Competing orders in pyrochlore magnets from a spin liquid perspective
Abstract
The pyrochlore materials have long been predicted to harbor a quantum spin liquid, an intrinsic longrangeentangled 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 spinorbitcoupled spinliquid states on the pyrochlore lattice by using the projective symmetry group (PSG) approach for bosonic spinons. For each spin liquid, we construct a meanfield 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 meanfield 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.
pacs:
Valid PACS appear hereI Introduction
Quantum spin liquids (QSLs) Savary and Balents (2016) are zerotemperature phases of interacting spin systems which possess intrinsic longrange 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 symmetrybreaking order. The theoretical understanding of QSLs is largely in terms of emergent gauge theory, which provides a convenient mathematical framework to describe longrange 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 twodimensional spin liquids, this statement is purely asymptotic; at any nonzero temperature , the putative QSL is adiabatically connected to a hightemperature paramagnet. However, some threedimensional spin liquids, particularly the socalled states with Isinglike emergent gauge fields, are more robust, and can persist in the form of a distinct lowtemperature 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 spinrotationinvariant Heisenberg systems on geometrically frustrated twodimensional lattices. However, it has recently been recognized that magnetic systems with strong spinorbit coupling provide a promising alternative avenue to QSLs WitczakKrempa 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 threedimensional lattices.
The most widely studied such threedimensional structure is the pyrochlore lattice, consisting of periodically arranged cornersharing tetrahedra. Experimentally, two large families of materials, the pyrochlore spinels and the rareearth pyrochlores, provide vast realworld 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 highenergy physics. In 2011/2012, theoretical applications of this idea to realistic models emerged, suggesting the presence of a U(1) spin liquid in the socalled “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 rareearth 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 24K, 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 symmetrybreaking 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 meanfield 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 symmetrybreaking 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 inversionbreaking “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 meanfield 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 realworld 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 meanfield theory for each PSG class. In Sec. IV, we analyze the meanfield 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 lowenergy limit  
Gapless at and  
Weak in the lowenergy limit  
Gapless along and  Unclear at NN level  
Lowenergy weight at all momenta  
  Gapless at  
Weak in the lowenergy limit  
Gapless along and  Unclear at NN level  
  Gapless at  
Weak in the lowenergy limit  
Gapless along and  Unclear at NN level  
  Gapless at  
Weak in the lowenergy limit  
Gapless at , , and  
Singular in the lowenergy limit  
Gapless at and  
Singular in the lowenergy limit  
  Gapless at  
Characteristic lower edge of the spectrum  
Gapless along and  Unclear at NN level  
  Gapless at  
Weak in the lowenergy limit  
Gapless at , , and  
Singular in the lowenergy limit  
Gapless at and  
Singular in the lowenergy limit 
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 meanfield Hamiltonian for the Schwinger bosons containing all onsite, nearestneighbor (NN), and nextnearestneighbor (NNN) terms allowed by symmetry. However, for simplicity, we focus on the flux QSLs and restrict the meanfield 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 lowenergy 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 powerlaw exponent of the lowtemperature 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 allinallout, antiferromagnetic, ferromagnetic, and PalmerChalker 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 inversionbreaking “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 FCCtype sublattices, which we label by . To index the sites of the lattice, we use two coordinate systems: the global cartesian coordinates (GCCs) and the sublatticeindexed 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
(1a)  
(1b)  
(1c) 
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
SIPC  
GCC 
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 nonsymmorphic screw operation , which is the composition of a twofold rotation around and a translation by . These spacegroup generators transform the SIPCs according to
(2)  
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 spacegroup generator by the lack of a subsequent translation along . A detailed description of the pointgroup structure is given in Appendix A.
In addition to the pyrochlore spacegroup symmetries, timereversal symmetry is also present in the pyrochlore materials. The corresponding timereversal operation commutes with all spacegroup operations and satisfies when acting on a halfinteger spin state. The complete list of independent group relations defining the symmetry group is then
(3) 
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
(4) 
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 meanfield level, they are governed by a quadratic Hamiltonian, commonly known as the meanfield 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 sitedependent U(1) phase transformation
(5) 
leaves the spins invariant. The physical Hilbert space can in principle be retained by enforcing the constraint
(6) 
at each site of the lattice.
Under a spacegroup 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
(7) 
However, due to the U(1) gauge redundancy, any operation is generally accompanied by a sitedependent U(1) phase transformation
(8) 
and the spinons thus actually transform as
(9) 
where the symbol “” indicates that the gaugeenriched 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,
(10) 
and the accompanying U(1) phase transformation,
(11) 
the spinons are found to transform as
(12) 
Note that because is real.
For a quantum spin liquid, the gaugeenriched operations and generate the symmetry group of the meanfield 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 gaugeinequivalent solutions for the gauge transformations and that are consistent with the symmetry group of the lattice, including spacegroup symmetries and timereversal symmetry. In particular, for each group relation [see Eq. (3)] taking the general form of
(13) 
we consider the gaugeenriched group relation
(14) 
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 meanfield 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 spacegroup operations, taking the form of Eq. (13), the gaugeenriched group relation in Eq. (14) can be rewritten as
Using the general conjugation rule
(16) 
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
(18)  
the last group relation in Eq. (3) translates into
(19) 
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 gaugeenriched group relations in Eq. (14) can be rewritten as
(20) 
transforming the phases according to
(21) 
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
(22a)  
(22b)  
(22c)  
(22d)  
(22e)  
(22f)  
where , , , and are four parameters, each being either or . Therefore, we find that there are gaugeinequivalent 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 AharonovBohm (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 meanfield ansätze
We are now in the position to construct the meanfield ansatz for each PSG class. The most general meanfield ansatz for bosonic spinons can be written as
(23) 
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
(24a)  
(24b) 
for spacegroup elements and as
(25a)  
(25b) 
for time reversal . The respective SU(2) matrices are
(26)  
where is the identity matrix. Suppressing the site indices for simplicity, we parameterize the matrices and in the general forms
(27a)  
(27b) 
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
(28) 
while the translations correspond to trivial SO(3) rotations: .
To reduce the number of parameters in the meanfield 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 spacegroup 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 meanfield ansatz in Eq. (23) can be constructed up to nextnearestneighbor level by specifying the real parameters for each of the following three representative bonds:

onsite “bond” :
(29) 
nearestneighbor (NN) bond :
(30) 
nextnearestneighbor (NNN) bond :
(31)
iii.4 Nontrivial parameter constraints
Class  Independent nonzero parameters  Constraints  Note  

  Onsite  NN  NNN  Onsite  NN  NNN  
0  ,  ,  U(1) at NN  
0  ,,  ,  ,  
0  ,  ,,  ,  , 