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arXiv:cond-mat/9712263 · Physical Review Letters

Novel Approach to Description of Spin-Liquid Phases in Low-Dimensional Quantum Antiferromagnets

Valeri N. Kotov, Oleg P. Sushkov, Zheng Weihong, J. Oitmaa

Published Jun 29, 1998• 127 citations
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Abstract

We consider quantum spin systems with dimerization, which at strong coupling have singlet ground states. To account for strong correlations, the $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ elementary excitations are described as a dilute Bose gas with infinite on-site repulsion. This approach is applied to the two-layer Heisenberg antiferromagnet at $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ with general couplings. Our analytic results for the triplet gap, the excitation spectrum, and the location of the quantum critical point are in excellent agreement with numerical results obtained by dimer series expansions.

DOI
10.1103/physrevlett.80.5790

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We consider quantum spin systems with dimerization, which at strong coupling have singlet ground states. To account for strong correlations, the S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1 elementary excitations are described as a dilute Bose gas with infinite on-site repulsion. This approach is applied to the two-layer Heisenberg antiferromagnet at T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 with general couplings. Our analytic results for the triplet gap, the excitation spectrum, and the location of the quantum critical point are in excellent agreement with numerical results obtained by dimer series expansions.
The two-layer Heisenberg antiferromagnet exhibits a zero temperature quantum phase transition from a disordered dimer phase to a collinear Neel phase, with long range order in the ground state. The spin-wave gap vanishes as ? ? ( J ? – J ? c ) &ngr; approaching the transition point. To account for strong correlations, the S = 1 elementary excitations triplets are described as a dilute Bose gas with infinite on-site repulsion. We apply the Brueckner diagram approach which gives the critical index &ngr; ˜ 0 . 5. We demonstrate also that the linearised in density Brueckner equations give the mean field result &ngr; = 1. Finally, an expansion of the Brueckner equations in powers of the density, combined with the scaling hypothesis, gives &ngr; ˜ 0 . 67. This value agrees reasonably with that of the nonlinear O(3) s model. Our approach demonstrates that for other quantum spin models the critical index can be different from that in the nonlinear s model. We discuss the conditions for this to occur.
We consider an exchange model describing two isotropic spin-1/2 Heisenberg antiferromagnets coupled by a quartic term on the square lattice. The model is relevant for systems with orbital degeneracy and strong electron-vibron coupling in the large Hubbard repulsion limit, and is known to show a spin-Peierls-like dimerization in one dimension. In two dimensions we calculate energy gaps, susceptibilities, and correlation functions with a Green's function Monte Carlo method. We find a finite triplet and singlet gap, with no evidence of order of any kind. We conclude that the ground state is, most likely, a spin liquid of resonating valence bonds.
We consider the two-layer Heisenberg antiferromagnet near a zero-temperature quantum phase transition from a disordered dimer phase to a N\'eel state. At approaching the transition point the spin-wave gap vanishes as $\ensuremath{\Delta}\ensuremath{\propto}{(J}_{\ensuremath{\perp}}\ensuremath{-}{J}_{\ensuremath{\perp}c}{)}^{\ensuremath{\nu}}.$ To account for strong correlations between the S=1S=1S=1 elementary excitations we apply the Brueckner diagram approach which gives the critical index $\ensuremath{\nu}\ensuremath{\approx}0.5.$ We demonstrate also that the linearized in density Brueckner equations give the mean field result $\ensuremath{\nu}=1.$ Finally an expansion of the Brueckner equations in powers of the density, combined with the scaling hypothesis, give $\ensuremath{\nu}\ensuremath{\approx}0.67.$ This value reasonably agrees with that of the nonlinear $O(3) \ensuremath{\sigma} \mathrm{model}.$ Our approach demonstrates that for other quantum spin models the critical index can be different from that in the nonlinear $\ensuremath{\sigma}$ model. We discuss the conditions for this to occur.
The quantum Heisenberg antiferromagnet is expressed in terms of physical electron operators, with a constraint to one electron per site, as it emerges in the large-UUU limit of the Hubbard model. It is then possible to carry out a systematic 1n\frac{1}{n}n1​ expansion built on a translation-invariant singlet ground state. Such an expansion yields a Fermi-liquid picture for the elementary excitations which should be appropriate for low-dimensional antiferromagnets. Explicit calculations in one dimension show that the excitation spectrum consists of a gapless continuum in qualitative agreement with exact Bethe ansatz results.
We numerically investigate the phase diagram of two-dimensional site-diluted coupled dimer systems in an external magnetic field. We show that this phase diagram is characterized by the presence of an extended Bose glass, not accessible to mean-field approximation, and stemming from the localization of two distinct species of bosonic quasiparticles appearing in the ground state. On the one hand, nonmagnetic impurities doped into the dimer-singlet phase of a weakly coupled dimer system are known to free up local magnetic moments. The deviations of these local moments from full polarization along the field can be mapped onto a gas of bosonic quasiparticles, which undergo condensation in zero and very weak magnetic fields, corresponding to transverse long-range antiferromagnetic order. An increasing magnetic field lowers the density of such quasiparticles to a critical value at which a quantum phase transition occurs, corresponding to the quasiparticle localization on clusters of local magnets (dimers, trimers, etc.) and to the onset of a Bose glass. Strong finite-size quantum fluctuations hinder further depletion of quasiparticles from such clusters, and thus lead to the appearance of pseudoplateaus in the magnetization curve of the system. On the other hand, site dilution hinders the field-induced Bose-Einstein condensation of triplet quasiparticles on the intact dimers, and it introduces instead a Bose glass of triplets. A thorough numerical investigation of the phase diagram for a planar system of coupled dimers shows that the two above-mentioned Bose glass phases are continuously connected, and they overlap in a finite region of parameter space, thus featuring a two-species Bose glass. The quantum phase transition from Bose glass to magnetically ordered phases in two dimensions is marked by novel universal exponents ($z\ensuremath{\approx}2$, $\ensuremath{\beta}\ensuremath{\approx}0.9$, and $\ensuremath{\nu}\ensuremath{\approx}1$). Hence we conclude that doped quantum antiferromagnets in a field represent an ideal setting for the study of fundamental dirty-boson physics.
We have studied the low-energy excitation spectrum of a dimerized and frustrated antiferromagnetic Heisenberg chain. We use an analytic approach, based on a description of the excitations as triplets above a strong-coupling singlet ground state. The quasiparticle spectrum is calculated by treating the excitations as a dilute Bose gas with infinite on-site repulsion. Additional singlet (S=0)(S=0)(S=0) and triplet (S=1)(S=1)(S=1) modes are found as two-particle bound states of the elementary triplets. We have also calculated the contributions of the elementary and collective excitations into the spin structure factor. Our results are in excellent agreement with exact diagonalizations and dimer series expansions data as long as the dimerization parameter \ensuremath{\delta} is not too small $(\ensuremath{\delta}>0.1),$ i.e., while the elementary triplets can be treated as localized objects.