Absence of zero-point entropy in a triangular Ising antiferromagnet

Frustrated Ising magnets host exotic excitations, such as magnetic monopoles in spin ice. The ground state (GS) in this case is characterized by an extensive degeneracy and associated residual entropy going back to the pioneering work by G. Wannier who established large residual entropy of nearly 50%Rln2 per mole spins in a triangular Ising antiferromagnet (TIAF) already in 1950. Here, we endeavor to verify this result experimentally using TmMgGaO4, a novel rare-earth-based frustrated antiferromagnet with Ising spins arranged on a perfect triangular lattice. Contrary to theoretical expectations, we find almost no residual entropy and ascribe this result to the presence of a weak second-neighbor coupling J2zz ~ 0.09J1zz that lifts the GS degeneracy and gives rise to several ordered states, the stripe order, 1/3-plateau, and 1/2-plateau. TmMgGaO4 gives experimental access to these novel phases of Ising spins on the triangular lattice.

It is also worth noting that the zero-temperature limit of Cm/T is nearly zero for pure TmMgGaO4 at odds with the highly diluted sample, where the value is finite. Therefore, interactions between Ising spins open a gap in the spectrum of TmMgGaO4. At 1.9 K and above 10 T, the magnetization of TmMgGaO4 shows a full polarization along the c-axis (see Fig. 1a), with a small Van Vleck susceptibility χ|| vv = 0.003 (1) cm 3 /mol Tm, and with an average pseudospin-1/2 g-factor g|| = 13.18 (1), which is close to the upper limit of 2JgJ = 14 for Tm 3+ , indicating the nearly classical CEF GS quasidoublet mainly formed by the |±6﹥states. Assuming pure classical Ising nature of the pesudospins, an effective NN Hamiltonian for TmMgGaO4 reads as 19,20 , (1) Through the Curie-Weiss fit to the susceptibility along the c-axis between 30 and 60 K (see Fig. 1c), we obtain an effective moment of μeff = 6.5(1)μB ~ g||μB/2 and θw = - 16.44(3) K. And we further get J1 zz ~ -2θw/3 ~ 10 K.
Absence of zero-point entropy. The magnetic heat capacity (Cm) of TmMgGaO4 can be determined accurately by subtracting the heat capacity of the non-magnetic LuMgGaO4 as phonon contribution (Supplementary Fig. 1) 8,9 . We further obtain the magnetic entropy by integrating Cm/T over T. The magnetic entropy of TmMgGaO4 shows a broad plateau of Rln2 between 30 and 60 K, and sharply decreases down to ~ 0.6%Rln2 at 0.1 K (see Fig. 1b), confirming the effective pseudospin-1/2 physics below ~ 60 K 9,18 .
Surprisingly, at ~ 0.1 K and 0 T the residual electronic spin entropy of TmMgGaO4 is measured to be almost zero (see Fig. 1b), Sm ≤ 0.6%Rln2 << S0 t , which conforms to the third law of thermodynamics. Whereas the heat capacity is smooth down to the lowest temperature of our measurement, a divergence of the field-cooled and zero-field-cooled susceptibilities indicates spin freezing below Tc ~ 0.27 K at 0.1 T (see Fig. 1e). These low-T observations clearly conflict to the NN TIAF model of Eq. (1), which predicts the macroscopically degenerate GS [5][6][7] . Therefore, other interactions or perturbations must be taken into account to fully understand the low-T magnetism of TmMgGaO4.
We explore them by studying thermodynamic properties in longitudinal magnetic field.
Low-T thermodynamic properties. The magnetization measured at low temperatures shows interesting features (see Fig. 3a). After taking derivative with respect to the field, magnetic susceptibility is obtained (see Fig. 3b). At 2 K, the susceptibility shows two very broad humps at μ0Hh ~ 0.4 and ~ 3.6 T respectively, which is consistent with the  . 3b). Further cooling to 40 mK has little effect, even though temperature and thermal fluctuations decrease by a factor of 5. Around the transition fields, the corresponding peaks are also clearly observed in the magnetic Grüneisen ratio (see Fig. 3c) and heat capacity (see Fig. 3d) measurements. At 0.3 K and above ~ 3.6 T, the spin system of TmMgGaO4 is almost fully polarized, so it contributes little to the heat capacity (see Fig. 3d).
Three field-induced transitions are unexpected in the NN TIAF model, where only two field-induced states, the 1/3-plateau and fully polarized, should occur 21 . On the other hand, adding the NNN coupling J2 zz allows for additional field-induced phases and may explain the occurrence of three transitions 22 . In the following, we use the modified Hamiltonian, to model the magnetization process of TmMgGaO4.
Phenomenologically, the susceptibility at 40 mK can be well fitted by three Lorentzian peaks (see Fig. 3b). This way, three transition fields and the associated changes in the magnetization are determined. On the other hand, the broadened nature of the transitions is not captured by Eq. (2), as the calculated magnetization curve at 0 K should be step-like (M||/M|| s = 0, 1/3, 1/2, or 1, see Fig. 2 and 3a), and three delta-peaks should be seen in the derivative. The broadening can not be caused by thermal fluctuations, as T = 40 mK is two orders of magnitude smaller than the energy scale of the broadening (> 0.8 T). Moreover, no significant differences are observed between the 0.3 K and 40 mK data (see Fig. 3b). Therefore, a distribution of g|| and magnetic couplings J1 zz & J2 zz due to structural randomness should be taken into account, and a much better agreement is indeed achieved by assuming Lorentzian distributions of these three parameters (see Fig. 3a). We thus obtain g|| = 12.6 (FWHM = 1.5), J1 zz = 9. In zero field, two broad humps are observed in the magnetic heat capacity at Th ~ 0.9 K and ~ 2 K (see Fig. 4a). Under magnetic field up to ~ 1.5 T, a very sharp λ-peak at Tc ~ 1 -2 K appears. Upon further increase in the field, the sharp peak gradually becomes a broad hump again (see Fig. 4a and Supplementary Fig. 3). The transition is most sharp at ~ 1.5 T (Tc ~ 1.6 K), it should be mainly driven by the strongest NN coupling J1 zz .
The low-T magnetic phase diagram of TmMgGaO4 is summarized in Fig. 4b.
According to the earlier study of the J1 zz -J2 zz TIAF model on the triangular lattice 22 , we identify the zero-field state as stripe order (affected by spin freezing), whereas the field-induced phases are the 1/3-plateau, 1/2-plateau, and the fully polarized state (see Fig. 2).

Discussion
We have shown that the random J1 zz -J2 zz TIAF model captures main features of the low-T magnetism of TmMgGaO4: 1) The increase in the magnetization around the transition fields equals to 0.37, 0.17, and 0.46M|| s according to the areas of Lorentzian peaks in Fig. 3b between 0 and 6 T. These values are consistent, respectively, with 1/3, 1/6, and 1/2M|| s expected for the classical J1 zz -J2 zz TIAF model (see Fig. 2b-e) 22 . Here, M|| s = g||μB/2 is the saturated magnetization.
3) At ~ 0 T, the transition temperature determined from susceptibility measurements, Tc ~ 0.27 K (see Fig. 1e), and the position of the lower temperature hump in the heat capacity, Th ~ 0.9 K (see Fig. 4a), are comparable to the median value of J2 zz , which supports our hypothesis that spin freezing toward the stripe state (see Fig. 2b) is driven by J2 zz . 4) By fitting the low-T part of zero-field Cm-Cn with a power-law function, Cm ~ T γ , we arrive at a large exponent of γ = 2.60(1) that exceeds γ = 2 in an ordered twodimensional antiferromagnet (see Fig. 1f). We conjecture that the low-T Cm shows gapped behavior up to Tc ~ 0.27 K, Cm ~ exp(-Δ0/T) (see Fig. 1f 19,20 . In TmMgGaO4, the average effective pseudospin-1/2 g-factor of g|| = 13.18(1) (see Fig. 1a) is slightly lower than the upper limit of 2JgJ = 14.
Thus, it is possible that the weights of smaller angular moment states in the GS CEF quasidoublet influence the low-temperature magnetism and trigger quantum fluctuations. Further insight into these effects can be obtained by studying magnetic excitations and dynamics via inelastic neutron scattering and muon spin relaxation, respectively.
In conclusion, TmMgGaO4 is an Ising antiferromagnet featuring the perfect triangular lattice of non-Kramer Tm 3+ ions that host robust Ising spins through the formation of the low-lying CEF quasidoublet as a result of structural randomness. Our comprehensive milli-Kelvin study reveals a weak NNN interaction, J2 zz ~ 0.09J1 zz , which is large enough to release all the zero-point entropy expected in the NN TIAF model. We propose that below 0.27 K a frozen stripe state is formed in zero field, whereas field-induced states include the 1/3-plateau, 1/2-plateau, and fully spin- Millikelvin measurement below 2 K. The heat capacity of the TmMgGaO4, Tm0.04Lu0.96MgGaO4, and Yb0.04Lu0.96MgGaO4 single crystals was measured by a home-built setup in a 3 He-4 He dilution refrigerator between 0.1 and 2.0 K at magnetic fields up to 5 T applied along the c-axis. Below ~ 0.3 K and in applied fields, the nuclear contribution becomes prominent, and the measured thermal relaxation slightly deviates from the two-tau model at short times 18 . We chose to exclude the 0.2 and 0.5 T heat capacity data below 0.12 and 0.2 K respectively, as the deviation is relatively large (adj. shows no size-effects and no small steps (see Fig. 3a). Four different phases with stripe, 1/3-plateau, 1/2-plateau, and ferromagnetic spin correlations (see Fig. 2) are separated by three critical/transition fields, μ0Hc1 = 6J2 zz /(μBg||), μ0Hc2 = 3(J1 zz -3J2 zz )/(μBg||), and μ0Hc3 = 3(J1 zz +J2 zz )/(μBg||), respectively. Our results are fully consistent with previous reports on the J1 zz > J2 zz case 21,22 .
Data availability. The data sets generated during and/or analysed during the current study are available from the corresponding author on request.