The view of TK-SVM on the phase hierarchy in the classical kagome Heisenberg antiferromagnet

The kagome Heisenberg antiferromagnet (KHAFM) is one of the most characteristic many-body systems where a simple Hamiltonian hosts strikingly rich physics. A kagome lattice is built from corner-shared triangular plaquettes, as illustrated in Figure 1. By placing Heisenberg spins on each lattice site and coupling the nearest-neighboring spins, one defines a kagome Heisenberg model, $H=J\sum_{\langle ij\rangle}\mathbf{S}_{i}\mathbf{S}_{j}$. When the interaction is antiferromagnetic, $J>0$, the system is highly geometrically frustrated and fails to find a unique ground state [1]. Such frustration has played a crucial role in the search for spin liquids and other exotic states of matter [2].

In the most general formulation of the problem one takes the interplay between geometric frustration, quantum fluctuations, and thermal fluctuations into account. However, it turns out that the thermodynamics of the classical KHAFM is already quite rewarding. In Ref. [3], Chalker et al. realized that there exists a hidden quadrupolar order characterized by a rank-$2$ tensorial order parameter. The emergence of this order is driven by an order-by-disorder phenomenon [4, 5], where coplanar states are entropically selected at low temperatures owing to the presence of soft modes [3]. Shortly afterwards, Reimers and Berlinsky [6] carried out a thorough investigation of excitations of the soft modes and the correlations of degenerate magnetic states. In spite of limited computational resources the authors showed, by classical Monte Carlo simulations, convincing signatures that the quadrupolar order indeed appears to be (quasi-)long ranged. At around the same time, Huse and Rutenberg [7] and Ritchey et al. [8] proposed that the physics in the selected plane might be effectively described by three-state Potts-like degrees of freedom, and the latter paper also discussed the associated topological defects which can support a generalized Berezinskii-Kosterlitz-Thouless (BKT) transition [9, 10, 11]. This is highly non-trivial, because the BKT transition is not realized in a typical two dimensional Heisenberg magnet as the fundamental group $\pi_{1}\left(SO(3)/SO(2)\right)=0$ is trivial [12, 13]. It, however, becomes possible with the effective Potts-like degrees of freedom, where the fundamental group is $\pi_{1}\left(O(3)/D_{3h}\right)=\bar{D}_{3}$ which gives rise to gapped topological defects [13], where $D_{3h}$ ($\bar{D}_{3}$) denotes the (binary) dihedral group. Although these results firmly evidenced that there is further structure in the selected coplanar states, it was only much later that Zhitomirsky pointed out that the structure is described by another hidden order, a rank-$3$ octupolar order [14, 15]. This author also showed that such order coexists with pinch points, which are usually seen in classical spin liquids such as spin ice [16, 17]. While the quadrupolar and octupolar orders are firmly established, they do not exhaust the rich hierarchy of phases in the classical KHAFM; specifically, the possibility of dipolar order is debated in the literature. Already in Ref. [7], the authors argued that fluctuations around the extensively degenerate KHAFM ground states interact in a non-linear way and will lead to unequally weighted Potts states. As a consequence, there may exist a critical point for a long-range antiferromagnetic $\sqrt{3}\times\sqrt{3}$ order in the $T\rightarrow 0$ limit. This scenario was further developed by Henley with a self-consistent effective Hamiltonian approach [18]. A recent Monte Carlo simulation, equipped with an ingenious cluster update, by Chern and Moessner also suggests that the $\sqrt{3}\times\sqrt{3}$ magnetization retains a finite, though remarkably small, value in the thermodynamical limit as $T\rightarrow 0$ [19].

Whereas establishing the $T\rightarrow 0$ magnetic order is very difficult because of the lack of a controlled theory [18] and efficient algorithms to simulate large system sizes from first principles at extremely low temperature [19, 20], a major challenge in understanding the hidden multipolar orders in the classical KHAFM is rooted in the complexity of those high-rank tensorial order parameters. It is therefore interesting to ask whether machine learning (ML), which is designed to analyze complex patterns, can assist in the analysis of such problems. This also concerns ML techniques as practical tools for physics. Although these techniques have proven useful for classifying phases and detecting phase transitions [21, 22, 23, 24, 25, 26, 27], representing quantum wave functions [28, 29, 30, 31, 32, 33], designing algorithms [34, 35, 36, 37], and predicting properties of materials [38, 39, 40, 41] (see Refs. [42, 43, 44] for recent reviews), the number of applications to intricate issues remains limited.

In the present paper, we apply our recently developed tensorial kernel support vector machine (TK-SVM) [45, 46, 47], which is an unsupervised and interpretable approach, to the classical KHAFM. We show that our machine readily picks up the hidden quadrupolar and octupolar orders and gives their tensorial order parameters in analytical form while its only inputs are real-space spin configurations. It identifies the ground-state constraint (GSC) and different temperature scales in the KHAFM. In addition, the machine also recognizes that the magnetic correlation of classical KHAFM at low-temperature is dominated by a $\sqrt{3}\times\sqrt{3}$ structure.

The manuscript is organized as follows. In Section 2, we review the method of TK-SVM. In Section 3, the finite-temperature phase diagram of KHAFM is discussed. Section 4 is devoted to the multipolar order parameters and GSC. Section 5 discusses magnetic correlations. We conclude in Section 6.

The tensorial kernel support vector machine (TK-SVM) is a numerical method to detect general symmetry-breaking orders [45, 46] and emergent local constraints [47, 48] in the problem of phase classifications; see also Ref. [49]. It inherits from support vector machines [50, 51], a well-known and successful classifying technique in machine learning, the property that it is interpretable: The decision function, which is the optimal classifier between two sets of data with a distinct property, can be shown to learn the square of order parameters when a quadratic kernel is used [21, 45, 46]. In TK-SVM the order parameter (or local constraint) can be any local tensor of a possibly high rank [45, 46].

Furthermore, the decision function also contains an offset known as the bias. Specifically for phase classification, the bias is sensitive to the presence of phase transitions or crossovers [46, 47]. It can be analyzed prior to and independent of the determination of the order parameters and allows one to perform an unsupervised graph partitioning of the phase diagram.

The main advantages of our approach are that the user does not need to devise suitable order parameters, which are typically very hard to construct for exotic states of matter, and that one gets near certainty about all phases with learned order parameters without supervision, resulting in enormous speedups for the analysis of an (unknown) phase diagram. Below we explain the most important concepts of TK-SVM.

In order to learn the phase diagram of the classical KHAFM in an unsupervised way, we start the analysis with the graph partitioning. For this purpose, we collect spin configurations ranging from high to extremely low temperatures: we choose $85$ logarithmically equidistant temperatures between $T/J=10^{3}$ and $10^{-5}$, and store $1000$ independent configurations at each temperature. The samples are obtained by classical Monte Carlo simulations utilizing parallel tempering and heat-bath updates for a lattice consisting of $3072$ spins ($32\times 32$ kagome unit cells) [49].

Next, we apply TK-SVM between any two temperature points and repeat this for different ranks and clusters, which both need to be sufficiently large in order to accommodate the correct orders. In practice, we typically examine clusters comprising a number of lattice unit cells and scrutinize the ranks that are compatible with a crystallographic system ($n=1,2,3,4,6$). For the kagome Heisenberg model, it turns out that the optimal choice for learning the quadrupolar and octupolar order is the kagome unit cell (i.e., a three-spin triangle) with ranks $n=2$ and $3$. The results will be compared with those using a single-spin cluster. In Section 5, we will also discuss the rank $n=1$ results of magnetic correlations.

Here are hence four different TK-SVM classification setups (the two types of clusters (a single unit cell and a single spin) combined with the two ranks ($n=2$ and $n=3$). Each leads to a graph of $85$ vertices and $3570$ edges, visualized by Figure 2. The edge weights are defined by a Lorentzian function

where $\rho$ is the bias parameter of the binary classification, and $\rho_{c}$ determines a characteristic value for “$\gg 1$” in the reduced $\rho$ criterion Eq. (6). The choice of $\rho_{c}$ is nevertheless not crucial because vertices in the same phase typically have stronger connections than those from different phases. The robustness of the graph partitioning follows from the fact that $\rho_{c}$ can be varied over several orders of magnitude without significant changes to the topology of the phase diagram [47].

Partitioning these graphs leads to four different Fiedler vectors, depicted in Figure 3. Clearly, the temperature axis is divided into three regions: A rank-$2$ quantity constructed from the triangle cluster (see also panel (c) in Figure 2) discriminates around a temperature $T/J\approx 1$ a high-temperature region (region III) from the rest. Both the single-spin and the triangle cluster (cf. panels (b) and (d) in Figure 2) define a rank-$3$ quantity which distinguishes the very low temperatures (region I, $T/J\leq 0.004$) from the intermediate temperature region II. The result of the single-spin cluster at rank-$2$ (cf. panel (a) in Figure 2) is in line with the results the rank-$3$ ones. The discrimination of these regions reproduces in fact the finite-temperature phase diagram of the kagome Heisenberg model [15, 3] – even before we visit the nature of those phases.

Having learned the topology of the KHAFM phase diagram, we now interpret the nature of each phase by their order parameters and possible local constraints. This will be done by extracting the analytical order parameters from the $C_{\mu\nu}$ matrices. In order to reduce the statistical errors on $C_{\mu\nu}$, we pool the data according to the phase diagram of Figure 3. In addition, we introduce $25,\!000$ fictitious configurations generated at $T=+\infty$ which represent completely disordered states. This sets up a reduced multi-classification problem among four classes: regime I ($28,\!000$ samples), regime II ($21,\!000$ samples), regime III ($21,\!000$ samples), and regime $T_{\infty}$. Consequently, we obtain six $C_{\mu\nu}(A\,|\,B)$ for each rank and cluster, with $A,B\in\{{\rm I,II,III,T_{\infty}}\}$ [49].

Not surprisingly, the high-temperature regime III is not distinct from the $T_{\infty}$ regime; the associated coefficient matrices are noise-like. Hence, only the two low-temperature regimes need further interpretation.

We now examine the magnetic correlations in the low-temperature regime, $T=10^{-5}J$, of the classical KHAFM, learned by rank-$1$ TK-SVM.

We first consider a cluster containing $3\times 3$ kagome unit cells, i.e. $27$ spins (see Figure 8), for a system defined on a lattice of linear size $L=36$. The resulting $C_{\mu\nu}$ matrix is shown in Figure 9. It has dimension $81\times 81$ and can be divided into small $3\times 3$ blocks. The structure of those blocks evinces in which way the correlation between two spins is defined. Because the small $3\times 3$ blocks have only entries on the diagonal, the correlation is simply captured by the usual inner product $S_{a}S_{b}\delta_{ab}$. (In general, the contraction can be more complicated even for magnetic orders). The change of color between the blocks indicates an antiferromagnetic arrangement of the spins: The blue blocks correspond to positive correlations, and their locations precisely reflect a $\sqrt{3}\times\sqrt{3}$ structure.

Next, in Figure 10, we show the result using a cluster of $12\times 12$ kagome unit cells ($432$ spins). Owing to the large dimensionality of the $C_{\mu\nu}$ matrix, $\dim C_{\mu\nu}=1296^{2}$, only the block structure is plotted, where each of its pixels corresponds to a $3\times 3$ block in Figure 9. In the presence of long-range order for an order parameter that can be defined on a small cluster, $C_{\mu\nu}$ learnt for a large cluster should display the same structure as for that small cluster. However, the pattern of Figure 10 does not repeat that of Figure 9: It fades rather fast, indicating that the linear correlation between two spins is not robustly established at longer distances. (Note that $C_{\mu\nu}$ does not directly measure the strength of spin-spin correlations. Instead, it probes the form of the correlations and order parameters. See Eqs. 14 and 4.2 for a concrete example.)

Therefore, consistent with observations in the literature [7, 6, 15, 19, 20], our machine detects that the antiferromagnetic $\sqrt{3}\times\sqrt{3}$ structure is the dominant type of correlation in the dipolar channel of the classical KHAFM at low temperature. However, to establish the $T\rightarrow 0$ critical point, a systematic analysis of temperature and finite-size effect is needed [19], which is beyond the scope of the present paper and current algorithms.

In summary, we have shown how TK-SVM can infer the phase diagram of the classical Heisenberg model defined on the kagome lattice in an unsupervised way. It has successfully learned that the spins are constrained to coplanar states owing to an emergent GSC and identified the hidden octupolar and quadrupolar tensor order parameters, which define a $D_{3h}$ biaxial-nematic state. Moreover, the machine also recognized the dominant $\sqrt{3}\times\sqrt{3}$ correlations in the low temperature regime.

As a data-driven approach, it does not require any prior knowledge or particular insight in the system. Instead, by revealing the phase diagram and analytical order parameters in an unsupervised setting, the underlying physics becomes immediately evident. We expect TK-SVM to become an indispensable tool in analyzing many-body spin systems, as it provides an enormous speedup when the order parameters and phase diagrams are complicated.

The TK-SVM library has been made openly available with documentation and examples [65].