Critical Behavior of Quantum Heisenberg Chains

Adriana Moreo, K. Hallberg, Peter Horsch

J. C. Bonner, Gerhard H. Muller, John B. Parkinson

J. C. Xavier

Using the density-matrix renormalization group, we investigate the critical behavior of the anisotropic Heisenberg chains with spins up to $s=9/2$. We show that through the relations arising from the conformal invariance and the DMRG technique it is possible to obtain accurate finite-size estimates of the conformal anomaly $c$, the sound velocity ${v}_{s}$, the anomalous dimension ${x}_{bulk}$, and the surface exponent ${x}_{s}$ of the anisotropic spin-$s$ Heisenberg chains with relatively good accuracy without fitting parameters. Our results indicate that the entanglement entropy $S(L,{l}_{A},s)$ of the spin-$s$ Heisenberg chains satisfies the relation $S(L,{l}_{A},s)\ensuremath{-}S(L,{l}_{A},s\ensuremath{-}1)=1/(2s+1)$ for $s>3/2$ in the thermodynamic limit.

Rainer Walter Kühne

We present a quantum Monte Carlo study for the isotropic antiferromagnetic Heisenberg chain coupled to Einstein phonons. By investigating the behavior of the correlation length we find the critical exponent to depend on both the phonon frequency and the spin-phonon coupling constant. Hence, these systems change the universality class continuously with the spin-phonon coupling.

Ernesto P. Raposo, Mauricio D. Coutinho-Filho

We present an extensive study of the ${\mathrm{AB}}_{2}$ polymeric Hubbard chains with experimental motivation in inorganic and organic compounds. In the half-filled strong-coupling limit, these systems can be mapped onto a quantum antiferromagnetic spin-$1/2 {\mathrm{AB}}_{2}$ Heisenberg model that presents a zero-temperature quantum phase transition to a ferrimagnetic ground state. We derive the spin-wave modes with the same correct ferromagnetic dispersion relation calculated through two different techniques. We also build an Euclidean integral formulation in the coherent state representation which naturally gives rise to Wess-Zumino terms of topological origin. The low-energy infrared behavior of the ${\mathrm{AB}}_{2}$ chains is properly described through a renormalization group analysis by the semiclassical fixed point of a one-dimensional quantum nonrelativistic nonlinear $\ensuremath{\sigma}$ model with critical dynamical exponent $z=2.$ Their low-temperature behavior is shown to be the same of a quantum ferromagnetic spin-1/2 Heisenberg chain, as evidenced by the temperature dependence of the correlation length, susceptibility, and specific heat. On the other hand, a quantum critical fixed point is also found, with the same zero-temperature critical behavior of a classical Heisenberg system in $D=d+z=3$ dimensions. Its low-temperature critical properties are related to the latter through the crossover exponent $\ensuremath{\varphi}=z{\ensuremath{\nu}}_{3},$ where ${\ensuremath{\nu}}_{3}$ is the correlation length exponent of the $D=3$ classical Heisenberg model. Sufficiently strong quantum critical fluctuations destroy the ordered ground state of the quantum $z=2$ nonlinear $\ensuremath{\sigma}$ model (but not that of the quantum ${\mathrm{AB}}_{2}$ Heisenberg spin systems, in agreement with a theorem by Lieb), giving rise to a quantum disordered phase at zero temperature. These results are supported by a connection between the quantum ${\mathrm{AB}}_{2}$ Heisenberg and generalized quantum rotor models.

J. Ricardo de Sousa