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Fermions and Solitons in the O(3) Nonlinear Sigma Model in 2 + 1 Space-Time Dimensions*

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Fermions and Solitons in the O(3) Nonlinear Sigma Model in 2 + 1 Space-Time Dimensions* - - SLAC-PUB-5027 c .- October 1989 CT) FERMIONS AND SOLITONS IN THE O(3) NONLINEAR SIGMA MODEL IN 2 + 1 SPACE-TIME DIMENSIONS* PURUSHOTHAM VORUGANTI~ Stanford Linear Accelerator Center Stanford University, Stanford, CA 94309 and Stanford University, Department of Physics .- Stanford, CA 94305 _. SEBASTIAN DONIACH Stanford University, Department of Applied Physics Stanford, CA 94305 ABSTRACT The field theory limit of antiferromagnetism with holes is described using the O(3) nonlinear sigma model. The minimal coupling to the spin-wave gives a Pauli term that couples the hole charge density to the topological charge density for solitons. This term leads to drastic consequences; an attractive potential for holes and solitons, spin-charge coupling for the holes and zero-momentum modes. The zero-momentum modes are exact nonperturbative solutions to the full coupled equations of motion for spin-waves and holes. The effect of a Chern-Simons term, connections to other mean-field approaches and the quantum boundstate problem are discussed. If the zero-modes are the states of lowest energy, then the holes in this field theory limit are attached to skyrmions. Submitted to Physical Review B. * Work supported by Department of Energy contract DE-AC03-76SF00515. o Now at University of Toronto, 60 St. George St., Toronto, Ontario, Canada M55 lA7. .- - I. MOTIVATION f Many of the new high-Tc superconductors exhibit a N&eel antiferromagnetic --I state at zero doping. After the addition of roughly one hole per ten Cu atoms, this ordering becomes short-ranged,’ and moreover, the system becomes supercon- ducting. The anisotropic magnetic and superconducting properties2 hint at the importance of dimensionality. If the feature of antiferromagnetism is at all crucial to the superconducting state, an adequate description of the magnetic state be- comes necessary. The unique feature of two-dimensional antiferromagnetism, once the connection to the nonlinear sigma model is made in the continuum limit, is the - _. finite energy soliton excitations. 3 If these are to play any role in the transition to the superconducting state, their interaction with holes must be fully addressed. If the holes and the solitons form bound pairs,* a natural explanation for the short- range magnetic order emerges, since each soliton presents a small area of disorder. Thus the disordered state could be equivalently viewed as the soliton condensed - state.5 If the soliton could bound two holes and no more, then the other half of the picture, namely Cooper pairs, provide an explanation for the superconductivity. -‘ -- Interestingly enough, many of these notions can be explicitly written as solutions to various equations. Although the solutions are difficult to extract, the classical and quantum regimes will be systematically discussed. Even though our conclusions are that soliton-hole energetics do not favor the above picture, the possibility remains that new methods may demonstrate the above scenario. The attractive feature of this description is its uniqueness to two dimensions and its deep connection to the magnetic properties. 2 xr. FIELD THEORY OF THE 2D HEISENBERG MODEL -. With the exception 6 of the 30K Bismuth-oxide superconductors, numerous f evidence exists that the most of the high-Tc compounds undergo a transition to --I the Neel state at low doping. The Neel state is a variational state for the S = l/2 - 2D Heisenberg model and is the exact ground state for spin S 2 1, due to a theorem by Dyson, Lieb and Simon. 7 Numerical simulations of the nonlinear sigma model and the Heisenberg model show that it is N&e1 ordered’ for S = l/2. Starting with the 2D Heisenberg model on the square lattice and expanding around the N&e1 State, Haldane was able to show that the relevant operators that appear in the long wavelength continuum limit are just given by the O(3) nonlinear sigma - -model. The assumption of this particular ground state then leads to a relativistic 2+1 field theory, which, after making the standard CP1 rnap,l’ is given by JkP = j(D,Z)+(D’Z) - q(Z+.Z’ - 1) 7 (1) with p = 0,1,2; D, = a,, + iA,; and 2 a complex two-spinor (a, /3). The theory - has one dimensionful coupling constant j and constraint fields ~(2) and A,(z). The Neel state corresponds to the field theory ground state of CI = constant and -‘ -- p2 = 1 -02. The field theory description has precisely the same number of massless spinwave modes as the lattice version, but unlike the lattice version, has a space- time Lorentz symmetry.lr The Hamiltonian in the presence of holes on the lattice is probably some- thing like the Hubbard Hamiltonian with onsite repulsion and nearest-neighbor hopping. I2 The field theory limit of the Hubbard Model is much more difficult to derive although many mean-field models exist.13 3 Writhe limit of large onsite repulsion, one can instead start with the t-J model - with hopping parameter t and a Heisenberg spin-stiffness term J. At zero doping, i the t term is irrelevant since it will lead to double occupancy which in the large I. U limit is energetically suppressed. The Heisenberg interaction is then mapped onto the sigma model by dividing the two-dimensional lattice into nonintersecting plaquettes of four sites each giving four mean-field variables for each plaquette. The massless modes in the large S limit are just the $ and n,#, fields which then describe the sigma model. The number of massless modes is two, even though four sites lead to four degrees of freedom in the continuum in agreement with the Goldstone theorem upon expanding around the Neel state. The expansion around - _. the N&e1 state thus describes the spin-spin interaction portion of the t - J mode. For the problem with nonzero doping, it is simplest to discuss the case of extremely small doping where the experimentally observed Ndel state is still a good description of the ground state and its symmetries. The mean-field descrip- tion of the small number of vacancies with spin-waves has attracted considerable attention.‘4t25 Like the two spin-wave modes, the single hole.ground states will identify the relevant number of degrees of freedom. Lattice calculations as well as other mean-field descriptions show that there are four minima in the Brillouin zone locatedI at z = (&n/2, f7r/2). Th ese four states should appear in any mean-field theory. Let us introduce two fermion operators Cl , Cl which live on each plaquette. From these two degrees of freedom, we can construct a Dirac Spinor, $J = (CT , CJ). The number of low-energy hole states corresponds to two such Dirac fermions but none of our results on soliton-hole physics will be affected if we just concentrate on one species of holes. Interactions of the spin-waves with the fermion degrees of 4 free&-m should flip down spins to up spins in the plaquette and this corresponds to the hopping of a hole from the up sublattice to the down sublattice. In order f to write the mean-field theory for the Dirac fermion coupled to the spin-waves, we -1 follow closely the symmetries of the spin-wave theory which like the Hamiltonian for the vacancies comes from the t - J model. In the limit of an extremely small number of holes, we expect the mean-field Lagrangian to have the same symme- tries as the system without any holes. The surprising symmetry of the N&e1 state is of course the Lorentz symmetry, a pseudorelativity with the speed of light be- ing the spin-wave velocity. At large doping, the NCel state is no longer the valid ground state and no symmetry arguments are valid in this regime, but luckily the - low-density regime may determine the nature of the spin-disordered state. The other symmetry of the nonlinear sigma model is the internal symmetry related to 2 -+ GZ where G is an element of SU(2). By restricting our attention to just the one Dirac fermion, we can consistently treat it as a singlet under this internal symmetry. Had we used the two “flavors” of fermions on each plaquette, other -. rich possibilities exist. l3 After considering various II, - 2 interactions, the simplest Lagrangian that preserves all the symmetries and contains the lowest number of derivatives and includes the spin-flips is given by -‘ -- L = j(D,Z)+(DV) + r&Z+2 - 1) + &“(ia, -id, + m)ti 9 (2) where we consider holes of any general coupling g and mass m (ti = 1, c = 1). The 1c, is a complex two-spinor and the yp is a 2 x 2 spin matrix that will be defined below. The coupling g begins as S but receives the usual renormalization. This Lagrangian for holes at low density has a rigorous formulation for the one-dimensional spin 5 chair?* Neither the sign of the mass nor charge nor the relativistic version of the - fermion Lagrangian is crucial to the behavior around solitons. We consider this c.- coupling for its generality, and the cases of same-sign fermions and nonrelativistic --I fermions are subcases of what follows. Having discussed the interactions of holes with spin-waves, we next examine the soliton sector of the theory. The solutions, however, come with arbitrary size and thus are not true minima of the energy functional. This property, in the absence of additional physics like holes, makes solitons unstable against quantum or l/S corrections to the nonlinear a-model. This problem in the presence of fermions is examined in Sec. III. III. SOLITONS AND SIZE INSTABILITY Solitons of the O(3) nonlinear sigma model in the CP’ language can be brought in cylindrical coordinates to the formI* (l+J&,2 ezPGw z sol = , (3) ( (1+&/2 ) where r ’ = (r/X)lNl, with X being the size and N the winding number.
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