By electron or hole doping quantum antiferromagnets may turn into high-temperature superconductors. The low-energy dynamics of antiferromagnets are governed by their Nambu-Goldstone bosons -the magnons -and are described by an effective field theory analogous to chiral perturbation theory for the pions in strong interaction physics. In analogy to baryon chiral perturbation theory -the effective theory for pions and nucleons -we construct a systematic low-energy effective theory for magnons and electrons or holes in an antiferromagnet. The effective theory is universal and makes model-independent predictions for the entire class of antiferromagnetic cuprates. We present a detailed analysis of the symmetries of the Hubbard model and discuss how these symmetries manifest themselves in the effective theory. A complete set of linearly independent leading contributions to the effective action is constructed. The coupling to external electromagnetic fields is also investigated.
In contrast to hole-doped systems which have hole pockets centered at (± π 2a , ± π 2a ), in lightly electron-doped antiferromagnets the charged quasiparticles reside in momentum space pockets centered at ( π a , 0) or (0, π a ). This has important consequences for the corresponding low-energy effective field theory of magnons and electrons which is constructed in this paper. In particular, in contrast to the hole-doped case, the magnonmediated forces between two electrons depend on the total momentum P of the pair. For P = 0 the one-magnon exchange potential between two electrons at distance r is proportional to 1/r 4 , while in the hole case it has a 1/r 2 dependence. The effective theory predicts that spiral phases are absent in electron-doped antiferromagnets.
The long-range forces between holes in an antiferromagnet are due to magnon exchange. The one-magnon exchange potential between two holes is proportional to cos(2ϕ)/ r 2 where r is the distance vector of the holes and ϕ is the angle between r and an axis of the square crystal lattice. One-magnon exchange leads to bound states of holes with antiparallel spins resembling d-wave symmetry. The role of these bound states as potential candidates for the preformed Cooper pairs of high-temperature superconductivity is discussed qualitatively.
Identifying the correct low-energy effective theory for magnons and holes in an antiferromagnet has remained an open problem for a long time. In analogy to the effective theory for pions and nucleons in QCD, based on a symmetry analysis of Hubbard and t-J-type models, we construct a systematic low-energy effective field theory for magnons and holes located inside pockets centered at lattice momenta (±). The effective theory is based on a nonlinear realization of the spontaneously broken spin symmetry and makes model-independent universal predictions for the entire class of lightly doped antiferromagnetic precursors of high-temperature superconductors. The predictions of the effective theory are exact, order by order in a systematic low-energy expansion. We derive the one-magnon exchange potentials between two holes in an otherwise undoped system. Remarkably, in some cases the corresponding two-hole Schrödinger equations can even be solved analytically. The resulting bound states have d-wave characteristics. The ground state wave function of two holes residing in different hole pockets has a d x 2 −y 2 -like symmetry, while for two holes in the same pocket the symmetry resembles d xy .
Based on a symmetry analysis of the microscopic Hubbard and t-J models, a systematic low-energy effective field theory is constructed for hole-doped antiferromagnets on the honeycomb lattice. In the antiferromagnetic phase, doped holes are massive due to the spontaneous breakdown of the SU(2) s symmetry, just as nucleons in QCD pick up their mass from spontaneous chiral symmetry breaking. In the broken phase the effective action contains a single-derivative term, similar to the Shraiman-Siggia term in the square lattice case. Interestingly, an accidental continuous spatial rotation symmetry arises at leading order. As an application of the effective field theory we consider one-magnon exchange between two holes and the formation of two-hole bound states. As an unambiguous prediction of the effective theory, the wave function for the ground state of two holes bound by magnon exchange exhibits f -wave symmetry.x s=↑,↓ c † xs c xs , (2.5)
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