\delta=0$ has the anomaly between the $\mathbb{Z}_{3}$ color-rotation symmetry and the $T_{1}$ translation symmetry on the $1/3$ plateau. As soon as $\delta$ is turned on, the $\mathbb{Z}_{3}$ symmetry breaks down because of the first three terms of Eq. (16) with the coupling $J_{1}$ break the $\mathbb{Z}_{3}$ color-rotation symmetry. On the basis of the anomaly argument, we can expect that the absence of the $\mathbb{Z}_{3}$ color-rotation symmetry makes the unique gapped ground state possible on the $1/3$ plateau of the kagome antiferromagnet.

In the end, we caught a glimpse of the anomaly of the kagome antiferromagnet on the $1/3$ plateau with the high $\mathbb{Z}_{3}$ color symmetry by fine-tuned parameters. However, we also found no impediment to the possibility of the unique gapped ground state in general kagome geometry because the $\mathbb{Z}_{3}$ color-rotation symmetry is broken unless parameters of the Hamiltonian are fine-tuned.

Our argument shows that there is no unique gapped ground state on the $1/3$ plateau of a model with the $\mathbb{Z}_{3}$ color-rotation symmetry, the $\mathrm{U}(1)$ spin-rotation symmetry, and the $T_{1}$ and $T_{2}$ translation symmetries beyond the quasi-one-dimensional limit. Unfortunately, it remains challenging to confirm the existence of this anomaly directly in the isotropic two-dimensional case. We leave it for future works.

To close this section, we need to mention an unlikely possibility of reducing the number of spins in the unit cell. In this paper, we took an upward triangle as a unit cell as a natural choice. We can choose, in principle, a unit cell with any other shape. One might expect that regardless of the unit cell’s shape, the unit cell of the kagome lattice will have three spins inside the unit cell.

This statement is actually nontrivial when we take into account general closed boundary conditions. For example, antiferromagnets on the checkerboard lattice have two spins inside the unit cell under the periodic boundary condition or the tilted boundary condition but have only one spin under the spatially twisted boundary condition Furuya and Horinouchi (2019). This reduction of the number of spins in the unit cell was a key to show the impossibility of the unique gapped ground state in the checkerboard antiferromagnet at zero magnetic field in Ref. Furuya and Horinouchi (2019). If we could construct a symmetric closed boundary condition with a unit cell containing only one spin on the kagome lattice, the lower bound $d_{\rm m}$ shown in Table. 1 would be raised.

The bottom line is that such closed boundary conditions with the reduced unit cell can indeed be constructed, but they break the $T_{1}$ translation symmetry. These closed boundary conditions with the reduced unit cell do not qualify as the symmetric closed boundary condition in the kagome lattice case different from the checkerboard lattice. We construct those closed boundary conditions in Appendix A. We thus close this section by concluding that $d_{\rm m}$ in Table. 1 gives the maximum value of the lower bound of the ground-state degeneracy under the $\mathrm{U(1)}$ spin-rotation symmetry and the translation symmetry.