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arXiv:2512.08768 · Physical review. B./Physical review. B

Triangular J 1 − J 2 Heisenberg antiferromagnet in a magnetic field

Thomas Bader, Shi Feng, Sasank Budaraju, Federico Becca, Johannes Knolle, Frank Pollmann

Published May 22, 2026• 3 citations
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Abstract

The behavior of the paradigmatic J 1 − J 2 triangular lattice Heisenberg antiferromagnet in a magnetic field remains unsettled despite decades of study. We map out the phase diagram using three complementary approaches, including self-consistent nonlinear spin-wave theory, density-matrix renormalization group, and variational Monte Carlo. This combined analysis resolves the competition among different field-induced magnetic orders and magnetization plateaus across the classically frustrated parameter range. In particular, there is a finite range in the parameter regime around J 2 / J 1 = 1 8 in which (i) upon the application of the external field, the gapless quantum spin liquid acquires a finite density of monopoles, and (ii) by further increasing the field, two plateaus are clearly obtained at m = 1 3 and m = 1 2 . We discuss the experimental importance of the consecutive magnetization plateau transitions as a signature of an underlying quantum spin-liquid phase.

DOI
10.1103/rwhz-jf5b

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The behavior of the paradigmatic J 1 − J 2 triangular lattice Heisenberg antiferromagnet in a magnetic field remains unsettled despite decades of study. We map out the phase diagram using three complementary approaches, including self-consistent nonlinear spin-wave theory, density-matrix renormalization group, and variational Monte Carlo. This combined analysis resolves the competition among different field-induced magnetic orders and magnetization plateaus across the classically frustrated parameter range. In particular, there is a finite range in the parameter regime around J 2 / J 1 = 1 8 in which (i) upon the application of the external field, the gapless quantum spin liquid acquires a finite density of monopoles, and (ii) by further increasing the field, two plateaus are clearly obtained at m = 1 3 and m = 1 2 . We discuss the experimental importance of the consecutive magnetization plateau transitions as a signature of an underlying quantum spin-liquid phase.
We investigate the spin-1/2 J_{1}-J_{2} triangular-lattice Heisenberg antiferromagnet in a magnetic field by combining large-scale density matrix renormalization group (DMRG) simulations with self-consistent spin-wave theory. The resulting field-coupling phase diagram reveals that quantum fluctuations stabilize coplanar order across the entire parameter range, giving rise to a characteristic sequence of magnetization plateaux. Near the quantum-spin-liquid window 0.06≲J_{2}/J_{1}≲0.16, which extends to magnetic field B∼J_{1}, we identify overlapping m=1/3 and m=1/2 plateaux-a distinctive hallmark of the system's proximity to the low-field spin-liquid regime. The excellent quantitative agreement between DMRG and self-consistent one-loop spin-wave calculations demonstrates that semiclassical approaches can reliably capture and parameterize the plateau phases of triangular quantum antiferromagnets.
Previous studies of the S=1/2 triangular-lattice J_{1}-J_{2} Heisenberg antiferromagnet have inferred the existence of a nonmagnetic ground-state phase for an intermediate range of J_{2}, but disagree concerning whether it is a gapped Z_{2} quantum spin liquid (QSL), a gapless (Dirac) QSL, or a weakly symmetry-broken phase. Using an improved dynamical density-matrix renormalization group method, we investigate the relevant intermediate J_{2} regime for cylinders with circumferences from 6 to 9. Depending on the initial state and boundary conditions, we find two distinct variational states. The higher energy state is consistent with a Dirac QSL. In the lower-energy state, both the static and dynamical properties are qualitatively similar to the magnetically ordered state at J_{2}=0, suggestive of either a weakly magnetically ordered non-QSL or a gapped QSL proximate to a continuous transition to such an ordered state.
The magnetic behavior of the spin-1/2 Heisenberg antiferromagnet on the extended Lieb lattice is examined in the presence of an external magnetic field. Using density matrix renormalization group calculations, we determine the zero-temperature magnetization curves and construct the ground-state phase diagram in the magnetic field vs. the interaction ratio plane. Depending on the interaction ratio, the zero-temperature magnetization curves exhibit field-driven quantum phase transitions between the gapped phases manifested as the intermediate plateaus at 1/51/51/5 and 3/53/53/5 of the saturation magnetization and a gapless spin-canted phase, in which the magnetization varies continuously with the applied magnetic field. Quantum Monte Carlo simulations reveal clear finite-temperatures signatures of the field-induced quantum phase transitions and the intermediate magnetization plateaus, while they provide no evidence for either continuous or discontinuous thermal phase transitions.
We use a combination of numerical density matrix renormalization group calculations and several analytical approaches to comprehensively study a simplified model for a spatially anisotropic spin-12\frac{1}{2}21​ triangular lattice Heisenberg antiferromagnet: the three-leg triangular spin tube (TST). The model is described by three Heisenberg chains, with exchange constant JJJ, coupled antiferromagnetically with exchange constant ${J}^{\ensuremath{'}}$ along the diagonals of the ladder system, with periodic boundary conditions in the shorter direction. Here, we determine the full phase diagram of this model as a function of both spatial anisotropy (between the isotropic and decoupled chain limits) and magnetic field. We find a rich phase diagram, which is remarkably dominated by quantum states: the phase corresponding to the classical ground state appears only in an exceedingly small region. Among the dominant phases generated by quantum effects are commensurate and incommensurate coplanar quasiordered states, which appear in the vicinity of the isotropic region for most fields, and in the high-field region for most anisotropies. The coplanar states, while not classical ground states, can at least be understood semiclassically. Even more strikingly, the largest region of phase space is occupied by a spin density wave phase, which has incommensurate collinear correlations along the field. This phase has no semiclassical analog, and may be ascribed to enhanced one-dimensional fluctuations due to frustration. Cutting across the phase diagram is a magnetization plateau, with a gap to all excitations and ``up-up-down'' spin order, with a quantized magnetization equal to 13\frac{1}{3}31​ of the saturation value. In the TST, this plateau extends almost but not quite to the decoupled chains limit. Most of the above features are expected to carry over to the two-dimensional system, which we also discuss. At low field, a dimerized phase appears, which is particular to the one-dimensional nature of the TST, and which can be understood from quantum Berry phase arguments.
After decades of debate, now there is a rough consensus that at zero temperature the spin-1/21/21/2 Heisenberg antiferromagnet on the triangular lattice is three-sublattice ${120}^{\ensuremath{\circ}}$ magnetically ordered, in contrast to a quantum spin liquid as originally proposed. However, there remains considerable discrepancy in the magnetization reported among various methods. To resolve this issue, in this work we revisit this model by the tensor-network state algorithm. The ground-state energy per bond Eb{E}_{b}Eb​ and magnetization per spin M0{M}_{0}M0​ in the thermodynamic limit are obtained with high precision. The former is estimated to be ${E}_{b}=\ensuremath{-}0.18334(10)$. This value agrees well with that from the series expansion. The three-sublattice magnetic order is firmly confirmed and the magnetization is determined as M0=0.161(5){M}_{0}=0.161(5)M0​=0.161(5). It is about 3232%32 of its classical value and slightly below the lower bound from the series expansion. In comparison with the best estimated value by Monte Carlo and density-matrix renormalization group, our result is about 2020%20 smaller. This magnetic order is consistent with further analysis of the three-body correlation. Our work thus provides benchmark results for this prototypical model.
By using Gutzwiller projected fermionic wave functions and variational Monte Carlo technique, we study the spin-1/21/21/2 Heisenberg model with the first-neighbor (J1{J}_{1}J1​), second-neighbor (J2{J}_{2}J2​), and additional scalar chiral interaction ${J}_{\ensuremath{\chi}}{\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}({\mathbf{S}}_{j}\ifmmode\times\else\texttimes\fi{}{\mathbf{S}}_{k})$ on the triangular lattice. In the nonmagnetic phase of the ${J}_{1}\ensuremath{-}{J}_{2}$ triangular model with $0.08\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.16$, recent density-matrix renormalization group (DMRG) studies [Zhu and White, Phys. Rev. B 92, 041105(R) (2015) and Hu, Gong, Zhu, and Sheng, Phys. Rev. B 92, 140403(R) (2015)] find a possible gapped spin liquid with the signal of a competition between a chiral and a Z2{Z}_{2}Z2​ spin liquid. Motivated by the DMRG results, we consider the chiral interaction ${J}_{\ensuremath{\chi}}{\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}({\mathbf{S}}_{j}\ifmmode\times\else\texttimes\fi{}{\mathbf{S}}_{k})$ as a perturbation for this nonmagnetic phase. We find that with growing ${J}_{\ensuremath{\chi}}$, the gapless U(1) Dirac spin liquid, which has the best variational energy for ${J}_{\ensuremath{\chi}}=0$, exhibits the energy instability towards a gapped spin liquid with nontrivial magnetic fluxes and nonzero chiral order. We calculate topological Chern number and ground-state degeneracy, both of which identify this flux state as the chiral spin liquid with fractionalized Chern number C=1/2C=1/2C=1/2 and twofold topological degeneracy. Our results indicate a positive direction to stabilize a chiral spin liquid near the nonmagnetic phase of the ${J}_{1}\ensuremath{-}{J}_{2}$ triangular model.