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Physica B Condensed Matter

Ferrimagnetic and Haldane-type phases in a mixed-spin 1-12-12 quantum trimer chain

A. Felinto, R. R. Montenegro-Filho

Published Aug 15, 2026• 0 citations
Publisher

Abstract

Bipartite Lieb-Mattis ferrimagnetism and the symmetry-protected Haldane phase are paradigmatic mechanisms in quasi-one-dimensional quantum magnets. Both emerge, in distinct regimes, in a mixed-spin $1$-$\tfrac{1}{2}$-$\tfrac{1}{2}$ Heisenberg trimer chain with antiferromagnetic backbone exchange $J$ and a side spin-$\tfrac{1}{2}$ coupled to each backbone spin by an exchange $J_t$ of either sign. Using the density matrix renormalization group, we compute magnetization curves and the entanglement spectrum and entropy. For$J_t>0$ a robust ferrimagnetic plateau forms at magnetization per unit cell $m=1$, whose multiplet entropy reflects how the conserved magnetization splits between the halves. For $J_t<0$ an $m=0$ plateau opens and grows with $|J_t|$, while the $m=1$ plateau closes. As$J_t\to-\infty$ the chain maps onto a spin-$1$ Heisenberg chain with coupling $J/2$: the $m=0$ width $\Delta h\simeq 0.196$ matches half the Haldane gap. Exponentially localized spin-$\tfrac{1}{2}$ edge states and the even-fold degeneracy of the entanglement spectrum confirm the Haldane character of the $m=0$ phase.

DOI
10.1016/j.physb.2026.419192

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Ferrimagnetic and Haldane-type phases in a mixed-spin $1$-$\tfrac{1}{2}$-$\tfrac{1}{2}$ quantum trimer chain

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Bipartite Lieb-Mattis ferrimagnetism and the symmetry-protected Haldane phase are paradigmatic mechanisms in quasi-one-dimensional quantum magnets. Both emerge, in distinct regimes, in a mixed-spin $1$-$\tfrac{1}{2}$-$\tfrac{1}{2}$ Heisenberg trimer chain with antiferromagnetic backbone exchange $J$ and a side spin-$\tfrac{1}{2}$ coupled to each backbone spin by an exchange $J_t$ of either sign. Using the density matrix renormalization group, we compute magnetization curves and the entanglement spectrum and entropy. For $J_t>0$ a robust ferrimagnetic plateau forms at magnetization per unit cell $m=1$, whose multiplet entropy reflects how the conserved magnetization splits between the halves. For $J_t<0$ an $m=0$ plateau opens and grows with $|J_t|$, while the $m=1$ plateau closes. As $J_t\to-\infty$ the chain maps onto a spin-$1$ Heisenberg chain with coupling $J/2$: the $m=0$ width $Δh\simeq 0.196$ matches half the Haldane gap. Exponentially localized spin-$\tfrac{1}{2}$ edge states and the even-fold degeneracy of the entanglement spectrum confirm the Haldane character of the $m=0$ phase.

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Bipartite Lieb-Mattis ferrimagnetism and the symmetry-protected Haldane phase are paradigmatic mechanisms in quasi-one-dimensional quantum magnets. Both emerge, in distinct regimes, in a mixed-spin 111-12\tfrac{1}{2}21​-12\tfrac{1}{2}21​ Heisenberg trimer chain with antiferromagnetic backbone exchange JJJ and a side spin-12\tfrac{1}{2}21​ coupled to each backbone spin by an exchange JtJ_tJt​ of either sign. Using the density matrix renormalization group, we compute magnetization curves and the entanglement spectrum and entropy. ForJt>0J_t>0Jt​>0 a robust ferrimagnetic plateau forms at magnetization per unit cell m=1m=1m=1, whose multiplet entropy reflects how the conserved magnetization splits between the halves. For Jt<0J_t<0Jt​<0 an m=0m=0m=0 plateau opens and grows with ∣Jt∣|J_t|∣Jt​∣, while the m=1m=1m=1 plateau closes. AsJt→−∞J_t\to-\inftyJt​→−∞ the chain maps onto a spin-111 Heisenberg chain with coupling J/2J/2J/2: the m=0m=0m=0 width Δh≃0.196\Delta h\simeq 0.196Δh≃0.196 matches half the Haldane gap. Exponentially localized spin-12\tfrac{1}{2}21​ edge states and the even-fold degeneracy of the entanglement spectrum confirm the Haldane character of the m=0m=0m=0 phase.
Ferrimagnetic and Haldane-type phases in a mixed-spin 111-12\tfrac{1}{2}21​-12\tfrac{1}{2}21​ quantum trimer chain
Bipartite Lieb-Mattis ferrimagnetism and the symmetry-protected Haldane phase are paradigmatic mechanisms in quasi-one-dimensional quantum magnets. Both emerge, in distinct regimes, in a mixed-spin 111-12\tfrac{1}{2}21​-12\tfrac{1}{2}21​ Heisenberg trimer chain with antiferromagnetic backbone exchange JJJ and a side spin-12\tfrac{1}{2}21​ coupled to each backbone spin by an exchange JtJ_tJt​ of either sign. Using the density matrix renormalization group, we compute magnetization curves and the entanglement spectrum and entropy. For Jt>0J_t>0Jt​>0 a robust ferrimagnetic plateau forms at magnetization per unit cell m=1m=1m=1, whose multiplet entropy reflects how the conserved magnetization splits between the halves. For Jt<0J_t<0Jt​<0 an m=0m=0m=0 plateau opens and grows with ∣Jt∣|J_t|∣Jt​∣, while the m=1m=1m=1 plateau closes. As Jt→−∞J_t\to-\inftyJt​→−∞ the chain maps onto a spin-111 Heisenberg chain with coupling J/2J/2J/2: the m=0m=0m=0 width Δh≃0.196Δh\simeq 0.196Δh≃0.196 matches half the Haldane gap. Exponentially localized spin-12\tfrac{1}{2}21​ edge states and the even-fold degeneracy of the entanglement spectrum confirm the Haldane character of the m=0m=0m=0 phase.
Magnetization curves of the ferrimagnetic mixed spin-(1/2,S) Heisenberg chains are calculated with the help of density-matrix renormalization group method for several quantum spin numbers S=1, 3/2, 2 and 5/2.It is shown that the ferrimagnetic mixed spin-(1/2,S) Heisenberg chains exhibit irrespective of the spin value S exactly one intermediate magnetization plateau, which can be identified with the gapped Lieb-Mattis ferrimagnetic ground state.The magnetization plateau due to the Lieb-Mattis ferrimagnetism breaks down at a quantum phase transition towards the Luttinger spin liquid, which is characterized by a continuous change of the magnetization with the magnetic field until another quantum critical point is reached at the saturation field.
The spin-1/2 Ising-Heisenberg branched chain composed of regularly alternating Ising spins and Heisenberg dimers involving an additional side branching is rigorously solved in a magnetic field by the transfer-matrix approach. The ground-state phase diagram, the magnetization process and the concurrence measuring a degree of bipartite entanglement within the Heisenberg dimers are examined in detail. Three different ground states were found depending on a mutual interplay between the magnetic field and two different coupling constants: the modulated quantum antiferromagnetic phase, the quantum ferrimagnetic phase, and the classical ferromagnetic phase. Two former quantum ground states are manifested in zero-temperature magnetization curves as intermediate plateaus at zero and one-half of the saturation magnetization, whereas the one-half plateau disappears at a triple point induced by a strong-enough ferromagnetic Ising coupling. The ground-state phase diagram and zero-temperature magnetization curves of the analogous spin-1/2 Heisenberg branched chain were investigated using density-matrix renormalization group calculations. The latter fully quantum Heisenberg model involves, besides two gapful phases manifested as zero and one-half magnetization plateaus, gapless quantum spin-liquid phase. The intermediate one-half plateau of the spin-1/2 Heisenberg branched chain vanishes at Kosterlitz-Thouless quantum critical point between gapful and gapless quantum ground states unlike the triple point of the spin-1/2 Ising-Heisenberg branched chain.
The density matrix renormalization group and quantum Monte Carlo methods are used to describe coupled trimer chains in a magnetic field hhh. The Hamiltonian contains exchange terms involving the intratrimer coupling J1{J}_{1}J1​ (taken as the unit of energy) and the intertrimer coupling J2{J}_{2}J2​, plus the Zeeman interaction for a magnetic field hhh along the zzz direction. Results for the magnetization per trimer mmm are calculated in regimes of positive and negative values of the ratio J=J2/J1J={J}_{2}/{J}_{1}J=J2​/J1​, from which the rich field-induced ground state phase diagram hhh versus JJJ is derived, with the presence of a Luttinger liquid, the 1/3 plateau (m=1/2m=1/2m=1/2), and the one of fully polarized magnetization (m=3/2m=3/2m=3/2). Also, zero-field Lanczos calculation of the spin-wave dispersion from the 1/3 plateau for Sz=1{S}^{z}=1Sz=1 is shown at the previous regimes of JJJ values. In addition, we also report on the decay of correlation functions of trimers along open chains, as well as the average two-magnon distribution. The ground state is ferrimagnetic for $0<J\ensuremath{\le}1$, and is a singlet for $\ensuremath{-}1\ensuremath{\le}J<0$. In the singlet phase, the spin correlation functions along the legs present an antiferromagnetic power-law decay, similar to the spin-1/2 linear chain, thus suggesting that the ground state is made of three coupled antiferromagnetically oriented chains. In the singlet phase, the dimensionless thermal magnetic susceptibility per site normalized by 1/∣J∣1/|J|1/∣J∣ gets closer to $1/{\ensuremath{\pi}}^{2}$ as the temperature $T\ensuremath{\rightarrow}0$. For the ferrimagnetic phase, we fit the susceptibility to the experimental data for the compound Pb3Cu3(PO4)4{\mathrm{Pb}}_{3}{\mathrm{Cu}}_{3}{({\mathrm{PO}}_{4})}_{4}Pb3​Cu3​(PO4​)4​ and estimate the model exchange couplings: J1=74.8K{J}_{1}=74.8\phantom{\rule{4pt}{0ex}}\text{K}J1​=74.8K and J=0.4J=0.4J=0.4. These values imply a range of energies for the magnon excitations that are in accord with the data from neutron scattering experiments on Pb3Cu3(PO4)4{\mathrm{Pb}}_{3}{\mathrm{Cu}}_{3}{({\mathrm{PO}}_{4})}_{4}Pb3​Cu3​(PO4​)4​ for two excitation modes. The 1/3 plateau closes only at 1/∣J∣=01/|J|=01/∣J∣=0 with J<0J<0J<0.
The mixed spin-1 and spin-1/2 Heisenberg octahedral chain with regularly alternating monomeric spin-1 sites and square-plaquette spin-1/2 sites is investigated using a variational technique, localized-magnon approach, exact diagonalization, and density-matrix renormalization-group method. The investigated model has, in a magnetic field, an extraordinarily rich ground-state phase diagram, which includes the uniform and cluster-based Haldane phases, two ferrimagnetic phases of Lieb-Mattis type, two quantum spin liquids, two bound magnon crystals, in addition to the fully polarized ferromagnetic phase. The lowest-energy eigenstates in a highly frustrated parameter region belong to flatbands and, hence, low-temperature thermodynamics above the bound magnon-crystal ground states can be satisfactorily described within the localized-magnon approach. The variational method provides exact evidence for the magnon-crystal phase with a character of the monomer-tetramer ground state at zero field, whereas another magnon-crystal phase with a single bound magnon at each square plaquette is found in a high-field region. A diversity of quantum ground states gives rise to manifold zero-temperature magnetization curves, which may involve up to four wide intermediate plateaus at zero, one-sixth, one-third, and two-thirds of the saturation magnetization, two quantum spin-liquid regions, and two tiny plateaus at one-ninth and one-twelfth of the saturation magnetization corresponding to the fragmentized cluster-based Haldane phases.
The ground-state properties of a spin S=1/2S=1/2S=1/2 tetrameric Heisenberg antiferromagnetic chain with alternating couplings AF1-AF2-AF1-F{\text{AF}}_{1}{\text{-AF}}_{2}{\text{-AF}}_{1}\text{-F}AF1​-AF2​-AF1​-F (AF and F denote antiferromagnetic and ferromagnetic couplings, respectively) are studied by means of the density-matrix renormalization-group method. Two plateaus of magnetization mmm are found at m=0m=0m=0 and 1/4. It is shown that in such a spin-1/2 AF system, there is a gap from the singlet ground state to the triplet excited states in the absence of a magnetic field. The spin-spin correlation function decays exponentially, and the gapped states have a nonvanishing string order, which measures a hidden symmetry in the system. By a dual transformation, the string order is transformed into a ferromagnetic order and the hidden symmetry is unveiled to be a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ discrete symmetry, which is fully broken in the gapped states. This half-integer spin Heisenberg AF chain is in a Haldane-like phase suggesting that the present findings extend the substance of Haldane's scenario. A valence-bond-solid state picture is also proposed for the gapped states.