Competing orders in pyrochlore magnets from a ${\mathbb{Z}}_{2}$ spin liquid perspective
Abstract
We report that 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 ${\mathbb{Z}}_{2}$ 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 z = 1 and z = 2 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. Lastly, this study provides a clear map of pyrochlore phases for future experiments and variational Monte Carlo studies in pyrochlore materials.
 Authors:

 Univ. of California, Santa Barbara, CA (United States)
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Univ. of California, Santa Barbara, CA (United States)
 Univ. of California, Santa Barbara, CA (United States) ; Canadian Inst.for Advanced Research, Toronto, ON (Canada)
 Publication Date:
 Research Org.:
 Univ. of California, Santa Barbara, CA (United States). Kavli Institute for Theoretical Physics
 Sponsoring Org.:
 USDOE Office of Science (SC), Basic Energy Sciences (BES)
 OSTI Identifier:
 1556863
 Alternate Identifier(s):
 OSTI ID: 1556842
 Grant/Contract Number:
 FG0208ER46524
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Physical Review B
 Additional Journal Information:
 Journal Volume: 100; Journal Issue: 7; Journal ID: ISSN 24699950
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 36 MATERIALS SCIENCE; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY
Citation Formats
Liu, Chunxiao, Halász, Gábor B., and Balents, Leon. Competing orders in pyrochlore magnets from a Z2 spin liquid perspective. United States: N. p., 2019.
Web. doi:10.1103/PhysRevB.100.075125.
Liu, Chunxiao, Halász, Gábor B., & Balents, Leon. Competing orders in pyrochlore magnets from a Z2 spin liquid perspective. United States. doi:10.1103/PhysRevB.100.075125.
Liu, Chunxiao, Halász, Gábor B., and Balents, Leon. Mon .
"Competing orders in pyrochlore magnets from a Z2 spin liquid perspective". United States. doi:10.1103/PhysRevB.100.075125. https://www.osti.gov/servlets/purl/1556863.
@article{osti_1556863,
title = {Competing orders in pyrochlore magnets from a Z2 spin liquid perspective},
author = {Liu, Chunxiao and Halász, Gábor B. and Balents, Leon},
abstractNote = {We report that 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 Z2 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 z = 1 and z = 2 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. Lastly, this study provides a clear map of pyrochlore phases for future experiments and variational Monte Carlo studies in pyrochlore materials.},
doi = {10.1103/PhysRevB.100.075125},
journal = {Physical Review B},
number = 7,
volume = 100,
place = {United States},
year = {2019},
month = {8}
}
Web of Science
Works referenced in this record:
Quantum spin liquids: a review
journal, November 2016
 Savary, Lucile; Balents, Leon
 Reports on Progress in Physics, Vol. 80, Issue 1
Correlated Quantum Phenomena in the Strong SpinOrbit Regime
journal, March 2014
 WitczakKrempa, William; Chen, Gang; Kim, Yong Baek
 Annual Review of Condensed Matter Physics, Vol. 5, Issue 1
Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets
journal, May 2014
 Gingras, M. J. P.; McClarty, P. A.
 Reports on Progress in Physics, Vol. 77, Issue 5
SpinOrbit Physics Giving Rise to Novel Phases in Correlated Systems: Iridates and Related Materials
journal, March 2016
 Rau, Jeffrey G.; Lee, Eric KinHo; Kee, HaeYoung
 Annual Review of Condensed Matter Physics, Vol. 7, Issue 1
Spin Liquid versus Spin Orbit Coupling on the Triangular Lattice
journal, January 2018
 Iaconis, Jason; Liu, Chunxiao; Halász, Gábor
 SciPost Physics, Vol. 4, Issue 1
Magnetic pyrochlore oxides
journal, January 2010
 Gardner, Jason S.; Gingras, Michel J. P.; Greedan, John E.
 Reviews of Modern Physics, Vol. 82, Issue 1
Pyrochlore photons: The $U\left(1\right)\mathrm{}$ spin liquid in a $S=\frac{1}{2}$ threedimensional frustrated magnet
journal, February 2004
 Hermele, Michael; Fisher, Matthew P. A.; Balents, Leon
 Physical Review B, Vol. 69, Issue 6
Quantum Excitations in Quantum Spin Ice
journal, October 2011
 Ross, Kate A.; Savary, Lucile; Gaulin, Bruce D.
 Physical Review X, Vol. 1, Issue 2
Coulombic Quantum Liquids in Spin $1/2$ Pyrochlores
journal, January 2012
 Savary, Lucile; Balents, Leon
 Physical Review Letters, Vol. 108, Issue 3
Large thermal Hall conductivity of neutral spin excitations in a frustrated quantum magnet
journal, April 2015
 Hirschberger, Max; Krizan, Jason W.; Cava, R. J.
 Science, Vol. 348, Issue 6230
Quantum Spin Ice Dynamics in the DipoleOctupole Pyrochlore Magnet ${\mathrm{Ce}}_{2}{\mathrm{Zr}}_{2}{\mathrm{O}}_{7}$
journal, May 2019
 Gaudet, J.; Smith, E. M.; Dudemaine, J.
 Physical Review Letters, Vol. 122, Issue 18
Candidate quantum spin ice in the pyrochlore ${\mathrm{Pr}}_{2}{\mathrm{Hf}}_{2}{\mathrm{O}}_{7}$
journal, July 2016
 Sibille, Romain; Lhotel, Elsa; Hatnean, Monica Ciomaga
 Physical Review B, Vol. 94, Issue 2
Candidate Quantum Spin Liquid in the ${\mathrm{Ce}}^{3+}$ Pyrochlore Stannate ${\mathrm{Ce}}_{2}{\mathrm{Sn}}_{2}{\mathrm{O}}_{7}$
journal, August 2015
 Sibille, Romain; Lhotel, Elsa; Pomjakushin, Vladimir
 Physical Review Letters, Vol. 115, Issue 9
Reentrant Phase Diagram of ${\mathrm{Yb}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$ in a $\u27e8111\u27e9$ Magnetic Field
journal, September 2017
 Scheie, A.; Kindervater, J.; Säubert, S.
 Physical Review Letters, Vol. 119, Issue 12
Quasiparticle Breakdown and Spin Hamiltonian of the Frustrated Quantum Pyrochlore ${\mathrm{Yb}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$ in a Magnetic Field
journal, August 2017
 Thompson, J. D.; McClarty, P. A.; Prabhakaran, D.
 Physical Review Letters, Vol. 119, Issue 5
Experimental signatures of emergent quantum electrodynamics in Pr2Hf2O7
journal, April 2018
 Sibille, Romain; Gauthier, Nicolas; Yan, Han
 Nature Physics, Vol. 14, Issue 7
Discovery of Emergent Photon and Monopoles in a Quantum Spin Liquid
journal, June 2018
 Tokiwa, Yoshifumi; Yamashita, Takuya; Terazawa, Daiki
 Journal of the Physical Society of Japan, Vol. 87, Issue 6
Experimental Insights into GroundState Selection of Quantum XY Pyrochlores
journal, March 2018
 Hallas, Alannah M.; Gaudet, Jonathan; Gaulin, Bruce D.
 Annual Review of Condensed Matter Physics, Vol. 9, Issue 1
Role of defects in determining the magnetic ground state of ytterbium titanate
journal, February 2019
 Bowman, D. F.; Cemal, E.; Lehner, T.
 Nature Communications, Vol. 10, Issue 1
Theory of multiplephase competition in pyrochlore magnets with anisotropic exchange with application to ${\mathrm{Yb}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7},{\mathrm{Er}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$ , and ${\mathrm{Er}}_{2}{\mathrm{Sn}}_{2}{\mathrm{O}}_{7}$
journal, March 2017
 Yan, Han; Benton, Owen; Jaubert, Ludovic
 Physical Review B, Vol. 95, Issue 9
Quantum orders and symmetric spin liquids
journal, April 2002
 Wen, XiaoGang
 Physical Review B, Vol. 65, Issue 16
Classification of spin liquids on the square lattice with strong spinorbit coupling
journal, November 2014
 Reuther, Johannes; Lee, ShuPing; Alicea, Jason
 Physical Review B, Vol. 90, Issue 17
Symmetric ${Z}_{2}$ spin liquids and their neighboring phases on triangular lattice
journal, April 2016
 Lu, YuanMing
 Physical Review B, Vol. 93, Issue 16
${\mathbb{Z}}_{2}$ spin liquids in the $S=\frac{1}{2}$ Heisenberg model on the kagome lattice: A projective symmetrygroup study of Schwinger fermion meanfield states
journal, June 2011
 Lu, YuanMing; Ran, Ying; Lee, Patrick A.
 Physical Review B, Vol. 83, Issue 22
Doping a spinorbit Mott insulator: Topological superconductivity from the KitaevHeisenberg model and possible application to (Na ${}_{2}$ /Li ${}_{2}$ )IrO ${}_{3}$
journal, August 2012
 You, YiZhuang; Kimchi, Itamar; Vishwanath, Ashvin
 Physical Review B, Vol. 86, Issue 8
Classification of quantum phases for the starlattice antiferromagnet via a projective symmetry group analysis
journal, August 2009
 Choy, TingPong; Kim, Yong Baek
 Physical Review B, Vol. 80, Issue 6
Interplay of nonsymmorphic symmetry and spinorbit coupling in hyperkagome spin liquids: Applications to ${\mathrm{Na}}_{4}{\mathrm{Ir}}_{3}{\mathrm{O}}_{8}$
journal, February 2017
 Huang, Biao; Kim, Yong Baek; Lu, YuanMing
 Physical Review B, Vol. 95, Issue 5
Ordering in a frustrated pyrochlore antiferromagnet proximate to a spin liquid
journal, April 2006
 Bergman, Doron L.; Fiete, Gregory A.; Balents, Leon
 Physical Review B, Vol. 73, Issue 13
“Magnetic monopole” condensation of the pyrochlore ice U(1) quantum spin liquid: Application to ${\mathrm{Pr}}_{2}{\mathrm{Ir}}_{2}{\mathrm{O}}_{7}$ and ${\mathrm{Yb}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$
journal, November 2016
 Chen, Gang
 Physical Review B, Vol. 94, Issue 20
Symmetry enriched U(1) topological orders for dipoleoctupole doublets on a pyrochlore lattice
journal, January 2017
 Li, YaoDong; Chen, Gang
 Physical Review B, Vol. 95, Issue 4
Kagome´ and triangularlattice Heisenberg antiferromagnets: Ordering from quantum fluctuations and quantumdisordered ground states with unconfined bosonic spinons
journal, June 1992
 Sachdev, Subir
 Physical Review B, Vol. 45, Issue 21
Spinliquid states on the triangular and Kagomé lattices: A projectivesymmetrygroup analysis of Schwinger boson states
journal, November 2006
 Wang, Fa; Vishwanath, Ashvin
 Physical Review B, Vol. 74, Issue 17
Schwinger boson mean field theories of spin liquid states on a honeycomb lattice: Projective symmetry group analysis and critical field theory
journal, July 2010
 Wang, Fa
 Physical Review B, Vol. 82, Issue 2
Schwinger boson spinliquid states on square lattice
journal, July 2016
 Yang, Xu; Wang, Fa
 Physical Review B, Vol. 94, Issue 3
Time reversal symmetry breaking chiral spin liquids: Projective symmetry group approach of bosonic meanfield theories
journal, March 2013
 Messio, Laura; Lhuillier, Claire; Misguich, Grégoire
 Physical Review B, Vol. 87, Issue 12
Monopole flux state on the pyrochlore lattice
journal, April 2009
 Burnell, F. J.; Chakravarty, Shoibal; Sondhi, S. L.
 Physical Review B, Vol. 79, Issue 14
Threedimensional nematic spin liquid in a stacked triangular lattice 6 $H$ B structure
journal, June 2013
 Hwang, Kyusung; Dodds, Tyler; Bhattacharjee, Subhro
 Physical Review B, Vol. 87, Issue 23
Interplay of uniform $U$ (1) quantum spin liquid and magnetic phases in rareearth pyrochlore magnets: A fermionic parton approach
journal, April 2019
 Sanyal, Sambuddha; Dhochak, Kusum; Bhattacharjee, Subhro
 Physical Review B, Vol. 99, Issue 13
BoseEinstein condensation in quantum magnets
journal, May 2014
 Zapf, Vivien; Jaime, Marcelo; Batista, C. D.
 Reviews of Modern Physics, Vol. 86, Issue 2
Order in the Heisenberg pyrochlore: The magnetic structure of ${\mathrm{Gd}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$
journal, September 2001
 Champion, J. D. M.; Wills, A. S.; Fennell, T.
 Physical Review B, Vol. 64, Issue 14
FluctuationDriven Selection at Criticality in a Frustrated Magnetic System: The Case of Multiple $\mathbf{k}$ Partial Order on the Pyrochlore Lattice
journal, April 2015
 Javanparast, Behnam; Hao, Zhihao; Enjalran, Matthew
 Physical Review Letters, Vol. 114, Issue 13
Breathing Pyrochlore Lattice Realized in $A$ Site Ordered Spinel Oxides ${\mathrm{LiGaCr}}_{4}{\mathrm{O}}_{8}$ and ${\mathrm{LiInCr}}_{4}{\mathrm{O}}_{8}$
journal, February 2013
 Okamoto, Yoshihiko; Nilsen, Gøran J.; Attfield, J. Paul
 Physical Review Letters, Vol. 110, Issue 9
Experimental realization of a quantum breathing pyrochlore antiferromagnet
journal, August 2014
 Kimura, K.; Nakatsuji, S.; Kimura, T.
 Physical Review B, Vol. 90, Issue 6
Anisotropic Exchange within Decoupled Tetrahedra in the Quantum Breathing Pyrochlore ${\mathrm{Ba}}_{3}{\mathrm{Yb}}_{2}{\mathrm{Zn}}_{5}{\mathrm{O}}_{11}$
journal, June 2016
 Rau, J. G.; Wu, L. S.; May, A. F.
 Physical Review Letters, Vol. 116, Issue 25
Putting competing orders in their place near the Mott transition
journal, April 2005
 Balents, Leon; Bartosch, Lorenz; Burkov, Anton
 Physical Review B, Vol. 71, Issue 14
Putting competing orders in their place near the Mott transition. II. The doped quantum dimer model
journal, April 2005
 Balents, Leon; Bartosch, Lorenz; Burkov, Anton
 Physical Review B, Vol. 71, Issue 14
A paritybreaking electronic nematic phase transition in the spinorbit coupled metal Cd _{2} Re _{2} O _{7}
journal, April 2017
 Harter, J. W.; Zhao, Z. Y.; Yan, J. Q.
 Science, Vol. 356, Issue 6335
Highpressure synthesis and characterization of the effective pseudospin $S=1/2$ XY pyrochlores ${\mathrm{R}}_{2}\mathrm{P}{\mathrm{t}}_{2}{\mathrm{O}}_{7}\phantom{\rule{0.16em}{0ex}}\left(R=\mathrm{Er},\phantom{\rule{0.16em}{0ex}}\mathrm{Yb}\right)$
journal, January 2016
 Cai, Y. Q.; Cui, Q.; Li, X.
 Physical Review B, Vol. 93, Issue 1
Fluctuations and AllIn–AllOut Ordering in DipoleOctupole ${\mathrm{Nd}}_{2}{\mathrm{Zr}}_{2}{\mathrm{O}}_{7}$
journal, November 2015
 Lhotel, E.; Petit, S.; Guitteny, S.
 Physical Review Letters, Vol. 115, Issue 19
Ordering due to disorder in a frustrated vector antiferromagnet
journal, April 1989
 Henley, Christopher L.
 Physical Review Letters, Vol. 62, Issue 17
Order by Quantum Disorder in ${\mathrm{Er}}_{2}{\mathrm{Ti}}_{2}{\mathbf{O}}_{7}$
journal, October 2012
 Savary, Lucile; Ross, Kate A.; Gaulin, Bruce D.
 Physical Review Letters, Vol. 109, Issue 16
Phase transition and thermal orderbydisorder in the pyrochlore antiferromagnet Er ${}_{2}$ Ti ${}_{2}$ O ${}_{7}$ : A hightemperature series expansion study
journal, December 2013
 Oitmaa, J.; Singh, R. R. P.; Javanparast, B.
 Physical Review B, Vol. 88, Issue 22
Orderbydisorder near criticality in $XY$ pyrochlore magnets
journal, May 2015
 Javanparast, Behnam; Day, Alexandre G. R.; Hao, Zhihao
 Physical Review B, Vol. 91, Issue 17
Order by virtual crystal field fluctuations in pyrochlore XY antiferromagnets
journal, May 2016
 Rau, Jeffrey G.; Petit, Sylvain; Gingras, Michel J. P.
 Physical Review B, Vol. 93, Issue 18
XY antiferromagnetic ground state in the effective $S=\frac{1}{2}$ pyrochlore ${\mathrm{Yb}}_{2}{\mathrm{Ge}}_{2}{\mathrm{O}}_{7}$
journal, March 2016
 Hallas, A. M.; Gaudet, J.; Wilson, M. N.
 Physical Review B, Vol. 93, Issue 10
Gapless quantum excitations from an icelike splayed ferromagnetic ground state in stoichiometric ${\mathrm{Yb}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$
journal, February 2016
 Gaudet, J.; Ross, K. A.; Kermarrec, E.
 Physical Review B, Vol. 93, Issue 6
Dynamical Splayed Ferromagnetic Ground State in the Quantum Spin Ice ${\mathrm{Yb}}_{2}{\mathrm{Sn}}_{2}{\mathbf{O}}_{7}$
journal, March 2013
 Yaouanc, A.; Dalmas de Réotier, P.; Bonville, P.
 Physical Review Letters, Vol. 110, Issue 12
Universal dynamic magnetism in Yb pyrochlores with disparate ground states
journal, March 2016
 Hallas, A. M.; Gaudet, J.; Butch, N. P.
 Physical Review B, Vol. 93, Issue 10
Projective symmetry group classification of chiral spin liquids
journal, March 2016
 Bieri, Samuel; Lhuillier, Claire; Messio, Laura
 Physical Review B, Vol. 93, Issue 9
Chiral Spin Liquid on a Kagome Antiferromagnet Induced by the DzyaloshinskiiMoriya Interaction
journal, June 2017
 Messio, Laura; Bieri, Samuel; Lhuillier, Claire
 Physical Review Letters, Vol. 118, Issue 26
Quantum spin ice on the breathing pyrochlore lattice
journal, August 2016
 Savary, Lucile; Wang, Xiaoqun; Kee, HaeYoung
 Physical Review B, Vol. 94, Issue 7
Works referencing / citing this record:
Threedimensional topological magnon systems
journal, October 2019
 Kondo, Hiroki; Akagi, Yutaka; Katsura, Hosho
 Physical Review B, Vol. 100, Issue 14