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Journal of Physics Conference Series

Phase diagrams of Heisenberg chains with different cell spins: role of the tree-site exchange interactions

Nedko B. Ivanov, Jürgen Schnack

Published Mar 1, 2019• 0 citations
Publisher

Abstract

Additional three-site exchange interactions, naturally arising, e.g., in the strong-coupling expansion of the two-band Hubbard model and in various effective spin Hamiltonians, may stabilize various exotic spin phases which are not typical for the pure Heisenberg models on frustrated lattices and/or with extra biquadratic exchange terms. The reason is hidden in some specific features of the isotropic three-site exchange interactions, such as the promotion of collinear spin configurations as well as the reinforcing tendency towards clustering of the quantum spins on the shortest exchange bonds. As a result, systems with integer and half­integer composite spins in the unit cell might be expected to support different quantum phase diagrams. In this work, we study this phenomenon by comparing the quantum phase diagrams of two Heisenberg chains with extra three-site exchange terms and two different alternating pairs of site spins S and σ [( S , σ) = and ( ], resulting in half-integer and integer cell spins S + σ , respectively.

DOI
10.1088/1742-6596/1186/1/012014

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Additional three-site exchange interactions, naturally arising, e.g., in the strong-coupling expansion of the two-band Hubbard model and in various effective spin Hamiltonians, may stabilize various exotic spin phases which are not typical for the pure Heisenberg models on frustrated lattices and/or with extra biquadratic exchange terms. The reason is hidden in some specific features of the isotropic three-site exchange interactions, such as the promotion of collinear spin configurations as well as the reinforcing tendency towards clustering of the quantum spins on the shortest exchange bonds. As a result, systems with integer and half­integer composite spins in the unit cell might be expected to support different quantum phase diagrams. In this work, we study this phenomenon by comparing the quantum phase diagrams of two Heisenberg chains with extra three-site exchange terms and two different alternating pairs of site spins S and σ [( S , σ) = and ( ], resulting in half-integer and integer cell spins S + σ , respectively.
The J1-J3 Heisenberg spin models with nearest-neighbor (J1) and additional isotropic three-site (J3) spin interactions remain relatively less explored, although such types of competing exchange terms can naturally emerge from different sources, including the strong-coupling expansion of the multiorbital Hubbard model. Below we present a short survey of the recently published research in this field, the emphasis being on the characteristics of the variety of quantum phases supported by a few generic uniform- and alternating-spin J1-J3 Heisenberg chains. For the reason that the positive (J3>0) three-site couplings tend towards the formation of local quantum dimers, the J1-J3 spin models typically experience some spontaneous dimerization upon increasing J3. Actually, it occurred that the established dimer phases in spin-S J1-J3 Heisenberg chains (S>1/2) serve as complete analogues of the famous gapped Majumdar-Ghosh dimer phase in the spin-1/2 Heisenberg chain with next-nearest-neighbor couplings. The same dimerizations have been observed in the alternating-spin (S,σ) J1-J3 chains (S>σ), provided that the cell spin S+σ= integer, whereas for half-integer cell spin, the local dimer formation produces gapless spin-liquid ground states. The alternating-spin J1-J3 chains also provide some typical examples of spin models supporting the so-called non-Lieb-Mattis magnetic phases.
The Heisenberg chain with alternating site spins ( S ,σ) = (1, 1/2) defines a realistic prototype model admitting extra isotropic three-body exchange terms which drive the chain into exotic quantum phases. In this paper, we focus on the non-magnetic part of the phase diagram. Based on numerical density-matrix renormalization group and exact-diagonalization calculations, we demonstrate that the nearest-neighbor three-body interaction stabilizes (i) a critical spin liquid phase described by two Gaussian conformal theories as well as (ii) a critical nematic-like phase characterized by dominant quadrupolar S-spin fluctuations. The emergence of these phases reflects some specific features of the three-body terms such as the promotion of local collinear spin configurations and the enhanced tendency towards nearest- neighbor clustering of the spins.
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.
Using exact numerical diagonalizations as well as the semiclassical spin-wave approach, we study the quantum phase diagram of the spin-12 Heisenberg kagome strip, which is a particular cut-out from the kagome lattice. The unit cell contains five spins placed on the central cite and on the four end sites building two spin-12 along the strip. The model can be characterized by three different constants for the nearest-neighbor exchange bonds between (i) the central and chain spins (J1), (ii) the end spins in a cluster (J2 > 0), and (ii) the spins of neighboring clusters (J′2>0) In the present work we extend the previous studies of the model, which were restricted to antiferromagnetic J1 bonds (J1 > 0), by including the region with ferromagnetic J1 couplings (J1<0). It is demonstrated that for dominating ferromagnetic J1 couplings there is another gapless phase, which can be related to the critical ground state of the spin-52 Heisenberg chain. The phase boundaries separating the discussed critical phases, as well as the behavior of their low-lying excitations close to these boundaries, are examined as well.
We study the low-energy properties of a sawtooth chain with spin-1's at the bases of the triangles and spin-12′s\frac{1}{2}\mathrm{'}\mathrm{s}21​′sat the vertices of the triangles. The spins have Heisenberg antiferromagnetic interactions between nearest neighbors, with a coupling J2{J}_{2}J2​ between a spin-1 and a spin-12,\frac{1}{2},21​, and a coupling J1=1{J}_{1}=1J1​=1 between two spin-1's. Analysis of the exact diagonalization data for periodic chains containing up to N=12N=12N=12 unit cells shows that the ground state is a singlet for exchange couplings up to approximately J2=3.8,{J}_{2}=3.8,J2​=3.8, whereas for larger J2{J}_{2}J2​ the system exhibits a ferrimagnetic ground state characterized by a net ferromagnetic moment per unit cell of 1/2. In the region of small interactions J2,{J}_{2},J2​, the mixed spin sawtooth chain maps on to an effective isotropic spin model representing two weakly interacting and frustrated spin-12\frac{1}{2}21​ Heisenberg chains composed of spin-12\frac{1}{2}21​ sites at odd and even vertices, respectively. Finally, we study the phenomenon of a macroscopic magnetization jump which occurs if a magnetic field is applied with a value close to the saturation field for J2=2.{J}_{2}=2.J2​=2.
In this article we study the ground-state properties of two square-lattice Heisenberg quantum spin models with competing bonds using a high-order coupled cluster treatment. The first model is a spin-half model with competing nearest-neighbour bonds with and without frustration. We discuss the influence of quantum fluctuations on the ground-state phase diagram and in particular on the nature of the zero-temperature phase transitions from phases with collinear magnetic order at small frustration to phases with noncollinear spiral order at large frustration. The second model is a highly frustrated ferrimagnet, which contains one sublattice (A) entirely populated with spin-one spins and an other sublattice (B) entirely populated with spin-half spins. Sublattice A sites are nearest-neighbours to sublattice B sites and vice versa and frustration is introduced by next-nearest-neighbour bonds. The model shows two collinear ordered phases and a noncollinear phase in which (classically) the spin-one spins are allowed to cant at an angle. Both examples show that the coupled-cluster method is able to describe the zero-temperature transitions well and provides a consistent description of collinear, noncollinear, and disordered phases, for cases in which other standard techniques (e.g. the quantum Monte Carlo technique for spin systems which are frustrated) are not applicable.