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Physical review. B, Condensed matter

Staggered flux phases in thet-Jmodel

Didier Poilblanc, Yasumasa Hasegawa

Published Apr 1, 1990• 25 citations
Publisher

Abstract

Antiferromagnetic flux states for the t-J model are investigated on a finite 4\ifmmode\times\else\texttimes\fi{}4 cluster with 2 holes. These wave functions consist of projected Slater determinants of noninteracting wave functions of electrons moving on a lattice with alternating positive and negative magnetic fluxes flowing through the two sublattices of unit cells. Following a recent approach used for commensurate flux phases, we also evaluate the exchange energy and the kinetic energy in the thermodynamic limit. At small J/t ratio the commensurate flux phase is shown to be unstable against the metallic staggered flux phase.

DOI
10.1103/physrevb.41.6989

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Physical review. B, Condensed matter•Mar 1, 1991•3 citations
Antiferromagnetic flux states for the t-J model are investigated on a finite 4\ifmmode\times\else\texttimes\fi{}4 cluster with 2 holes. These wave functions consist of projected Slater determinants of noninteracting wave functions of electrons moving on a lattice with alternating positive and negative magnetic fluxes flowing through the two sublattices of unit cells. Following a recent approach used for commensurate flux phases, we also evaluate the exchange energy and the kinetic energy in the thermodynamic limit. At small J/t ratio the commensurate flux phase is shown to be unstable against the metallic staggered flux phase.
We investigate flux phases of the three-dimensional (3D) t-J model for a layer structure. For some range of parameters, the system is stabilized by a uniform fictitious flux, which is commensurate to the electron number density. This state corresponds in 2D lattices to the commensurate flux phase, although in the 3D system we have the flux orientation as another variational parameter to lower the energy. Exchange of two carriers gives an extra phase factor to the wave function; this is not allowed in a continuous space but is in a lattice structure. The system exhibits 2D superconductivity confined in a plane perpendicular to the commensurate flux.
We calculate the energy of flux states described by Slater determinants of single-particle levels in a uniform magnetic field with p/q flux quanta per plaquette and with no double occupancy. We find the exchange energy of flux states is lower than that of the Gutzwiller wave function by 10%--20%. The minimum kinetic energy is obtained by introducing phases associated with the holes. The phases are calculated by mapping onto a classical frustrated XY model. At 10% doping, the 1/2 flux state (\ensuremath{\pi} phase) is stabilized for the parameter regime J/t\ensuremath{\gtrsim}0.2. At about 20% doping, the time-reversal-breaking (1/3 flux state has the lowest energy for J/t\ensuremath{\gtrsim}1.0.
Commensurate flux states for the t-J model are investigated on a finite 4\ifmmode\times\else\texttimes\fi{}4 cluster with two holes. Qualitative agreement is found with a renormalized mean-field theory; in particular, the magnetic energy exhibits an absolute minimum when the flux exactly equals the filling. Competition with the kinetic energy favors a smaller commensurate flux at intermediate J/t values. We investigate the effect of a nearest-neighbor hole-hole repulsion V. Comparing the exact ground-state results shows that the V causes an effective reduction in the value of J/t. The commensurate flux phases (CFP's) do not exhibit hole-hole attraction in real space, contrary to the t-J model for J/t\ensuremath{\ge}0.4. Therefore, the stability of the CFP's would be enhanced by a short-range repulsion between the holes.
We study wave functions for generalized flux phases by the variational Monte Carlo method. Flux tubes are attached to the holes, which produce additional phase factors besides those due to the motion of electrons. The phase factors cancel each other when the electrons and holes interchange position at some special electron densities and, as a result, there are no orbital currents in these states. We investigate especially the one-third filled case and find a slightly lower kinetic energy than the original commensurate flux phase. We also calculate the chirality to confirm that time-reversal symmetry is also broken in these states. The analogy to the quantum-Hall-effect wave functions leads us to study modified forms of off-diagonal long-range order in the pair-correlation function. The existence of this long-range order is shown explicitly by variational Monte Carlo calculations and renormalized mean-field approximations. The flux phases do not have the usual off-diagonal long-range order but do have a modified form similar to that in the quantum Hall effect. The relationships to the slave-boson mean-field theories and anyon systems are also discussed.
Kinetic and magnetic energies of the staggered flux phase, with a fictitious flux of equal magnitude but opposite direction in adjacent square plaquettes, are calculated for the two-dimensional t-J model using the variational Monte Carlo method. They are compared to the energies of the resonating-valence-bond state, the flux phase with half a quantum per plaquette, and the projected Fermi-liquid state. For about 10% hole concentration the staggered flux phase has the lowest energy of the nonsuperconducting states, but its energy is still higher than that of the superconducting d-wave state.
We propose a variational approach for the study of chiral flux-phase states. We present a class of variational wave functions of the generalized Laughlin-type. The generalization consists of the introduction of Gutzwiller projections and of a fictitious magnetic field. We classify antiferromagnetic chiral flux-phase states and evaluate analytically the corresponding expectation values of the t-J Hamiltonian at any value of filling. There are two types of chiral flux-phase states: commensurate and fractional. We have found that the minimal energy has a commensurate-flux-phase state for which the value of the flux of the orbital magnetic field equals the filling.