Generic spiral spin liquids

Abstract

Spiral spin liquids are unique classical spin liquids that occur in many frustrated spin systems, but do not comprise a new phase of matter. Owing to extensive classical ground-state degeneracy, the spins in a spiral spin liquid thermally fluctuate cooperatively from a collection of spiral configurations at low temperatures. These spiral propagation wavevectors form a continuous manifold in reciprocal space, i.e., a spiral contour or a spiral surface, that strongly governs the low-temperature thermal fluctuations and magnetic physics. In this paper, the relevant spin models conveying the spiral spin liquid physics are systematically explored and the geometric origin of the spiral manifold is clarified in the model construction. The spiral spin liquids based on the dimension and the codimension of the spiral manifold are further clarified. For each class, the physical properties are studied both generally and for specific examples. The results are relevant to a wide range of frustrated magnets. A survey of materials is given and future experiments are suggested.

Graphical abstract

Keywords

spiral spin liquids / thermal order-by-disorder / Monte Carlo simulation

1 Introduction

As an intriguing field in modern correlation physics, frustrated magnetism has attracted much interest in both theoretical and experimental efforts [1]. In general, frustration describes various competing interactions in a system that cannot be satisfied simultaneously. For frustrated spin systems, these interactions refer to exchange interactions between the local magnetic moments. It is expected that strong magnetic frustration could prevent the development of conventional magnetic orders and strongly suppress the ordering temperature. As a result, a wide range of exotic quantum phenomena may emerge, including quantum phases such as quantum spin liquids [2–4] and the spin nematic phase [5]. To understand these novel physics, new concepts represented by the internal gauge structure and fractionalized excitations have been introduced and developed in the past decades [1, 2, 4]. Among these phenomena, quantum fluctuations play a significant role even in the absence of geometric frustration [6].

Surprisingly, when quantum fluctuation is not important or even absent, strong frustration still has the ability to realize rich and interesting physics. One such example is the spiral spin liquid that was first proposed for the J₁–J₂ Heisenberg model on the geometrically unfrustrated diamond lattice [7]. In this spiral spin liquid, the classical ground states are coplanar spin spirals and have macroscopic degeneracy. The propagation vectors of the spiral spin configurations form a continuous manifold dubbed the spiral surface or spiral contour in reciprocal space. At finite temperatures, strong thermal fluctuations on the spiral manifold can stand out from the trivial paramagnetic regime. Hence, there could be a broad spiral spin liquid regime that smoothly connects to the trivial paramagnet in the high-temperature regime, and this spiral spin liquid regime can manifest itself in the spin correlations and thermodynamic properties. Despite being quite interesting, the spiral spin liquid is in the thermal paramagnetic regime and is not really a new state of matter. In Fig. 1, a representative behavior of the magnetic susceptibility of a typical frustrated magnet is depicted in which the spiral spin liquid would be located in the regime from the ordering temperature T_c to the Curie- Weiss temperature |Θ_CW|. At very low temperatures, the spiral orderings may develop from so-called thermal order by disorder or from other sub-leading interactions [7, 8]. The external magnetic field could further select a combination of equivalent and coplanar magnetic wavevectors and stabilize non-trivial topological spin structures such as vortexes, merons, and skyrmion lattices [9–13].

Recently, the degenerate spiral surface has been confirmed directly by neutron-scattering experiments in a long-promised spin-5/2 diamond-lattice compound MnSc₂S₄ [11, 13] with the single-crystal sample nine years after the theoretical proposal [7]. This caused a new wave of attention to be paid to the spiral spin liquids, the proximate orders, and the underlying novel physics [12, 14–22]. Although many relevant models and materials have been well studied in past decades, a systematic study for the flourishing family of the spiral spin liquids is still lacking. A recent theoretical work by Attig et al. gave a quite complete list of the spin models conveying the spiral manifold [23]. What the authors of Ref. [23] focused on are the spin-fermion correspondence and patterns of spiral manifolds at zero temperature. The resulting behaviors at finite temperatures still remain to be understood. The purpose of this work is to classify the spiral spin liquids, as well as to illustrate the characteristics of their finite-temperature behaviors by constructing the minimal models. The basic idea is to consider the dimensions and codimensions of the spiral manifold (see Table 1). The spin interactions are then designed accordingly and the massively degenerate ground-state manifold effectively manipulated. For both spiral contour and spiral surface cases, a large regime is identified in which spiral spin liquid physics dominates up to a high temperature compared to interactions. The robustness of the spiral spin liquid features is emphasized, which is important for the realization and identification in the realistic materials. Also discussed are the thermal order-by-disorder mechanism [24] and its consequences in three-dimensional systems. It is shown that thermodynamic quantities such as the specific heat would be modified by the induced spiral ordering, and that a distinct temperature dependence is exhibited for spiral manifolds with different dimensions. The scenario of the energy–entropy competition might be more intricate than as demonstrated herein, but is believed to be ubiquitous in real materials [24–26].

The rest of this paper is organized as follows. The spiral spin liquids are reviewed and classified in Section 2. For each class, a general Heisenberg Hamiltonian is constructed to capture the essential physics of a spiral manifold at zero temperature. As quintessences, the square, honeycomb, and AB-stacked multilayered triangular lattices are analyzed in detail. In Section 3, phase diagrams with the temperature and identity of the spiral spin liquid regimes for each case are further established. The difference in the low-temperature specific heat is discussed. Finally, in Section 4, this work concludes with a survey of the existing and potential materials with spiral spin liquid physics, and a discussion about open theoretical questions.

2 Spiral spin liquids: Classification and model construction

The basic structure underlying spiral spin liquids is the continuous spiral manifold formed by the propagation wavevectors of the classical ground states in reciprocal space. The continuity of this spiral manifold reflects the massive ground-state degeneracy and promises a spiral spin liquid. As will be shown later, the relationship between the dimensions and codimensions of the spiral manifold is crucial for the physical properties of the systems. The codimension d_c of the spiral manifold is defined as the difference between the dimension of the system and the dimension (d_s) of the spiral manifolds, and the spiral spin liquids are classified based on (d_s, d_c). For example, the well-known J₁–J₂ model on the diamond lattice for MnSc₂S₄ has a degenerate spiral surface with a dimension of d_s = 2, and the codimension of this spiral surface is d_c = 1. This physical classification is listed in Table. 1. In this section, specific models are proposed to explicitly demonstrate the existence of other classes and study their physical properties. The square, honeycomb, and AB-stacked multilayered triangular lattices are analyzed in detail.

2.1 Class with d_s = 1 and d_c = 1

First, a generic J₁–J₂ Heisenberg model on the bipartite lattices with the following form is considered:

\begin{array}{}\text{(1)}& H=\frac{1}{2}{\displaystyle \sum _{i\mathrm{,}j}\left[J_1{\mathit{M}}_{ij}+J_2{({\mathit{M}}^2-z\mathbb{I})}_{ij}\right]{\mathit{S}}_i\cdot {\mathit{S}}_j\mathrm{,}}\end{array}

where S_i is the classical three-component spin vector with unit length. M is a symmetric adjacency matrix of the underlying lattice, defined as

\begin{array}{}\text{(2)}& {\mathit{M}}_{ij}={\mathit{M}}_{ji}=\{\begin{array}{cc}1& \text{for}\langle ij\rangle \mathrm{,}\\ 0& \text{otherwise,}\end{array}\end{array}

where 〈ij〉 refers to the bond connecting the nearest-neighbor sites i and j, \mathbb{I} represents the identity matrix, and z is the coordination number of the lattice. The geometric meaning of M² − z\mathbb{I} is clear. Since (M 2)_ij =P_k M_ik M_kj, (M²)_ij essentially counts the number of different paths hopping from site i to site j via two nearest-neighbor bonds. When (M²)_ii = z\mathbb{I} , this represents the processes hopping back to the starting site. Therefore, these returning processes are subtracted from M². Hereafter, it is assumed that the antiferromagnetic interactions are J₁ > 0 and J₂ > 0. The regime with J₁ < 0 and J₂ > 0 can be achieved by only flipping spins in one sublattice, and there is no frustration when J₂ < 0. It will be seen that this model encodes the essential ingredients of the spiral manifold, and can be constructed on different lattices.

The well-known Luttinger–Tisza method is employed to explore the ground states of the classical J₁–J₂ Heisenberg model in Eq. (1). The core of this method is softening the local unit-length constraint on classical spin vectors S_i^2=1 by a global constraint Σ_i |S_i|² = N, where N is the total number of spins. After this replacement, the original problem is reduced to minimizing the energy

\begin{array}{}\text{(3)}& E_{\text{LT}}=H-\lambda \left({\displaystyle \sum _i|{\mathit{S}}_i{|}^2-N}\right)\mathrm{,}\end{array}

where λ is a Lagrange multiplier. Fourier transformation is implemented on each spin vector with the sublattice index µ and unit-cell index i as

\begin{array}{}\text{(4)}& {\mathit{S}}_i^{\mu }=\frac{1}{\sqrt{N/2}}{\displaystyle \sum _{\mathit{k}}{\mathit{S}}_{\mathit{k}}^{\mu }{\text{e}}^{\text{i}\mathit{k}\cdot {\mathit{r}}_i}\mathrm{.}}\end{array}

Then, the energy in reciprocal space reads

\begin{array}{}\text{(5)}& E_{\text{LT}}={\displaystyle \sum _{\mathit{k}}\left[\frac{1}{2}{\mathcal{J}}_{\mu \nu }(\mathit{k})-\lambda {\delta }_{\mu \nu }\right]{\mathit{S}}_{\mathit{k}}^{\mu }{\mathit{S}}_{-\mathit{k}}^{\nu }+\lambda N\mathrm{,}}\end{array}

where {\mathcal{J}}_{\mu \nu } (k) is the Fourier transformation of the interaction matrix defined by the adjacency matrix M in real space. For the model proposed here, it has the simple form

\begin{array}{}\text{(5)}& \mathcal{J}(\mathit{k})=\left(\begin{array}{cc}{J}_2(\text{Λ}{(\mathit{k})}^2-z)& J_1\text{Λ}(\mathit{k}){\text{e}}^{\text{i}\theta (\mathit{k})}\\ J_1\text{Λ}(\mathit{k}){\text{e}}^{-\text{i}\theta (\mathit{k})}& J_2(\text{Λ}{(\mathit{k})}^2-z)\end{array}\right)\mathrm{,}\end{array}

where the parameter Λ(k) = |{\displaystyle {\sum }_{{\mathit{R}}_{\text{NN}}}}{\text{e}}^{-\text{i}\mathit{k}\cdot {\mathit{R}}_{NN}}| . R_NN denotes the nearest-neighbor vectors and θ(k) refers to the phase shift between two sublattices. Now, minimizing the energy E_LT is equivalent to finding the minimum eigenvalue of a quadratic form \mathcal{J} (k). It is found that the eigenvalues form two bands with dispersions as

\begin{array}{}\text{(6)}& {\omega }_{\pm }=J_2{\left[\text{Λ}(\mathit{k})\pm \frac{J_1}{2J_2}\right]}^2-\frac{J_1^2+4z{J}_2^2}{4J_2}\mathrm{.}\end{array}

A phase transition occurs at J₂/J₁ = 1/(2z). When J₂/J₁ < 1/(2z), ω₋ takes its minimum at the Γ point in reciprocal space, corresponding to the antiferromagnetic Neél ordered state for the bipartite lattices. With increasing J₂/J₁, a massive ground-state degeneracy emerges and manifests itself as a spiral manifold determined by

\begin{array}{}\text{(7)}& \text{Λ}(\mathit{k})=\frac{J_1}{2J_2}\mathrm{.}\end{array}

The spin configuration is then decoded from the corresponding eigenvectors. Given a wavevector Q that resides on the degenerate manifold, the ground-state spin configuration in real space takes a spiral form,

\begin{array}{}\text{(8)}& \begin{array}{l}{\mathit{S}}_i^{\mu }={(-1)}^{\mu }\cos \left[\mathit{Q}\cdot {\mathit{r}}_i+{(-1)}^{\mu }\theta (\mathit{Q}/2)\right]{\mathit{e}}_1\\ +{(-1)}^{\mu }\sin \left[\mathit{Q}\cdot {\mathit{r}}_i+{(-1)}^{\mu }\theta (\mathit{Q}/2)\right]{\mathit{e}}_2\mathrm{,}\end{array}\end{array}

where µ can take 0 and 1 for two sublattices, respectively. e_1,2 are any two orthogonal unit vectors.

In this subsection, the class with d_s = 1 and d_c = 1 is studied, where the spiral manifold is a contour in reciprocal space. In this class, the dimension of the system is 2. To achieve this, only one constraint on the spiral wavevector is needed to force the spins onto the spiral manifold for the classical ground states. The square and honeycomb lattices are explicitly presented as examples that are shown in Figs. 2(a) and (b). The square and honeycomb lattices are both bipartite and have the coordination numbers z = 4 and z = 3, respectively. The nearest-neighbor interaction J₁ and the next-nearest-neighbor interaction J₂ are considered for both lattices. In addition, a third nearest-neighbor interaction with the same strength of J ₂ for the square lattice is involved to satisfy the definition of the adjacency matrix on the square lattice. The resulting spiral contour parameters are

\begin{array}{}\text{(9)}& \text{Λ}(\mathit{k})=4|\cos ({\mathit{a}}_1\cdot \frac{\mathit{k}}{2})\cos ({\mathit{a}}_2\cdot \frac{\mathit{k}}{2})|\end{array}

for the square lattice and

\begin{array}{}\text{(10)}& \begin{array}{ll}\text{Λ}(\mathit{k})=\hfill & \left[\begin{array}{c}3+2\cos ({\mathit{a}}_1\cdot \mathit{k})+2\cos ({\mathit{a}}_2\cdot \mathit{k})\end{array}\right]\hfill \\ \hfill & +2\cos [({\mathit{a}}_1-{\mathit{a}}_2)\cdot \mathit{k}{]]}^{\frac{1}{2}}\hfill \end{array}\end{array}

for the honeycomb lattice. Here, a₁ and a₂ are the lattice vectors defined in Appendix. A. In Figs. 3(a) and 4(a), the spiral contours for the two lattices and their evolution with increasing J₂/J₁ are plotted. It is shown that both two spiral contours emerge at J₂/J₁ = 1/(2z) from the Γ point and expand towards the boundary of the respective Brillouin zones. Despite this similarity, the spiral contours of two lattices behave dramatically differently in the large-J₂/J₁ regime. For the square lattice, the spiral contour approaches the boundary continuously, but does not touch it until the J₂/J₁→∞limit, where the spiral contour completely coincides with the Brillouin-zone boundary. Physically, in this limit the proposed generic J₁–J₂ Heisenberg model on the square lattice reduces to a conventional one with the largest frustration, and the massive degeneracy of the ground states remains. On the contrary, the spiral contour of the honeycomb lattice [18] touches the M point and other symmetry-related points in reciprocal space at J₂/J₁ = 1/2, and then splits into six arcs. These arcs are symmetry equivalent and can be viewed as closed loops centered at the K points. They shrink with increasing J₂/J₁ and finally become isolated points residing at point K. This means that the ground-state degeneracy disappears and Néel ordering is restored. Actually, in the J₂/J₁→∞limit, the two sublattices of the honeycomb lattice can be decoupled into two antiferromagnets on the triangular lattice, which is consistent with the present results. Therefore, the most important difference of these two types of lattices is whether the largest frustration state can be achieved at finite J₂/J₁, where the spiral contour has the largest circumference.

2.2 Class with d_s = 1 and d_c = 2

The class with d_s = 1 and d_c = 2 leads to a spiral contour on a three-dimensional lattice. Such a model can be constructed from a two-dimensional bipartite lattice. Specifically, the honeycomb lattice is decoupled into two triangular sub-systems according to its sublattices. The sublattice layers are then stacked alternatively into a three-dimensional multilayered system, as shown in Fig. 2(c). This is known as AB stacking. Here, the projection of a site from one layer onto its adjacent layer is located at the center of the triangles. The spin Hamiltonian retains the J₁–J₂ Heisenberg form

\begin{array}{}\text{(11)}& H=J_1{\displaystyle \sum _{i\mathrm{,}j}}{\mathit{S}}_i\cdot {\mathit{S}}_j+J_2{\displaystyle \sum _{i\mathrm{,}j}}{\mathit{S}}_i\cdot {\mathit{S}}_j\mathrm{,}\end{array}

where J₁ > 0 (J₂ > 0) then becomes the interlayer (intralayer) interaction. Repeating the process of the Luttinger–Tisza method, it is found that the interaction matrix is

\begin{array}{}\text{(12)}& \mathcal{J}(\mathit{k})=\left[\begin{array}{cc}{J}_2(\frac{1}{4}\text{Λ}{({\mathit{k}}_{\parallel })}^2-3)& J_1^{\prime }\text{Λ}({\mathit{k}}_{\parallel }){\text{e}}^{\text{i}\theta (\mathit{k})}\\ J_1^{\prime }\text{Λ}({\mathit{k}}_{\parallel }){\text{e}}^{-\text{i}\theta (\mathit{k})}& J_2(\frac{1}{4}\text{Λ}{({\mathit{k}}_{\parallel })}^2-3)\end{array}\right]\mathrm{,}\end{array}

where k_∥= {k_x, k_y, 0} and J_1^{\prime }=J_1\cos \left(\frac{1}{2}{\mathit{a}}_3\cdot \mathit{k}\right) with the lattice vectors a_i defined in Appendix A. The explicit form of Λ(k_∥) is expressed as

\begin{array}{}\text{(13)}& \begin{array}{l}\begin{array}{ll}\text{Λ}(\mathit{k})=& 2[3+2\left[\begin{array}{c}\cos ({\mathit{a}}_1\cdot {\mathit{k}}_{\parallel })+\cos ({\mathit{a}}_2\cdot {\mathit{k}}_{\parallel })\end{array}\right]\hfill \\ & {+\cos \left[({\mathit{a}}_1-{\mathit{a}}_2)\cdot {\mathit{k}}_{\parallel }\right]]}^{\frac{1}{2}}\mathrm{.}\hfill \end{array}\hfill \end{array}\end{array}

The minimum eigenvalue of \mathcal{J} (k),

\begin{array}{}\text{(14)}& {\omega }_{-}=\frac{J_2}{4}{\left[\text{Λ}({\mathit{k}}_{\parallel })-\frac{2J_1^{\prime }}{J_2}\right]}^2-\frac{{J_1^{\prime }}^2+3J_2^2}{J_2}\mathrm{,}\end{array}

defines a spiral contour

\begin{array}{}\text{(15)}& \text{Λ}({\mathit{k}}_{\parallel })=\frac{2J_1}{J_2}\mathrm{,}\end{array}

on the k_z = 0 plane when J₂/J₁ > 1/3. Before the spiral contour emerges, an antiferromagnetic Néel ordering is favored. As depicted in Fig. 5, the evolution of the spiral contour is very similar to that of the honeycomb lattice shown in Fig. 4(a). Nevertheless, the critical value at which the spiral contour touches the M points and is reconstructed becomes J₂/J₁ = 1. This can be attributed to the layer structure with doubling interlayer (inter-sublattice) bonds compared to the honeycomb lattice.

2.3 Examples with d_c = 0

In addition to the classes listed in Table 1, there exist examples with d_c = 0. The well-known cases are the classical Kagomé lattice Heisenberg model and classical pyrochlore lattice Heisenberg model. Because of the local zero-energy modes, the classical ground state has extensive degeneracy. The lowest eigenvalue of the exchange matrix in reciprocal space has no dispersion. The flat-ness arises from the existence of local zero-energy modes on each hexagon plaquette in the ground-state manifold. It is better to undersand the classical ground state in the real-space picture. The classical ground state requires that each triangular plaquette of the Kagomé lattice has a total spin equal to zero. This is like a Gauss’s law constraint for electromagnetism. In fact, the low-temperature magnetic properties can be well captured by an emergent classical electromagnetism that gives rise to a power-law spin correlation with a dipolar-like angular dependence. The classical pyrochlore lattice Heisenberg model was understood in a similar fashion. For the above reason, these two d_c = 0 examples are not included in the current study of spiral spin liquids.

3 Finite-temperature behaviors

In the preceding section, the similarity and difference between different classes of spiral spin liquids were discussed. Next, the finite-temperature behaviors of the spiral spin liquid regime are explored. A crucial difference here is the existence of ordered states due to the thermal order-by-disorder in the three spatial dimensions, which is prohibited in two spatial dimensions by the well-known Mermin– Wagner theorem. These ordered states can only persist in the rather low-temperature regime in three dimensions. In general, the increasing thermal fluctuation will eventually favor a disordered state as temperature increases. It will be shown that a broad range of the spiral spin liquid regime can persist before crossing over to the trivial paramagnetic regime.

3.1 Spiral spin liquid regime

The presence of the massive ground-state degeneracy, i.e., the spiral surface manifold, will affect the finite-temperature behavior of the system by revealing itself in different ways in different temperature regimes. For two spatial dimensions, the Mermin–Wagner theorem implies the absence of long-range order at finite temperatures, and there should be a wide regime in which the spiral spin liquid dominates the physics down to zero temperature when J₂/J₁ > (J₂/J₁)_c. The stability of the spiral spin liquid regime can be extended to the three-dimensional case, except that ordered states may exist at a sufficiently low temperature, which will be explained later. Here, the spiral spin liquid regime is characterized by the strong peaks of the spin-spin correlation in reciprocal space, located exactly on the spiral manifold. As the thermal fluctuation increases, these peaks will gradually become flat and broad, and will eventually become invisible in the trivial paramagnetic limit. To elucidate the regime of the spiral spin liquid, the spin correlation function at high temperatures is calculated by the self-consistent Gaussian approximation. Under the global constraint as introduced in Section 2, the partition function of the spin system is given by

\begin{array}{}\text{(16)}& \mathbb{Z}={\displaystyle \int }\mathcal{D}S\mathcal{D}\lambda {\text{e}}^{-\beta H-\text{i}\lambda ({\displaystyle \sum _i{S}_i^2-N})}\mathrm{,}\end{array}

where λ is the Lagrange multiplier to enforce {\displaystyle {\sum }_i}{\mathit{S}}_i^2=N . This is also known as the spherical approximation [7]. The saddle-point solution can be obtained by integrating out the classical spins, and the effective action scales as N that naturally leads to the saddle-point solution for λ. The saddle-point solution is taken by replacing iλ→βλ′(T)/2, where λ′(T) is a saddle-point parameter to be determined self-consistently. Equivalently, the saddle-point equation can be obtained from the spherical constraint for the spins,

\begin{array}{}\text{(17)}& \begin{array}{ll}{\displaystyle \sum _i}{\mathit{S}}_i^2\hfill & ={\displaystyle \sum _{\mu }}{\displaystyle \sum _{\mathit{k}}}{S}_{\mathit{k}}^{\mu }{S}_{-\mathit{k}}^{\mu }\hfill \\ \hfill & ={\displaystyle \sum _{\mathit{k}}}\frac{3}{\beta }\left[\frac{1}{{\omega }_{-}(\mathit{k})+{\lambda }^{\prime }(T)}+\frac{1}{{\omega }_{+}(\mathit{k})+{\lambda }^{\prime }(T)}\right]\hfill \\ \hfill & \equiv N\mathrm{,}\hfill \end{array}\end{array}

where the spin-correlation function in reciprocal space for the spins from the same sublattice µ is

\begin{array}{}\text{(18)}& \langle {\mathit{S}}_{\mathit{k}}^{\mu }{\mathit{S}}_{-\mathit{k}}^{\mu }\rangle =\frac{1}{2\beta }\left[\frac{1}{{\omega }_{-}(\mathit{k})+{\lambda }^{\prime }(T)}+\frac{1}{{\omega }_{+}(\mathit{k})+{\lambda }^{\prime }(T)}\right]\mathrm{,}\end{array}

and ω± refers to the two branches of eigenvalue dispersions of the exchange-interaction matrix \mathcal{J} (k) defined in Section 2. One can obtain the parameter λ′(T) by solving the above saddle-point equation. In Figs. 3(b) and 4(b), the temperature dependence of the spin correlation \langle {\mathit{S}}_{\mathit{k}}^{\mu }{\mathit{S}}_{-\mathit{k}}^{\mu }\rangle is depicted as a function of Λ(k) for the square and honeycomb lattices. The peaks correspond to points near the spiral contour that are apparent and remain discernible up to a crossover temperature. The spiral spin liquid regime can then be roughly identified by tracing this feature. The crossover between the spiral spin liquid regime and conventional high-temperature paramagnet regime is outlined in Figs. 3(c) and 4(c). It is shown that the frustration from the competing spin interactions can manifest itself qualitatively through this crossover temperature. The enhancement of frustration will strongly suppress the spiral spin liquid physics and make it easier for the system to enter a trivial paramagnetic state under the thermal fluctuation. In proximity to ordered states, a spiral spin liquid can persist up to a higher temperature. Since the largest frustration for the square lattice appears in the J₂/J₁→∞limit, a monotonically decreasing crossover temperature is obtained, in contrast to the honeycomb-lattice case in which there is a minimal crossover temperature near J₂/J₁ = 1/2. The spiral spin liquid regime is a special paramagnetic phase in which the spiral manifold governs the thermal fluctuations.

The finite-temperature behaviors for three dimensions have no qualitative difference. For the proposed AB-stacked multilayered triangular lattice, the spiral spin liquid regime and its crossover to the trivial paramagnet resemble those of the honeycomb lattice. The difference is that, in the low-temperature regime, the spiral ordered states can emerge through the thermal order-by-disorder mechanism. This is discussed in the next subsection.

3.2 Thermal order-by-disorder

Although the ground-state configuration can fluctuate on the degenerate spiral manifold without any energy cost at zero temperature, a global selection among these ground states will occur when weak thermal fluctuation is considered. At finite temperatures, the free energies of the degenerate ground states are usually different. To minimize the free energy, certain spiral ordered states associated with large entropy are selected. This stabilization mechanism of long-range order driven by the entropy is termed thermal order-by-disorder. This effect for the AB-stacked multilayered triangular lattice model is studied in the low-temperature regime.

First, a spiral spin order at momentum q with a spin configuration \overline{S} lying in the x–y plane is assumed and then its local stability is analyzed. The fluctuations can be parameterized as

\begin{array}{}\text{(19)}& \varphi ={\varphi }^{\text{o}}\widehat{\mathit{z}}+{\varphi }^{\text{i}}[\widehat{\mathit{z}}\times \overline{\mathit{S}}]\mathrm{,}\end{array}

where ϕ^o and ϕⁱ are out-of-plane and in-plane fluctuations, respectively. Physical spins can be expressed in fluctuation variables around the assumed order as

\begin{array}{}\text{(20)}& \mathit{S}=\varphi +\overline{\mathit{S}}{(1-{\varphi }^2)}^{1/2}\mathrm{.}\end{array}

The unit-vector constraint remains satisfied since \varphi \cdot \overline{\mathbf{S}}= 0. For ϕ²≪1, one can safely expand the spin Hamiltonian (11) in a fluctuation field up to second order,

\begin{array}{}\text{(21)}& H=\frac{1}{2}{\displaystyle \sum _{ij}}{\tilde{J}}_{ij}{\varphi }_i^{\text{o}}{\varphi }_j^{\text{o}}+{\tilde{J}}_{ij}({\overline{\mathit{S}}}_i\cdot {\overline{\mathit{S}}}_j){\varphi }_i^{\text{i}}{\varphi }_j^{\text{i}}\mathrm{.}\end{array}

Here, {\tilde{J}}_{ij}=J_{ij}-{\delta }_{ij}{\omega }_{-}^{\text{min}} . The shift makes both eigenvalues non-negative and only scales the partition function by a constant. There is no change in free energy, except for a zero-point movement. The zeroth- and first-order Hamiltonians have also been ignored because the former is just the ground-state energy and the latter should vanish in an ordered state. The resulting partition function is

\begin{array}{}\text{(22)}& \mathbb{Z}={\displaystyle \int }\mathcal{D}\varphi {\text{e}}^{-\beta H}{(1-{\varphi }^2)}^{-1/2}\propto {\displaystyle \int }\mathcal{D}{\varphi }^{\text{o}}\mathcal{D}{\varphi }^{\text{i}}{\text{e}}^{-\mathcal{S}}\mathrm{,}\end{array}

where the action is \mathcal{S} = βH. In the last step, a Jacobian factor is dropped since (1−ϕ²)^−1/2≈ {\text{e}}^{({\varphi }^{\text{o}2}+{\varphi }^{\text{i}2})/2} up to second order. That means that this factor only contributes diagonal terms to the action and does not affect the relative magnitude of free energy. At the Gaussian level, two components of the fluctuation field are decoupled. The out-of-plane fluctuation modes ϕ^o have zero modes for all momenta on the spiral manifold and thus are responsible for the massive ground-state degeneracy. The in-plane fluctuation modes ϕⁱ, however, have only one single gapless mode and are well-behaved at low temperature. The gapless mode is just the Goldstone mode required by symmetries and indicates a long-range order.

For the proposed J₁–J₂ Heisenberg model on the AB-stacked multilayered triangular lattice, these ϕⁱ fluctuation modes with spiral ordering wavevector q are

\begin{array}{}\text{(23)}& \begin{array}{l}{\gamma }_{\pm }(\mathit{k}\mathrm{,}\mathit{q})=\frac{J_2}{8}\left(|{\xi }_{+}{|}^2+|{\xi }_{-}{|}^2-24\right)-{\omega }_{-}^{\min }\\ \pm \frac{J_1}{2}|{\xi }_{+}\cos \frac{{\mathit{a}}_3\cdot (\mathit{k}+\mathit{q})}{2}+{\xi }_{-}\cos \frac{{\mathit{a}}_3\cdot (\mathit{k}-\mathit{q})}{2}|\mathrm{,}\end{array}\end{array}

where ξ_± = Λ(k_∥ ± q)e^{∓iθ(q)+iθ(k±q)}. Therefore, one can integrate the fluctuation fields to obtain the q-dependent part of free energy,

\begin{array}{}\text{(24)}& F_{{\varphi }^{\text{i}}}(\mathit{q})=T{\displaystyle {\int }_{\mathit{k}}}[\ln {\gamma }_{+}(\mathit{k}\mathrm{,}\mathit{q})+\ln {\gamma }_{-}(\mathit{k}\mathrm{,}\mathit{q})]/(2\pi T)\mathrm{.}\end{array}

The free-energy distribution on the spiral contour is visualized as shown in Fig. 5, in which the color reflects the relative magnitude of F_{{\varphi }^{\text{i}}} (q). Once the spiral contour emerges, the free-energy minima arise at (0, q, 0) and symmetry-related points, indicating a phase transition from Néel order to spiral order. When the spiral contour touches the Brillouin-zone boundary at J₂ = J₁, the minima locate at M points and move towards K points as J₂/J₁ increases. In the J₂/J₁→∞ limit, two decoupled antiferromagnetic magnets on triangular lattices are obtained, as expected.

The spiral ordering selected by entropy will melt when the system leaves the low-temperature regime. Above a critical temperature T_c, the spiral spin liquid behavior prevails again. To construct a finite-temperature phase diagram for spin orderings, classical Monte Carlo simulations are carried out on the AB-stacked multilayered triangular lattice with L³ unit cells. Owing to strong frustrations, a simple Metropolis–Hastings update will have a very long auto-correlation time that would lead to non-ergodicity. Here, a generally applicable method, parallel tempering [27], is employed to resolve this issue. Essentially, in parallel tempering the simulated space consists of several replicas of the systems while they are at different temperatures. Exchanging configurations in different replicas will decrease or increase the total energy according to the detailed balance, and there is an acceptance probability for this update. Therefore, the configurations of the high-temperature replicas can be tunneling to the lower-temperature replicas and vice versa. This diffusion process will significantly improve the efficiency. Furthermore, the over-relaxation method is employed to further decrease the auto-correlation time as well. The finite-temperature phase boundary can be determined by the grown peak of the specific heat with increasing system size, and the numerical results are shown in Fig. 6. Compared to the rapidly declining boundary of Néel order, T_c has a slower growth in the spiral order phases. The lowest critical temperature also occurs near the largest frustration area. Moreover, it is found that thermal fluctuation stabilizes the Néel phase even slightly above J₂/J₁ = 1/3, and thus induces re-entrant behavior. This subtle boundary exposes the intricate competitions between the spin interactions and thermal fluctuation in spiral spin liquid systems.

3.3 Specific heat

As mentioned previously, different classes of spiral spin liquids could exhibit distinct behaviors at low temperatures. In addition to the spin correlations, these distinctions can be directly reflected in the thermodynamic quantities. To demonstrate, here the specific heats C_v for the spiral spin liquids with (d_s, d_c) = (1, 1) and (1, 2) respectively, are calculated.

In two-dimensional systems, the spiral spin liquid can persist down to zero temperature. In the low-temperature regime, fluctuations near the spiral contour govern the physics and the minimum eigenvalue mode vanishes for momenta on the spiral contour. Therefore, the self-consistent equation (17) can be approximated as

\begin{array}{}\text{(25)}& {\displaystyle {\int }_{\mathit{k}}}\frac{1}{{\omega }_{-}(\mathit{k})+T^{\alpha }\overline{\lambda }}\sim \frac{2}{3}\beta \mathrm{.}\end{array}

Here, it is assumed that the saddle-point solution has a power-law temperature dependence as {\lambda }^{\prime }(T)\sim T^{\alpha }\overline{\lambda } . At low temperatures, the integral can be well estimated by considering momenta near the spiral contour. For a momentum q located on the spiral contour, a coordination framework spanned by the normal vector \widehat{n} and the tangent vector \widehat{t} can be associated. Along the normal direction, the dispersion of the minimum eigenvalue mode can be replaced by ω₋(k) = v(δp)², where δp = (k−q)·\widehat{\mathit{n}} and k·\widehat{\mathit{t}} = 0. The integral can be taken over δp with a proper cutoff and over the contour itself. By modifying the variable δp = T^α/2x, one can scale out the power α = 2 for the saddle-point solution λ′(T).

The free energy F = −T ln \mathbb{Z} in this approximation is

\begin{array}{}\text{(26)}& F\approx -\frac{1}{2}{T}^2\overline{\lambda }+\frac{1}{2}T{\displaystyle {\int }_{\mathit{k}}}\ln [({\omega }_{-}(\mathit{k})+T^2\overline{\lambda })/(2\pi T)]\mathrm{.}\end{array}

To calculate the specific heat C_v = −T(∂²F/∂T²)_V,N, it is convenient to consider

\begin{array}{}\text{(27)}& \frac{\partial (F/T)}{\partial T}=-\frac{\overline{\lambda }}{2}-\frac{1}{4\pi T}+{\displaystyle {\int }_{\mathit{k}}}\frac{T\overline{\lambda }}{{\omega }_{-}(\mathit{k})+T^2\overline{\lambda }}\approx \frac{c_1}{T}+c_2\mathrm{.}\end{array}

In the last step, the self-consistent equation is used to finish the integral. Thus, the free energy has the form F = c₁T ln T + c₂T² + +c₃T, and the specific heat is expressed as

\begin{array}{}\text{(28)}& C_v=c_1+c_{2}T\mathrm{.}\end{array}

Here, c₁ and c₂ are new constants absorbing all coefficients.

For spiral spin liquids in three spatial dimension, the development of spiral orderings in the low-temperature regime has significant effects on the thermodynamic quantities. In the absence of the thermal order-by-disorder, the dispersions of fluctuation modes from out-of-plane field ϕ^o are ω_±(k)−{\omega }_{-}^{\min } at the Gaussian level and the lowest one vanishes for any momentum on the spiral contour. This inevitably leads to a divergence of the fluctuation amplitude proportional to T{\displaystyle {\int }_{\mathit{k}}} 1/G₀(k), where G₀(k) is the propagator of ϕ^o. To cure the divergence, one must consider higher-order terms in the expression of fluctuation fields ϕⁱ and ϕ^o. Their contributions can be formally treated as a self-energy correction Σ(k), giving rise to the Green’s function

\begin{array}{}\text{(29)}& G(\mathit{k})=\frac{1}{{\omega }_{\pm }(\mathit{k})-{\omega }_{-}^{\min }+\text{Σ}(\mathit{k})}\mathrm{.}\end{array}

With the aid of the diagrammatic perturbation theory, Σ(k) can be expressed self-consistently as

\begin{array}{}\text{(30)}& \text{Σ}(\mathit{k})=T{\displaystyle {\int }_{\mathit{p}}}D(\mathit{k}\mathrm{,}\mathit{p})G(\mathit{p})\mathrm{,}\end{array}

where D(k, p) is a well-behaved function and vanishes only at finite momenta corresponding to the Goldstone modes. Therefore, the dominant contribution of the integral still comes from momenta near the spiral contour. The numerator of the integrand is thus replaced by D(k, q), where q denotes the momentum on the contour. Similarly, one can associate the point q with a coordination framework spanned by the normal vector \widehat{\mathit{n}} , the tangent vector \widehat{\mathit{t}} , and the original z-axis vector \widehat{\mathit{z}} . In the \widehat{\mathit{n}}-\widehat{\mathit{z}} plane, the dispersion of the eigenvalue mode ω−(p)−ω_min is still quadratic. One can parameterize momentum p in this plane as p·\widehat{\mathit{n}} = δp cos θ and p·\widehat{\mathit{z}} = δp sin θ, where p·\widehat{\mathit{t}} = 0 and δp = |p−q|. The dispersion of the minimum eigenvalue mode can then be expressed as v²(δp)² approximately. It is further assumed that the self-energy has a temperature dependence with the leading form Σ(k) = T^αΣ′(k) due to the thermal splitting. The self-consistent equation reduces to

\begin{array}{}\text{(31)}& T^{\alpha }{\text{Σ}}^{\prime }(\mathit{k})=T{\displaystyle {\int }_{\mathit{p}}}\frac{D(\mathit{k}\mathrm{,}\mathit{q})}{v^2{(\delta p)}^2+T^{\alpha }{\text{Σ}}^{\prime }(\mathit{q})}\mathrm{.}\end{array}

The integral is taken on the \widehat{\mathit{n}}-\widehat{\mathit{z}} plane with respect to θ and δp with a proper cutoff, and over the contour itself. Modifying the variables as δr = δp/T^α/2, the stiffness T^α can be scaled out of the integral and one obtains α = 1.

The full free energy should include both the ϕⁱ and ϕ^o parts, and be corrected by the self-energy as well. Up to the order of T, it takes the form

\begin{array}{}\text{(32)}& \begin{array}{ll}F\hfill & =F_{{\varphi }^{\text{i}}}(\mathit{q})+T{\displaystyle {\int }_{\mathit{k}}}\ln [({\omega }_{\pm }(\mathit{k})-{\omega }_{-}^{\min }+T{\text{Σ}}^{\prime }(\mathit{k}))/(2\pi T)]\hfill \\ \hfill & =c_{1}T\ln T+T{\displaystyle {\int }_{\mathit{k}}}\ln [{\omega }_{\pm }(\mathit{k})-{\omega }_{-}^{\min }+T{\text{Σ}}^{\prime }(\mathit{k})]+c_{3}T\mathrm{.}\hfill \end{array}\end{array}

To calculate the specific heat, the following equation is evaluated:

\begin{array}{}\text{(33)}& \begin{array}{ll}\frac{\partial (F/T)}{\partial T}\hfill & =\frac{c_1}{T}+{\displaystyle {\int }_{\mathit{k}}}\frac{{\text{Σ}}^{\prime }(\mathit{k})}{{\omega }_{\pm }(\mathit{k})-{\omega }_{-}^{\min }+T{\text{Σ}}^{\prime }(\mathit{k})}\mathrm{.}\hfill \end{array}\end{array}

Repeating the same scaling trick, it is found that the temperature dependence of free energy F = c₁T ln T + c₂T² + c₃T, and thus the low-temperature specific heat is

\begin{array}{}\text{(34)}& C_v=c_1+c_{2}T\mathrm{.}\end{array}

We have absorbed all coefficients into c₁ and c₂ and they should not be confused with the previous constants.

4 Discussion

4.1 Survey of materials

A survey of existing materials is presented in which spiral spin liquid physics can be relevant. The A-site spinels AB₂X₄ represent a large family of compounds for the diamond-lattice antiferromagnets. The magnetism comes from the A-site ions that form a diamond lattice. Among them, MnSc₂S₄ and CoAl₂O₄ manifest the strongest frustration of classical spins and are suggested to support spiral spin liquids by early experimental and theoretical works [7, 28–30]. Although later measurements ruled out the possibility of CoAl₂O₄ as a candidate [31], in recent experiments the spiral surface in MnSc₂S₄ was directly observed by neutron-scattering measurements and proved the existence of a spiral spin liquid regime for the spin-5/2 Mn²⁺ ions [11]. Instead of an ordering transition via the thermal order-by-disorder, a multi-step ordering behavior was observed at low temperatures. This reveals the limitation of the J₁–J₂ Heisenberg model and suggests that more interactions should be considered, such as the third-nearest-neighbor coupling in real materials [8, 32]. Other spinels like ARh₂O₄ (A = Ni, Co, and Cu) and MgCr₂O₄ have also been synthesized recently [14, 33–35]. For NiRh₂O₄ with frustrated spin-1 Ni²⁺ ions, the negative Curie–Weiss temperature and absence of the magnetic ordering transition down to 0.1 Kare noted. This material has spin S = 1 and the quantum effects seem to be strong. Quantum effects on the spiral manifold were seri-ously considered [35, 36]. As the material has a tetragonal symmetry rather than cubic symmetry and the Ni²⁺ has a partially filled t_2g shell, the single-ion anisotropy as well as the on-site spin–orbit coupling were thus also considered in addition to the superexchange interactions [35–38]. More experiments are needed to resolve the physics in NiRh₂O₄. The Co²⁺ ion in CoRh₂O₄ has S = 3/2, and a large nearest-neighbor exchange was found in CoRh₂O₄, so an Néel order was observed [34]. The Cu²⁺ ion in CuRh₂O₄ and CuAl₂O₄ has a d⁹ configuration with five electrons on the upper t_2g shell. The spin-orbit coupling is active. It is not purely spin physics, and one would necessarily consider the orbital degree of freedom. It was noted that this Cu²⁺ ion in the tetrahedral environment is similar to the well-known Ir⁴⁺ ion in the octahedral environment and gives a J_eff = 1/2 local moment [37, 39, 40]. The model for these Cu²⁺ moments in CuRh₂O₄ would not be a simple J₁–J₂ model, so spiral spin liquid physics may not be directly relevant. This system would be better described by an anisotropic Heisenberg-compass model on the diamond lattice. This will be analyzed in future works. Recently, a rare-earth compound LiYbO₂ was proposed to realize a diamond lattice [15]. Its theoretical understanding was based on the effective spin-1/2 J₁–J₂ Heisenberg model with the proximity to spiral spin liquid physics and the spiral manifold despite a large spin-orbit coupling for the Yb³⁺ ions. The possible success in LiYbO₂ may point to the possibility of realizing a spiral spin liquid in other Yb-based systems, such as the Yb-based honeycomb-lattice magnets.

The 6H-B-Ba₃NiSb₂O₉ compound with a spin S = 1 local moment has the same structure of the AB-stacked multilayered triangular lattice model studied herein, and this material was proposed to be a spin liquid candidate [41]. It has been suggested that the quantum effect in addition to spiral contour physics plays an important role in the spin-liquid phenomenology [42]. This may inspire the study of isostructural materials with d_s = 1 spiral manifolds in three dimensions. The compound MgCr₂O₄ was shown to have line degeneracies in reciprocal space, and the spiral spin liquid was proposed to be relevant [14]. This again corresponds to the d_s = 1 case.

The relevant materials in two dimensions can be Bi₃Mn₄O₁₂(NO₃) and Ca₁₀Cr₇O₂₈. Bi₃Mn₄O₁₂(NO₃) is a bilayer honeycomb-lattice magnet with S = 3/2 spins and exhibits a spin-liquid-like behavior down to 0.4 K [43]. Theoretically, this observed behavior falls into the situation of massive ring-like degeneracy and an order-by-disorder mechanism in a J₁–J₂ Heisenberg model [43, 44]. Recently, a novel ripple state under the external magnetic field has been reported and can be explained by the infinite ring-like degeneracy [12]. Ca₁₀Cr₇O₂₈ is another frustration magnets in which Cr⁵⁺ ions form a Kagomé bilayer structure. As the interaction between the Cr⁵⁺ spin-1/2 moment inside the triangular unit is dominantly ferromagnetic, one can treat the triangular unit as a total spin S = 3/2. The physics of this bilayer system can then be modeled as a semiclassical Heisenberg Hamiltonian on a honeycomb lattice with effective spin S = 3/2 moments and the intralayer and interlayer couplings are antiferromagnetic and ferromagnetic, respectively [16]. A ring-like pattern in the neutron-scattering signal and ordering transition at low temperature are also suggested for these materials [16, 17]. These studies suggest a particular relevance to the spiral spin liquid. However, there is still a lack of experimental results to reach a conclusion at this stage.

4.2 Open theoretical questions

Several open theoretical questions about the physics related to the spiral spin liquids are discussed here. In our calculation, the spin is treated classically. However, in reality, the spins always have quantum effects, especially for small spins. It is well known that the quantum fluctuations could break the spiral degeneracy for the ground state via the mechanism of the quantum order by disorder, and specific spiral states could be selected. The ordered states via the quantum order by disorder can sometimes be different from the thermal order by disorder. The thermal order by disorder is an entropic effect and requires a finite temperature, while the quantum order by order results from the quantum zero-point energy and works well for zero and low temperatures. This seems to suggest that there could be two thermal transitions as one cools the system from the high-temperature paramagnetic phase. The system first enters the ordered state via the thermal order by disorder and then develops another ordered phase due to the quantum order by disorder at a lower temperature. This is possible. Nevertheless, the separation of quantum order by disorder and thermal order by disorder is an artificial procedure, while nature blends these two effects together in the system. The blending of these effects may even favor another ordered state that differs from that favored by either. These issues may be resolved in future theoretical works.

Another important question concerns the internal spin space. All the models considered in this work have a global rotation symmetry with Heisenberg interactions and spins. The question is whether the spiral spin liquid regime can persist for XY spins or even for Ising spins. For XY spins with a U (1) symmetry, it is still possible to construct spiral states with two components of the XY spins. Thus, it is natural to expect that the spiral spin liquid exists for the XY or Heisenberg spins with an easy-plane anisotropy. Moreover, the senario for sprial spin liquids with XY spins resembles the bosonic models with frustrated hoppings, in which bosons could condense in a closed line instead of discrete points. Such phenomena have been discussed in various systems including the frustrated systems [45], high T_c superconductors [46], and driven-dissipative systems [47]. For Ising spins, it is not obvious if such a spiral spin liquid could survive as there is no way that one can construct spiral spin states with Ising spins. In contrast, the self-consistent Gaussian approximation that ignores the individual spin-magnitude constraint works rather well for the spiral spin liquid, and was known to work reasonably for certain Ising systems. Taken together, it is uncertain whether Ising spins could support the spiral spin liquid regime; this question will be addressed in a forthcoming paper.

Finally, the interplay of the quantum effect and degenerate spiral manifold may require a more systematic study in the future. Only in a few previous works, including Refs. [35–37, 48] in the diamond lattice antiferromagnet and Refs. [18, 21, 42] in the honeycomb lattice, and the AB-stacking multilayered triangular lattice, have explorations along this line been conducted. In fact, many of the spin moments in the existing materials already involve significant quantum fluctuations and could provide a good experimental platform for studying these physics.

4.3 Summary

In summary, the spiral spin liquids based on the dimension d_s and codimension d_c of the degenerate spiral manifold are classified. For each class, a minimal model based on the J₁–J₂ Heisenberg Hamiltonian is constructed. These constructions are not unique, but essentially capture the physics of spiral spin liquids. Systematic studies are performed on the square, honeycomb, and AB-stacked multilayered triangular lattices. These lattices, together with the well-studied diamond lattice, cover all three classes in the classification presented in Table 1. The present model construction takes the bipartite lattice as a starting point. The non-Bravais lattices, like the honeycomb and diamond lattices, have been studied and incorporated into this work and the classification. Many Bravais lattices, such as the square lattice in two dimensions, simple cubic, face-centered-cubic, and body-centered cubic lattices in three dimensions, can be treated as bipartite lattices as well once their adjacency matrices are carefully dealt with. The penalty of this straightforward extension is the requirement of further and more complicated interactions that may be rather unnatural in real materials. It is likely that some materials may be located near the unnatural parameter regime, and the additional interactions can be treated as a perturbation to break the spiral manifold. Despite this, the current classification and theoretical results depend only on the dimensions and can be immediately applied once the spiral manifold emerges.

References

[1] L. Balents, Spin liquids in frustrated magnets, Nature 464, 199 (2010)

[2] L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80, 016502 (2016)

[3] P. A. Lee, From high temperature superconductivity to quantum spin liquid: Progress in strong correlation physics, Rep. Prog. Phys. 71, 012501 (2007)

[4] Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liquid states, Rev. Mod. Phys. 89, 025003 (2017)

[5] Y. Kohama, H. Ishikawa, A. Matsuo, K. Kindo, N. Shannon, and Z. Hiroi, Possible observation of quantum spin-nematic phase in a frustrated magnet, Proc. Nat. Acad. Sci. 116, 10686 (2019)

[6] A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006)

[7] D. Bergman, J. Alicea, E. Gull, S. Trebst, and L. Balents, Order-by-disorder and spiral spin-liquid in frustrated diamond-lattice antiferromagnets, Nature Phys. 3, 487 (2007)

[8] S. Lee and L. Balents, Theory of the ordered phase in A-site antiferromagnetic spinels, Phys. Rev. B 78, 144417 (2008)

[9] L. Seabra, P. Sindzingre, T. Momoi, and N. Shannon, Novel phases in a square-lattice frustrated ferromagnet: 13-magnetization plateau, helicoidal spin liquid, and vortex crystal, Phys. Rev. B 93, 085132 (2016)

[10] T. Shimokawa, T. Okubo, and H. Kawamura, Multiple-q states of the J₁–J₂ classical honeycomb-lattice Heisenberg antiferromagnet under a magnetic field, Phys. Rev. B 100, 224404 (2019)

[11] S. Gao, O. Zaharko, V. Tsurkan, Y. Su, J. S. White, G. S. Tucker, B. Roessli, F. Bourdarot, R. Sibille, D. Chernyshov, T. Fennell, A. Loidl, and C. Rüegg, Spiral spin-liquid and the emergence of a vortex-like state in MnSc₂S₄, Nature Phys. 13, 157 (2016)

[12] T. Shimokawa and H. Kawamura, Ripple state in the frustrated honeycomb-lattice antiferromagnet, Phys. Rev. Lett. 123, 057202 (2019)

[13] S. Gao, H. D. Rosales, F. A. Gomez Albarracn, V. Tsurkan, G. Kaur, T. Fennell, P. Steens, M. Boehm, P. Cermak, A. Schneidewind, et al., Fractional antiferromagnetic skyrmion lattice induced by anisotropic couplings, Nature 586, 37 (2020)

[14] X. Bai, J. A. M. Paddison, E. Kapit, S. M. Koohpayeh, 12 J.-J. Wen, S. E. Dutton, A. T. Savici, A. I. Kolesnikov, G. E. Granroth, C. L. Broholm, J. T. Chalker, and M. Mourigal, Magnetic excitations of the classical spin liquid MgCr₂O₄, Phys. Rev. Lett. 122, 097201 (2019)

[15] M. M. Bordelon, C. Liu, L. Posthuma, E. Kenney, M. J. Graf, N. P. Butch, A. Banerjee, S. Calder, L. Balents, and S. D. Wilson, Frustrated Heisenberg J 1–J 2 model within the stretched diamond lattice of LiYbO₂, arXiv: 2009.04043 (2020)

[16] S. Biswas and K. Damle, Semiclassical theory for liquidlike behavior of the frustrated magnet Ca₁₀Cr₇O₂₈, Phys. Rev. B 97, 115102 (2018)

[17] R. Pohle, H. Yan, and N. Shannon, How many spin liquids are there in Ca₁₀Cr₇O₂₈? arXiv: 1711.03778 (2017)

[18] A. Mulder, R. Ganesh, L. Capriotti, and A. Paramekanti, Spiral order by disorder and lattice nematic order in a frustrated heisenberg antiferromagnet on the honeycomb lattice, Phys. Rev. B 81, 214419 (2010)

[19] P. Balla, Y. Iqbal, and K. Penc, Ane lattice construction of spiral surfaces in frustrated Heisenberg models, Phys. Rev. B 100, 140402 (2019)

[20] P. Balla, Y. Iqbal, and K. Penc, Degenerate manifolds, helimagnets, and multi-Q chiral phases in the classical Heisenberg antiferromagnet on the face-centered-cubic lattice, Phys. Rev. Res. 2, 043278 (2020)

[21] N. Niggemann, M. Hering, and J. Reuther, Classical spiral spin liquids as a possible route to quantum spin liquids, J. Phys.: Condens. Matter 32, 024001 (2019)

[22] Z. Nussinov, Commensurate and incommensurate o(n) spin systems: Novel even-odd eects, a generalized mermin-wagner-coleman theorem, and ground states, arXiv: 0105253 (2004)

[23] J. Attig and S. Trebst, Classical spin spirals in frustrated magnets from free-fermion band topology, Phys. Rev. B 96, 085145 (2017)

[24] J. Villain, R. Bidaux, J.-P. Carton, and R. Conte, Order as an eect of disorder, Journal de Physique 41, 1263 (1980)

[25] C. L. Henley, Ordering due to disorder in a frustrated vector antiferromagnet, Phys. Rev. Lett. 62, 2056 (1989)

[26] J. N. Reimers and A. J. Berlinsky, Order by disorder in the classical Heisenberg Kagomé antiferromagnet, Phys. Rev. B 48, 9539 (1993)

[27] K. Hukushima and K. Nemoto, Exchange Monte Carlo method and application to spin glass simulations, J. Phys. Soc. Japan 65, 1604 (1996)

[28] N. Tristan, J. Hemberger, A. Krimmel, H.-A. Krug von Nidda, V. Tsurkan, and A. Loidl, Geometric frustration in the cubic spinels M Al₂O₄ (M = Co, Fe, and Mn), Phys. Rev. B 72, 174404 (2005)

[29] T. Suzuki, H. Nagai, M. Nohara, and H. Takagi, Melting of antiferromagnetic ordering in spinel oxide CoAl₂O₄, J. Phys.: Condens. Matter 19, 145265 (2007)

[30] V. Fritsch, J. Hemberger, N. Büttgen, E.-W. Scheidt, H.- A. Krug von Nidda, A. Loidl, and V. Tsurkan, Spin and orbital frustration in MnSc₂S₄ and FeSc₂S₄, Phys. Rev. Lett. 92, 116401 (2004)

[31] O. Zaharko, N. B. Christensen, A. Cervellino, V. Tsurkan, A. Maljuk, U. Stuhr, C. Niedermayer, F. Yokaichiya, D. N. Argyriou, M. Boehm, and A. Loidl, Spin liquid in a single crystal of the frustrated diamond lattice antiferromagnet CoAl₂O₄, Phys. Rev. B 84, 094403 (2011)

[32] Y. Iqbal, T. Müller, H. O. Jeschke, R. Thomale, and J. Reuther, Stability of the spiral spin liquid in MnSc₂S₄, Phys. Rev. B 98, 064427 (2018)

[33] J. R. Chamorro, L. Ge, J. Flynn, M. A. Subramanian, M. Mourigal, and T. M. McQueen, Frustrated spin one on a diamond lattice in NiRh₂O₄, Phys. Rev. Mater. 2, 034404 (2018)

[34] L. Ge, J. Flynn, J. A. M. Paddison, M. B. Stone, S. Calder, M. A. Subramanian, A. P. Ramirez, and M. Mourigal, Spin order and dynamics in the diamond-lattice Heisenberg antiferromagnets CuRh₂O₄ and CoRh₂O₄, Phys. Rev. B 96, 064413 (2017)

[35] G. Chen, Quantum paramagnet and frustrated quantum criticality in a spin-one diamond lattice antiferromagnet, Phys. Rev. B 96, 020412 (2017)

[36] F. L. Buessen, M. Hering, J. Reuther, and S. Trebst, Quantum spin liquids in frustrated spin-1 diamond antiferromagnets, Phys. Rev. Lett. 120, 057201 (2018)

[37] F.-Y. Li and G. Chen, Spin-orbital entanglement in d⁸ Mott insulators: Possible excitonic magnetism in diamond lattice antiferromagnets, Phys. Rev. B 100, 045103 (2019)

[38] S. Das, D. Nafday, T. Saha-Dasgupta, and A. Paramekanti, NiRh₂O₄: A spin–orbit entangled diamond-lattice paramagnet, Phys. Rev. B 100, 140408 (2019)

[39] S. A. Nikolaev, I. V. Solovyev, A. N. Ignatenko, V. Y. Irkhin, and S. V. Streltsov, Realization of the anisotropic compass model on the diamond lattice of Cu²⁺ in CuAl₂O₄, Phys. Rev. B 98, 201106 (2018)

[40] X.-P. Yao, C.-J. Huang, C. Liu, F.-Y. Li, and G. Chen, The eects of spin–orbit coupling in diamond lattice magnets: A study of heisenberg-compass model on a diamond lattice (2020) (in preparation)

[41] J. G. Cheng, G. Li, L. Balicas, J. S. Zhou, J. B. Goode-nough, C. Xu, and H. D. Zhou, High-pressure sequence of Ba₃NiSb₂O₉ structural phases: New S = 1 quantum spin liquids based on Ni²⁺, Phys. Rev. Lett. 107, 197204 (2011)

[42] G. Chen, M. Hermele, and L. Radzihovsky, Frustrated quantum critical theory of putative spin-liquid phenomenology in 6H–B–Ba₃NiSb₂O₉, Phys. Rev. Lett. 109, 016402 (2012)

[43] S. Okumura, H. Kawamura, T. Okubo, and Y. Motome, Novel spin-liquid states in the frustrated Heisenberg antiferromagnet on the honeycomb lattice, J. Phys. Soc. Japan 79, 114705 (2010)

[44] M. Matsuda, M. Azuma, M. Tokunaga, Y. Shimakawa, and N. Kumada, Disordered ground state and magnetic eld-induced long-range order in an S = 3/2 antiferromagnetic honeycomb lattice compound Bi₃Mn₄O₁₂(NO₃), Phys. Rev. Lett. 105, 187201 (2010)

[45] T. A. Sedrakyan, L. I. Glazman, and A. Kamenev, Spontaneous formation of a nonuniform chiral spin liquid in a moat-band lattice, Phys. Rev. Lett. 114, 037203 (2015)

[46] S. Jiang, L. Zou, and W. Ku, Non-Fermi-liquid scattering against an emergent Bose liquid: Manifestations in the kink and other exotic quasiparticle behavior in the normal-state cuprate superconductors, Phys. Rev. B 99, 104507 (2019)

[47] Z. Wang, C. Navarrete-Benlloch, and Z. Cai, Pattern formation and exotic order in driven-dissipative Bose– Hubbard systems, Phys. Rev. Lett. 125, 115301 (2020)

[48] J.-S. Bernier, M. J. Lawler, and Y. B. Kim, Quantum order by disorder in frustrated diamond lattice antiferromagnets, Phys. Rev. Lett. 101, 047201 (2008)

Rights & permissions

Higher Education Press 2021