ArticleField-tuned quantum effects in a triangular-lattice Ising magnet
Graphical abstract
Introduction
Ising model is a stereotype model in modern statistical physics and has revolutionarily advanced our understanding of phase transitions [1]. It can be realized in quantum magnets with a strong easy-axis anisotropy that pins the spin moments towards a fixed axis. Due to the pinning effect, the quantum effect is suppressed, and the physics becomes rather classical [2]. To introduce quantum effects, one usually applies an external magnetic field along the transverse direction and creates a quantum Ising model [3], [4], [5], [6], [7]. Representative examples of this type include the quasi-one-dimensional magnets CoNb2O6 (Ref. [8]), BaCo2V2O8 (Refs. [9], [10], [11]), and SrCo2V2O8 (Refs. [12], [13]), in which quantum criticality and novel transitions have been reported. Nevertheless, the experimental realization of quantum Ising model in two dimensional systems is rare despite decades of theoretical efforts [4], [14], [15], [16], [17], [18]. In contrast to this conventional route, nature provides a distinct example of quantum Ising magnets that build quantum mechanics intrinsically in the system. The observation is that, although the exchange interaction is primarily Ising like, the intrinsic crystal electric field (CEF) splitting of the Ising moment naturally creates quantum effects out of these classically interacting degrees of freedom [19], [20], [21]. These systems are quoted as “intrinsic quantum Ising magnets”.
The triangular-lattice antiferromagnet TmMgGaO4 (Ref. [22]) is a promising candidate for such requirements. Due to the strong spin–orbit coupling (SOC) and CEF splitting, the CEF ground state wave function of Tm3+ ions is dominated by , leading to a large magnetic moment of 6.59 /Tm3+ and Ising spin nature [23]. Thus, quantum effects are expected to be significantly suppressed. Interestingly, however, the TmMgGaO4 CEF ground state is a quasi-doublet composed of two singlets separated by a small energy gap that can be mapped into an effective transverse field (Fig. 1a). In this case, the transverse field, which is intrinsic in origin and homogeneous in general, leads to quantum tunneling effects among various Ising spin configurations and strongly competes with the Ising-type interactions [19], [20], [21]. Different from the coherent quantum fluctuations in quantum spin liquids that lead to long-range entanglement [24], [25], [26], [27], the quantum effects here are of single-ion level and will potentially stabilize three-sublattice ordering. Meanwhile, short-range quantum entanglement emerges from the non-commutativeness between transverse field and Ising interactions, which may renormalize the magnetic interactions. The magnetic properties can be effectively described by the transverse field Ising model (TFIM) [23], [28]where the Ising exchange interactions can be kept within the first few neighbors and are geometrically frustrated. Intriguingly, due to the large g-factor in TmMgGaO4 () [29], the contribution from external longitudinal field, , is comparable to the Ising interactions and quantum fluctuations so that the interplay between quantum and classical contributions can be easily tuned by external field. Therefore, TmMgGaO4 provides a unique platform to manipulate the quantum effects in a controlled manner.
Moreover, in rare-earth materials, due to the complex SOC and CEF splitting, the pseudo-spins can host multipolar behaviors [30], [31], [32]. In the case of TmMgGaO4, we find that the transverse components of the pseudo-spins, and , behave as multipoles that cannot be directly detected by neutron diffraction while the longitudinal one, , remains dipolar. Its zero-field magnetic ground state is an intertwined dipolar and multipolar order in which the dipolar forms a three-sublattice clock phase and the multipolar components are ferro-aligned [23], [28]. The observed spin excitations are in a reasonable agreement with the linear spin wave (LSW) theory, in which, however, only channel is detectable, arising from the coherent spin wave excitations and fluctuations of the multipolar components [32], [33]. In this paper, we continue this research and study the evolution of intrinsic quantum properties of TmMgGaO4 in longitudinal fields.
Section snippets
Heat capacity and neutron diffraction
We start by reviewing the phase transitions of TmMgGaO4 in absence of external field. Although no -shaped transition is present in the heat capacity data (Fig. 1c), well-defined magnetic Bragg peak associated with the three-sublattice spin order is observed at the K point, Q = (1/3, 1/3, 0), at low temperature (Fig. 2c). With temperature decreasing from 20 K, the intensity gradually increases with reduced peak width (Fig. 2a, b). The most abrupt change takes place around 1 K, which corresponds
Discussion and conclusion
It has been suggested theoretically that Phase I will melt in a two-step manner through two Berezinskii-Kosterlitz-Thouless (BKT) transitions and the intermediate BKT phase hosts an emergent U(1) symmetry [15], [16], [17], [28], [42]. Furthermore, it is predicted that in the BKT phase the magnetic susceptibility will diverge in the small longitudinal field limit with a unique scaling behavior [16], [17]. However, this is not observed in our magnetization measurements (Supplementary materials).
Conflict of interest
The authors declare that they have no conflict of interest.
Acknowledgments
This work was supported by the Innovation Program of Shanghai Municipal Education Commission (2017–01-07–00-07-E00018), the National Key R&D Program of the MOST of China (2016YFA0300203, 2016YFA0300500, 2016YFA0301001, and 2018YFE0103200), the National Natural Science Foundation of China (11874119), Shanghai Municipal Science and Technology Major Project (2019SHZDZX04), and the Hong Kong Research Grants Council (17303819 and 17306520). Y.F. and X.T. were supported by the National Natural
Yayuan Qin received her Bachelor degree of Science from Central South University. Now she is a Ph.D. candidate at Department of Physics, Fudan University. Her research mainly focuses on various strong correlated systems, including unconventional superconductors and frustrated systems.
References (44)
- et al.
Experimental observation of Bethe strings
Nature
(2018) Crystal statistics. I. A two-dimensional model with an order-disorder transition
Phys Rev
(1994)Strongly geometrically frustrated magnets
Annu Rev Mater Sci
(1994)Quantum phase transitions
(2011)- et al.
Two-dimensional periodic frustrated Ising models in a transverse field
Phys Rev Lett
(2000) - et al.
Disorder-induced quantum spin liquid in spin ice pyrochlores
Phys Rev Lett
(2017) - et al.
Solitary excitations in one-dimensional magnets
Adv Phys
(1991) - et al.
Quantum critical behavior for a model magnet
Phys Rev Lett
(1996) - et al.
Quantum criticality in an Ising chain: Experimental evidence for emergent symmetry
Science
(2010) - et al.
Topological quantum phase transition in the Ising-like antiferromagnetic spin chain BaCo2V2O8
Nat Phys
(2018)
Magnetic structure and dispersion relation of the S=1/2 quasi-one-dimensional Ising-like antiferromagnet BaCo2V2O8 in a transverse magnetic field
Phy Rev B
Quantum criticality of an Ising-like spin-1/2 antiferromagnetic chain in a transverse magnetic field
Phys Rev Lett
Ising models of quantum frustration
Phys Rev B
Interplay of quantum and thermal fluctuations in a frustrated magnet
Phys Rev B
Melting of three-sublattice order in easy-axis antiferromagnets on triangular and Kagome lattices
Phys Rev Lett
Singular ferromagnetic susceptibility of the transverse-field Ising antiferromagnet on the triangular lattice
Phys Rev B
Tuning the two-step melting of magnetic order in a dipolar Kagome spin ice by quantum fluctuations
Phys Rev B
Collective excitations and magnetic ordering in materials with singlet crystal-field ground state
Phys Rev
Intrinsic transverse field in frustrated quantum Ising magnets: physical origin and quantum effects
Phys Rev Res
Quantum versus classical spin fragmentation in dipolar Kagome ice Ho3Mg2Sb3O14
Phys Rev X
Anisotropic magnetic properties of the triangular plane lattice material TmMgGaO4
Mater Res Bull
Cited by (7)
Exchange-renormalized crystal field excitations in the quantum Ising magnet KTmSe2
2023, Physical Review B
Yayuan Qin received her Bachelor degree of Science from Central South University. Now she is a Ph.D. candidate at Department of Physics, Fudan University. Her research mainly focuses on various strong correlated systems, including unconventional superconductors and frustrated systems.
Gang Chen is a faculty at Department of Physics, University of Hong Kong. He received the B.S. degree from University of Science and Technology of China in 2004, and Ph.D. degree from University of California, Santa Barbara in 2010. He has ever worked as a postdoctoral fellow in University of Colorado Boulder and University of Toronto. Then, he moved to Fudan University as a professor in early 2015 and then University of Hong Kong in 2018. His current research interest includes theoretical study on various condensed matters systems, including quantum materials, topological phases, and strongly correlated systems.
Jun Zhao received the B.S. degree from Tsinghua University in 2002, and the M.S. degree from Institute of Physics, Chinese Academy of Sciences in 2005. He received his Ph.D. degree in Physics from the University of Tennessee, Knoxville in 2010. Then, he worked at UC Berkeley as a Miller Fellow. He is currently the Xie Xide Junior Chair Professor at Department of Physics, Fudan University, where he has been a faculty member since 2012. His research mainly focuses on using various neutron scattering techniques to study the phase transitions and spin dynamics of strongly correlated systems, including unconventional superconductors and quantum magnets.
- 1
These authors contributed equally to this work.
- 2
Current address: Heinz Maier-Leibnitz Zentrum (MLZ), Technische Universität München, Garching 85748, Germany.