Pyrochlore U(1) spin liquid of mixed symmetry enrichments in magnetic fields

We point out the experimental relevance and the detection scheme of symmetry enriched U(1) quantum spin liquids (QSLs) outside the perturbative spin-ice regime. Recent experiments on Ce-based pyrochlore QSL materials suggest that the candidate QSL may not be proximate to the well-known spin ice regime, and thus differs fundamentally from other pyrochlore QSL materials. We consider the possibility of the $\pi$-flux U(1) QSL favored by frustrated transverse exchange interactions rather than the usual quantum spin ice. It was previously suggested that both dipolar U(1) QSL and octupolar U(1) QSL can be realized for the generic spin model for the dipole-octupole doublets of the Ce$^{3+}$ local moments on the pyrochlore magnets Ce$_2$Sn$_2$O$_7$ and Ce$_2$Zr$_2$O$_7$. We explain and predict the experimental signatures especially the magnetic field response of the octupolar $\pi$-flux U(1) QSL. Fundamentally, this remarkable state is a mixture of symmetry enrichments from point group symmetry and from translational symmetry. We discuss the relevant experiments for pyrochlore U(1) QSLs and further provide some insights to the pyrochlore Heisenberg model.

Introduction.-Symmetry is the key that underlies the traditional Landau's paradigm of many-body phases and phase transitions. It is almost so in the classification and understanding of topological and exotic quantum phases of matter 1 . In last decade or so, a tremendous progress has been made theoretically to classify various symmetry enriched topological phases, where symmetry creates much more topological phases [2][3][4][5][6][7][8] . These symmetry enriched topological phases are described by the same topological quantum field theory, but they are distinct by the realization of symmetries for example on the fractionalized excitations. These beautiful theories so far do not have strong experimental connections. It is thus of a great interest to find an experimental relevance and establish the connection.
In last decade or so, various quantum spin liquid (QSL) candidate materials have been proposed, and the rareearth pyrochlore magnets comprise an important and large family of materials  . In these materials, the rareearth ions carry a spin-orbital-entangled effective spin-1/2 local moment that interact with highly anisotropic superexchange interactions 10,[30][31][32][33] Due to the proximity to the classical spin ice regime where the classical Ising interaction dominates, many pyrochlore materials develop a spin ice type of Pauling entropy plateau at low but finite temperatures 9,34-40 . Introducing quantum fluctuations and/or perturbations to the extensively degenerate spin ice manifold could then convert the system into a QSL state, and this state is often quoted as quantum spin ice U(1) QSL or pyrochlore ice U(1) QSL 9,10,37,[41][42][43][44] . Is the proximity to the spin ice regime necessary to produce a U(1) QSL? In our opinion, this condition was merely a theoretical convenience to access the interesting and exotic state in early theory works 41,44 . It is now established that, the pyrochlore U(1) QSL is much more robust in the so-called frustrated regime where the spinon experiences an emergent background π flux [45][46][47][48] . Since this π-flux U(1) QSL is expected to extend much beyond the perturbative spin ice regime 45 , it is natural to expect that the proximity to the spin ice regime is not quite necessary to obtain the pyrochlore U(1) QSL. We refer the U(1) QSL in this regime as non-spin-ice pyrochlore U(1) QSL or simply as pyrochlore U(1) QSL instead of pyrochlore spin ice U(1) QSL.
In the actual experiments on the Ce-based pyrochlore QSL materials (in particular, Ce 2 Zr 2 O 7 ) 49,50 , there is no signature of spin ice type of Pauling entropy plateau down to very low temperatures while the magnetic entropy is almost completely exhausted. This is a clear indication that the system is not in the spin ice regime. Another interesting aspect is that the Ce 3+ local moment in both Ce 2 Sn 2 O 7 and Ce 2 Zr 2 O 7 is a dipole-octupole doublet 33,51,52 . It is thus natural for us to consider the possibility of non-spin-ice pyrochlore U(1) QSL with the dipole-octupole doublets. It was previously suggested that, the anisotropic interaction between the dipoleoctupole doublets on the pyrochlore lattice could stabilize two symmetry enriched U(1) QSLs, i.e. dipolar U(1) QSL and octupolar U(1) QSL 51 . We explore the keen distinction and the experimental signatures between the dipolar U(1) QSL and the octupolar U(1) QSL in the frustrated non-spin-ice regime. As the spinon experiences a π flux in the frustrated regime, we refer these states as dipolar U(1) π QSL and octupolar U(1) π QSL in this paper.
The effective spin model.-We start with the generic effective spin model for dipole-octupole doublets on the pyrochlore lattice. The model is known as the XYZ model 33,51,52 , where microscopically S x and S y are magnetic octupole arXiv:1902.07075v2 [cond-mat.str-el] 20 Feb 2019 moments while S z is magnetic dipole moment. We have also introduced the Zeeman coupling that only acts on the magnetic dipole moment S z , andn is the field direction andẑ i defines the local z direction of each sublattice (see Appendix A for definition of these conventions). Only the nearest-neighbor interaction is considered, which is expected to be reasonable for the localized 4f electrons 53 . The XYZ form is obtained by applying a rotation around the y-direction by θ to eliminate the J xz term; the resulting Hamiltonian reads whereS x,z i are related to S x,z i by the θ-rotation, and S y ≡ S y . In the phase diagram of H XYZ without the magnetic field, the system supports three disconnected U(1) QSLs 33,51 . WhenJ z (J y ) is antiferromagnetic and dominant while the remaining two couplings are not large enough to drive a magnetic order, the ground state is a dipolar (octupolar) U(1) QSL. In the case whenJ x is antiferromagnetic and large, the relevant U(1) QSL is regarded as dipolar U(1) QSL and shares the same universal and qualitatively similar physics with the dipolar U(1) QSL becauseS x andS z transform identically under the point group symmetry. The dipolar U(1) QSL and octupolar U(1) QSL are symmetry enriched U(1) QSLs and are enriched by the point group symmetry.
Octupolar U(1) π QSL.-Since the experiments suggest that Ce 2 Zr 2 O 7 is not in the spin ice regime 50 , we would like to understand the physical properties of both dipolar and octupolar U(1) QSLs in the non-spin-ice regime. From the previous argument and early results 45 , the nonspin-ice regime for a XXZ model would be in the frustrated regime with a frustrated transverse exchange interaction and support the U(1) QSL with π flux for spinons.
The XYZ model with zero field can be rewritten with two different but equivalent forms below, where , and the couplings (J z ± , J z ±± ) and (J y ± , J y ±± ) can be read off from the expansion the above two Hamiltonians into the original form. For convenience, we focus on the regime where the groud state of H z (H y ) is the dipolar (octupolar) U(1) QSL, i.e. whenJ z (J y ) is antiferromagnetic and dominant. It is known that, as long as J z ± > 0 (J y ± > 0), the model for either sign of J z ±± (J y ±± ) does not have a fermion sign problem for quantum Monte carlo simulation 33 . In this unfrustrated regime, numerics shows that the system has the classical spin ice phenomena such as the Pauling entropy plateau at low and finite temperatures even when the system is located in the QSL phase at zero temperature 54 . It means that the frustrated regime J z ± < 0 (J y ± < 0) should carry the QSL physics for the Ce-based pyrochlore magnets. Since the frustrated regime for the U(1) QSL generates an emergent π-flux for the spinons, it is then natural to understand the physical properties of the dipolar and octupolar U(1) π QSLs. It is interesting to note that the π flux for the spinons is a signature of symmetry enrichments in the translational symmetry of the spinon sector. This is a translational symmetry enrichment on top of the point group symmetry enrichments. Due to the π flux, we expect the spinon continuum develops an enhanced spectral periodicity in the reciprocal space with a folded Brillouin zone 46,55 . Although certain generic properties may be established from the model level, there is still a gap to establishing a quantitative connection to the actual physical observables of the dipole-octupole doublets.
To make connection with the experiments, it is important to notice that only S z in Eq. (1) is magnetic 33,51,52 , and only S z -S z correlation is measurable in a neutron scattering experiment. From Eq. (2), S z i = cos θσ z i + sin θσ x i . Thus, inelastic neutron scattering would measure both σ z -σ z and σ x -σ x correlators. For the dipolar U(1) π QSL of H z with a large and antifer-romagneticJ z (for the dipolar U(1) π QSL with a large and antiferromagneticJ x ), the spinon continuum is contained in the σ x -σ x (σ z -σ z ) correlator, and the "magnetic monopole" continuum and the gauge photon are contained in the σ z -σ z (σ x -σ x ) correlator. The inclusion of the "magnetic monopole" continuum was only understood quite recently 56 . Due to the background π flux for the spinons in the dipolar U(1) π QSL, the spinon continuum develops an enhanced spetral periodicity with a folded Brillouin zone 45,46,56 . For the "magnetic monopoles", the continuum should always have an enhanced spectral periodicity with a folded Brillouin zone due to the effective spin-1/2 nature of the local moment 46,56,57 . As for the octupolar U(1) π QSL, because τ y is not directly measurable, the S z -S z correlator only detects the gapped spinon continuum, and the continuum has an enhanced spectral periodicity 46,51,56 .
The effect of external magnetic fields.-To access the ground state and illustrated the emergent gauge structure of the XYZ Hamiltonian, we implement the mapping introduced in Refs. 44 and 45 of the spin model to an Abelian-Higgs model with compact U(1) gauge field and bosonic spinon matter. Focusing on the octupolar QSL regime (whenJ y is positive and dominant in Eq. (4)), we express the spin operators as where r belongs to the I diamond sublattice (our convention is summarized in Appendix A). Here s y r,r is the emergent electric field in the octupolar U(1) QSL phase, s ± r,r is the gauge string operator ending at sites r and r , and Φ r (Φ † r ) is the spinon annihilation (creation) operator at the diamond lattice site r. The physical Hilbert space is obtained by imposing the following constraints where η r = ±1 for r in sublattice I and II, respectively, and Q r is the operator measuring the local gauge charge through the "Gauss law", and is canonically conjugate to Φ r , Under this mapping the Hamiltonian becomes Within the U(1) π QSL regime, we choose a gauge to take care of the background π-flux 45,46 , such that the spinons hop on the diamond lattice with modulated signs of hoppings (see Appendix A). In the absence of the field, the spinon continuum, that is measurable via inelastic neutron scattering measurement in the octupolar U(1) π QSL, shows a spectral periodicity enhancement with a folded Brillouin zone. As we calculate explicitly and show in the left panel of Fig. 1), both the upper and lower excitation edges of the two-spinon continuum develop the spectral periodicity enhancement. The external magnetic field, that directly couples to the spinon matters, modifies the spinon band structures. This modification can be directly measured by the inelastic neutron scattering. This provides an interesting example to manipulate or control the emergent fractionalized spinon degrees of freedom with external means that is the external magnetic field here. More importantly, such a manipulability could be recorded and tested. We apply the fields along three high symmetry directions, i.e.
[001], [110] and [111] crystallographic directions. In the central panel and the right panel of Fig. 1, we plot the upper and lower excitations of the spinon continuum under two different magnetic fields along the [110] direction. Because of the background π flux, the spinon continuum in these plots shows an enhanced spectral periodicity with a folded Brillouin zone. The detailed calculation scheme and the results for the fields along the [001] and [111] directions are displayed in Appendix B and Appendix C.
It is interesting to notice that there is a (hidden) competition between the transverse spin exchange interaction and the magnetic field. Our observation is as follows. The strong magnetic field would simply favor an uniform polarized state that preserves translations, while the simple spinon condensation of the U(1) π QSL would favor a state that breaks the lattice translational symmetry 46 . This frustration could enhance the stability of U(1) QSL under the field. This might also be the reason for the more stability of the antiferromagnetic Kitaev QSL in the magnetic field over the ferromagnetic one 58 . Perturbatively, the magnetic field favors a zero-flux state. One may wonder if the field can drive a phase transition between two symmetry enriched U(1) QSL, i.e., from U(1) π to U(1) 0 QSLs, and then from U(1) 0 QSL to spinon condensed state. This may be examined numerically or experimentally.
For the fully or nearly polarized state, the spins (or the local z components) are aligned along the preferred direction by the external magnetic field. Since the transverse spin components that create coherent spin excitations are octupolar moments, the neutron spin does not couple linearly with the transverse spin component and thus the inelatic neutron scattering signal would be strongly suppressed. There can still be some intensity for the nearly polarized state due to the crossing coupling J xz (S x i S z j + S z i S x j ). terials whose models are provided theoretically 33,51 . The major task would be to establish connections between the theoretical results/understanding and experiments. The main result in this paper is based on the U(1) QSLs with dipole-octupole doublets, and the experimental predictions are various spectroscopic properties. It has been shown that the spinon spectrum could have an enhanced periodicity with a folded Brillouin zone and the proximate orders could break translational symmetry by doubling the unit cell. Another set of experiment would be thermal Hall transports. As we will explain in a separate paper 59 that focuses on thermal Hall effect, we predict that there should be a non-trivial topological thermal Hall effect for "magnetic monopoles" due to the dual Berry phase effect in the dipolar U(1) QSL (or any other spin-ice based U(1) QSL materials) while there is no such topological thermal Hall effect for "magnetic monopoles" in the octupolar U(1) QSL. The possibility of Z 2 QSL is not considered here. Although the region of possible Z 2 QSL is tiny on the unfrustrated (sign-problem-free) side, the presence of Z 2 QSL on the frustrated side is not so clear. Thus, Z 2 QSL may still be possible, and the spectrum would be fully gapped. This may be examined carefully with specific heat measurements.
In the frustrated regime where the spinon experiences a π flux or Berry phase, there exists a fermion sign problem for quantum Monte carlo simulation. In contrast, in the XXZ limit and in the unfrustrated regime 41,44,54,[60][61][62][63][64][65] where the spinon experiences a zero flux, there is no a fermion sign problem. If one introduces extra (small) interactions, however, one could generate a fermion sign problem. Thus, it is reasonable to conjecture that there exist two types of fermion sign problems. One is intrinsic sign problem and associated with the intrinsic property of the underlying quantum phase. One cannot get rid of this sign problem without crossing the phase boundary. For this π flux state, the spectral periodicity enhancement of the spinon continuum is an intrinsic property associated with the π-flux U(1) QSL and comes together with this fermion sign problem. The fermion sign problem here is manifested as a π Berry phase for the spinons. More generally, we think this fermion sign problem in pure interacting fermion systems may manifest in the dynamics of certain collective modes via a non-trivial Berry phase effect. The other is extrinsic sign problem in the sense that one could tune the parameters such state the system can get rid of the sign problem. This happens for example when one removes extra small interactions and gets back to the unfrustrated regime of the XXZ model.
Returning to the XYZ model, we know that the model reduces to the Heisenberg model when three couplings are equal. The ground state of the pyrochlore lattice Heisenberg model is one of the hardest problems in quantum magnetism. From the property of the XYZ model, one could at least conclude that the ground state for the Heisenberg model cannot be the π-flux U(1) QSL for the XXZ model in the frustrated regime. This is because the three spin components have different physical meanings in the emergent spinon-gauge description while the three spin components are symmetrically related by the SU (2) spin rotation at the Heisenberg point. The centers of the corner-sharing tetrahedra in the pyrochlore lattice constitute a diamond structure with two sublattices, which we denote I and II; see Fig. 2. We choose the origins of the two sublattices as follows, The basis vectors of the diamond lattices are taken to be For each site of the I (II) sublattice there are four vertices of the II (I) sublattice that are nearest neighbors to it, with displacement vectors where η r = ±1 for r in sublattice I and II, respectively. At the midpoint of each of such bonds, there is a vertex of the pyrochlore lattice. Correspondingly, we define the local coordinate systems on the four sublattices of the pyrochlore lattice, as summarized in the Table I.
Appendix B: Gauge pattern for octupolar U(1)π QSL state and Bloch hamiltonian As pointed out in the literature 45,56 , in the frustrated regime J ± < 0 of the XYZ model (see Eqs. (3,4)) the ground state has π-flux within an elementary hexagon. Within the gauge mean field theory, recall that the s ± operators are gauge string operators, s ± r,r = 1 2 e ±iA r,r . We take the following gauge choice for the π-flux state, where = (0110), Q = 2π(100), and r belongs to the I sublattice, as illustrated in Fig. 2.
Because of the π-flux and the choice of Q, the unit cell doubles in the x-direction. Correspondingly, there are four sublattices of the system, which we term I, ± and II, ±. Specifically, a site on the origional I sublattice at r belongs to the I, + (I, −) sublattice if Q · (r − O I ) is an even (odd) multiple of π; and this is similarly for a site in the II sublattice.
Here λ is a Lagrange multiplier to ensure the (relaxed) spinon occupation number constraint, r (|Φ r | 2 −1) = 0. Now all Φ-Φ correlation functions (including the dynamic spin structure factor) can be computed from this action.