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Journal of Physics Condensed Matter

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

Manodip Routh, Sayan Ghosh, Manoranjan Kumar

Published Sep 12, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1361-648x/ae0670

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Motivated by the largely unexplored domain of multipolar ordered spin states in the Kitaev-Heisenberg (KH) systems we investigate the ground state dynamics of the spin-12\frac{1}{2}21​ KH model, focusing on quadrupolar (QP) order in two-leg ladder and two-dimensional honeycomb lattice geometries. Employing exact diagonalization and density-matrix renormalization group methods, we analyze the QP order parameter and correlation functions. Our findings reveal a robust QP order across a wide range of the phase diagram, influenced by the interplay between Heisenberg and Kitaev interactions. Notably, we observe an enhancement of QP order near Kitaev quantum spin-liquid (QSL) phases, despite the absence of long-range spin-spin correlations. This highlights a complex relationship between QP order and QSLs, offering different insights into quantum magnetism in low-dimensional systems. Our findings provide a rational explanation for the observed nonlinear magnetic susceptibility in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$.
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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 J1−J2J_1-J_2J1​−J2​ 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.
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We study the ground state and low-lying excited states of the Kitaev-Heisenberg model on a two-leg ladder geometry using the density-matrix renormalization-group and Lanczos exact diagonalization methods. The Kitaev and Heisenberg interactions are parametrized as $K=sin\ensuremath{\phi}$ and $J=cos\ensuremath{\phi}$ with an angle parameter $\ensuremath{\phi}$. Based on the results for several types of order parameters, excitation gaps, string order parameter, and entanglement spectra, the $\ensuremath{\phi}$-dependent ground state phase diagram is determined. Remarkably, the phase diagram is quite similar to that of the Kitaev-Heisenberg model on a honeycomb lattice, exhibiting the same long-range-ordered states, namely rung singlet (analog to N\'eel in three dimensions), zigzag, ferromagnetic, and stripy, and the presence of gapped spin liquids around the exactly solvable Kitaev points $\ensuremath{\phi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$. We also calculate the expectation value of a plaquette operator corresponding to a $\ensuremath{\pi}$-flux state in order to establish how the gapped Kitaev spin liquid extends away from the $\ensuremath{\phi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$. Furthermore, we determine the dynamical spin structure factor and discuss the effect of the Kitaev interaction on the spin-triplet dispersion.
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