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Physical review. B, Condensed matter

Magnetization of the S=2 antiferromagnetic Heisenberg chain near the critical magnetic field

Jizhong Lou, Shaojin Qin, Zhao-Bin Su

Published Dec 1, 2000• 3 citations
Publisher

Abstract

Using the density-matrix renormalization-group (DMRG) technique, we carry out a large scale numerical calculation for the $S=2$ antiferromagnetic Heisenberg (AFH) chain. We obtain the magnetization curve of $S=2$ chain near the critical field. The corrections to the magnetization due to the magnon-magnon interaction are the same order for $S=2$ and $S=1$ chains. By comparing the experimental magnetization with our calculation, we find that the gap obtained experimentally is about $0.08J$ which is close to the accurate DMRG estimation for the $S=2$ AFH chain. Numerical errors in our large scale calculation are also discussed.

DOI
10.1103/physrevb.62.13832

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We report on density-matrix renormalization-group and analytical work on S=1S=1S=1 antiferromagnetic Heisenberg spin chains. We study the finite-size behavior within the framework of the nonlinear $\ensuremath{\sigma}$ model. We study the effect of magnon-magnon interactions on the finite-size spectrum and on the magnetization curve close to the critical magnetic field, determine the magnon scattering length, and compare it to the prediction from the nonlinear $\ensuremath{\sigma}$ model.
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Using the recently proposed density matrix renormalization group technique we show that the magnons in the S=1 antiferromagnetic Heisenberg chain effectively behave as bosons that condense at a critical field hc{\mathit{h}}_{\mathit{c}}hc​. We determine the spin-wave velocity, v=2.49(1), as well as the gap \ensuremath{\Delta}=0.4107(1)J.
We study thermodynamic behaviors of the antiferromagnetic zigzag spin chain in magnetic fields, using the density-matrix renormalization-group method for the quantum transfer matrix. We focus on the thermodynamics of the system near the critical fields in the ground-state magnetization process $(M\ensuremath{-}H$ curve): the saturation field, the lower critical field associated with excitation gap, and the field at the middle-field cusp singularity. We calculate magnetization, susceptibility, and specific heat of the zigzag chain in magnetic fields at finite temperatures, and then discuss how the calculated quantities reflect the low-lying excitations of the system related with the critical behaviors in the $M\ensuremath{-}H$ curve.
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We study the zero-temperature magnetization curve $(M\ensuremath{-}H$ curve) of the one-dimensional quantum antiferromagnet of spin one. The Hamiltonian H we consider is of the bilinear-biquadratic form: $H={\ensuremath{\sum}}_{i}f({s}_{i}\ensuremath{\cdot}{s}_{i+1})$ (+Zeeman term) where si{s}_{i}si​ is the spin operator at site i and $f(X)=X+\ensuremath{\beta}{X}^{2}$ with $0<~\ensuremath{\beta}<1.$ We focus on validity of the $\ensuremath{\delta}$-function Bose-gas picture near the two critical fields: upper-critical field Hs{H}_{s}Hs​ above which the magnetization saturates and the lower-critical field Hc{H}_{c}Hc​ associated with the Haldane gap. As for the behavior near Hs,{H}_{s},Hs​, we take ``low-energy effective S matrix'' approach, where the correct effective Bose-gas coupling constant c is extracted from the two down-spin S matrix in its low-energy limit. We find that the resulting value of c differs from the spin-wave value. We draw the $M\ensuremath{-}H$ curve by using the resultant Bose gas, and compare it with numerical calculation where the product-wave-function renormalization-group (PWFRG) method, a variant of White's density-matrix renormalization group method, is employed. Excellent agreement is seen between the PWFRG calculation and the correctly mapped Bose-gas calculation. We also test the validity of the Bose-gas picture near the lower-critical field Hc.{H}_{c}.Hc​. Comparing the PWFRG-calculated $M\ensuremath{-}H$ curves with the Bose-gas prediction, we find that there are two distinct regions, I and II, of $\ensuremath{\beta}$ separated by a critical value ${\ensuremath{\beta}}_{c}(\ensuremath{\approx}0.41).$ In region I, $0<\ensuremath{\beta}<{\ensuremath{\beta}}_{c},$ the effective Bose coupling c is positive but rather small. The small value of c makes the ``critical region'' of the square-root behavior $M\ensuremath{\sim}\sqrt{H\ensuremath{-}{H}_{c}}$ very narrow. Further, we find that in the $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\beta}}{\ensuremath{\beta}}_{c}\ensuremath{-}0,$ the square-root behavior transmutes to a different one, $M\ensuremath{\sim}(H\ensuremath{-}{H}_{c}{)}^{\ensuremath{\theta}}$ with $\ensuremath{\theta}\ensuremath{\approx}1/4.$ In region II, ${\ensuremath{\beta}}_{c}<\ensuremath{\beta}<1,$ the square-root behavior is more pronounced as compared with region I, but the effective coupling c becomes negative.