Competing gauge fields and entropically driven spin liquid to spin liquid transition in non-Kramers pyrochlores
Edited by J. C. Davis, University of Oxford, Oxford, United Kingdom; received February 20, 2024; accepted June 10, 2024
Significance
Competing interactions in magnetic systems frustrate the development of long-range order, possibly down to absolute zero temperature, giving rise to a spin liquid. Gauge fields, akin to those used to describe the fundamental forces in the universe, are the mathematical objects of choice to describe spin liquids. In a model relevant to real magnetic pyrochlore materials, we uncover a spin liquid described by combined vector-like and matrix-like gauge fields. Classically, similarly to the liquid-to-liquid transition observed in some molecular liquids, this state disappears at low temperatures and gives way to a spin liquid with extensive entropy and with only the vector-like field thermally fluctuating. Quantum mechanically, the ground state is a spin liquid described by both gauge fields.
Abstract
Gauge theories are powerful theoretical physics tools that allow complex phenomena to be reduced to simple principles and are used in both high-energy and condensed matter physics. In the latter context, gauge theories are becoming increasingly popular for capturing the intricate spin correlations in spin liquids, exotic states of matter in which the dynamics of quantum spins never ceases, even at absolute zero temperature. We consider a spin system on a three-dimensional pyrochlore lattice where emergent gauge fields not only describe the spin liquid behavior at zero temperature but crucially determine the system’s temperature evolution, with distinct gauge fields giving rise to different spin liquid phases in separate temperature regimes. Focusing first on classical spins, in an intermediate temperature regime, the system shows an unusual coexistence of emergent vector and tensor gauge fields where the former is known from classical spin ice systems while the latter has been associated with fractonic quasiparticles, a peculiar type of excitation with restricted mobility. Upon cooling, the system transitions into a low-temperature phase where an entropic selection mechanism depopulates the degrees of freedom associated with the tensor gauge field, rendering the system spin-ice-like. We further provide numerical evidence that in the corresponding quantum model, a spin liquid with coexisting vector and tensor gauge fields has a finite window of stability in the parameter space of spin interactions down to zero temperature. Finally, we discuss the relevance of our findings for non-Kramers magnetic pyrochlore materials.
Data, Materials, and Software Availability
Data in addition to those presented in the main text and in the SI Appendix, along with computer codes developed for the present study, are available in the following database: https://github.com/daniel-lozano/R1R2_spin_liquid_data (70).
Acknowledgments
We acknowledge useful discussions with Kai Chung, Alex Hickey, and Peter Holdsworth. The work at the University of Waterloo was supported by the Natural Sciences and Engineering Research Council of Canada and the Canada Research Chair (Tier 1, M.J.P.G.) program. Numerical simulations done at Waterloo were performed thanks to the computational resources of the Digital Research Alliance of Canada. D.L.-G. acknowledges the computing time provided by the Digital Research Alliance of Canada and the financial support from the Deutsche Forschungsgemeinschaft through the Hallwachs-Röntgen Postdoc Program of the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter—ct.qmat (EXC 2147, project-id 390858490) and through Sonderforschungsbereich 1143 (project-id 247310070). J.R. and V.N. gratefully acknowledge the computing time provided to them on the high-performance computer Noctua 2 at the Verbund für Nationales Hochleistungsrechnen Center PC2. This is funded by the Federal Ministry of Education and Research and the state governments participating on the basis of the resolutions of the Gemeinsame Wissenschaftskonferenz for the national high-performance computing at universities (https://www.nhr-verein.de/unsere-partner). Some of the computations for this research were performed using computing resources under project hpc-prf-pm2frg. V.N. would like to thank the High Performance Computing Service of Zentraleinrichtung für Datenverarbeitung and Tron cluster service at the Department of Physics, Freie Universität Berlin, for computing time. The work of Y.I. and M.J.P.G. was performed, in part, at the Aspen Center for Physics, which is supported by NSF grant PHY-2210452. The work of R.R.P.S. was supported by the NSF grant DMR-1855111. J.O. acknowledges computing support provided by the Australian National Computation Infrastructure program. J.R. thanks Indian Institute of Technology (IIT) Madras for a Visiting Faculty Fellow position under the Institute of Eminence program. The participation of Y.I. at the Aspen Center for Physics was supported by the Simons Foundation. This research was supported in part by the NSF under Grant No. NSF PHY-1748958. Y.I. acknowledges support by the International Centre for Theoretical Physics through the Associates Programme and from the Simons Foundation through grant number 284558FY19, IIT Madras through the QuCenDiEM Centre of Excellence (Project No. SP22231244CPETWOQCDHOC), the International Centre for Theoretical Sciences (ICTS), Bengaluru, India during a visit for participating in the program “Frustrated Metals and Insulators” (Code: ICTS/frumi2022/9). Y.I. acknowledges the use of the computing resources at High Performance Computing Environment, IIT Madras.
Author contributions
Y.I., J.R., and M.J.P.G. designed research; D.L.-G., V.N., J.O., and R.R.P.S. performed research; D.L.-G., V.N., J.O., R.R.P.S., J.R., and M.J.P.G. analyzed data; and D.L.-G., V.N., R.R.P.S., Y.I., J.R., and M.J.P.G. wrote the paper.
Competing interests
The authors declare no competing interest.
Supporting Information
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References
1
F. J. Wegner, Duality in generalized Ising models and phase transitions without local order parameters. J. Math. Phys. 12, 2259–2272 (1971).
2
P. A. Lee, N. Nagaosa, X. G. Wen, Doping a Mott insulator: Physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).
3
X. G. Wen, Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons (Oxford University Press, 2007).
4
C. Lacroix, P. Mendels, F. Mila, Introduction to Frustrated Magnetism (Springer, 2011).
5
L. Balents, Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).
6
Y. Zhou, K. Kanoda, T. K. Ng, Quantum spin liquid states. Rev. Mod. Phys. 89, 025003 (2017).
7
M. J. P. Gingras, P. A. McClarty, Quantum spin ice: A search for gapless quantum spin liquids in pyrochlore magnets. Rep. Prog. Phys. 77, 056501 (2014).
8
C. Broholm et al., Quantum spin liquids. Science 367, eaay0668 (2020).
9
L. Savary, L. Balents, Quantum spin liquids: A review. Rep. Prog. Phys. 80, 016502 (2016).
10
G. Misguich, D. Serban, V. Pasquier, Quantum dimer model on the Kagome lattice: Solvable dimer-liquid and Ising gauge theory. Phys. Rev. Lett. 89, 137202 (2002).
11
O. Benton, R. Moessner, Topological route to new and unusual coulomb spin liquids. Phys. Rev. Lett. 127, 107202 (2021).
12
A. Wietek, S. Capponi, A. M. Läuchli, Quantum electrodynamics in 2 + 1 dimensions as the organizing principle of a triangular lattice antiferromagnet Phys. Rev. X 14, 021010 (2024).
13
X. Y. Song, C. Wang, A. Vishwanath, Y. C. He, Unifying description of competing orders in two-dimensional quantum magnets. Nat. Commun. 10, 4254 (2019).
14
C. L. Henley, The coulomb phase in frustrated systems. Annu. Rev. Condens. Matter Phys. 1, 179 (2010).
15
C. Castelnovo, R. Moessner, S. L. Sondhi, Spin ice, fractionalization, and topological order. Annu. Rev. Condens. Matter Phys. 3, 35 (2012).
16
H. Yan, O. Benton, L. D. C. Jaubert, N. Shannon, Rank-2 U(1) spin liquid on the breathing pyrochlore lattice. Phys. Rev. Lett. 124, 127203 (2020).
17
O. Benton, L. D. C. Jaubert, H. Yan, N. Shannon, A spin-liquid with pinch-line singularities on the pyrochlore lattice. Nat. Commun. 7, 11572 (2016).
18
H. D. Zhou et al., Chemical pressure effects on pyrochlore spin ice. Phys. Rev. Lett. 108, 207206 (2012).
19
T. Fennell et al., Magnetic coulomb phase in the spin ice Ho2Ti2O7. Science 326, 415–417 (2009).
20
D. J. P. Morris et al., Dirac strings and magnetic monopoles in the spin ice Dy2Ti2O7. Science 326, 411–414 (2009).
21
C. L. Henley, Power-law spin correlations in pyrochlore antiferromagnets. Phys. Rev. B 71, 014424 (2005).
22
K. T. K. Chung et al., Probing flat band physics in spin ice systems via polarized neutron scattering. Phys. Rev. Lett. 128, 107201 (2022).
23
D. Bergman, J. Alicea, E. Gull, S. Trebst, L. Balents, Order-by-disorder and spiral spin-liquid in frustrated diamond-lattice antiferromagnets. Nat. Phys. 3, 487–491 (2007).
24
T. Mizoguchi, L. D. C. Jaubert, R. Moessner, M. Udagawa, Magnetic clustering, half-moons, and shadow pinch points as signals of a proximate Coulomb phase in frustrated Heisenberg magnets. Phys. Rev. B 98, 144446 (2018).
25
F. L. Buessen, M. Hering, J. Reuther, S. Trebst, Quantum spin liquids in frustrated spin-1 diamond antiferromagnets. Phys. Rev. Lett. 120, 057201 (2018).
26
Y. Iqbal et al., Quantum and classical phases of the pyrochlore Heisenberg model with competing interactions. Phys. Rev. X 9, 011005 (2019).
27
N. Niggemann, Y. Iqbal, J. Reuther, Quantum effects on unconventional pinch point singularities. Phys. Rev. Lett. 130, 196601 (2023).
28
F. Kolley, S. Depenbrock, I. P. McCulloch, U. Schollwöck, V. Alba, Phase diagram of the J1-J2 Heisenberg model on the Kagome lattice. Phys. Rev. B 91, 104418 (2015).
29
M. Pretko, X. Chen, Y. You, Fracton phases of matter. Int. J. Mod. Phys. A 35, 2030003 (2020).
30
S. Onoda, Y. Tanaka, Quantum fluctuations in the effective pseudospin- model for magnetic pyrochlore oxides. Phys. Rev. B 83, 094411 (2011).
31
S. Lee, S. Onoda, L. Balents, Generic quantum spin ice. Phys. Rev. B 86, 104412 (2012).
32
J. G. Rau, M. J. P. Gingras, Frustrated quantum rare-earth pyrochlores. Annu. Rev. Condens. Matter Phys. 10, 357–386 (2019).
33
H. Yan, A. Sanders, C. Castelnovo, A. H. Nevidomskyy, Experimentally tunable QED in dipolar-octupolar quantum spin ice. arXiv [Preprint] (2023). http://arxiv.org/abs/2312.11641 (Accessed 10 May 2024).
34
A. Sanders, C. Castelnovo, Vison crystal in quantum spin ice on the breathing pyrochlore lattice. Phys. Rev. B 109, 094426 (2024).
35
J. Villain, Insulating spin glasses. Z. Phys. B 33, 31 (1979).
36
Y. Katayama et al., A first-order liquid-liquid phase transition in phosphorus. Nature 403, 170 (2000).
37
L. Henry et al., Liquid-liquid transition and critical point in sulfur. Nature 584, 382 (2020).
38
J. N. Hallén, C. Castelnovo, R. Moessner, Thermodynamics and fractal dynamics of a nematic spin ice: A doubly frustrated pyrochlore Ising magnet. Phys. Rev. B 109, 014438 (2024).
39
J. S. Gardner et al., Cooperative paramagnetism in the geometrically frustrated pyrochlore antiferromagnet Tb2Ti2O7. Phys. Rev. Lett. 82, 1012–1015 (1999).
40
H. Takatsu et al., Order in the frustrated pyrochlore Tb2+xTi2−xO7+y. Phys. Rev. Lett. 116, 217201 (2016).
41
H. Yan, O. Benton, L. Jaubert, N. Shannon, Theory of multiple-phase competition in pyrochlore magnets with anisotropic exchange with application to Yb2Ti2O7, Er2Ti2O7, and Er2Sn2O7. Phys. Rev. B 95, 094422 (2017).
42
V. Noculak et al., Classical and quantum phases of the pyrochlore magnet with Heisenberg and Dzyaloshinskii-Moriya interactions. Phys. Rev. B 107, 214414 (2023).
43
M. Taillefumier, O. Benton, H. Yan, L. D. C. Jaubert, N. Shannon, Competing spin liquids and hidden spin-nematic order in spin ice with frustrated transverse exchange. Phys. Rev. X 7, 041057 (2017).
44
Y. Kato, S. Onoda, Numerical evidence of quantum melting of spin ice: Quantum-to-classical crossover. Phys. Rev. Lett. 115, 077202 (2015).
45
R. Moessner, J. T. Chalker, Properties of a classical spin liquid: The Heisenberg pyrochlore antiferromagnet. Phys. Rev. Lett. 80, 2929–2932 (1998).
46
R. Moessner, J. T. Chalker, Low-temperature properties of classical geometrically frustrated antiferromagnets. Phys. Rev. B 58, 12049–12062 (1998).
47
C. L. Henley, Effective Hamiltonians and dilution effects in Kagome and related anti-ferromagnets. Can. J. Phys. 79, 1307 (2001).
48
P. H. Conlon, J. T. Chalker, Absent pinch points and emergent clusters: Further neighbor interactions in the pyrochlore Heisenberg antiferromagnet. Phys. Rev. B 81, 224413 (2010).
49
M. Elhajal, B. Canals, R. Sunyer, C. Lacroix, Ordering in the pyrochlore antiferromagnet due to Dzyaloshinsky-Mriya interactions. Phys. Rev. B 71, 094420 (2005).
50
H. Kadowaki, H. Takatsu, T. Taniguchi, B. Fak, J. Ollivier, Composite spin and quadrupole wave in the ordered phase of Tb2+xTi2−xO7+y SPIN 5, 540003 (2015).
51
H. Yan, O. Benton, A. H. Nevidomskyy, R. Moessner, Classification of classical spin liquids: Detailed formalism and suite of examples. Phys. Rev. B 109, 174421 (2024).
52
D. A. Garanin, B. Canals, Classical spin liquid: Exact solution for the infinite-component antiferromagnetic model on the kagomé lattice. Phys. Rev. B 59, 443–456 (1999).
53
S. V. Isakov, K. Gregor, R. Moessner, S. L. Sondhi, Dipolar spin correlations in classical pyrochlore magnets. Phys. Rev. Lett. 93, 167204 (2004).
54
C. L. Henley, Ordering due to disorder in a frustrated vector antiferromagnet. Phys. Rev. Lett. 62, 2056–2059 (1989).
55
A. Prem, S. Vijay, Y. Z. Chou, M. Pretko, R. M. Nandkishore, Pinch point singularities of tensor spin liquids. Phys. Rev. B 98, 165140 (2018).
56
J. T. Chalker, P. C. W. Holdsworth, E. F. Shender, Hidden order in a frustrated system: Properties of the Heisenberg kagomé antiferromagnet. Phys. Rev. Lett. 68, 855–858 (1992).
57
L. R. Walker, R. E. Walstedt, Computer model of metallic spin-glasses. Phys. Rev. B 22, 3816–3842 (1980).
58
B. Javanparast, A. G. R. Day, Z. Hao, M. J. P. Gingras, Order-by-disorder near criticality in xy pyrochlore magnets. Phys. Rev. B 91, 174424 (2015).
59
J. Reuther, P. Wölfle, J1-J2 frustrated two-dimensional Heisenberg model: Random phase approximation and functional renormalization group. Phys. Rev. B 81, 144410 (2010).
60
T. Müller et al., Pseudo-fermion functional renormalization group for spin models. IOP Science 87, 036501 (2024).
61
J. Oitmaa, C. Hamer, W. Zheng, Series Expansion Methods for Strongly Interacting Lattice Models (Cambridge University Press, 2006).
62
L. Ye, M. E. Sorensen, M. D. Bachmann, I. R. Fisher, Measurement of the magnetic octupole susceptibility of PrV2Al20. arXiv [Preprint] (2023). https://doi.org/10.48550/arXiv.2309.04633 (Accessed 20 October 2023).
63
A. W. C. Wong, Z. Hao, M. J. P. Gingras, Ground state phase diagram of generic xy pyrochlore magnets with quantum fluctuations. Phys. Rev. B 88, 144402 (2013).
64
J. D. Alzate-Cardona, D. Sabogal-Suárez, R. F. L. Evans, E. Restrepo-Parra, Optimal phase space sampling for Monte Carlo simulations of Heisenberg spin systems. J. Phys. Condens. Matter 31, 095802 (2019).
65
M. E. Zhitomirsky, M. V. Gvozdikova, P. C. W. Holdsworth, R. Moessner, Quantum order by disorder and accidental soft mode in Er2Ti2O7. Phys. Rev. Lett. 109, 077204 (2012).
66
M. Creutz, Overrelaxation and Monte Carlo simulation. Phys. Rev. D 36, 515–519 (1987).
67
H. Shinaoka, Y. Tomita, Y. Motome, Loop algorithm for classical antiferromagnetic Heisenberg models with biquadratic interactions. J. Phys. Conf. Ser. 320, 012009 (2011).
68
H. Shinaoka, Y. Motome, Loop algorithm for classical Heisenberg models with spin-ice type degeneracy. Phys. Rev. B 82, 134420 (2010).
69
M. Galassi et al., GNU Scientific Library Reference Manual (Network Theory Ltd., ed. 3, 2009).
70
D. Lozano-Gómez et al., R1R2_spin_liquid_data. GitHub. https://github.com/daniel-lozano/R1R2_spin_liquid_data. Deposited 26 July 2024.
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Copyright © 2024 the Author(s). Published by PNAS. This article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).
Data, Materials, and Software Availability
Data in addition to those presented in the main text and in the SI Appendix, along with computer codes developed for the present study, are available in the following database: https://github.com/daniel-lozano/R1R2_spin_liquid_data (70).
Submission history
Received: February 20, 2024
Accepted: June 10, 2024
Published online: August 28, 2024
Published in issue: September 3, 2024
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Acknowledgments
We acknowledge useful discussions with Kai Chung, Alex Hickey, and Peter Holdsworth. The work at the University of Waterloo was supported by the Natural Sciences and Engineering Research Council of Canada and the Canada Research Chair (Tier 1, M.J.P.G.) program. Numerical simulations done at Waterloo were performed thanks to the computational resources of the Digital Research Alliance of Canada. D.L.-G. acknowledges the computing time provided by the Digital Research Alliance of Canada and the financial support from the Deutsche Forschungsgemeinschaft through the Hallwachs-Röntgen Postdoc Program of the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter—ct.qmat (EXC 2147, project-id 390858490) and through Sonderforschungsbereich 1143 (project-id 247310070). J.R. and V.N. gratefully acknowledge the computing time provided to them on the high-performance computer Noctua 2 at the Verbund für Nationales Hochleistungsrechnen Center PC2. This is funded by the Federal Ministry of Education and Research and the state governments participating on the basis of the resolutions of the Gemeinsame Wissenschaftskonferenz for the national high-performance computing at universities (https://www.nhr-verein.de/unsere-partner). Some of the computations for this research were performed using computing resources under project hpc-prf-pm2frg. V.N. would like to thank the High Performance Computing Service of Zentraleinrichtung für Datenverarbeitung and Tron cluster service at the Department of Physics, Freie Universität Berlin, for computing time. The work of Y.I. and M.J.P.G. was performed, in part, at the Aspen Center for Physics, which is supported by NSF grant PHY-2210452. The work of R.R.P.S. was supported by the NSF grant DMR-1855111. J.O. acknowledges computing support provided by the Australian National Computation Infrastructure program. J.R. thanks Indian Institute of Technology (IIT) Madras for a Visiting Faculty Fellow position under the Institute of Eminence program. The participation of Y.I. at the Aspen Center for Physics was supported by the Simons Foundation. This research was supported in part by the NSF under Grant No. NSF PHY-1748958. Y.I. acknowledges support by the International Centre for Theoretical Physics through the Associates Programme and from the Simons Foundation through grant number 284558FY19, IIT Madras through the QuCenDiEM Centre of Excellence (Project No. SP22231244CPETWOQCDHOC), the International Centre for Theoretical Sciences (ICTS), Bengaluru, India during a visit for participating in the program “Frustrated Metals and Insulators” (Code: ICTS/frumi2022/9). Y.I. acknowledges the use of the computing resources at High Performance Computing Environment, IIT Madras.
Author contributions
Y.I., J.R., and M.J.P.G. designed research; D.L.-G., V.N., J.O., and R.R.P.S. performed research; D.L.-G., V.N., J.O., R.R.P.S., J.R., and M.J.P.G. analyzed data; and D.L.-G., V.N., R.R.P.S., Y.I., J.R., and M.J.P.G. wrote the paper.
Competing interests
The authors declare no competing interest.
Notes
This article is a PNAS Direct Submission.
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Competing gauge fields and entropically driven spin liquid to spin liquid transition in non-Kramers pyrochlores, Proc. Natl. Acad. Sci. U.S.A.
121 (36) e2403487121,
https://doi.org/10.1073/pnas.2403487121
(2024).
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