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arXiv:cond-mat/9904155 · Physical review. B, Condensed matter

Middle-field cusp singularities in the magnetization process of one-dimensional quantum antiferromagnets

Kouichi Okunishi, Yasuhiro Hieida, Yasuhiro Akutsu

Published Sep 1, 1999• 58 citations
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Abstract

We study the zero-temperature magnetization process $(M\ensuremath{-}H$ curve) of one-dimensional quantum antiferromagnets using a variant of the density-matrix renormalization group method. For both the $S=1/2$ zig-zag spin ladder and the $S=1$ bilinear-biquadratic chain, we find clear cusp-type singularities in the middle-field region of the $M\ensuremath{-}H$ curve. These singularities are successfully explained in terms of the double-minimum shape of the energy dispersion of the low-lying excitations. For the $S=1/2$ zig-zag spin ladder, we find that the cusp formation accompanies the Fermi-liquid to non-Fermi-liquid transition.

DOI
10.1103/physrevb.60.r6953

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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.
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.
We investigate magnetization process $(M\ensuremath{-}H\phantom{\rule{0ex}{0ex}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{e})$ of S=1S=1S=1 bilinear-biquadratic (BLBQ) spin chain near critical fields at finite temperatures. We use the density matrix renormalization group (DMRG) for the two-dimensional classical lattice model mapped by the Trotter decomposition, with a help of the Baxter's variational principle. By comparing the DMRG result of S=1/2XYS=1/2 \mathrm{XY}S=1/2XY chain with the exact solution, we show that the DMRG is an efficient tool to calculate the $M\ensuremath{-}H$ curve at finite temperatures. Further, we compute the $M\ensuremath{-}H$ curve of the BLBQ chain. We compare the DMRG curve of the magnetization process of the BLBQ chain with those obtained by analytic approaches: correctly mapped $\ensuremath{\delta}$-function Bose-gas approach and ``Bethe-ansatz approximation'' approach. Near the saturation field, we show that the $\ensuremath{\delta}$-function Bose gas and the Bethe-ansatz approximation describe the $M\ensuremath{-}H$ curve well in both at zero temperature and finite temperatures. Near the lower critical field, we find that the $\ensuremath{\delta}$-function Bose gas is a good effective model at the Heisenberg point. We further find that the $\ensuremath{\delta}$-function Bose gas cannot describe the $M\ensuremath{-}H$ curve at finite temperatures, near the special point where the $M\ensuremath{-}H$ curve changes qualitatively.