## This content will become publicly available on August 12, 2020

# 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 long-rangeentangled 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 ${\mathbb{Z}}_{2}$ 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 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) (SC-22)

- OSTI Identifier:
- 1556863

- Alternate Identifier(s):
- OSTI ID: 1556842

- Grant/Contract Number:
- FG02-08ER46524

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review B

- Additional Journal Information:
- Journal Volume: 100; Journal Issue: 7; Journal ID: ISSN 2469-9950

- 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.
```

```
@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 long-rangeentangled 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 Z2 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 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}

}

*Citation information provided by*

Web of Science

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 Spin-Orbit Regime

journal, March 2014

- Witczak-Krempa, 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

##
Spin-Orbit Physics Giving Rise to Novel Phases in Correlated Systems: Iridates and Related Materials

journal, March 2016

- Rau, Jeffrey G.; Lee, Eric Kin-Ho; Kee, Hae-Young
- 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 $S=\frac{1}{2}$ three-dimensional 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 Dipole-Octupole 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 Ground-State 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 multiple-phase 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, Xiao-Gang
- Physical Review B, Vol. 65, Issue 16

##
Classification of spin liquids on the square lattice with strong spin-orbit coupling

journal, November 2014

- Reuther, Johannes; Lee, Shu-Ping; 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, Yuan-Ming
- 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 symmetry-group study of Schwinger fermion mean-field states

journal, June 2011

- Lu, Yuan-Ming; Ran, Ying; Lee, Patrick A.
- Physical Review B, Vol. 83, Issue 22

##
Doping a spin-orbit Mott insulator: Topological superconductivity from the Kitaev-Heisenberg model and possible application to (Na ${}_{2}$ /Li ${}_{2}$ )IrO ${}_{3}$

journal, August 2012

- You, Yi-Zhuang; Kimchi, Itamar; Vishwanath, Ashvin
- Physical Review B, Vol. 86, Issue 8

##
Classification of quantum phases for the star-lattice antiferromagnet via a projective symmetry group analysis

journal, August 2009

- Choy, Ting-Pong; Kim, Yong Baek
- Physical Review B, Vol. 80, Issue 6

##
Interplay of nonsymmorphic symmetry and spin-orbit 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, Yuan-Ming
- 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 dipole-octupole doublets on a pyrochlore lattice

journal, January 2017

- Li, Yao-Dong; Chen, Gang
- Physical Review B, Vol. 95, Issue 4

##
Kagome´- and triangular-lattice Heisenberg antiferromagnets: Ordering from quantum fluctuations and quantum-disordered ground states with unconfined bosonic spinons

journal, June 1992

- Sachdev, Subir
- Physical Review B, Vol. 45, Issue 21

##
Spin-liquid states on the triangular and Kagomé lattices: A projective-symmetry-group 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 spin-liquid 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 mean-field 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

##
Three-dimensional 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 rare-earth pyrochlore magnets: A fermionic parton approach

journal, April 2019

- Sanyal, Sambuddha; Dhochak, Kusum; Bhattacharjee, Subhro
- Physical Review B, Vol. 99, Issue 13

##
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

##
Fluctuation-Driven 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 parity-breaking electronic nematic phase transition in the spin-orbit 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

##
High-pressure 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 All-In–All-Out Ordering in Dipole-Octupole ${\mathrm{Nd}}_{2}{\mathrm{Zr}}_{2}{\mathrm{O}}_{7}$

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 order-by-disorder in the pyrochlore antiferromagnet Er ${}_{2}$ Ti ${}_{2}$ O ${}_{7}$ : A high-temperature series expansion study

journal, December 2013

- Oitmaa, J.; Singh, R. R. P.; Javanparast, B.
- Physical Review B, Vol. 88, Issue 22

##
Order-by-disorder 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 Dzyaloshinskii-Moriya 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, Hae-Young
- Physical Review B, Vol. 94, Issue 7