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Physical review. B, Condensed matter

Renormalized mean-field theory of flux phases

Didier Poilblanc

Published Mar 1, 1990• 12 citations
Publisher

Abstract

A renormalized Hamiltonian approach is applied to construct commensurate flux phases for the t-J model. The assumption of nonuniform phases and modulus for the bond hopping amplitudes is necessary and (sufficient) condition of self-consistency. The supercell of the modulation is closely related to the filling. The self-consistent coupled equations are solved numerically on finite square clusters of sizes as large as 20\ifmmode\times\else\texttimes\fi{}20. The differences in kinetic energy and in spin-spin correlation with respect to the t=0 uniform commensurate flux state show \ensuremath{\sim}-(t/J${)}^{1/2}$ and \ensuremath{\sim}t/J behaviors, respectively. When t/J\ensuremath{\ge}1, time-reversal symmetry is only ``weakly broken'' and the Fermi level lies in a pseudogap with a depressed density of state.

DOI
10.1103/physrevb.41.4827

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arXiv:cond-mat/0608441•Aug 19, 2006•0 citations
A renormalized Hamiltonian approach is applied to construct commensurate flux phases for the t-J model. The assumption of nonuniform phases and modulus for the bond hopping amplitudes is necessary and (sufficient) condition of self-consistency. The supercell of the modulation is closely related to the filling. The self-consistent coupled equations are solved numerically on finite square clusters of sizes as large as 20\ifmmode\times\else\texttimes\fi{}20. The differences in kinetic energy and in spin-spin correlation with respect to the t=0 uniform commensurate flux state show \ensuremath{\sim}-(t/J)1/2{)}^{1/2})1/2 and \ensuremath{\sim}t/J behaviors, respectively. When t/J\ensuremath{\ge}1, time-reversal symmetry is only ``weakly broken'' and the Fermi level lies in a pseudogap with a depressed density of state.
We investigate T=0 properties of the commensurate flux phases, a class of variational wave functions for the t-J model proposed recently by Anderson, Shastry, and Hristopoulos and which generalize the (1/2-flux state of Affleck and Marston away from half-filling. These states can be formally written in the quantum spin liquid form, the singlet bond amplitudes of which break the lattice translational symmetry. The supercell of the electromagnetic gauge (defining the fictitious flux) imposes the actual periodicities of the correlated wave function. Their order of commensurability with the lattice unit length is closely related to the hole density.
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 have studied the stability of the Fermi-liquid phase of a square-lattice tight-binding model with respect to the spontaneous generation of magnetic flux and orbital magnetization. An energy density of the form 1/2KB2{\mathit{KB}}^{2}KB2 is associated with the magnetic field B, and the phase diagram is investigated as a function of K, temperature T, and density or chemical potential \ensuremath{\mu}. For K\ensuremath{\ne}0 and density near half filling, the system condenses below a critical temperature into a \ensuremath{\surd}2 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}2 antiferromagnetic flux phase in which the spontaneously generated flux alternates in sign in a checkerboard pattern. Farther from half filling and for K\ensuremath{\ne}0, the \ensuremath{\surd}2 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}2 pattern is replaced by an incommensurate flux phase in which the spatial periodicity of the spatially modulated flux is in general an irrational multiple of the lattice constant. At zero temperature for densities off half filling, the commensurate flux phase with uniform flux density equal to the particle density is the lowest-energy state of those considered at sufficiently small values of K. We also relate the Fermi-liquid instabilities of this gauge-invariant tight-binding (t-\ensuremath{\varphi}) model at nonzero K/t to those of the large-N t-J model at nonzero t/J, and thereby predict the existence of similar flux-phase instabilities in the latter model.
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.
By using a mean-field Hartree-Fock treatment, models of strongly correlated fermions on a square lattice are reduced to the problem of fermions propagating in a magnetic field. We compute the kinetic energy of the Fermi sea for any value of the fermion density, magnetic flux, and frustration. From these we obtain, among other things, the optimal flux associated with the global energy minimum and the location of the local energy minima. Every cusp corresponds to an integer number of filled Landau levels, and the minimum-energy cusp corresponds to the one-level case. The breaking of time-reversal symmetry, when the diagonal coupling is turned on, produces a family of asymmetric cusps around \ensuremath{\Phi}=${\mathrm{\ensuremath{\Phi}}}_{0}$/2, in the one-particle energy. We use perturbation theory, valid for low fermion density, in order to analyze quantitatively the behavior of the cusplike energy minima; these minima are due to the Landau-level structure when the flux is close to a rational value. We have derived a phase diagram indicating regions of similar behavior and the points that exhibit anomalous dispersion induced by frustration.
We propose a new form of inhomogeneous phases consisting of out-of-phase staggered flux domains separated by diagonal charged domain walls centered on bonds or on sites. Remarkably, such domain flux phases are spin-rotationally symmetric and exhibit cone-like quasiparticle dispersion as well as incommensurate order of orbital currents. Such features are consistent with the pseudogap behavior and the diagonal stripes observed experimentally in lightly doped cuprates. A renormalized mean field theory shows that such solutions are competitive candidates within the ttt--JJJ model.