Abstract
Actively sought since the turn of the century, twodimensional quantum spin liquids (QSLs) are exotic phases of matter where magnetic moments remain disordered even at zero temperature. Despite ongoing searches, QSLs remain elusive, due to a lack of concrete knowledge of the microscopic mechanisms that inhibit magnetic order in materials. Here we study a model for a broad class of frustrated magnetic rareearth pyrochlore materials called quantum spin ices. When subject to an external magnetic field along the [111] crystallographic direction, the resulting interactions contain a mix of geometric frustration and quantum fluctuations in decoupled twodimensional kagome planes. Using quantum Monte Carlo simulations, we identify a set of interactions sufficient to promote a groundstate with no magnetic longrange order, and a gap to excitations, consistent with a Z_{2} spin liquid phase. This suggests an experimental procedure to search for twodimensional QSLs within a class of pyrochlore quantum spin ice materials.
Introduction
In a twodimensional (2D) quantum spin liquid (QSL) state, strong quantum fluctuations prevent the ordering of magnetic spins, even at zero temperature. The resulting disordered phase can potentially be a remarkable state of matter, supporting a range of exotic quantum phenomena. Some, such as emergent gauge structures and fractional charges, are implicated in a wide range of future technologies like hightemperature superconductivity^{1,2} and topological quantum computing^{3}. It is therefore remarkable that, despite extensive examination of the basic theoretical ingredients required to promote a 2D QSL in microscopic models^{4,5}, the state remains elusive, with only a few experimental candidates existing today^{6,7}.
Recently, the search for QSL states has turned to consider quantum fluctuations in the socalled spin ice compounds^{8}. In these systems, magnetic ions reside on a pyrochlore lattice—a nonBravais lattice consisting of cornersharing tetrahedra. Classical magnetic moments (described by Ising spins) on the pyrochlore lattice can be geometrically frustrated at low temperatures, leading to spin configurations that obey the socalled ‘ice rules’, a mapping to the protondisorder problem in water ice^{9}. The ice rules result in a large set of degenerate ground states—a classical spin liquid with a finite thermodynamic entropy per spin^{10,11}. Two canonical materials, Ho_{2}TiO_{7} and Dy_{2}Ti_{2}O_{7}, have been demonstrated to manifest spin ice behaviour, and experiments and theory enjoy a healthy dialogue due to the existence of classical microscopic models capable of describing a wide range of experimental phenomena^{10}.
Classical spin ice pyrochlores are conjectured to lead to QSLs in the presence of the inevitable quantum fluctuations at low temperatures^{4,8}. The effects of certain types of quantum fluctuations on the spin ice state have been investigated theoretically^{12} and numerically^{13,14}, where they have been demonstrated to lift the classical degeneracy and promote a threedimensional (3D) QSL phase with lowenergy gapless excitations that behave like photons^{12,13}. In several related pyrochlore compounds, particularly Tb_{2}Ti_{2}O_{7}, Yb_{2}Ti_{2}O_{7}, Pr_{2}Zr_{2}O_{7} and Pr_{2}Sn_{2}O_{7}, quantum effects have been observed, which make them natural candidates to search for such 3D QSLs^{15,16,17}. In an attempt to elucidate the microscopic underpinnings of these and related materials, recent theoretical studies have produced a general lowenergy effective spin1/2 model for magnetism in rare earth pyrochlores^{18,19,20}. In an important development, Huang, Chen and Hermele^{21} have shown that, on the pyrochlore lattice, strong spinorbit coupling can lead to Kramers doublets with dipolaroctupolar character in d and felectron systems. This allows for a specialization of the general effective model to one without the debilitating ‘sign problem’—amenable to solution through quantum Monte Carlo (QMC) methods—thus admitting a systematic search for QSL phases via largescale computer simulations. Using largescale QMC simulations, we show that a twodimensional model on the kagome lattice descending from the quantum Hamiltonian discussed by Huang, Chen and Hermele^{21} exhibits an exotic disordered phase—a quantum kagome ice state—in a wide range of Hamiltonian parameters. Such a state displays exponentially decaying correlations and is consistent with a gapped QSL phase. These results suggest an alternative experimental route to search for the longsought QSL phase in two dimensions starting from quantum spin ice pyrochlore materials subject to an external field along the [111] direction.
Results
A quantum kagome ice model
While the possibility for 3D QSLs in the above compounds is intriguing, spin ice materials offer a compelling mechanism for dimensional reduction to 2D, since singleion anisotropy constrains magnetic moments to point along the local tetrahedral symmetry axes in the pyrochlore lattice. This mechanism consists of the application of an external magnetic field along the global [111] crystallographic direction that partially lifts the spin ice degeneracy by pinning one spin per tetrahedron. As illustrated in Fig. 1a, this [111] magnetic field effectively decouples spins between the alternating kagome and triangular layers of the original pyrochlore structure. To simplest approximation, the system becomes a twodimensional system of stacked kagome planes^{22,23,24,25}, where spins on the intervening triangular planes align in the direction of the field (becoming energetically removed from the problem), while those in the kagome plane (Fig. 1b) remain partially disordered. These kagome spins retain a fraction of the zerofield spin ice entropy, though still preserving the spin ice rules (twoin, twoout) of the parent pyrochlore system. This leads to classically disordered state, termed kagome ice^{22,24,25,26}, evidenced to date in several experimental studies on spin ice materials^{27,28,29}.
The above observations lead to a natural microscopic mechanism to search for 2D QSL behaviour^{30}. First, one begins with classical nearestneighbour spin ice in an applied [111] field, so as to promote the aforementioned kagome ice state. This model maps to a projected pseudospin Ising model with a symmetrybreaking Zeeman field h, arising from a combination of the physical [111] field and the original pyrochlore spin exchange interaction (Fig. 1c). For moderate h, the classical ground state retains an extensive degeneracy, before becoming a fully polarized ferromagnetic state for h/J_{z} >2. Next, to include the effect of quantum fluctuations, one may add exchanges from the recent quantum spin ice models^{18,19,20,21}. We consider only those quantum fluctuations discussed by Huang et al.^{21}, to obtain a pseudospin Hamiltonian on the kagome lattice,
Here, S_{r} are spin1/2 operators, with a global z axis ( and in Fig. 1c). This Hamiltonian cannot be solved exactly by analytical techniques; however largescale QMC simulations are possible in a parameter regime (J_{±}≥0) devoid of the prohibitive sign problem.
One can imagine a 2D QSL state arising conceptually by considering the quantum fluctuations J_{±} and J_{±±} as perturbations on the classical kagome ice limit, where only diagonal terms J_{z}>0 and are present. Previously, largescale QMC simulations have been performed on the kagome model in the limit J_{±}>0 and J_{±±}=0 (refs 31, 32) (a parameter regime where the Hamiltonian retains U(1) invariance). In that case, quantum fluctuations promote an inplane ferromagnetic (FM) phase for h=0, and a valence bondsolid (VBS—a conventional symmetry broken phase) for h>0. Thus, it happens that fluctuations of the form induced by J_{±} are not sufficient to promote a 2D QSL state.
However, there remains the theoretical possibility of a gapped Z_{2} QSL phase promoted by the J_{±±} quantum fluctuations. As detailed in the Supplementary Note 1 and Supplementary Fig. 1, the local constraints of classical kagome ice can be translated into a chargefree condition on the dual honeycomb lattice. Then, the full Hamiltonian (1) can be recast as a system of interacting bosonic spinons coupled to a compact U(1) gauge field on the dual lattice. In the limit of J_{±}=0, this theory is expected to exhibit two distinct phases. One is a confined phase, corresponding to a conventional spinordered state; the other is a deconfined Z_{2} QSL phase^{20,21,33}. From these simple arguments it is conceivable that these two phases exist in the phase diagram of equation (1). In the next section, we set J_{±}=0 and explore this possibility for all parameter regimes J_{±±}/J_{z} and h/J_{z}, using nonperturbative, unbiased QMC simulations. Before proceeding with the description of the QMC results, we emphasize that neither our simulations nor the analysis of the resulting lattice gauge theory are restricted to states within the ice manifold.
Quantum Monte Carlo results
We implement a finitetemperature Stochastic Series Expansion^{34,35,36} (SSE) QMC algorithm with directed loop updates in a 2+1 dimensional simulation cell, designed specifically to study the Hamiltonian equation (1) with J_{±}=0 (for details, see the Methods Summary). Note, this Hamiltonian explicitly breaks U(1) invariance, retaining global Z_{2} symmetries. By a canonical transformation, S^{±}→±iS^{±}; we simulate only J_{±±}<0, without loss of generality^{21}. Various measurements are possible in this type of QMC simulation. Simplest are the standard SSE estimators for energy, magnetization , and uniform spin susceptibility . The latter two allow us to map out the broad features of the phase diagram. Further, we measure the offdiagonal spin structure factor^{37}
Here, r_{i} points to the sites of the underlying triangular lattice (containing N_{s} sites) of the kagome lattice (containing V=3 × N_{s} sites). The vectors α are the position of each site within the unit cell with respect to the vector r_{i}. This quantity allows us to define, for this spin Hamiltonian, the analogue of a condensate fraction in bosonic systems^{38,39}, which detects transverse magnetic ordering. We define as the ratio of largest eigenvalue n_{M} of the onebody density matrix to the number of sites V. The eigenvalues of ρ_{i,j} coincide with for a translationally invariant system.
Figure 2a shows the QMC phase diagram for the J_{±}=0 model of equation (1), using data for the condensate fraction f_{0}. Careful finitetemperature and finitesize scaling, performed up to lattice sizes of V=L × L × 3=39 × 39 × 3 and β=J_{z}/T=96, is detailed in the Supplementary Information (Supplementary Figs 2 and 3 and Supplementary Note 2). The magnetization curve and the uniform spin susceptibility across the phase boundary at fixed h/J_{z}=0.833 are presented in Fig. 2a,b. The data clearly indicate the existence of two magnetized lobes on the phase diagram for J_{±±}/J_{z}<0.5 and h/J_{z}≠0, where the zeromomentum condensate fraction of a surrounding FM phase is destroyed by a phase transition (which appears to be first order). The lobes have magnetizations of m≈−1/6 and m≈+1/6 for h/J_{z}<0 and h/J_{z}>0, respectively. The FM phase has a finite uniform susceptibility χ_{z}, while the lobe phases retain a small but finite χ_{z} that can be understood by the nature of the quantum fluctuation as a spin pair interaction, which does not conserve the total magnetization . As discussed above, the phase in these lobes is a candidate for supporting a 2D QSL state.
To examine this hypothesis, we perform a detailed search for ordered structures in the lobes. In related models, particularly the spin1/2 XXZ model on kagome (that is, J_{±±}=0 and J_{±}>0)^{31,32}, the analogous lobes support a conventional VBS phase, which is evident in the diagonal structure factor: , where
If there is longrange order then will scale with system size for at least one value of q. We also measure the bondbond structure factor using a fourpoint correlation function.
where . Nearest neighbour sites i_{aα} and j_{aα} belong to bond α in a unit cell located at position r_{a}. Again, if there is pair longrange order then should scale with system size for at least one value of q, with which we define B_{q}=BB_{q}/V.
Figure 3 illustrates the various qdependent structure factors for spin and bond order. These structure factors display diffuse peaks at various wave vectors, notably q=0, q=K=(2π/3,0), and symmetryrelated momenta. Such peaks would indicate the presence of longrange order, should they sharpen, and survive in intensity in the infinitesize limit, where S/V would correspond to an order parameter squared. In Fig. 3d through Fig. 3f, we examine this through a standard finitesize scaling analysis, for several candidate peaks for each of the structure factors. Further scaling analysis, including larger system sizes, is presented and analysed in the light of perturbative arguments in the Supplementary Information (Supplementary Note 3 and Supplementary Figs 2 and 4). In each case, the QMC data indicates a scaling of each order parameter to zero in the limit V→∞. Note in particular, the largest value of B_{q} corresponds to q=0, which remains finite as V→∞, meaning that the bond expectation values is finite in the lobes. This is expected as represents the kinetic energy of quantum fluctuations in the system, and thus it should be finite in all phases. More importantly, the data indicate that in the limit of V→∞ this quantity is the same on all bonds of the unit cell of the kagome lattice, meaning that there is no breaking of spacegroup symmetry (see Supplementary Note 4 and Supplementary Figs 5–7).
Finally, as the above data suggest the existence of a phase that is homogeneous, disordered and quantummechanically fluctuating at extremely low temperatures, one should also examine whether the energy for excitations out of this ground state is gapped or gapless. Although a direct measurement of the gap is not possible in this type of SSE QMC method, we can indirectly probe its existence by looking at the decay of realspace correlations. In Fig. 4, we compare the decay of singleparticle correlations between the m_{z}=±1/6 magnetization lobes and the adjacent FM ordered phase. For the system size studied, it is clear that correlations in the lobe are consistent with exponential decay, and therefore indicative of a gap. In contrast, in the FM phase the correlations quickly reach a finite value, indicating symmetry breaking (additional details about the nature the FM phase are discussed in Supplementary Figs 8–10 and Supplementary Note 5). Similarly, the diagonal part of the spin correlation function is consistent with exponential decay both in the lobes and in the FM phase (not illustrated).
Thus, our QMC results have elucidated a phase diagram for our kagome pseudospin XYZ model (with J_{±}=0), which contains a predominant FM phase, surrounding lobes of an exotic disordered m_{z}=±1/6 magnetization phase. As our dual gauge theory (detailed in the Supplementary Note 1) indicates that these lobes may realize a QSL with an emergent Z_{2} gauge symmetry, it is clear that further simulation work should be carried out to address this hypothesis. To confirm the presence of a Z_{2} QSL, one requires evidence of either excitations consistent with this gauge structure (for example, magnetic spinons or nonmagnetic visons at nonzero temperature) or a smoking gun such as the topological entanglement entropy^{40,41}. Such evidence, although demonstrated in the past with SSE QMC^{42,43}, is resourceintensive to obtain, requiring high numerical precision at very low temperatures, and thus outside of the scope of the present manuscript.
However, we also note that, due to the presence of only a discrete symmetry in our kagome XYZ model, an emergent Z_{2} structure is not strictly required by the LiebSchultzMattisHastings (LSMH) theorem^{44,45,46}, which states that a system with halfoddinteger spin in the unit cell cannot have a gap and a unique ground state. In highersymmetry Hamiltonians, the requirements of the LSMH theorem are satisfied in a gapped QSL phase by the topological degeneracy, which is a consequence of the emergent discrete gauge symmetry. For our Hamiltonian with a gapped QSL arising in a model with only global discrete symmetries, an emergent gauge structure is not required. Rather, it is possible that the groundstate is a quantum paramagnet. In contrast, other types of emergent gauge structure, topological order or other exotic phenomena are theoretically possible. Fortunately, the nature of this Hamiltonian, which is among the first to show a 2D QSL phase with only nearestneighbour interactions, lends itself exceedingly well to study by signproblemfree QMC simulations. We therefore expect a large number of studies in the near future will help elucidate the precise nature of this QSL phase.
Discussion
Through extensive quantum Monte Carlo simulations, we have studied a signproblemfree model of frustrated quantum spins interacting on a twodimensional kagome lattice. This model is descendent from a more general quantum XYZ Hamiltonian discussed by Huang, Chen and Hermele^{21}, derived for the threedimensional pyrochlore lattice, when subject to a magnetic field along the [111] crystallographic direction. For a large range of Hamiltonian parameters, the QMC data uncover an exotic disordered phase that breaks no symmetries, has strong quantum mechanical fluctuations and exponentially decaying correlations—a candidate gapped QSL phase. This discovery is consistent with an analytical dual gauge theory (detailed in the Supplementary Note 1), which indicates that, in the limit of small quantum fluctuations, the phase could be a 2D QSL with an emergent Z_{2} gauge symmetry.
Our work suggests a new experimental avenue to search for the highly coveted QSL phase in two dimensions. Previous efforts have focused largely on SU(2) Hamiltonians on kagome or triangular lattice materials^{6,7}. In contrast, we propose to concentrate the search on the quantum spin ice pyrochlore materials, subject to an external field along the [111] direction. Such kagome ice phases have been identified in various materials in the past. A closer look at several quantum spin ice candidates is warranted, particularly in materials where strong quantum fluctuations are known to exist, such as Tb_{2}Ti_{2}O_{7}, Yb_{2}Ti_{2}O_{7}, Pr_{2}Zr_{2}O_{7} and Pr_{2}Sn_{2}O_{7}. While Hamiltonians describing some of these materials may include additional interactions^{18} not considered in equation (1), our study highlights the importance of the of J_{±±} term, which is present for instance in Yb_{2}Ti_{2}O_{7} (ref. 47), and promotes spinon pairing required in the formation of a Z_{2} spin liquid. As the phase found in this study has a gap, it is necessarily protected against any local perturbation as long as the effective energy scale of the perturbation is smaller than the gap, and thus it may still be realized even in the presence of additional perturbations. Other candidate materials where the Hamiltonian in equation (1) may be explicitly relevant include Nd_{2}B_{2}O_{7} (where B=Zr and Sn), rare earth spinels like CdEr_{2}Se_{4}, and Dy pyrochlores, though in the latter the effect of dipolar interactions need be considered^{21}. In light of recent experiments^{48} that suggest the oftstudied classical spin ice state is only metastable in Dy_{2}Ti_{2}O_{7}, it would seem prudent to reexamine the kagome ice state of this material using similar longtimescale techniques, to ascertain whether evidence of a QSL state may be present yet dynamically inhibited in the shorttimescale studies performed to date.
Finally, we note that recent work by Glaetzle et al. have demonstrated that precisely our Hamiltonian (equation (1)) can be realized experimentally with cold alkali atoms stored in optical or magnetic trap arrays^{49}. Such a proposal suggests that Quantum Kagome Ice may actually be realized in engineered systems of laserdressed Rydberg atoms in the very near future.
Methods
Computational details
We developed a Stochastic Series Expansion^{34} (SSE) QMC algorithm in the global S^{z} basis designed to study the Hamiltonian equation (1) with J_{±}=0 at finite temperature, using a 2+1dimensional simulation cell. Within the SSE, the Hamiltonian was implemented with a triangular plaquette breakup^{36}, which helps ergodicity in the regime where J_{z}/J_{±±} is large. Using this Hamiltonian breakup, the standard SSEdirected loop equations^{35} were modified to include sampling of offdiagonal operators of the type . The resulting algorithm is highly efficient, scaling linearly in the number of lattice sites V and the inverse temperature β. This scaling is modified to V^{2}β in the cases where a full qdependent structure factor measurement is required. The program was implemented in Fortran and verified by comparing results for small clusters with exact diagonalization data (See Supplementary Fig. 11 for an energy comparisons with ED and temperature convergence of the energy for larger system sizes). For each set of parameters in equation (1), the simulation typically requires 10^{7} QMC steps, with ∼10% additional equilibration steps. The data presented in this paper required computational resources equivalent to 155 CPU coreyears, run on a highperformance computing (HPC) cluster with Intel Xeon CPUs running at 2.83 GHz clock speed.
Additional information
How to cite this article: Carrasquilla, J. et al. A twodimensional spin liquid in quantum kagome ice. Nat. Commun. 6:7421 doi: 10.1038/ncomms8421 (2015).
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Acknowledgements
We thank F. Becca, A. Burkov, L. Cincio, T. Senthil, M. Stoudenmire and W. WitczakKrempa for enlightening discussions, and M. Gingras for critical reading of the manuscript. We are particularly indebted to Gang Chen for bringing the models discussed in ref. 21 to our attention and for stimulating discussions motivating this study. In addition, we are grateful to M. Heremele and A. Lauchli for emphasizing the role of perturbation theory. This research was supported by NSERC of Canada, the Perimeter Institute for Theoretical Physics, and the John Templeton Foundation. R.G.M. acknowledges support from a Canada Research Chair. Research at Perimeter Institute is supported through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation. Numerical simulations were carried out on the Shared Hierarchical Academic Research Computing Network (SHARCNET).
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J.C. and R.M. conceived the numerical strategy. J.C. designed and coded the algorithm, ran the computations, and performed the data analysis. Z.H. conceived the gauge theory mapping on the dual lattice. All the authors contributed to the interpretation of results, and the writing of the manuscript.
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Supplementary Figures 111, Supplementary Notes 15 and Supplementary References. (PDF 1899 kb)
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Carrasquilla, J., Hao, Z. & Melko, R. A twodimensional spin liquid in quantum kagome ice. Nat Commun 6, 7421 (2015). https://doi.org/10.1038/ncomms8421
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DOI: https://doi.org/10.1038/ncomms8421
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