We present a brief survey of the recent theoretical work related to generic Heisenberg spin models describing quasi-one-dimensional quantum ferrimagnets. The emphasis is on quantum chains and ladders with strong competing interactions, such as the frustrated $J_1-J_2$ chain with alternating (1,1/2) spins, the spin-1/2 diamond chain with four-spin cyclic couplings, and some generic types of mixed-spin ladders with geometric frustration. As a rule, discussed models exhibit rich quantum phase diagrams and provide some interesting examples of one-dimensional magnetic-paramagnetic quantum phase transitions. A number of open problems in the reviewed research area are discussed.We present a brief survey of the recent theoretical work related to generic Heisenberg spin models describing quasi-one-dimensional quantum ferrimagnets. The emphasis is on quantum chains and ladders with strong competing interactions, such as the frustrated $J_1-J_2$ chain with alternating (1,1/2) spins, the spin-1/2 diamond chain with four-spin cyclic couplings, and some generic types of mixed-spin ladders with geometric frustration. As a rule, discussed models exhibit rich quantum phase diagrams and provide some interesting examples of one-dimensional magnetic-paramagnetic quantum phase transitions. A number of open problems in the reviewed research area are discussed.

arXiv:0909.2182Sep 11, 20093 citations

Magnetic phase diagram of a frustrated ferrimagnetic ladder: Relation to the one-dimensional boson Hubbard model

Nedko B. Ivanov, Johannes Richter

We study the magnetic phase diagram of two coupled mixed-spin $(1,\frac{1}{2})$ Heisenberg chains as a function of the frustration parameter related to diagonal exchange couplings. The analysis is performed by using spin-wave series and exact numerical diagonalization techniques. The obtained phase diagram---containing the Luttinger liquid phase, the plateau phase with a magnetization per rung $M=1∕2$ , and the fully polarized phase---is closely related to the generic $(J∕U,\ensuremath{\mu}∕U)$ phase diagram of the one-dimensional boson Hubbard model.We study the magnetic phase diagram of two coupled mixed-spin $(1,\frac{1}{2})$ Heisenberg chains as a function of the frustration parameter related to diagonal exchange couplings. The analysis is performed by using spin-wave series and exact numerical diagonalization techniques. The obtained phase diagram---containing the Luttinger liquid phase, the plateau phase with a magnetization per rung $M=1∕2$, and the fully polarized phase---is closely related to the generic $(J∕U,\ensuremath{\mu}∕U)$ phase diagram of the one-dimensional boson Hubbard model.

arXiv:cond-mat/0604495Physical Review BApr 19, 20063 citations

Frustrated spin ladder with alternating spin-1 and spin- 12 rungs

Vinod Chandra, Nedko B. Ivanov, Johannes Richter

We study the impact of the diagonal frustrating couplings on the quantum phase diagram of a two-leg ladder composed of alternating spin-1 and spin- $\frac{1}{2}$ rungs. As the coupling strength is increased the system successively exhibits two gapped paramagnetic phases (a rung-singlet and a Haldane-type nondegenerate states) and two ferrimagnetic phases with different ferromagnetic moments per rung. The first two states are similar to the phases studied in the frustrated spin- $\frac{1}{2}$ ladder, whereas the magnetic phases appear as a result of the mixed-spin structure of the model. A detailed characterization of these phases is presented using density-matrix renormalization-group calculations, exact diagonalizations of periodic clusters, and an effective Hamiltonian approach inspired by the analysis of numerical data. The present theoretical study was motivated by the recent synthesis of the quasi-one-dimensional ferrimagnetic material ${\text{Fe}}^{\text{II}}{\text{Fe}}^{\text{III}}$ (trans-1,4-cyclohexanedicarboxylate) exhibiting a similar ladder structure.We study the impact of the diagonal frustrating couplings on the quantum phase diagram of a two-leg ladder composed of alternating spin-1 and spin-$\frac{1}{2}$ rungs. As the coupling strength is increased the system successively exhibits two gapped paramagnetic phases (a rung-singlet and a Haldane-type nondegenerate states) and two ferrimagnetic phases with different ferromagnetic moments per rung. The first two states are similar to the phases studied in the frustrated spin-$\frac{1}{2}$ ladder, whereas the magnetic phases appear as a result of the mixed-spin structure of the model. A detailed characterization of these phases is presented using density-matrix renormalization-group calculations, exact diagonalizations of periodic clusters, and an effective Hamiltonian approach inspired by the analysis of numerical data. The present theoretical study was motivated by the recent synthesis of the quasi-one-dimensional ferrimagnetic material ${\text{Fe}}^{\text{II}}{\text{Fe}}^{\text{III}}$ (trans-1,4-cyclohexanedicarboxylate) exhibiting a similar ladder structure.

arXiv:0911.1216Physical Review BJan 20, 201012 citations

Exploring quantum phases in frustrated spin- 1 2 chain and ladders: a detailed review

Manodip Routh, Sayan Ghosh, Manoranjan Kumar

In this topical review, we explore low-dimensional spin- $\frac{1}{2}$ systems, focusing on key models such as the Majumdar-Ghosh (MG) model on a spin- $\frac{1}{2}$ chain and the Heisenberg spin- $\frac{1}{2}$ model applied to various ladder geometries, including normal, zigzag, two-leg honeycomb, skewed, and diamond. We discuss the key physical features and experimental realizations of these models in real materials, providing a detailed overview. The review also includes the Kitaev-Heisenberg model on a two-leg ladder, a recent development in quantum magnetism. It discusses emerging quantum phases, such as spin liquids and magnetization plateaus, and highlights open research areas, including the investigation of unexplored exotic ground states. Additionally, we briefly discuss how quantum simulators based on ion traps, superconducting circuits, optical lattices, etc., are being used as powerful tools to explore the ground-state phases of spin- $\frac{1}{2}$ systems. Furthermore, the review emphasizes the potential for new material design, offering an exciting avenue for future experimental and theoretical advancements in low-dimensional quantum systems.In this topical review, we explore low-dimensional spin-$\frac{1}{2}$ systems, focusing on key models such as the Majumdar-Ghosh (MG) model on a spin-$\frac{1}{2}$ chain and the Heisenberg spin-$\frac{1}{2}$ model applied to various ladder geometries, including normal, zigzag, two-leg honeycomb, skewed, and diamond. We discuss the key physical features and experimental realizations of these models in real materials, providing a detailed overview. The review also includes the Kitaev-Heisenberg model on a two-leg ladder, a recent development in quantum magnetism. It discusses emerging quantum phases, such as spin liquids and magnetization plateaus, and highlights open research areas, including the investigation of unexplored exotic ground states. Additionally, we briefly discuss how quantum simulators based on ion traps, superconducting circuits, optical lattices, etc., are being used as powerful tools to explore the ground-state phases of spin-$\frac{1}{2}$ systems. Furthermore, the review emphasizes the potential for new material design, offering an exciting avenue for future experimental and theoretical advancements in low-dimensional quantum systems.

Journal of Physics Condensed MatterSep 12, 20251 citation

Quantum rotors on the AB2 chain with competing interactions

Antônio Sandoildo Freitas Tenório, R. R. Montenegro-Filho, Mauricio D. Coutinho-Filho

We present the ground-state phase diagram of $q=1/2$ quantum-rotor chains with competing interactions (frustration) calculated through cluster variational mean field approaches. We consider two interaction patterns, named ${\text{F}}_{1}$ and ${\text{F}}_{2}$ models, between the quantum-rotor momentum and position operators, which follow exchange patterns of known one-dimensional spin-1/2 systems with a ferrimagnetic state in their phase diagrams. The spin-1/2 ${\text{F}}_{1}$ model is known as the diamond chain and is related to the azurite compound, while the spin-1/2 ${\text{F}}_{2}$ model was recently shown to present a frustration-induced condensation of magnons. We provide a detailed comparison between the quantum-rotor phase diagrams, in single- and multisite mean-field approaches, and known results for the spin-1/2 models, including exact diagonalization and density matrix renormalization group data for these systems, as well as phase diagrams of the associated classical models.We present the ground-state phase diagram of $q=1/2$ quantum-rotor chains with competing interactions (frustration) calculated through cluster variational mean field approaches. We consider two interaction patterns, named ${\text{F}}_{1}$ and ${\text{F}}_{2}$ models, between the quantum-rotor momentum and position operators, which follow exchange patterns of known one-dimensional spin-1/2 systems with a ferrimagnetic state in their phase diagrams. The spin-1/2 ${\text{F}}_{1}$ model is known as the diamond chain and is related to the azurite compound, while the spin-1/2 ${\text{F}}_{2}$ model was recently shown to present a frustration-induced condensation of magnons. We provide a detailed comparison between the quantum-rotor phase diagrams, in single- and multisite mean-field approaches, and known results for the spin-1/2 models, including exact diagonalization and density matrix renormalization group data for these systems, as well as phase diagrams of the associated classical models.

arXiv:1602.05383Physical Review BAug 17, 20099 citations

Quantum Phase Diagram of a Frustrated Spin‐1/2 Ferro‐Antiferromagnetic Normal Ladder

Monalisa Chatterjee, Debasmita Maiti, Manoranjan Kumar

Abstract In this work, we consider a frustrated two‐leg spin‐1/2 ladder composed of Heisenberg ferromagnetic and antiferromagnetic spin‐1/2 chains, and nearest spins from different legs interact via Heisenberg type rung exchange interactions that can be either ferromagnetic or antiferromagnetic in nature. The competing exchange interactions in the system lead to five different quantum phases like ferromagnetic, non‐collinear ferrimagnetic (NCF), , antiferromagnetic and dimer phases. The quantum phase diagram is constructed for the Heisenberg spin‐1/2 model and the phases are characterized using the correlation functions which are calculated by the density matrix renormalization group method. We also analyze the stability of phase and calculate the pitch angle in the NCF phase.Abstract In this work, we consider a frustrated two‐leg spin‐1/2 ladder composed of Heisenberg ferromagnetic and antiferromagnetic spin‐1/2 chains, and nearest spins from different legs interact via Heisenberg type rung exchange interactions that can be either ferromagnetic or antiferromagnetic in nature. The competing exchange interactions in the system lead to five different quantum phases like ferromagnetic, non‐collinear ferrimagnetic (NCF), , antiferromagnetic and dimer phases. The quantum phase diagram is constructed for the Heisenberg spin‐1/2 model and the phases are characterized using the correlation functions which are calculated by the density matrix renormalization group method. We also analyze the stability of phase and calculate the pitch angle in the NCF phase.

ChemPhysChemOct 31, 20220 citations

J_1-J_2

chain with alternating (1,1/2) spins, the spin-1/2 diamond chain with four-spin cyclic couplings, and some generic types of mixed-spin ladders with geometric frustration. As a rule, discussed models exhibit rich quantum phase diagrams and provide some interesting examples of one-dimensional magnetic-paramagnetic quantum phase transitions. A number of open problems in the reviewed research area are discussed.

We study the magnetic phase diagram of two coupled mixed-spin

(1,\frac{1}{2})

Heisenberg chains as a function of the frustration parameter related to diagonal exchange couplings. The analysis is performed by using spin-wave series and exact numerical diagonalization techniques. The obtained phase diagram---containing the Luttinger liquid phase, the plateau phase with a magnetization per rung

M=1∕2

, and the fully polarized phase---is closely related to the generic $(J∕U,\ensuremath{\mu}∕U)$ phase diagram of the one-dimensional boson Hubbard model.

We study the impact of the diagonal frustrating couplings on the quantum phase diagram of a two-leg ladder composed of alternating spin-1 and spin-

\frac{1}{2}

rungs. As the coupling strength is increased the system successively exhibits two gapped paramagnetic phases (a rung-singlet and a Haldane-type nondegenerate states) and two ferrimagnetic phases with different ferromagnetic moments per rung. The first two states are similar to the phases studied in the frustrated spin-

\frac{1}{2}

ladder, whereas the magnetic phases appear as a result of the mixed-spin structure of the model. A detailed characterization of these phases is presented using density-matrix renormalization-group calculations, exact diagonalizations of periodic clusters, and an effective Hamiltonian approach inspired by the analysis of numerical data. The present theoretical study was motivated by the recent synthesis of the quasi-one-dimensional ferrimagnetic material

{\text{Fe}}^{\text{II}}{\text{Fe}}^{\text{III}}

(trans-1,4-cyclohexanedicarboxylate) exhibiting a similar ladder structure.

In this topical review, we explore low-dimensional spin-

\frac{1}{2}

systems, focusing on key models such as the Majumdar-Ghosh (MG) model on a spin-

\frac{1}{2}

chain and the Heisenberg spin-

\frac{1}{2}

model applied to various ladder geometries, including normal, zigzag, two-leg honeycomb, skewed, and diamond. We discuss the key physical features and experimental realizations of these models in real materials, providing a detailed overview. The review also includes the Kitaev-Heisenberg model on a two-leg ladder, a recent development in quantum magnetism. It discusses emerging quantum phases, such as spin liquids and magnetization plateaus, and highlights open research areas, including the investigation of unexplored exotic ground states. Additionally, we briefly discuss how quantum simulators based on ion traps, superconducting circuits, optical lattices, etc., are being used as powerful tools to explore the ground-state phases of spin-

\frac{1}{2}

systems. Furthermore, the review emphasizes the potential for new material design, offering an exciting avenue for future experimental and theoretical advancements in low-dimensional quantum systems.

We present the ground-state phase diagram of

q=1/2

quantum-rotor chains with competing interactions (frustration) calculated through cluster variational mean field approaches. We consider two interaction patterns, named

{\text{F}}_{1}

and

{\text{F}}_{2}

models, between the quantum-rotor momentum and position operators, which follow exchange patterns of known one-dimensional spin-1/2 systems with a ferrimagnetic state in their phase diagrams. The spin-1/2

{\text{F}}_{1}

model is known as the diamond chain and is related to the azurite compound, while the spin-1/2

{\text{F}}_{2}

model was recently shown to present a frustration-induced condensation of magnons. We provide a detailed comparison between the quantum-rotor phase diagrams, in single- and multisite mean-field approaches, and known results for the spin-1/2 models, including exact diagonalization and density matrix renormalization group data for these systems, as well as phase diagrams of the associated classical models.