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Progress of Theoretical Physics Supplement

Excitations and Dynamics of Spin-Orbital Chains

U. Schollwöck, Alexei K. Kolezhuk

Published Jan 1, 2002• 0 citations
Publisher

Abstract

The low-energy excitations and dynamics of a spin-orbital chain are calculated within a variational framework based on dimer states, exact diagonalization and the dynamical Density Matrix Renormalization Group (DMRG). In its dimerized phase, the spin-orbital chain exhibits a rich interplay of solitonic and magnonic excitations. While a solitonic excitation is the lowest excitation in a narrow region around the exactly solvable dimer point, magnonic excitations govern the phase transitions to neighbouring phases. Numerical calculations confirm that the curvature of the magnonic excitations is exactly flat on the so-called dimer line, changing sign across the line. This statement is not true for the solitonic excitations, which determine the lower edge of the excitation spectrum in the vicinity of the dimer line. DMRG calculations suggest that low-lying solitonic excitations may be experimentally hard to observe, with most spectral weight in higher-lying magnons.

DOI
10.1143/ptps.145.259

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The low-energy excitations and dynamics of a spin-orbital chain are calculated within a variational framework based on dimer states, exact diagonalization and the dynamical Density Matrix Renormalization Group (DMRG). In its dimerized phase, the spin-orbital chain exhibits a rich interplay of solitonic and magnonic excitations. While a solitonic excitation is the lowest excitation in a narrow region around the exactly solvable dimer point, magnonic excitations govern the phase transitions to neighbouring phases. Numerical calculations confirm that the curvature of the magnonic excitations is exactly flat on the so-called dimer line, changing sign across the line. This statement is not true for the solitonic excitations, which determine the lower edge of the excitation spectrum in the vicinity of the dimer line. DMRG calculations suggest that low-lying solitonic excitations may be experimentally hard to observe, with most spectral weight in higher-lying magnons.
We study, both numerically and variationally, the interplay between different types of elementary excitations in the model of a spin chain with anisotropic spin-orbit coupling, in the vicinity of the ``dimer line'' with an exactly known dimerized ground state. Our variational treatment is found to be in a qualitative agreement with the exact diagonalization results. Soliton pairs are shown to be the lowest excitations only in a very narrow region of the phase diagram near the dimer line, and the phase transitions are always governed by magnon-type excitations which can be viewed as soliton-antisoliton bound states. It is shown that when the anisotropy exceeds certain critical value, an additional phase boundary appears. In the doped model on the dimer line, the exact elementary charge excitation is shown to be a hole bound to a soliton. Bound states of those ``charged solitons'' are studied; exact solutions for N-hole bound states are presented.
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.
The excitation spectrum of the one-dimensional spin-orbital model in a magnetic field is studied, using a recently developed dynamical density-matrix renormalization-group technique. The method is applied to chains with up to 80 sites, and examined for test cases such as the one-dimensional spin-1/2 Heisenberg antiferromagnet whose excitation spectrum is known exactly from the Bethe ansatz. In the spin-orbital chain, the characteristic dynamical response depends strongly on the model parameters and the applied magnetic field. The coupling between the spin and orbital degrees of freedom is found to influence the incommensuration at finite magnetizations. In the regions of the phase diagram with only massive spin and orbital excitations, a finite field is required to overcome the spin gap. An orbital mode is found to become massless in this partially spin-polarized regime, indicating a strong coupling between the two degrees of freedom. In the critical region with three elementary gapless excitations, a prominent particle-hole excitation is observed at higher energies, promoted by the biquadratic term in the model Hamiltonian of the spin-orbital chain.
The dynamical spin structure factor and the Raman response are calculated for structurally dimerized and spin-Peierls chains in a magnetic field, using exact diagonalization techniques. In both cases there is a spin liquid phase composed of interacting singlet dimers at small fields h<hc1,h<{h}_{c1},h<hc1​, an incommensurate regime (hc1<h<hc2){(h}_{c1}<h<{h}_{c2})(hc1​<h<hc2​) in which the modulation of the triplet excitation spectra adapts to the applied field, and a fully spin-polarized phase above an upper critical field hc2.{h}_{c2}.hc2​. For structurally dimerized chains, the spin gap closes in the incommensurate phase, whereas spin-Peierls chains remain gapped. In the spin liquid regimes, the dominant feature of the triplet spectra is a one-magnon bound state, separated from a continuum of states at higher energies. There are also indications of a singlet bound state above the one-magnon triplet.
We study the dynamical structure factor of the frustrated spin-$3/2\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chains, with particular focus on the partially dimerized phase that emerges between two Kosterlitz-Thouless transitions. Using a valence bond solid Ansatz corroborated by density-matrix renormalization-group simulations, we investigate the nature of magnon and spinon excitations through the single-mode approximation. We show that the magnon develops an incommensurate dispersion at ${J}_{2}\ensuremath{\approx}0.32{J}_{1}$, while the spinons, viewed as domain walls between degenerate valence bond solid states, become incommensurate at ${J}_{2}\ensuremath{\approx}0.4{J}_{1}$ beyond the Lifshitz point $({J}_{2}\ensuremath{\approx}0.388{J}_{1})$. The dynamical structure factor exhibits rich spectral features shaped by the interplay between these excitations, with magnons appearing as resonances embedded in the spinon continuum. The spinon gap shows a nonmonotonic behavior, reaching a peak near the center of the partially dimerized phase and closing at the boundaries, suggesting the appearance of a floating phase as a result of the condensation of incommensurate spinons. Comparative analysis with the spin-5/25/25/2 case confirms the universality of these phenomena across half-integer higher-spin systems. Our results provide detailed insight into how fractionalization and incommensurate condensation govern the spectral properties of frustrated spin chains, offering a unified picture across different spin magnitudes.
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