Competing orders in pyrochlore magnets from a \mathbb{Z}_{2} spin liquid perspective

Competing orders in pyrochlore magnets from a spin liquid perspective

Chunxiao Liu Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA    Gábor B. Halász Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, USA    Leon Balents Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, USA Canadian Institute for Advanced Research, 661 University Ave., Toronto, ON M5G 1M1 Canada
July 4, 2019

The pyrochlore materials have long been predicted to harbor a quantum spin liquid, an intrinsic long-range-entangled state supporting fractionalized excitations. Existing pyrochlore experiments, on the other hand, have discovered several weakly ordered states and a tendency of close competition amongst them. Motivated by these facts, we give a complete classification of spin-orbit-coupled spin-liquid states on the pyrochlore lattice by using the projective symmetry group (PSG) approach for bosonic spinons. For each spin liquid, we construct a mean-field Hamiltonian that can be used to describe phase transitions out of the spin liquid via spinon condensation. Studying these phase transitions, we establish phase diagrams for our mean-field Hamiltonians that link magnetic orders to specific spin liquids. In general, we find that seemingly unrelated magnetic orders are intertwined with each other and that the conventional spin orders seen in the experiments are accompanied by more exotic hidden orders. Our critical theories are categorized into and types, based on their spinon dispersion and Hamiltonian diagonalizability, and are shown to give distinct signatures in the heat capacity and the spin structure factor. This study provides a clear map of pyrochlore phases for future experiments and variational Monte Carlo studies in pyrochlore materials.

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I Introduction

Quantum spin liquids (QSLs) Savary and Balents (2016) are zero-temperature phases of interacting spin systems which possess intrinsic long-range entanglement and support nonlocal excitations carrying fractionalized quantum numbers. Typically, they respect all symmetries of the underlying lattice, i.e., they exhibit a lack of conventional symmetry-breaking order. The theoretical understanding of QSLs is largely in terms of emergent gauge theory, which provides a convenient mathematical framework to describe long-range entanglement, along with the nonlocal nature of the fractionalized excitations.

In frustrated magnetic systems Balents (2010), QSL ground states may control the physics even at (small) finite temperatures, as long as energy dominates over entropy. For two-dimensional spin liquids, this statement is purely asymptotic; at any nonzero temperature , the putative QSL is adiabatically connected to a high-temperature paramagnet. However, some three-dimensional spin liquids, particularly the so-called states with Ising-like emergent gauge fields, are more robust, and can persist in the form of a distinct low-temperature phase up to a nonzero critical temperature.

While QSLs are extremely interesting from a conceptual perspective, it is far from obvious to realize them in experimental materials, or even realistic spin Hamiltonians. Traditionally, most studies considered spin-rotation-invariant Heisenberg systems on geometrically frustrated two-dimensional lattices. However, it has recently been recognized that magnetic systems with strong spin-orbit coupling provide a promising alternative avenue to QSLs Witczak-Krempa et al. (2014); Gingras and McClarty (2014); Rau et al. (2016a); Iaconis et al. (2018). In general, these systems have a large number of magnetically anisotropic terms, leading to exchange frustration as well as an extended parameter space, and are thus expected to harbor QSL ground states on a wide range of two- and three-dimensional lattices.

The most widely studied such three-dimensional structure is the pyrochlore lattice, consisting of periodically arranged corner-sharing tetrahedra. Experimentally, two large families of materials, the pyrochlore spinels and the rare-earth pyrochlores, provide vast real-world possibilities Gardner et al. (2010) to test theoretical predictions on the pyrochlore lattice. In the 2000s, it was predicted that certain antiferromagnetic pyrochlore models could support a U(1) QSL phase Hermele et al. (2004) [the “U(1)” means that the gauge field belongs to the Lie algebra of the U(1) group and that the emergent charges are characterized by integers related to the generating charge of U(1)], which is a simulacrum of electromagnetic gauge theory in high-energy physics. In 2011/2012, theoretical applications of this idea to realistic models emerged, suggesting the presence of a U(1) spin liquid in the so-called “quantum spin ice” pyrochlore materials Ross et al. (2011); Savary and Balents (2012). So far, these predictions remain to be confirmed in experiments, even though there are some promising recent developments Hirschberger et al. (2015, 2019); Gaudet et al. (2019); Sibille et al. (2016, 2015); Scheie et al. (2017); Thompson et al. (2017); Sibille et al. (2018); Tokiwa et al. (2018).

Another thread recurring in the experimental study of rare-earth pyrochlores is the close competition amongst several weakly ordered states Hallas et al. (2017). Several hints at this competition are present in the family of Yb pyrochlores, YbBO, which have a systematic structural evolution across the series B = Ge, Ti, Pt, Sn. While the germanate orders antiferromagnetically, the remaining members of the family have ferromagnetic ground states, suggesting the close proximity of at least these two phases. In each material, the specific heat is peaked at a temperature of 2-4K, while the maximum ordering temperature is 0.6K in the germanate and half or less than that in the rest of the family. These findings indicate the onset of strong spin correlations well above the ordering temperature, but an inability of the system to decide upon its ground state. The weak ferromagnetic ground state in YbTiO is also famously mercurial, changing its character substantially with sample variations Bowman et al. (2019). Theoretically, a classical analysis indeed finds close competition amongst several distinct phases Yan et al. (2017), but a quantum picture of this phase competition is not yet available.

In this work, we combine the two threads of phase competition and QSL physics by utilizing the connection of symmetry to emergent gauge structure. This connection is mathematically described by the projective symmetry group (PSG), proposed by Wen in 2002 Wen (2002), which encapsulates the fact that, in a QSL, the group operations of the physical symmetry group are interleaved with those of the emergent gauge group. The embedding of the physical symmetries into the PSG can then lead to a unification of distinct symmetry-breaking orders that are unrelated in classical physics. Such a unified description of seemingly unrelated magnetic orders is the main motivation behind the present study.

The PSG also offers a straightforward method to classify QSLs in the presence of symmetry. Concretely, the PSG specifies a distinct set of transformation rules for the emergent matter and gauge fields in each QSL phase, corresponding to a given PSG class. Employing the PSG method, an entire zoo of QSLs has been found on the square Reuther et al. (2014), triangular Lu (2016), kagome Lu et al. (2011), honeycomb You et al. (2012), star Choy and Kim (2009), and hyperkagome Huang et al. (2017) lattices, to give a few notable examples. Generally, these QSLs can be connected to magnetically ordered states by considering the condensation patterns that emerge when the energy of a bosonic QSL excitation is brought to zero Bergman et al. (2006); Chen (2016); Li and Chen (2017).

In this paper, we employ the PSG method to obtain a full classification of QSLs with gauge structure on the pyrochlore lattice using Schwinger bosons Sachdev (1992); Wang and Vishwanath (2006); Wang (2010); Yang and Wang (2016). While standard parton constructions also allow U(1) and SU(2) gauge structures, we consider the gauge structure for two reasons. First, it is the simplest one: quasiparticles in a QSL are weakly interacting because the gauge field itself is gapped. Second, it is also the richest one: a single U(1) PSG class can be further split into several PSG classes if the gauge symmetry is lowered from U(1) to . We use Schwinger bosons rather than Abrikosov fermions Sanyal et al. (2019) to immediately obtain a bosonic excitation, the elementary Schwinger boson itself, that can condense at the phase transition out of the QSL.

As a result of our PSG analysis, we find different QSLs on the pyrochlore lattice. We use a standard mean-field description to study the -flux QSLs, in which translation symmetry acts linearly (i.e., as in classical physics) on the Schwinger bosons. The PSG method also allows us to describe phase transitions from these QSLs to magnetically ordered phases. Condensing the Schwinger bosons, we identify different ordering patterns, and call them “paraphases”, since each of them actually unifies several distinct symmetry-breaking orders. We find that, generically, these orders are intertwined, necessarily appearing together at the phase transition out of the QSL, and that conventional spin orders are in many cases accompanied by inversion-breaking “hidden” orders.

The phase transitions corresponding to these paraphases fall into two dynamical classes of and quantum criticality, exhibiting critical modes with linear and quadratic dispersions, respectively. We uncover the mathematical structure discriminating between these two classes, related to Hamiltonian diagonalizability, and derive their effective field theories, along with their most important experimental signatures. In particular, we use mean-field theory to compute static and dynamic spin structure factors for each of the paraphases. Finally, by comparing the magnetic orders associated with each paraphase to those observed in experiments, we identify a set of likely QSL phases that might be relevant to real-world pyrochlore materials.

The rest of the paper is organized as follows. First, in Sec. II, we summarize our main results on the different QSL phases and the corresponding phase transitions out of them (“paraphases”). In Sec. III, we employ the PSG method, deriving the PSG classes, and constructing a mean-field theory for each PSG class. In Sec. IV, we analyze the mean-field theories of our QSL phases, describing phase transitions out of them, and establishing the two dynamical classes with critical exponents . In Sec. V, we move on to the experimental signatures of our phase transitions, describing the heat capacity and the spin structure factors, and also introducing the concept of intertwined and hidden orders. Finally, in Sec. VI, we discuss our results and connect them to existing experimental data. Detailed derivations and lengthy formulas are given in the Appendices for reference.

Ii Main Results

Spin- Critical “paraphases” Magnetically ordered phases
liquid Condensation Dynamical Heat capacity: Dynamic spin structure factor Spin orders Hidden
phases momenta exponent AIAO AFM FM PC orders
- Gapless at
Weak in the low-energy limit
Gapless at and
Weak in the low-energy limit
Gapless along and Unclear at NN level
Low-energy weight at all momenta
- Gapless at
Weak in the low-energy limit
Gapless along and Unclear at NN level
- Gapless at
Weak in the low-energy limit
Gapless along and Unclear at NN level
- Gapless at
Weak in the low-energy limit
Gapless at , , and
Singular in the low-energy limit
Gapless at and
Singular in the low-energy limit
- Gapless at
Characteristic lower edge of the spectrum
Gapless along and Unclear at NN level
- Gapless at
Weak in the low-energy limit
Gapless at , , and
Singular in the low-energy limit
Gapless at and
Singular in the low-energy limit
Table 1: Most important characteristics of the critical “paraphases” corresponding to phase transitions between the six -flux spin-liquid phases [labeled as -] and a rich variety of magnetically ordered phases. Each paraphase is labeled by the condensation momenta (see Table IV for notation) where the spinons become gapless and condense at the phase transition. For each critical theory, experimental signatures are given in terms of the dynamical critical exponent, the low-temperature behavior of the heat capacity, and the low-energy features of the dynamic spin structure factor (see Fig. 4). For each magnetically ordered phase obtained by spinon condensation, it is specified whether various orders are generically present () or absent (), including conventional spin orders, such as all-in-all-out (AIAO) order, XY antiferromagnetic (AFM) order [], ferromagnetic (FM) order, and Palmer-Chalker (PC) order [], as well as inversion-breaking “hidden” orders.

From our PSG classification scheme, we find that there are different PSG classes of Schwinger bosons, corresponding to inequivalent QSL phases, on the pyrochlore lattice. Out of these different QSLs, there are eight -flux QSLs and eight -flux QSLs. For each QSL, we construct a general quadratic mean-field Hamiltonian for the Schwinger bosons containing all onsite, nearest-neighbor (NN), and next-nearest-neighbor (NNN) terms allowed by symmetry. However, for simplicity, we focus on the -flux QSLs and restrict the mean-field Hamiltonian to onsite and NN terms. At such a NN level, two out of eight -flux Hamiltonians have an enlarged U(1) gauge symmetry, and we thus concentrate on the remaining six -flux Hamiltonians with gauge symmetry.

In each of the six corresponding QSL phases, the Schwinger bosons can be identified as elementary spinon excitations carrying fractionalized quantum numbers. If the chemical potential is tuned to its critical value, there is a phase transition driven by the condensation of these bosonic spinons. Depending on the particular patterns of spinon condensation, we describe different critical “paraphases” out of the six QSL phases. The most important characteristics of these paraphases, labeled by their parent QSL phases and the condensation momenta of the spinons, are tabulated in Table 1.

For each paraphase, the spinon spectrum is gapless at the critical point by construction. The effective field theory of the critical point is characterized by the low-energy spinon dispersion, , in terms of the dynamical critical exponent, which is either or . These two dynamical classes give rise to distinct sets of experimental signatures. For a start, the power-law exponent of the low-temperature heat capacity, , is determined by the dynamical exponent and the dimensionality of the condensation manifold, i.e., if the spinons condense at points or along lines in the Brillouin zone (BZ). Also, the dynamical exponent gives rise to universal features in the static and dynamic spin structure factors, which appear on top of more detailed characteristics specific to given paraphases. In particular, when approaching zero energy, the spectral weight in the dynamic structure factor vanishes for but diverges for ; the divergence in the case is also observable as a nonanalytic behavior in the static structure factor.

To establish a connection between spinon condensation and the resulting magnetic orders, restricted to zero momentum for simplicity, we investigate the transformation rules of the possible order parameters under the point group of the pyrochlore lattice. For each paraphase, we determine which magnetic orders generically appear, concentrating in particular on the conventional spin orders seen in the experiments: the all-in-all-out, antiferromagnetic, ferromagnetic, and Palmer-Chalker orders. In doing so, we learn two important general lessons on magnetic orders obtained by spinon condensation. First, several distinct orders may be intertwined, i.e., they necessarily accompany each other, even though they are completely unrelated on the classical level. Second, the conventional spin orders may emerge together with more exotic inversion-breaking “hidden” orders.

Iii Projective symmetry group

iii.1 Lattice symmetries

We first introduce the symmetries of the pyrochlore lattice; the convention and notation we establish here is used throughout the rest of the paper. The pyrochlore lattice consists of four FCC-type sublattices, which we label by . To index the sites of the lattice, we use two coordinate systems: the global cartesian coordinates (GCCs) and the sublattice-indexed pyrochlore coordinates (SIPCs). The GCCs are the standard frame coordinates for the FCC cube of edge length . The SIPCs are spanned by the lattice vectors , , and , which are expressed in GCCs as


We define () to be the displacement vectors of the sublattices from the sublattice, where we understand . The relation between the SIPCs and the GCCs is then


The space group of the pyrochlore lattice is the cubic space group (No. 227), minimally generated by the translations , , and along the lattice vectors , , and , a sixfold rotoreflection around the axis (i.e., around ), and a non-symmorphic screw operation , which is the composition of a twofold rotation around and a translation by . These space-group generators transform the SIPCs according to


Note that we can write the rotoreflection as , where is an inversion with respect to the origin and is a threefold rotation around the axis. The generators are therefore equivalent to the generator ; we choose a single generator to reduce the number of generators and group relations.

The point group of the pyrochlore lattice, formally defined as the quotient group of the space group and the group of pure translations, is the cubic group . This group is minimally generated by and , where is a twofold rotation around , distinguished from the space-group generator by the lack of a subsequent translation along . A detailed description of the point-group structure is given in Appendix A.

In addition to the pyrochlore space-group symmetries, time-reversal symmetry is also present in the pyrochlore materials. The corresponding time-reversal operation commutes with all space-group operations and satisfies when acting on a half-integer spin state. The complete list of independent group relations defining the symmetry group is then


The notation in Eq. (3) is understood as .

iii.2 Projective symmetry group

In this subsection, we classify all possible quantum spin liquids that are compatible with the symmetries of the pyrochlore lattice. We first write the spins in terms of Schwinger boson bilinears as


where , and are the Pauli matrices (also denoted by , respectively). Physically, the Schwinger bosons describe the deconfined spinon excitations of the quantum spin liquid and, on the mean-field level, they are governed by a quadratic Hamiltonian, commonly known as the mean-field ansatz.

It is important to emphasize that the transformation in Eq. (4) is not faithful as it enlarges the local Hilbert space at each site . Consequently, there is a local gauge redundancy for the Schwinger bosons. Indeed, any site-dependent U(1) phase transformation


leaves the spins invariant. The physical Hilbert space can in principle be retained by enforcing the constraint


at each site of the lattice.

Under a space-group operation , the spins transform as , where is the SU(2) rotation matrix associated with the operation . We therefore naïvely expect that the spinons transform as


However, due to the U(1) gauge redundancy, any operation is generally accompanied by a site-dependent U(1) phase transformation


and the spinons thus actually transform as


where the symbol “” indicates that the gauge-enriched operation is a composition of the pure symmetry operation and the gauge transformation .

Under a time reversal of the system, the spins transform as , where , while applies complex conjugation to everything on its right. Once again, combining the naïve transformation rule for the spinons,


and the accompanying U(1) phase transformation,


the spinons are found to transform as


Note that because is real.

For a quantum spin liquid, the gauge-enriched operations and generate the symmetry group of the mean-field ansatz, commonly known as the projective symmetry group (PSG). To enumerate all quantum spin liquids, we need to find all distinct PSG solutions, i.e., all gauge-inequivalent solutions for the gauge transformations and that are consistent with the symmetry group of the lattice, including space-group symmetries and time-reversal symmetry. In particular, for each group relation [see Eq. (3)] taking the general form of


we consider the gauge-enriched group relation


where is a pure gauge transformation, thus corresponding to the identity operation for the spins. Being an element of the PSG by definition, is also an element of the invariant gauge group (IGG), the group of all gauge transformations that leave the mean-field ansatz invariant. In most cases, such gauge transformations are exclusively “global” (i.e., site independent), and the IGG is thus a subgroup of U(1), typically or U(1), corresponding to and U(1) spin liquids, respectively. Since we are interested in classifying spin liquids, we consider IGG in the following. The only two elements of the IGG are then with .

For any group relation in terms of exclusively space-group operations, taking the form of Eq. (13), the gauge-enriched group relation in Eq. (14) can be rewritten as

Using the general conjugation rule


following directly from Eqs. (7) and (8), this group relation then becomes a pure phase equation:

For group relations involving time reversal, special care must be taken due to the presence of the complex conjugation . Using the modified conjugation rule


the last group relation in Eq. (3) translates into


while the penultimate group relation gives rise to a trivial equation due to the cancellation between the phase factors and .

The PSG classification is obtained by listing all group relations and finding all solutions of the corresponding phase equations [see Eqs. (III.2) and (19)] for the parameters as well as the phases and . We emphasize that distinct solutions, describing distinct spin liquids, must be gauge inequivalent. Indeed, by means of a general gauge transformation [see Eq. (5)], the gauge-enriched group relations in Eq. (14) can be rewritten as


transforming the phases according to


and thus indicating that two seemingly distinct solutions for the phases might in fact be equivalent.

The detailed solution of the PSG equations is presented in Appendix B. The PSG results for the phases are


where , , , and are four parameters, each being either or . Therefore, we find that there are gauge-inequivalent PSG classes, corresponding to distinct quantum spin liquids, which we label by the notation -. The four parameters have concrete interpretations:

  • The parameter comes from the three PSG equations corresponding to , which are required by the PSG to share the same parameter. Physically, it quantifies the Aharonov-Bohm (AB) phase a spinon accumulates while moving on the closed edge of a plaquette, which is traversed by such a sequence of translations. In the case of (), the AB phase is (), corresponding to a -flux (-flux) spin liquid.

  • The parameter comes from the PSG equation corresponding to . Physically, it describes the AB phase a spinon accumulates after completing six subsequent sixfold rotoreflections. Together with , it determines whether or not the sixfold rotoreflection acts projectively.

  • The parameter comes from the PSG equation corresponding to . Physically, it describes the AB phase a spinon accumulates after completing the operation sequence . Together with and it determines whether or not the screw operation acts projectively.

  • The parameter comes from the PSG equation corresponding to . Physically, it describes the AB phase a spinon accumulates after completing the operation sequence .

iii.3 Construction of mean-field ansätze

We are now in the position to construct the mean-field ansatz for each PSG class. The most general mean-field ansatz for bosonic spinons can be written as


where and are matrices acting on spin space, and the labels “h” and “p” indicate hopping and pairing terms, respectively.

The PSG operators and are the symmetry operators of the Hamiltonian , meaning and . Since the spinons transforms under and according to Eqs. (9) and (12), the matrices and must transform as


for space-group elements and as


for time reversal . The respective SU(2) matrices are


where is the identity matrix. Suppressing the site indices for simplicity, we parameterize the matrices and in the general forms


where are all complex. The additional factor appearing in ensures that and transform in the same way under the respective unitary conjugations and for any SU(2). In both cases, the singlet parameters and transform as scalars, while the triplet parameters and transform as SO(3) vectors. Indeed, any SU(2) rotation leaves the singlet parameters invariant and performs the corresponding SO(3) rotation on the triplet vectors: and . For the generators and , these SO(3) rotations are


while the translations correspond to trivial SO(3) rotations: .

To reduce the number of parameters in the mean-field ansatz, we first consider the effect of time reversal. Substituting Eq. (27) into Eq. (25), and taking from Eq. (22d), we obtain as well as and deduce that all parameters of and are real.

Turning to space-group symmetries and using Eq. (24), we can then establish relations between the respective parameters of and that correspond to different bonds of the lattice. In fact, the entire mean-field ansatz in Eq. (23) can be constructed up to next-nearest-neighbor level by specifying the real parameters for each of the following three representative bonds:

  • onsite “bond” :

  • nearest-neighbor (NN) bond :

  • next-nearest-neighbor (NNN) bond :


iii.4 Nontrivial parameter constraints

Class Independent nonzero parameters Constraints Note
- Onsite NN NNN Onsite NN NNN
0- , , U(1) at NN
0- ,, , ,
0- , ,, , ,