The 2018 study by Biswas and Damle of the triangular-lattice transverse-field Ising antiferromagnet HIsing=J1∑σzσz−B∑σz−Γ∑σxH_{\text{Ising}}=J_1\sum\sigma^z\sigma^z-B\sum\sigma^z-\Gamma\sum\sigma^x left four open issues: (i) the predicted singular divergence of the uniform easy-axis susceptibility, χu(B)∼∣B∣−(4−18η)/(4−9η)\chi_u(B)\sim|B|^{-(4-18\eta)/(4-9\eta)} in the intermediate floating (power-law ordered) phase, had been tested only through its B=0B=0 finite-size analogue because the available quantum cluster algorithm operates at zero longitudinal field; (ii) the finite-size scaling collapse of χu\chi_u at the lower phase boundary T1T_1 was blocked by long Monte Carlo autocorrelation times; (iii) the nature of the transition between antiferromagnetic and ferrimagnetic three-sublattice order (weakly first order versus a fine-tuned power-law intermediate phase) was unsettled; and (iv) no real frustrated easy-axis triangular magnet realizing the Hamiltonian was known. Since then, (iv) has been largely answered by the TmMgGaO4_4 program and other platforms, and nonzero-field quantum Monte Carlo has been demonstrated; one experimental report of the predicted dM/dB∼∣B∣−2/3dM/dB\sim|B|^{-2/3} scaling exists but is contested by independent measurements that find no divergence. The surviving open core is the direct, uncontested verification of the predicted nonzero-field singularity (numerically in the clean model and experimentally), the χu\chi_u finite-size scaling at T1T_1, and the thermodynamic-limit nature of the antiferromagnetic–ferrimagnetic transition.

Background

The transverse-field Ising antiferromagnet on the triangular lattice is defined by HIsing=J1∑⟨RR′⟩σRzσR′z−B∑RσRz−Γ∑RσRx,J1>0,H_{\text{Ising}}=J_1\sum_{\langle\mathbf{R}\mathbf{R}'\rangle}\sigma^z_{\mathbf{R}}\sigma^z_{\mathbf{R}'}-B\sum_{\mathbf{R}}\sigma^z_{\mathbf{R}}-\Gamma\sum_{\mathbf{R}}\sigma^x_{\mathbf{R}},\qquad J_1>0, where σx,z\sigma^{x,z} are Pauli matrices representing S=1/2S=1/2 moments on the sites R\mathbf{R} of the triangular lattice, BB is a uniform magnetic field along the easy axis z^\hat z, and Γ\Gamma is a transverse field, i.e. a field perpendicular to the easy axis that does not couple to the Ising energy but induces quantum flips of the spins. At Γ=B=0\Gamma=B=0 this is the classical triangular-lattice Ising antiferromagnet studied by Wannier: its minimum-energy configurations are macroscopically degenerate (in one-to-one correspondence with dimer covers of the dual honeycomb lattice), and the system remains paramagnetic down to T=0T=0 with spin correlations that decay as a power law at the three-sublattice wavevector Q=(2π/3,2π/3)\mathbf{Q}=(2\pi/3,2\pi/3). For any nonzero Γ\Gamma, quantum fluctuations stabilize — by the order-by-disorder mechanism, in which fluctuations select an ordered state out of a classically degenerate manifold — a ground state with long-range antiferromagnetic three-sublattice order, i.e. a modulation of ⟨σz⟩\langle\sigma^z\rangle at Q\mathbf{Q} carrying no net moment (Moessner and Sondhi 2001; Isakov and Moessner 2003). This order persists up to Γc≈1.7J1\Gamma_c\approx1.7J_1, beyond which the ground state is a quantum paramagnet polarized along x^\hat x.

On heating at fixed Γ<Γc\Gamma<\Gamma_c the three-sublattice order melts in two steps (a scenario established earlier for easy-axis triangular antiferromagnets with larger spin by Heidarian and Damle 2018). Between an upper transition temperature T2T_2 and a lower one T1T_1 the system is in a floating phase with quasi-long-range order, ⟨σz(R)σz(0)⟩∼cos⁡(Q⋅R)/∣R∣η(T),\langle\sigma^z(\mathbf{R})\sigma^z(0)\rangle\sim\cos(\mathbf{Q}\cdot\mathbf{R})/|\mathbf{R}|^{\eta(T)}, where the temperature-dependent exponent η(T)\eta(T) grows from η(T1)=1/9\eta(T_1)=1/9 to η(T2)=1/4\eta(T_2)=1/4. In the coarse-grained description the complex three-sublattice order parameter ψ=∣ψ∣eiθ=∑RσRzeiQ⋅R\psi=|\psi|e^{i\theta}=\sum_{\mathbf{R}}\sigma^z_{\mathbf{R}}e^{i\mathbf{Q}\cdot\mathbf{R}} has a freely fluctuating phase θ\theta in this window; the leading symmetry-allowed anisotropy λ6∣ψ∣6cos⁡6θ\lambda_6|\psi|^6\cos 6\theta (a six-fold clock term distinguishing the six discrete orientations of θ\theta) becomes marginal at η=1/9\eta=1/9 and locks θ\theta below T1T_1, while vortices of θ\theta proliferate at η=1/4\eta=1/4; both boundaries are of Kosterlitz–Thouless (KT) type, topological transitions driven by binding and unbinding of vortex–antivortex pairs, with correlation lengths diverging exponentially rather than as power laws.

The uniform magnetization m=∑RσRz/L2m=\sum_{\mathbf{R}}\sigma^z_{\mathbf{R}}/L^2 couples to the translation-invariant, spin-flip-odd combination ∼cos⁡3θ\sim\cos 3\theta of the order parameter, whose correlations decay with exponent 9η9\eta. From this, a field-theoretical analysis by Damle (2015) predicted that the uniform easy-axis susceptibility χu=∂m/∂B\chi_u=\partial m/\partial B is singular in the floating phase, with the nonzero-field divergence confined to η(T)∈(1/9, 2/9)\eta(T)\in(1/9,\,2/9): at B=0B=0 its finite-size analogue scales as χu(L)∼L2−9η\chi_u(L)\sim L^{2-9\eta}, and at nonzero field χu(B)∼∣B∣−4−18η4−9η,η(T)∈(1/9, 2/9),\chi_u(B)\sim|B|^{-\frac{4-18\eta}{4-9\eta}},\qquad\eta(T)\in(1/9,\,2/9), with exponent 2/32/3 at η=1/9\eta=1/9 decreasing to 00 at η=2/9\eta=2/9; equivalently the induced magnetization vanishes nonanalytically as m∼∣B∣9η/(4−9η)m\sim|B|^{9\eta/(4-9\eta)} when B→0B\to0. The prediction is counter-intuitive — the Hamiltonian has no ferromagnetic coupling and the low-temperature phase has no net moment — and any nonzero BB is a relevant perturbation that pins the floating phase into long-range up-up-down (UUD, two sublattices up and one down) order with a net moment, so the singular law describes the small-BB approach within the field-pinned phase.

Biswas and Damle (2018) tested the B=0B=0 finite-size form using a quantum cluster algorithm (Biswas, Rakala and Damle 2016) within the stochastic series expansion (SSE) framework of quantum Monte Carlo, an exact sampling method that expands the Boltzmann weight in powers of the Hamiltonian in imaginary time. Their cluster update relies on the spin-flip symmetry that BB breaks and therefore operates only at B=0B=0; at Γ=0.8J1\Gamma=0.8J_1 the data confirmed χu∼L2−9η\chi_u\sim L^{2-9\eta} in the floating phase. With an added ferromagnetic second-neighbour coupling J2<0J_2<0 the low-temperature order becomes ferrimagnetic (UUD-type) beyond a threshold J2c≈−0.03J1J_2^c\approx-0.03J_1, and the same singular χu\chi_u was found in the floating phase above it. Landau theory of the free energy of ψ\psi permits, for the transition between antiferromagnetic and ferrimagnetic three-sublattice order, a generic first-order transition or an intermediate mixed phase; an intervening power-law ordered phase requires the higher clock anisotropy λ12\lambda_{12} and beyond to vanish, a fine-tuned limit.

Experimentally, the rare-earth magnet TmMgGaO4_4, with Tm3+^{3+} dipolar-multipolar doublet states on a triangular lattice, has been proposed and widely accepted as a realization of this model with an intrinsic transverse field (Li et al., Nat. Commun. 11, 1111 (2020), DOI: 10.1038/s41467-020-14907-8; Liu, Huang and Chen, Phys. Rev. Research 2, 043013 (2020), DOI: 10.1103/PhysRevResearch.2.043013; Chen, Phys. Rev. Research 1, 033141 (2019), DOI: 10.1103/PhysRevResearch.1.033141), which converts the last question of the 2018 paper — whether any real frustrated easy-axis triangular magnet realizes HIsingH_{\text{Ising}} — into a quantitative question about model fidelity for specific compounds, and makes the singular susceptibility a concrete experimental target.

Problem Statement

Settle the unresolved part of the open-issues list left by Biswas and Damle (2018) for the triangular-lattice transverse-field Ising antiferromagnet HIsing=J1∑⟨RR′⟩σRzσR′z−B∑RσRz−Γ∑RσRxH_{\text{Ising}}=J_1\sum_{\langle\mathbf{R}\mathbf{R}'\rangle}\sigma^z_{\mathbf{R}}\sigma^z_{\mathbf{R}'}-B\sum_{\mathbf{R}}\sigma^z_{\mathbf{R}}-\Gamma\sum_{\mathbf{R}}\sigma^x_{\mathbf{R}} with J1>0J_1>0 (the published quantum Monte Carlo results are for Γ=0.8J1\Gamma=0.8J_1):

  1. Nonzero-field test of the singular susceptibility. Provide a direct test of the predicted divergence χu(B)∼∣B∣−4−18η(T)4−9η(T),η(T)∈(1/9, 2/9),\chi_u(B)\sim|B|^{-\frac{4-18\eta(T)}{4-9\eta(T)}},\qquad\eta(T)\in(1/9,\,2/9), of the uniform easy-axis susceptibility at nonzero longitudinal field BB for temperatures in the part of the floating phase where η(T)∈(1/9, 2/9)\eta(T)\in(1/9,\,2/9) — either (a) a numerical measurement on the clean nearest-neighbour model in the asymptotic small-BB regime, with η(T)\eta(T) determined independently from the Q\mathbf{Q}-wavevector correlations and system sizes large enough to control the crossover to the field-pinned up-up-down ordered state, or (b) an experimental measurement in a material realization of the model, reporting either a confirmation or a measured alternative behavior. Since then, reported dM/dB∼∣B∣−2/3dM/dB\sim|B|^{-2/3} scaling near the lower transition and independent reports of no divergence in the small-field limit disagree, so an experimental test would also have to resolve this post-source disagreement, which currently leaves the experimental status unsettled.

  2. Finite-size scaling of χu\chi_u at the lower boundary. Perform the finite-size scaling collapse of χu\chi_u at the lower phase boundary T1T_1 (where η=1/9\eta=1/9), which the original study could not carry out because the Monte Carlo autocorrelation times of χu\chi_u at low temperature greatly exceed those of the staggered susceptibility χQ\chi_{\mathbf{Q}}.

  3. Nature of the antiferromagnetic-ferrimagnetic transition. Determine the thermodynamic-limit nature of the transition between antiferromagnetic and ferrimagnetic three-sublattice order, for example as driven through J2c≈−0.03J1J_2^c\approx-0.03J_1 by a ferromagnetic second-neighbour coupling J2<0J_2<0 at Γ=0.8J1\Gamma=0.8J_1: weakly first order, an intermediate mixed phase, or an intervening power-law ordered phase arising only under fine-tuning in which the λ12\lambda_{12} and higher clock anisotropies vanish.

These three questions are jointly the unresolved remainder of the source's open items — the direct nonzero-BB test of χu(B)\chi_u(B), the scaling collapse of χu\chi_u at T1T_1, and the nature of the antiferromagnetic–ferrimagnetic transition — and share one objective: completing the characterization of the singular uniform susceptibility and the adjacent phase boundaries associated with the two-step melting of three-sublattice order in this model. The fourth issue of the source — whether any real frustrated easy-axis triangular magnet realizes HIsingH_{\text{Ising}} — has been largely answered by the TmMgGaO4_4 program and other platforms.

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