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arXiv:cond-mat/0606766 · Physical Review B

Two-hole bound states from a systematic low-energy effective field theory for magnons and holes in an antiferromagnet

Christoph Brügger, F. Kämpfer, M. Moser, Michele Pepe A, U.‐J. Wiese

Published Dec 27, 2006• 41 citations
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Abstract

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 $t\text{\ensuremath{-}}J$-type models, we construct a systematic low-energy effective field theory for magnons and holes located inside pockets centered at lattice momenta $(\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a},\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a})$. 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 Schr\"odinger 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 ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-like symmetry, while for two holes in the same pocket the symmetry resembles ${d}_{xy}$.

DOI
10.1103/physrevb.74.224432

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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\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J-type models, we construct a systematic low-energy effective field theory for magnons and holes located inside pockets centered at lattice momenta (\ifmmode±\else\textpm\fi\ensuremathπ2a,\ifmmode±\else\textpm\fi\ensuremathπ2a)(\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a},\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a})(\ifmmode±\else\textpm\fi2a\ensuremathπ​,\ifmmode±\else\textpm\fi2a\ensuremathπ​). 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 Schr"odinger equations can even be solved analytically. The resulting bound states have ddd-wave characteristics. The ground state wave function of two holes residing in different hole pockets has a dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​-like symmetry, while for two holes in the same pocket the symmetry resembles dxy{d}_{xy}dxy​.
Based on a symmetry analysis of the microscopic Hubbard and ttt-JJJ 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\text{SU}{(2)}_{s}SU(2)s​ symmetry, just as nucleons in Quantum Chromodynamics (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 fff-wave symmetry.
We consider a microscopic model for a doped quantum ferromagnet as a test case for the systematic low-energy effective field theory for magnons and holes, which is constructed in complete analogy to the case of quantum antiferromagnets. In contrast to antiferromagnets, for which the effective field theory approach can be tested only numerically, in the ferromagnetic case, both the microscopic and the effective theory can be solved analytically. In this way, the low-energy parameters of the effective theory are determined exactly by matching to the underlying microscopic model. The low-energy behavior at half-filling as well as in the single- and two-hole sectors is described exactly by the systematic low-energy effective field theory. In particular, for weakly bound two-hole states the effective field theory even works beyond perturbation theory. This lends strong support to the quantitative success of the systematic low-energy effective field theory method not only in the ferromagnetic but also in the physically most interesting antiferromagnetic case.
The ttt-JJJ model in the spinless-fermion representation is studied. An effective Hamiltonian for the quasiparticles is derived using a canonical transformation approach. It is shown that the rather simple form of the transformation generator allows one to take into account the effect of hole interactions with the short-range spin waves and to describe the single-hole ground state. Obtained results are very close to ones of the self-consistent Born approximation. Further accounting of the long-range spin-wave interaction is possible on a perturbative basis. Spin-wave exchange and an effective interaction due to minimization of the number of broken antiferromagnetic bonds are included in the effective quasiparticle Hamiltonian. The two-hole bound state problem is solved using a Bethe-Salpeter equation. The only bound state found to exist in the region of 1<(t/J)<51<(t/J)<51<(t/J)<5 is the ddd wave. Both types of the hole-hole interaction are important for its formation. A discussion of the possible relation of the obtained results to the problem of superconductivity in real systems is presented.
We study the hole and magnon spectral functions as a function of hole doping in the two-dimensional $t\ensuremath{-}J$ and $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{'}\ensuremath{'}}\ensuremath{-}J$ models working within the limits of spin-wave theory by linearizing the hole-spin-deviation interaction and by adapting the noncrossing approximation. We find that the staggered magnetization decreases rather rapidly with doping and it goes to zero at a few percent of hole concentration in both $t\ensuremath{-}J$ and $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{'}\ensuremath{'}}\ensuremath{-}J$ models. Furthermore, our results show that the residue of the quasiparticle peak at $\stackrel{P\vec}{G}=(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2)$ decreases very rapidly with doping. We also find pockets centered at $\stackrel{P\vec}{G}$, (i) with an elliptical shape with large eccentricity along the antinodal direction in the case of the $t\ensuremath{-}J$ model and (ii) with an almost circular shape in the case of the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{'}\ensuremath{'}}\ensuremath{-}J$ model. Last, we show that the spectral intensity distribution in the doped antiferromagnet has a waterfall-like pattern along the nodal direction of the Brillouin zone, a feature that is also seen in angle-resolved photoemission spectroscopy measurements.
A two-chain ladder model is considered that is described by the strong-coupling t - t ´- J - J ´ Hamiltonian. For the case of two holes moving in a background of antiferromagnetically interacting spins, exact, analytical results are derived for the ground-state energy and low-lying excitation spectrum. The ground state is a bound state of two holes with total spin S = 0. The charge excitation is gapless and the spin excitation has a gap. The corresponding wave functions are also exactly determined. The bound hole pair is found to have symmetry of the d-wave type. In the limit of strong rung coupling, the model maps onto an effective hard-core-boson model which exhibits dominant superconducting pairing correlations.