We investigate frustration-induced instabilities in a trimerized quantum spin chain motivated by recent experimental findings for the compound Na$_2$Cu$_3$Ge$_4$O$_{12}$. Employing a cluster mean-field approach combined with Lanczos exact diagonalization, we analyze the ground-state and quantum-information properties of a Heisenberg model with competing interactions in a magnetic field. In the weakly frustrated regime, the system exhibits a robust $1/3$ magnetization plateau associated with a collective ferrimagnetic-like trimer state. Increasing the next-nearest-neighbor intratrimer coupling drives a pronounced reorganization of spin correlations, leading to a crossover toward a doublon-like correlation regime and providing a static ground-state picture consistent with the composite excitations observed dynamically in trimerized chains. The resulting low-energy behavior can be interpreted in terms of weakly interacting emergent spins, offering a microscopic explanation for the extended stability of the magnetization plateau. Furthermore, through finite-size scaling analyses of the energy gap, von Neumann entanglement entropy, and fidelity susceptibility, we characterize the zero-field criticality of the model. Ultimately, our results suggest that frustration gives rise to qualitatively distinct quantum states and provide a microscopic framework for understanding the emergence of fractionalized excitations in trimerized quantum spin systems.
We investigate frustration-induced instabilities in a trimerized quantum spin chain motivated by recent experimental findings for the compound Na2Cu3Ge4O12. Employing a cluster mean-field approach combined with Lanczos exact diagonalization, we analyze the ground-state and quantum-information properties of a Heisenberg model with competing interactions in a magnetic field. In the weakly frustrated regime, the system exhibits a robust 1/3 magnetization plateau associated with a collective ferrimagnetic-like trimer state. Increasing the next-nearest-neighbor intratrimer coupling drives a pronounced reorganization of spin correlations, leading to a crossover toward a doublon-like correlation regime and providing a static ground-state picture consistent with the composite excitations observed dynamically in trimerized chains. The resulting low-energy behavior can be interpreted in terms of weakly interacting emergent spins, offering a microscopic explanation for the extended stability of the magnetization plateau. Furthermore, through finite-size scaling analyses of the energy gap, von Neumann entanglement entropy, and fidelity susceptibility, we characterize the zero-field criticality of the model. Ultimately, our results suggest that frustration gives rise to qualitatively distinct quantum states and provide a microscopic framework for understanding the emergence of fractionalized excitations in trimerized quantum spin systems.
Theoretical and experimental studies of melting of the 1/3 magnetization plateau in a frustrated S=1/2 antiferromagnetic trimerized quantum Heisenberg spin chain compound Na2Cu3Ge4O12
Abstract We report the magnetization process of S=1/2 S = 1 / 2 Heisenberg antiferromagnetic J1−J1−J2 J 1 - J 1 - J 2 trimerized spin chain system (with frustration due to the next-nearest-neighbor interaction J3 J 3 within the trimer) under applied magnetic field by means of density-matrix renormalization group method. Here, J′ J ′ s are the antiferromagnetic exchange interaction constants. Specifically, we explore the phenomenon of 1/3 magnetization plateau for the exchange coupling ratios α=0.18,β=0.18 α = 0.18 , β = 0.18 , and J1=235 J 1 = 235 K, where α=J2/J1 α = J 2 / J 1 , and β=J3/J1 β = J 3 / J 1 as found for the model compound Na2Cu3Ge4O12 Na 2 Cu 3 Ge 4 O 12 . It is found that a wide 1/3 quantum magnetization plateau state (observed in ground state magnetization) gradually disappears with increasing temperature beyond a crossover temperature Tco T co due to thermal fluctuations. Based on numerical data, the width w(T) of the magnetization plateau is found to follow an exponential behave as ∼e−B/Tco−T ∼ e - B / T co - T , where B is a constant and Tco T co is estimated to be ≈60 ≈ 60
Abstract Motivated by recent advancements in theoretical and experimental studies of the high-energy excitations on an antiferromagnetic trimer chain, we numerically investigate the quantum phase transition and composite dynamics in this system by applying a magnetic field. The numerical methods we used include the exact diagonalization, density matrix renormalization group, time-dependent variational principle, and cluster perturbation theory. From calculating the entanglement entropy, we have revealed the phase diagram which includes the XY-I, 1/3 magnetization plateau, XY-II, and ferromagnetic phases. Both the critical XY-I and XY-II phases are characterized by the conformal field theory with a central charge c ≃ 1. By analyzing the dynamic spin structure factor, we elucidate the distinct features of spin dynamics across different phases. In the regime with weak intertrimer interaction, we identify the intermediate-energy and high-energy modes in the XY-I and 1/3 magnetization plateau phases as internal trimer excitations, corresponding to the propagating of doublons and quartons, respectively. Notably, applying a magnetic field splits the high-energy spectrum into two branches, labeled as the upper quarton and lower quarton. Furthermore, we explore the spin dynamics of a frustrated trimerized model closely related to the quantum magnet Na 2 Cu 3 Ge 4 O 12 . In the end, we extend our discuss on the possibility of the quarton Bose-Einstein condensation in the trimer systems. Our results are expected to be further verified through the inelastic neutron scattering and resonant inelastic X-ray scattering, and also provide valuable insights for exploring high-energy exotic excitations.
Trimerized Spin-1/2 Chain: Emergent Low-Energy Hamiltonian, Higher-Energy Excitations, and Magnetic and Thermodynamic Responses
We investigate a spin-1/2 antiferromagnetic chain with trimerized exchange couplings relevant to Na2Cu3Ge4O12. We show that the low-energy Hilbert space maps onto an effective Heisenberg chain of composite spin-1/2 trimer degrees of freedom with positive coupling, enabling an exact spinon excitations via the Bethe ansatz. To access higher-energy dynamics, we develop a self-consistent Jordan-Wigner mean-field theory that yields three fermionic bands reflecting the underlying trimer structure. Remarkably, this approach reproduces the exact low-energy spinon continuum while predicting two additional higher-energy excitation bands consistent with the experimental observations and previous numerical simulations. The theory further captures the 1/3 magnetization plateau under an applied magnetic field, and provides testable predictions in magnetic susceptibility and specific heat. Our results establish a unified framework connecting low- and high-energy excitations in trimerized quantum spin chains.
Topologically nontrivial 1/3-magnetization plateau state in a spin-1/2 trimer chain
Topologically nontrivial Haldane phase is theoretically proposed to be realized in the 1/3-magnetization (M) plateau of spin-1/2 trimer systems. However, the spin excitation gap, typical characteristic of Haldane phase, is not yet experimentally verified. Here, we report the nuclear magnetic resonance investigations into the low-energy spin dynamics in the S=1/2 spin-trimer antiferromagnetic chain compound Na2Cu3Ge4−xSixO12 (x=0,0.1∼1.5). In the parent compound (x=0), the spin-lattice relaxation rate (1/T1) shows significantly different temperature dependence when the external magnetic field is increased above the critical field of μ0Hc = 29 T. The spin excitation gap is evidenced from the thermally activated behavior of 1/T1(T) in the 1/3-M plateau state. By substituting Ge4+ with Si4+, the critical field for the 1/3-M plateau significantly decreases, e.g. μ0Hc=17 T in x=1.0 samples, which results from the suppressed inter-trimer coupling J2. The gapped spin excitation is confirmed again above 17 T, whose size shows temperature-dependent behavior for μ0H≥25.72 T. These observations provide further insights into the Haldane physics.
Quantum spin-1/2 antiferromagnetic Heisenberg trimerized chain with strong intradimer and weak monomer-dimer coupling constants is studied using the many-body perturbation expansion developed about the exact solution of the spin-1/2 Ising-Heisenberg diamond chain. The Heisenberg trimerized chain shows the intermediate one-third plateau, which was also observed in the magnetization curve of the polymeric compound Cu3(P2O6OH)2 affording its experimental realization. Within the strong-coupling method we have obtained the effective Hamiltonians for the magnetic-field range from zero to one-third plateau, and from one-third plateau to the saturation magnetization. The second-order perturbation theory shows reliable results even for the moderate ratio between weaker dimer-monomer and stronger intradimer coupling constants. We have also examined thermodynamic properties and recovered the low-temperature peak in the specific heat. The accuracy of the developed method is tested through a comparison with numerical density-matrix renormalization group and quantum Monte Carlo simulations. Using the results for the effective Hamiltonian we suggest straightforward procedure for finding the microscopic parameters of one-dimensional trimerized magnetic compounds with strong intradimer and weak monomer-dimer couplings. We found the refined values for the coupling constants of Cu3(P2O6OH)2 by matching the theoretical results with the available experimental data for the magnetization and magnetic susceptibility in a wide range of temperatures and magnetic fields.
A quantum spin-21 antiferromagnetic Heisenberg trimerized chain with strong intradimer and weak monomer-dimer coupling constants is studied using the many-body perturbation expansion, which is developed from the exactly solved spin-21 Ising-Heisenberg diamond chain preserving correlations between all interacting spins of the trimerized chain. It is evidenced that this perturbation approach is superior with respect to the standard perturbation scheme developed from a set of noninteracting spin monomers and dimers, and its accuracy even coincides, up to a moderate ratio of the coupling constants, with state-of-the-art numerical techniques. The Heisenberg trimerized chain shows the intermediate one-third plateau, which was also observed in the magnetization curve of the polymeric compound Cu3(P2O6OH)2 affording its experimental realization. Within the modified strong-coupling method we have obtained the effective Hamiltonians for the magnetic field range from zero to the one-third plateau, and from the one-third plateau to the saturation magnetization. The unconventional second-order perturbation theory provides extremely accurate results for both critical fields of the intermediate one-third plateau up to a moderate ratio of the coupling constants as convincingly evidenced through a comparison with numerical density-matrix renormalization group data. It is shown that the derived effective Hamiltonian also provides at low enough temperatures sufficiently accurate results for magnetization curves and thermodynamic properties as corroborated through a comparison with quantum Monte Carlo simulations. Using the results for the effective Hamiltonian, we additionally suggest a straightforward procedure for finding the microscopic parameters of one-dimensional trimerized magnetic compounds with strong intradimer and weak monomer-dimer couplings. We found the refined values for the coupling constants of Cu3(P2O6OH)2 by matching the theoretical results with the available experimental data for the magnetization and magnetic susceptibility in a wide range of temperatures and magnetic fields.