Dynamic critical behavior of random spin chains in a field

The dynamic critical behavior of one-dimensional random spin systems in a field at zero temperature is investigated by a generalized transfer-matrix scaling technique. Three spin models are considered: Heisenberg in a longitudinal field, XY in a transverse field, and Ising in a transverse field, with either spin-glass or random-field disorder. Near the transition, i.e., for small values of the reduced field Δ=(H-$H_c$), where $H_c$ is the critical field, one finds that the dispersion relation for the low-frequency ω, small-wave-vector k spin-wave excitations takes the scaling form ω=$k^z$f(Δ/$k^{\mathrm{cphi}}$), and exact results are given for the dynamic exponent z and crossover exponent cphi. This form contains a crossover of the frequency ω between two different asymptotic behaviors: $k^z$ for Δ≪$k^{\mathrm{cphi}}$ and $h^{\delta }$ for Δ≫$k^{\mathrm{cphi}}$, where the field exponent is δ=z/cphi. In the case of the Heisenberg systems the spin-glass disorder gives rise to the nontrivial dynamic exponent z=(3/2, whereas in the random-field case it is the exponent associated to the field that becomes nontrivial, taking the value (4/3. For the transverse XY systems the random-field disorder implies nontrivial values for both the dynamic and field exponents, (3/2 and (3/4, respectively, whereas the spin-glass disorder does not affect the dynamics of the system, which behaves like a pure transverse XY ferromagnet. Finally, for the transverse Ising systems neither the random field nor the spin-glass disorder affects the dynamics, which is the same as for a pure transverse Ising ferromagnet.