Quantum impurities in two-dimensional antiferromagnets and d-wave superconductors

A very large number of recent experiments have studied the consequences of replacing the magnetic Cu ions in various correlated electron systems by non-magnetic Zn or Li ions. Unlike the Cu ions they replace, the non-magnetic impurity ions have no active unpaired d electron with a fluctuating spin, and so each impurity effectively creates a static hole in a background of fluctuating spins: this turns out to be an exquisitely sensitive probe of the spin wavefunction of the original system and an important playground for the confrontation of theory and experiment. In the high temperature superconductors, there is very strong evidence that each non-magnetic impurity ion actually binds a net S=1/2 moment of Cu spins in its vicinity: this is most directly evident from site-specific NMR experiments which have now been carried out at very low temperatures within the superconducting phase (J. Bobroff et al cond-mat/0010234).

We have argued that this moment formation implies that the high temperature superconductors are best understood as doped Mott insulators with confinement of S=1/2 excitations in the Mott insulating state . Such confining Mott insulators are discussed under a separate category, and we can use similar cartoon pictures to understand the confinement of a S=1/2 near each non-magnetic impurity.

Confinement of a moment near a non-magnetic impurity in a confining Mott insulator. The X's represent the non-magnetic ion. In the top picture, we sketch a trial wavefunction for the ground state within the subspace of short-range singlet bonds: there is a line of defect bonds which costs additional energy per unit length. Eventually the system will prefer to break a singlet bond and create free S=1/2 spins near each impurity, as in the lower picture. See also the cartoon pictures of states without impurities .

The Mott insulator above has a broken translational symmetry associated with bond order. This order is expected to survive for a finite range of doping, and experimental evidence for such ordering is discussed under a separate category. The quantum transition where translational symmetry is restored need not coincide with the transition where the moment is eventually screened by the S=1/2 fermionic quasiparticles. These ideas have been reviewed in a non-technical manner in paper 6.

This picture has also allowed us to develop quantitative theories of neutron scattering (H.F. Fong et al, Physical Review Letters 82, 1939 (1999) and Y. Sidis et al, Physical Review Letters 84, 5900 (2000)) and Scanning Tunneling Microscopy (E.W. Hudson et al, Science 285, 88 (1999), S.H. Pan et al, Nature 403, 746 (2000)) experiments of Zn/Li/Ni impurities, and the results are in good accord with the experiments.


  1. Quantum impurity in a nearly-critical two dimensional antiferromagnet, S. Sachdev, C. Buragohain, and M. Vojta, Science 286, 2479 (1999); cond-mat/0004156.
  2. Quantum impurity dynamics in two-dimensional antiferromagnets and superconductors, M. Vojta, C. Buragohain, and S. Sachdev, Physical Review B 61, 15152 (2000); cond-mat/9912020.
  3. Impurity spin dynamics in 2D antiferromagnets and superconductors, M. Vojta, C. Buragohain. and S. Sachdev, Proceedings of the M2S-HTSC-VI conference, Physica C 341-348, 327 (2000); cond-mat/0002316.
  4. Damping of collective modes and quasiparticles in d-wave superconductors, S. Sachdev and M. Vojta, New Theoretical Approaches to Strongly Correlated Systems, NATO ASI Series, Kluwer Academic, Dordrecht (2000); cond-mat/0005250.
  5. Static hole in a critical antiferromagnet: field-theoretic renormalization group, S. Sachdev, Physica C 357, 78 (2001); cond-mat/0011233.
  6. Impurity in a d-wave superconductor: Kondo effect and STM spectra, A. Polkovnikov, S. Sachdev, and M. Vojta, Physical Review Letters 86, 296 (2001); cond-mat/0007431.
  7. Non-magnetic impurities as probes of insulating and doped Mott insulators in two dimensions, S. Sachdev and M. Vojta, Proceedings of the XIII International Congress on Mathematical Physics, July 2000, London, A. Fokas, A. Grigoryan, T. Kibble, and B. Zegarlinski eds, International Press, Boston (2001); cond-mat/0009202.
  8. Spin and charge order in Mott insulators and d-wave superconductors, S. Sachdev, Journal of Physics and Chemistry of Solids 63, 2269 (2002); cond-mat/0108238.
  9. Quantum impurity in an antiferromagnet: non-linear sigma model theory, S. Sachdev, and M. Vojta, Physical Review B 68, 064419 (2003); cond-mat/0303001.
  10. Quantum impurity in a magnetic environment, S. Sachdev, Proceedings of the conference on Field Theory and Statistical Mechanics, Rome, June 2002 in honor of G. Jona-Lasinio, Journal of Statistical Physics 115, 47 (2004); cond-mat/0304171.
  11. Theory of quantum impurities in spin liquids, A. Kolezhuk, S. Sachdev, R. R. Biswas, and P. Chen, Physical Review B 74, 165114 (2006); cond-mat/0606385.
  12. Impurity induced spin texture in quantum critical 2D antiferromagnets, K. H. Hoglund, A. W. Sandvik, and S. Sachdev, Physical Review Letters 98, 087203 (2007); cond-mat/0611418.
  13. Impurity spin textures across conventional and deconfined quantum critical points of two-dimensional antiferromagnets, M. A. Metlitski and S. Sachdev, Physical Review B 76, 064423 (2007); cond-mat/0703790.
  14. Valence bond solid order near impurities in two-dimensional quantum antiferromagnets, M. A. Metlitski and S. Sachdev, Physical Review B 77, 054411 (2008); arXiv:0710.0626.
  15. Imaging bond order near non-magnetic impurities in square lattice antiferromagnets, R. K. Kaul, R. G. Melko, M. A. Metlitski, and S. Sachdev, Physical Review Letters 101, 187206 (2008); arXiv:0808.0495.
  16. Edge and impurity response in two-dimensional quantum antiferromagnets, M. A. Metlitski, and S. Sachdev, Physical Review B 78, 174410 (2008); arXiv:0808.0496.