The Heisenberg Antiferromagnet With a Low Concentration of Static Defects

The static and dynamic response associated with a low concentration, x, of static defects in a Heisenberg antiferromagnet at zero temperature is analyzed within linearized spin‐wave theory. We obtain the dispersion relation for long‐wavelength spin waves in the form ω(q )= c ( x ) q + iγ( x ) q τ. Our results for c(x) agree with previous work, and in particular give c(x) = c(0)[1 + αx + O(x 2)] where the coefficient α, which can be related to the helicity modulus and the uniform perpendicular susceptibility, diverges in the limit d→2, where d is the spatial dimensionality. One major new result is that τ=d−1 for defects whose spin, S’, is different from that (S) of the host lattice and τ=d+1 when S’=S. Disciplines Physics | Quantum Physics This journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/433 The Heisenberg antiferromagnet with a low concentration of static defects C. C. Wan and A. B. Harris Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 The static and dynamic response associated with a low concentration, X, of static defects in a Heisenberg antiferromagnet at zero temperature is analyzed within linearized spin-wave theory. We obtain the dispersion relation for long-wavelength spin waves in the form w(q) = c(x)q + iy(x)qY Our results for c(x) agree with previous work, and in particular give c(x) = c(O)[l + ax + 0(x2>] where the coefficient a, which can be related to the helicity modulus and the uniform perpendicular susceptibility, diverges in the limit d-+2, where d is the spatial dimensionality. One major new result is that r=d 1 for defects whose spin, S’, is different from that (S) of the host lattice and T=d + 1 when S’=S. The transition from a magnetic insulator, through doping, to a nonmagnetic superconductor is familiar in the high-T, materials like lanthanum cuprate.’ It is interesting to understand how impurities in such a system eventually destroy the magnetic long-range order and induce the transition to superconducting phase. Here we treat the simpler problem of a low concentration of static defects at zero temperature. Our model is a Heisenberg spin system with spin S and nearest neighbor coupling constant .J, and the defects have spin S’ and a coupling constant J’. The Hamiltonian of such a system can be written as H2 S(R)*S(R + 8) + c e(R) V(R) RJ R =Ho+ c E(R)UR), (1) R where S is summed over nearest-neighbor vectors and E(R) is unity at defect sites, which are assumed not to be adjacent to one another and e(R) = 0 otherwise. For a defect at site R we have V(R)= z (J’S’(R) -JS(R)).S(R+S). (2) s Throughout we use dimensionless parameters, such as s=S’/S, and j=J/J. We introduce boson operators in the usual way. For spins on the A sublattice we set SR = S a$’ a< ; SL = @an; S, = @a:. For the B sublattice we have Sk = -S + bz b,; SL = $%bL ; S< = @bR Within linear spin-wave theory, the Hamiltonian of the pure system is