I Introduction
Kitaev materials Jackeli and Khaliullin (2009); Chaloupka et al. (2010); Takagi et al. (2019) are Mott insulators featuring very strong spin-orbit coupling, a necessary ingredient to realize quantum Kitaev spin liquids (KSLs) Kitaev (2006). Experimental signatures of the half-quantized thermal Hall effect, a key characteristic of spin- KSLs, in - Kasahara et al. (2018); Yokoi et al. (2020), and the absence of noticeable magnetic orders in the irridates Kitagawa et al. (2018) and Takahashi et al. (2019) explain why these materials are considered among the most prominent candidates for hosting spin liquids. Theoretical studies have put forward an even greater variety of spin liquids and other exotic states Song et al. (2016); Kimchi and You (2011); Singh et al. (2012); Price and Perkins (2012); Li et al. (2015); Sears et al. (2015); Janssen et al. (2016, 2017); Jiang et al. (2019); Chern et al. (2020); Wang et al. (2019); Lee et al. (2020); Gohlke et al. (2018, 2020); Gordon et al. (2019); Osorio Iregui et al. (2014); Gohlke et al. (2017); Motome and Nasu (2020); Zhu et al. (2018); Hickey and Trebst (2019); Hickey et al. (2020); Zhu et al. (2020); Dong and Sheng (2019); Berke et al. (2020); Khait et al. (2020) and generalized the family of Kitaev materials to high-spin systems Stavropoulos et al. (2019); Xu et al. (2020) and three dimensional systems Modic et al. (2014); Takayama et al. (2015). Nevertheless, this enormous progress pales in comparison with the open questions. The role of non-Kitaev interactions, which generically exist in real materials, is yet to be understood. The microscopic model of prime candidate compounds including - remains unclear Kim et al. (2015); Kim and Kee (2016); Winter et al. (2016, 2017); Ran et al. (2017); Hou et al. (2017); Eichstaedt et al. (2019); Laurell and Okamoto (2020); Sears et al. (2020); Banerjee et al. (2016, 2017); Lampen-Kelley et al. (2018); Balz et al. (2019); Gass et al. (2020). Moreover, conceptual understanding beyond the exactly solvable Kitaev limit largely relies on mean-field and spin-wave methods Rau et al. (2014, 2016); Chaloupka and Khaliullin (2015); Rusnačko et al. (2019); Janssen and Vojta (2019); Okamoto (2013), as different numerical calculations of the same model Hamiltonian predict phase diagrams that are qualitatively in conflict with each other Jiang et al. (2019); Wang et al. (2019); Gordon et al. (2019); Lee et al. (2020); Chern et al. (2020); Gohlke et al. (2018, 2020).
A data driven approach such as Machine learning may open an alternate route to research in Kitaev materials. Over the past years it has begun realizing its potential in physics Carleo et al. (2019). Successful applications include representing quantum wave functions Carleo and Troyer (2017), learning order parameters Ponte and Melko (2017); Wang (2016)
, classifying phases
Carrasquilla and Melko (2017); van Nieuwenburg et al. (2017), designing algorithms Liao et al. (2019); Liu et al. (2017), analyzing experiments Zhang et al. (2019) and optimizing material searches Schmidt et al. (2019). Most of these advances are focused on algorithmic developments and resolving known problems. Instead, whether a hard, longstanding, and otherwise insoluble problem in physics can be solved this way is still an open question.In this article, we employ our recently developed tensorial kernel support vector machine (TK-SVM) Greitemann et al. (2019a); Liu et al. (2019); Greitemann et al. (2019b), an unsupervised and interpretable machine-learning scheme, to learn the phase diagram of the classical Kitaev- model in a magnetic field, by which we provide the first instance of machine identifying new phases of matter. We summarize our main findings below.
First, KSLs can survive non-Kitaev interactions in the large- limit. The classical phase diagram shows remarkable similarities to its quantum counterpart in the subregion intensively investigated for spin- systems, including a field-induced suppression of magnetic orders.
Second, the explicit ground-state constraints for classical spin liquids (SLs) are found, and their local transformations are formulated.
Third, cooperation and competition between classical Kitaev and spin liquids lead to two orders and two orders; the latter features a spin-lattice entangled modulation. One of those modulated states occurs in the parameter region corresponding to -, offering a viable alternative to the usual understanding that the ground state of this material at zero fields is a zigzag magnet.
This article is built as follows. We define in Section II the -- Hamiltonian and explain the essential ingredients of TK-SVM. Section III is devoted to an overview of the machine-learned phase diagram. Section IV discusses the ground state constraints (GSCs) of classical Kitaev and spin liquids and their local symmetries. The exotic and orders are elaborated in Section V. We conclude in Section VI.
Ii Model and method
We subject the honeycomb Kitaev- model, one major candidate model of Kitaev materials, in a uniform field to the analysis of TK-SVM. The spins will be treated as classical vectors to achieve a large system size which is important to capture competing orders induced by the interaction.
Hamiltonian. The -- Hamiltonian is defined as
(1) |
where and denote the strength of Kitaev and off-diagonal interactions, respectively; labels the three different nearest-neighbor (NN) bonds ; are mutually orthogonal; . We parameterize the interactions as , , with . The region corresponds to parameters of / transition metals with ferromagnetic (FM) Takagi et al. (2019), while relates to -electron based systems with anti-ferromagnetic (AFM) Jang et al. (2019).
The Hamiltonian Eq. (1) features a global symmetry which acts simultaneously on the real and spin space, where rotates the six spins on a hexagon (anti-)clockwise, and (anti-)cyclically permutates . In the absence of magnetic fields, the Hamiltonian is also symmetric under a sublattice transformation by sending , , and meanwhile for either of the honeycomb sublattices. This sublattice symmetry indicates equivalence between the - model of FM and AFM Kitaev interaction, which is respected by the phase diagram Figure 1 (a) and the associated order parameters.
Machine learning. The TK-SVM is defined by the decision function
(2) |
Here, denotes a spin configuration of spins, which is the only required input. No prior knowledge of the phase diagram is required.
denotes a feature vector mapping to an auxiliary feature space. When orders are detected, they are encoded in the coefficient matrix . The first term in captures both the form and the magnitude of orders in the system, regardless of whether they are unconventional magnets, hidden nematics Greitemann et al. (2019a); Liu et al. (2019) or classical spin liquids Greitemann et al. (2019b). The extraction of analytical order parameters is straightforward in virtue of strong interpretability of SVM (see Appendix A for details).
The second term, , in the decision function reflects an order-disorder hierarchy between two sample sets, allowing one to infer if two states belong to the same phase Liu et al. (2019); Greitemann et al. (2019b). This property of the parameter leads to a graph analysis. By treating points in the physical parameter space as vertices and assigning an edge to any two vertices, one can create a graph with the edge weights determined by . Computing the phase diagram is then realized by an unsupervised graph partitioning (see Appendix B).
The concrete application of TK-SVM consists of several steps. First, we collect samples from the parameter space of interest. For the classical -- model, large-scale parallel-tempering Monte Carlo simulations Hukushima and Nemoto (1996); Landau and Binder (2005) are utilized to generate those configurations, with system sizes up to spins. As major parts of the phase diagram are unknown, we distribute the phase points (almost) uniformly in the - space. In total, distinct -points at low temperature are collected; each has samples. Then, we perform a SVM multi-classification on the sampled data. From the obtained ’s, we build a graph of vertices and
edges and partition it by Fiedler’s theory of spectral clustering
Fiedler (1973, 1975). The outcome is the so-called Fiedler vector reflecting clustering of the graph, which plays the role of the phase diagram [see Figure 1 (c)]. In the next step, based on the learned phase diagram, we collect more samples (typically a few thousands) for each phase and perform a separate multiclassification. The goal here is to learn the matrices of high quality in order to extract analytical quantities. The dimension of this reduced classification problem depends on the number of phases (subgraphs). Finally, we measure the learned quantities to validate that they are indeed the correct order parameters.Iii Global view of the phase diagram
The -- model shows a rich phase diagram, including a variety of classical spin liquids and exotic magnetic orders. In the vicinity of the ferromagnetic Kitaev limit with (i.e. ), which has been intensively studied for spin- systems, the classical phase diagram shares a number of important features with the quantum counterpart. We will focus here on the topology of the machine-learned phase diagram. The specific properties of each phase are analyzed in subsequent sections.
We first discuss the phase diagram at , depicted in Figure 1 (a). In the absence of external fields the Hamiltonian Eq. (1) has four limits at and , corresponding to two classical KSLs and two SLs. These particular limits divide the - phase diagram into four regions. When both the Kitaev and interactions are ferromagnetic or antiferromagnetic, the system is unfrustrated, while when they are of different sign, the system stays highly frustrated.
In the two unfrustrated regions, when and are both finite, the system immediately changes from a spin liquid to a magnetically ordered phase. We find that the corresponding order parameters have six sublattices and can be described by the symmetric group . We therefore refer to them as the FM phase and the AFM phase, respectively. As we shall see in Section V, these two orders can be understood as the result of cooperation between the Kitaev and spin liquids.
The physics is profoundly different in the frustrated regions. The two KSLs can extend to a finite value of for . There has been mounting evidence suggesting that quantum KSLs survive in some non-Kitaev interactions Kasahara et al. (2018); Yokoi et al. (2020); Gohlke et al. (2018, 2020); Lee et al. (2020); Wang et al. (2019); Gordon et al. (2019); Osorio Iregui et al. (2014); Gohlke et al. (2017). It is quite remarkable that such an extension already manifests itself in the classical large-
limit. Using the corresponding ground state constraint (GSC), we estimate
(Appendix C). This value is comparable but (slightly) greater than the one proposed for the spin- - model Gohlke et al. (2018, 2020); Wang et al. (2019); Lee et al. (2020), which is consistent with the fact that the ground states of a KSL are more extensively degenerate for large Baskaran et al. (2008).By contrast, the classical SLs is found to only exist in the limit , as they have much smaller extensive ground-state degeneracy (exGSD) (Cf. Section IV).
The majority of the frustrated regions are occupied by two exotic orders. The one on the ferromagnetic sector may relate to the magnetic order in the spin-liquid candidate -. Based on the static structure factor from neutron scattering Banerjee et al. (2016, 2017), it has been commonly considered a zigzag magnet. However, by learning the explicit order parameter (Section V), our machine reveals that this order, as well as the counterpart on the antiferromagnetic sector, have a more intriguing structure. They possess threefolds of the magnetic structure discussed for the FM and AFM phase, leading to eighteen sublattices. The three sectors mutually cancel via a novel modulation, and we henceforth refer to them as modulated phase. We also find out that competition between a Kitaev and a spin liquid induces these orders.
Between each modulated phase and the corresponding KSL, there is a window of another correlated disordered region. It may be understood as a crossover between the two phases, as we are considering
spins at two dimensions and finite temperature. We refer to such regions as correlated paramagnet (CP) and expect them to be squeezed out in quantum cases where sharp phase transitions can take place.
When the magnetic field is turned on, the fate of each phase strongly depends on the sign of its interactions, as is shown in Figure 1 (c). Those featuring only ferromagnetic interactions, including the FM phase, the FM Kitaev and spin liquids, immediately polarize. However, the phases with one or both antiferromagnetic interactions are robust against finite . Specifically, the AFM KSL persists up to . And before trivial polarization occurs at , there exists an intermediate region, dubbed , where the magnetic field induces two novel correlations with a global symmetry (Section IV). Interestingly, this region appears to coincide with a gapless spin liquid phase recently proposed for quantum spin- and spin- systems Motome and Nasu (2020); Zhu et al. (2018); Hickey and Trebst (2019); Hickey et al. (2020); Zhu et al. (2020).
The frustrated regions are again richest in physics. The FM KSL extends to a small, but finite, field thanks to an antiferromagnetic , while the AFM KSL extends over a much greater area. At intermediate , there are disordered regions separating a phase from a spin liquid or a trivially polarized state. We refer to them as partially-polarized correlated paramagnets (s) to distinguish them from the parent spin liquid. In particular, the and regimes erode the modulated phase, reminiscent of the experimental observation of the field-induced suppression of magnetic order in - Kasahara et al. (2018); Yokoi et al. (2020); Lampen-Kelley et al. (2018); Balz et al. (2019); Gass et al. (2020). It is worth mentioning that a field-induced unconventional paramagnet has also recently been proposed for quantum spin- in the region Lee et al. (2020); Gohlke et al. (2020). These common features indicate that some critical properties of - and the quantum - model may already be understood at the classical level.
Before delving deeper into each phase, we comment on the distinctions between the graph partitioning in TK-SVM and traditional approaches of computing phase diagrams. In learning the finite- phase diagram Figure 1 (c), we did not use particular order parameters, nor any form of supervision. Instead, distinct decision functions are implicitly utilized; each serves as a classifier between two points. Moreover, all phases are identified at once, rather than individually scanning each phase boundaries. These make TK-SVM an especially efficient framework to explore phase diagrams with complex topology and unknown order parameters.
Iv Emergent Local constraints
A common feature of classical spin liquids is the existence of a non-trivial GSC which is an emergent local quantity that defines the ground-state manifold and controls low-lying excitations. A system can be considered as a classical spin liquid if it breaks no orientation symmetry, and meanwhile its GSC has a local symmetry. We now discuss the GSCs learned by TK-SVM for the classical Kitaev and spin liquids.
Our machine learns a distinct constraint for each spin liquid in the phase diagram Figure 1. These constraints can be expressed in terms of quadratic correlations on a hexagon. We classify six types of such correlations at and another two field-induced correlations for the AFM KSL, as tabulated in Table 1.
Symmetry | |||
Correlations | Global | Local | |
Cov. | |||
For KSLs, we reproduce the GSCs previously obtained by a Jordan-Wigner construction Baskaran et al. (2008),
(3) |
where corresponds to the FM and AFM interaction, respectively (the same convention used below); denotes the thermal average on a hexagon. As discussed in Refs. Baskaran et al., 2008; Sela et al., 2014, these constraints impose degenerate dimer coverings on a honeycomb lattice, which are precisely the ground states of classical KSLs.
In case of classical SLs, our machine identifies two new constraints,
(4) |
which directly differentiate between the FM and AFM case, and satisfying them will naturally lead to the ground-state flux pattern for every three hexagon plaquettes Rousochatzakis and Perkins (2017); Saha et al. (2019), where .
Aside from manifesting ground state configurations, knowing the explicit GSC will make clear the symmetry properties and the extensive degeneracy of a spin liquid. The above Kitaev and constraints preserve the global symmetry of the Hamiltonian Eq. (1), and more importantly, possess a different local symmetry, representing distinct classical spin liquids.
The Kitaev constraints Eq. (3) are invariant by locally flipping the component of a NN bond ,
(5) |
For a given dimer covering configuration, this will give rise to
redundant degrees of freedom on each hexagon. Together with
dimer coverings on a honeycomb lattice Wu (2006); Baxter (1970); Kasteleyn (1963), it enumerates extensively degenerate ground states Baskaran et al. (2008), resulting in a residual entropy at zero temperature.The local invariance of the SL constraints Eq. (IV) takes a different form, defined on a hexagon,
(6) |
Here, are the components normal to ; “” denotes the second nearest-neighbor bonds with corresponding to the two connecting NN bonds; “” denotes the third nearest-neighbor bonds, and equals the on a parallel NN bond; as depicted in Figure 2. This symmetry is considerably involved but also evident once the explicit GSC is identified.
The corresponding exGSD can again be counted by the local redundancy on a hexagon, giving with a residual entropy . This degeneracy is exponentially less than that of KSLs. As a result, SLs are more prone to fluctuations (see Figure 1 and 4).
Furthermore, in addition to the constraints for ground states, in the region in the phase diagram Figure 1 (c), we identify two field-induced quadratic correlations. The two correlations, denoted as and in Table 1, are invariant under global rotations about the direction of the fields. From general symmetry principle, a continuous global symmetry will naturally support gapless modes. Hence, aside from being novel local observables in the classical AFM Kitaev model, they may also shine light on the nature of the corresponding gapless quantum spin liquid Motome and Nasu (2020); Zhu et al. (2018); Hickey and Trebst (2019); Hickey et al. (2020); Zhu et al. (2020).
Note that the GSCs and other quadratic correlations learned by TK-SVM are not limited to classical spins. Their formalism holds for general spin- and can be directly measured in the quantum - model. Comparing to other quantities (such as plaquette fluxes, Wilson/Polyakov loops, and spin structure factors), which may exhibit similar behaviors in different spin liquids, GSCs can uniquely define a ground-state manifold and hence may be more distinctive. Moreover, their violation provides a natural way to measure the breakdown of a spin liquid, which is what we use to estimate the extension of KSLs (Appendix C).
V Cooperative and competitive constraints induced ordering
, | , | , | , | , | |
---|---|---|---|---|---|
, | , | , | , | , | |
, | , | , | , | , | |
, | , | , | , | , |
A standard protocol to devise spin liquids is to introduce competing orders. In contrast to this familiar scenario, the emergence of the and the modulated orders are caused here by cooperation and competition between two spin liquids.
Unfrustrated orders. We first discuss the two phases in the unfrustrated regions . The discussion will also facilitate the understanding of the more exotic phases.
From the learned matrices, we identify that both orders have six magnetic sublattices with an order parameter
(7) |
where are ordering matrices, given in Table 2, and the FM and AFM order differ by a global sign in , , and . The six ordering matrices form the symmetric group . Its cyclic subgroup, , are three-fold rotations about the direction in spin space, while and correspond to reflection planes , respectively.
We find that these two orders feature the same static spin structure factor (SSF). Both develop magnetic Bragg peaks at the K points of honeycomb Brillouin zone (Figure 3), as the well-known order. This example highlights significance of knowing the explicit order parameter, as different phases may have the same SSF.
Furthermore, we identify the other two novel GSCs,
(8) |
which equally comprise and in Eqs. (3)-(IV), with additional and terms owing to the normalization .
As we measure in Figure 4 (a), (c), in the spin-liquid limits , Kitaev and GSCs satisfy, as or with other correlations vanishing. However, when both and interactions are present and of the same sign, the two characteristic correlations and will lock together. This eliminates the local symmetries of Kitaev and spin liquids and gives way to the orders.
It is worth noting that the two phases also represent rare instances where magnetic states possess non-trivial GSCs, which normally exist in cases of classical spin liquids and multipolar orders Greitemann et al. (2019b).
Mod phases. The modulated orders have a more intricate structure. Their order parameters take the form
(9) |
where are eighteen ordering matrices given in Table 2, and distinguish three different sectors as illustrated in Figure 1 (b). The and order differ by a global sign for all even ’s.
Spins in the two phases are organized by a delicate spin-lattice entangled modulation,
(10) |
In concrete terms, remain three-fold rotations along the direction, but there is an additional factor entering some, but not all, spin components. The location of this factor, as shown in Table 2, alternates among the three sectors, to achieve the cancellation in Eq. (10). Furthermore, mirror reflections, with even ’s are decorated by a factor , in such a way that a cancellation with the mirror of the same type occurs, as . The value of , which TK-SVM also identifies, strongly depends on the relative strength , while the reflection planes remain locked on .
This modulation is very different from those in multiple- orders and spin-density-wave (SDW) orders where phase factors universally act on all spin components. Moreover, since this modulation does not preserve spin length, the magnetization will not saturate to unity, but to a reduced value , reflecting an intrinsic frustration.
The SSF of the two phases, shown in Figure 3 (b), exhibits a very similar pattern to the neutron scattering result of - Banerjee et al. (2016, 2017). The magnetic Bragg peaks appear at the points of a reduced honeycomb Brillouin zone. Given the experimental and numerical evidence that the microscopic interactions in - are dominated by , Kim et al. (2015); Kim and Kee (2016); Winter et al. (2016, 2017); Ran et al. (2017); Hou et al. (2017); Eichstaedt et al. (2019); Laurell and Okamoto (2020); Sears et al. (2020), which falls into the modulated phase, our result provides a viable competitor to the usually thought zigzag order that should be checked in experiment and quantum calculations.
To better understand the nature of the modulated orders, we show their magnetization along with the and correlations in Figure 4 (b) and (d). To exclude the
-dependence in the order parameter, we defined an alternative magnetization by including only odd
’s in Eq. (9), . Clearly, in the frustrated regions, the characteristic Kitaev and correlations develop toward opposite directions. Near the Kitaev limits, , dominates; the system stays disordered, either in an extended KSL phase or a CP region. When is sufficiently strong to compete with , at , an order emerges from the two conflicting correlations, and expands till the large limits owing to the small exGSD of a SL.From the machine learning point of view, the modulated orders provides as far as we know the first instance of a machine-learning algorithm identifying totally unknown phases. And in light of the explicit order parameters, the essence of these complicated phases immediately becomes transparent.
Vi Conclusions
To summarize, we performed a large-scale analysis to the honeycomb - model, which governs the microscopic physics of a wide array of Kitaev materials, utilizing the unsupervised and interpretable machine-learning method TK-SVM.
We found that the classical phase diagram of the - model in a magnetic field is exceptionally rich (Figure 1), with several unconventional symmetry-broken phases and a multitude of disordered states at . The phase diagram clearly shows the finite extent of KSLs, an intermediate disordered phase at AFM Kitaev limit, and a field-induced suppression of magnetic orders, which were previously only reported for quantum systems. These common features strongly suggest that certain aspects of Kitaev materials can be understood from a semi-quantitive classical picture and call for systematic investigations of general spin- systems.
On the top of the phase diagram, two novel phases, the modulated magnets, with an unknown type of modulations were detected. The implications are twofold. (i) These states represent the first success that machine learning identifies new phases. Their structure is so complicated and close to impossible to find for humans, but is picked up without difficulty by TK-SVM. (ii) The phase provides a potential explanation to the ground state of - in zero magnetic field, as it shares a very similar spin-structure factor with this material and occurs in the same parameter region and Although the precise modeling of - remains an open issue and requires considering other symmetry-allowed couplings, such as the Heisenberg interactions and possibly longer-range interactions, there is a consensus that the Kitaev and interactions play a significant role.
We discovered the GSCs of classical SLs and reproduced the known GSCs of KSLs. Not only did this enhance our understanding of SLs, but it is also critical to comprehend the ordering in the - model. The two magnets emerge as the characteristic Kitaev and correlation cooperatively eliminate the extensive degeneracy of a KSL and SL. By contrast, the two modulated magnets can be understood as consequences of the competition between these two spin liquids. Theses mechanisms may also enrich protocols of searching for exotic phases.
Open source. The TK-SVM library has been made openly available with documentation and examples Greitemann et al. .
Acknowledgements.
We wish to thank Hong-Hao Tu and Simon Trebst for fruitful discussions. KL, NS, NR, JG, and LP acknowledge support from FP7/ERC Consolidator Grant QSIMCORR, No. 771891, and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2111 – 390814868. Our simulations make use of the -SVM formulation Schölkopf et al. (2000), the LIBSVM library Chang and Lin (2001, 2011), and the ALPSCore library Gaenko et al. (2017).Appendix A Setting up of TK-SVM
The TK-SVM method has been introduced in our previous work Greitemann et al. (2019a); Liu et al. (2019); Greitemann et al. (2019b). Here we review its essential ingredients for completeness.
For a sample , the feature vector maps to degree- monomials
(11) |
where represents a lattice average up to a cluster of spins; label spins in the cluster; are collective indices.
TK-SVM constructs from a tensorial feature space (-space) to host potential orders Greitemann et al. (2019a); Liu et al. (2019). The capacity of the -space depends on the degree () of monomials and the size () of the cluster. As the minimal and are unknown parameters, in practice, we choose large clusters according to the Bravais lattice and , where detects magnetic orders and probes multipolar orders and emergent local constraints. In learning the phase diagram Figure 1, we constructed -spaces using clusters up to spins ( honeycomb unit-cells) at rank- and clusters up to spins at rank-, much beyond the needed capacity. We also confirmed the results are consistent when varying the size and shape of clusters and found ranks to be irrelevant.
The coefficient matrix measures correlations of , defined as
(12) |
where the Lagrange multiplier denotes the weight of the -th sample and is solved in the underlying SVM optimization problem Greitemann et al. (2019a); Liu et al. (2019)
. Its non-vanishing entries identify the relevant basis tensors of the
-space, and their interpretation leads to order parameters.Appendix B Details of Graph Partitioning
Not all matrices need to be interpreted. In the graph partitioning, where the goal is to learn the topology of the phase diagram, it suffices to analyze the bias parameter . When are two phase points where spin configurations are generated, the bias parameter in the corresponding binary classification problem behaves as
(13) |
Thus, as demonstrated in our previous work, can detect phase transitions and crossovers Liu et al. (2019); Greitemann et al. (2019b). (Though the sign of also has physical meaning and can reveal which phase is in the (dis-)ordered side, the absolute value is sufficient for the graph partitioning; see Ref. Greitemann et al., 2019b for details.)
The graph partitioning in TK-SVM is a systematic application of the criteria Eq. (13). The graph is built from vertices, each corresponding to a point , and connecting edges; as exemplified in Figure 6. The weight of an edge is defined by in the SVM classification between the two endpoints, with a Lorentzian weighting function
(14) |
Here sets a characteristic scale for “” in Eq. (13), as a larger tends to suppress weight of the edges. The choice of is not critical since points in the same phase are always more connected than those from different phases. In computing the phase diagram Figure 1, is applied, but we also verified that the results are robust when is changed over an interval ranging from a small to a large , where all edge weights are almost eliminated.
A graph with edges is considered a small problem in graph theory and may be partitioned with different methods. We have applied Fiedler’s theory of spectral clustering Fiedler (1973, 1975). The result is a so-called Fiedler vector of the dimensionality , corresponding to the vertices. Strongly connected vertices, namely those in the same phase, share equal or very close Fiedler-entry values, while those in different phases have substantially different Fiedler entries. In this sense, the Fiedler vector can act as a phase diagram.
Appendix C Extension of Classical KSLs
As a GSC, , characterizes a classical spin liquid, we can accordingly define a susceptibility to measure how sharp it is defined,
(15) |
where is a thermal average, and denotes the volume of the system. Such a susceptibility was first introduced in Ref. Greitemann et al. (2019b), and we showed with various examples its high sensitivity to the breakdown of an associated classical spin liquid.
To estimate the extension of classical KSLs, we define for the susceptibility and measure it in Figure 8 against a competing interaction. At a fixed , develops two peaks/bumps, reflecting a dramatic violation of the GSC. The sharper peak at a smaller is responsible for the crossover between a KSL and a non-Kitaev correlated paramagnet. The broad bump at a larger signals the second crossover to a modulated phase. (The optimal measure to this second crossover is the order parameter instead of . However, the location of the bump qualitatively agrees with the results based on the magnetization, see Figure 4 for example.)
Furthermore, to examine field effects on the AFM KSL, we measure the field-induced correlations, and in Table 1, over a large scope of . As shown in Figure 9, at small field, both and stay close to their value at . However, under intermediate and large fields, they experience sudden jumps and form plateaus, further dividing the finite- phase diagram into a (), a partially polarized (), and a trivially polarized region (). In the main text (Section III and IV), we discussed that the intermediate region coincides with a gapless spin liquid proposed for quantum spin- and spin- AFM Kitaev models Motome and Nasu (2020); Zhu et al. (2018); Hickey and Trebst (2019); Hickey et al. (2020); Zhu et al. (2020). Beyond this, a similar segmentation in the finite- phase diagram is also observed in the quantum case Zhu et al. (2018, 2020), marking another common feature between the quantum and classical Kitaev model.
The behaviors of , , and are used to determine the boundary [the dash lines in Figure 1 (c)] between KSLs and other correlated paramagnets, supplementing the graph partitioning. This is needed because, in the graph partitioning shown in Figure 1, we only employed a rank- TK-SVM which is designed for detecting the presence and absence of magnetic order. To classify different spin liquids, we use rank- TK-SVM to identify their GSCs. In principle, we can also perform a separate graph partitioning with rank- TK-SVM. Nevertheless, given the rank- results, there are only particular regions to examine, and it is more convenient to directly measure the learned GSCs and associated quantities.
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