Translational symmetry breaking and valence-bond-solid (VBS) order in doped antiferromagnets
The theoretical strategy of these works is summarized in Figure 1.
Figure 1. Experimentally, the high temperature superconductors are produced
by doping the insulating compound La2CuO4 with mobile holes. However, in this process, one crosses a quantum phase transition at which long-range magnetic order disappears. Theoretically it is more
convenient to first destroy the magnetic long-range order in the insulator by some other mechanism (e.g. by adding frustrating magnetic interactions), and to then dope with mobile carriers: this two-step theoretical process is sketched above. Details of the first step are discussed elsewhere in these web pages, while the second step is described in the papers below.
The first theoretical step in Figure 1 was discussed elsewhere in these web pages: it
was predicted in early work that destroying Neel
order in insulating, square lattice antiferromagnets leads to the appearance
of spin-Peierls order (or more generally valence-bond-solid (VBS) order). In addition, such paramagnetic insulators also necessarily possess a sharp S=1 collective spin resonance, and the confinement of a S=1/2 moment near each non-magnetic impurity (discussed elsewhere). The papers below discuss the consequences
of doping such a VBS state by mobile holes: VBS order, confinement of S=1/2 moments near non-magnetic impurities, and the sharp S=1 resonance survive for a finite range of doping, and co-exist with d-wave-like superconductivity.
A first analysis of the doping of the non-magnetic VBS-ordered Mott insulator
appeared in paper 2, and the results are summarized in Figure 2 below.
Figure 2. Phase diagram from paper 2 of the doping of a VBS-ordered
Mott insulator, as described by the t-J model. It is instructive to follow the phases as
a function of the doping d for large values of t/J. Initially, there
is a superconducting state which coexists with VBS ordering.
Here the superconducting order
is d-wave like (that is, the pairing amplitudes in the x
and y directions have opposite signs) and it coexists with VBS order:
such a state was first discussed
in paper 2. With increasing
doping, the amplitude of the VBS order decreases and gapless fermionic
excitations appear at nodal points, which are first generated at the edges
of the Brillouin zone. Eventually there is a phase transition to an isotropic
The computation also found an instability to phase
separation below the dotted line.
A more complete analysis of the phase diagram requires a more careful consideration
of the influence of the long-range
Coulomb interactions, and of the possibility of VBS-ordered states with
larger periods. Such an analysis was carried out
in papers 3,4, and the results are summarized in Figure 3 below.
Figure 3. Phase diagram analogous to Figure 1 above from papers 3,4. The vertical axis represents
an arbitrary short-range exchange interaction which can destroy the long-range Neel order in
the insulator. A specific example is the study of A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Physical Review Letters 89, 247201 (2002), which used a ring exchange interaction: as expected from the arguments in this web page, the large ring exchange phase had
VBS order, as shown in the figure above.
The computations of papers 3,4 were carried out in the unhatched region (which has no long range
magnetic order) and sample configurations of the VBS order are shown; the states shown pick one
directions as special, but "checkerboard" states also appear (M. Vojta, Physical Review B 66, 104505 (2002)). The physics of the transition from the hatched to the unhatched regions (involving restoration of spin rotation
invariance) is discussed in the web pages on collinear magnetic order, non-collinear magnetic order, quantum criticality, and the influence of an applied magnetic field . While the mean-field studies of VBS order
lead to a bond-centered modulation of the lattice spacings (shown above), there can be "resonance"
between different mean-field configurations, and the symmetry of bond modulations can
also be such that the reflection symmetry is about a site.
Notice the more elaborate types of VBS order in the non-magnetic superconductor. It is expected that
the primary modulation in these VBS-ordered states will be in the exchange, kinetic, and pairing energies
between sites, rather than the on-site charge densities. This issue should be distinguished from the issue
of the lattice symmetry of the modulation, which could have a plane of reflection symmetry about
either the bonds or the sites.
While the VBS states discussed above break the symmetry between the x and y
lattice directions, they are not necessarily ``quasi one-dimensional'' i.e. the basic instability leading to VBS ordering in the undoped
paramagnet is genuinely two-dimensional. Furthermore, "checkerboard" states can also appear
(M. Vojta, Physical Review B 66, 104505 (2002)).
See also the article by Barbara Levi in Physics Today, volume 57, number 9, page 24 (2004).
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