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arXiv:cond-mat/9707148 · Physical review. B, Condensed matter

Alternating spin chains with singlet ground states

Takahiro Fukui, Norio Kawakami

Published Oct 1, 1997• 34 citations
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Abstract

We investigate low-energy properties of the alternating-spin-chain model composed of spins ${s}_{1}$ and ${s}_{2}$ with a singlet ground state. After examining the spin-wave spectrum in detail, we map low-energy spin excitations to the O(3) nonlinear \ensuremath{\sigma} model in order to take into account quantum fluctuations. Analyzing the topological term in the resulting \ensuremath{\sigma} model, we discuss how the massless or massive excitations are developed, especially according to the topological nature of the alternating spin system.

DOI
10.1103/physrevb.56.8799

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We investigate low-energy properties of the alternating-spin-chain model composed of spins s1{s}_{1}s1​ and s2{s}_{2}s2​ with a singlet ground state. After examining the spin-wave spectrum in detail, we map low-energy spin excitations to the O(3) nonlinear \ensuremath{\sigma} model in order to take into account quantum fluctuations. Analyzing the topological term in the resulting \ensuremath{\sigma} model, we discuss how the massless or massive excitations are developed, especially according to the topological nature of the alternating spin system.
We investigate a spin-chain model composed of a periodic array of two kinds of spins S1{\mathrm{S}}_{1}S1​ and S2{\mathrm{S}}_{2}S2​, which allows us to study the spin chains with impurities, as well as the alternating spin chains, in a unified fashion. By using the Lieb-Shultz-Mattis theorem, we first study the model rigorously, and then by mapping it to the nonlinear \ensuremath{\sigma} model, we extensively investigate low-energy properties with particular emphasis on the competition between the massive and massless phases.
We study a period-4 antiferromagnetic mixed quantum spin chain consisting of three kinds of spins. When the ground state is singlet, the spin magnitudes in a unit cell are arrayed as (s-t, s, s+t, s) with integer or half-odd integer s and t (0 <= t < s). The spin Hamiltonian is mapped onto a nonlinear sigma model (NLSM) in a previously developed method. The resultant NLSM includes only two independent parameters originating from four exchange constants for fixed s and t. The topological angle in the NLSM determines the gapless phase boundaries between disordered phases in the parameter space. The phase diagrams for various s and t shows rich structures. We systematically explain the phases in the singlet-cluster-solid picture.
We study the properties of the ground state and the low-lying excited states of an alternating spin-1 and spin-12\frac{1}{2}21​ antiferromagnetic chain with ferromagnetic next-nearest-neighbor coupling with a variety of methods, namely, spin-wave analysis, density-matrix renormalization-group (DMRG) method, and exact-diagonalization method. The ground state of this model with 2N2N2N spins is ferrimagnetic with the total spin SG=N∕2{S}_{G}=N∕2SG​=N∕2 for all parameter values. In the long chain limit, a gapless excitation is found with characteristics of goldstone mode of the ferromagnetic order with total spin $S={S}_{G}\ensuremath{-}1$. The lowest gapped excitation, however, exists in the total spin S=SG+1S={S}_{G}+1S=SG​+1 sector. Interestingly, we find that with the increase in ferromagnetic coupling, spin-wave gapped excitation becomes lower than the massless mode for the large wave numbers. Correspondingly, from DMRG analysis we find that the excitation in the total spin S=SG+1S={S}_{G}+1S=SG​+1 is lower than that in the total spin $S={S}_{G}\ensuremath{-}1$ when the size of the chain is short and the ferromagnetic interaction is large. Characteristics of these excitations are also reflected in the low temperature thermodynamic quantities, where the low-energy properties are strongly affected by the system size and the coupling strength.
Microscopic models of quantum antiferromagnets are investigated on the basis of a mapping onto effective low-energy Hamiltonians. Lattice effects are carefully taken into account and their role is discussed. We show that the presence of an external staggered magnetic field modifies in a nontrivial way the usual mapping onto the nonlinear sigma model, leading to the appearance of terms neglected in previous works. Our analysis is compared with Lanczos diagonalizations of S=1S=1S=1 Heisenberg chains in a staggered field, confirming the validity of the single-mode approximation for the evaluation of the dynamical structure factor. The results are relevant for the interpretation of experiments in quasi-one-dimensional compounds. Microscopic realizations of SU(4) spin chains are also discussed in the framework of spin-orbital lattice systems. The low-energy physics is shown to be described by sigma models with topological angle $\ensuremath{\theta}$ in one dimension. This mapping strongly suggests that the one-dimensional CP3{\mathrm{CP}}^{3}CP3 model (with $\ensuremath{\theta}=\ensuremath{\pi})$ undergoes a second-order phase transition as a function of the coupling.
We prove a theorem on the O(3) nonlinear \ensuremath{\sigma} model with the topological \ensuremath{\theta} term which states that the grround-state energy at \ensuremath{\theta}=\ensuremath{\pi} is always higher than the ground-state energy at \ensuremath{\theta}=0, for the same value of the coupling constant g. Provided that the nonlinear \ensuremath{\sigma} model gives the correct description for the Heisenberg spin chains in the large-s limit, this theorem makes a definite prediction relating the ground-state energies of the half-integer- and the integer-spin chains. The ground-state energies obtained from the exact Bethe Ansatz solution for the spin-(1/2 chain and the numerical diagonalizaton on the spin-1,- (3/2, and -2 chains support this prediction.
We investigate a model for the regularly depleted two-leg spin-ladder systems. By using the Lieb-Schultz-Mattis theorem, it is rigorously shown that this model realizes massless excitations or, alternatively, a degenerate ground state, although the original spin-ladder system has a spin gap and a unique ground state. The ground state of the depleted model is either a spin singlet or partially ferromagnetic reflecting topological properties of the depleted sites. In order to show that the low-energy excitations are indeed massless, we proceed in our analysis in two different ways by resorting to effective field theories. We first investigate an effective weak-coupling model in terms of renormalization-group methods. Although the tendency to massless-spin excitations is seen in the strong-coupling regime, it turns out that the model is still massive for any finite coupling, implying that a conventional weak-coupling approach is not efficient to describe massless modes in our model. To overcome this difficulty, we further study low-energy properties of the depleted spin model by mapping on the nonlinear sigma model, and confirm that the massless-spin excitation indeed occurs.