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Physical review. B, Condensed matter

Large-Nlimit of the square-latticet-Jmodel at (1/4 and other filling fractions

Subir Sachdev A

Published Mar 1, 1990• 20 citations
Publisher

Abstract

We present a mean-field theory of the t-J model on a square lattice that becomes exact in a particular large-N limit. A detailed study of the free-energy manifold is carried out for the case of filling fraction \ensuremath{\nu}=(1/4 as a function of t/J. Over the manifold examined, the ground state is either an insulating crystal of an N-hole bound state (for small t/J) or a normal Fermi liquid (for large t/J). For (1/4\ensuremath{\le}\ensuremath{\nu}\ensuremath{\le} 1) / 2 , and t=${0}^{+}$, states with the localized holes, and nonzero spin correlations restricted to clusters of 4 or smaller, are global free-energy minima; in particular they are lower in energy than the state with uniform flux \ensuremath{\Phi}=\ensuremath{\nu}, with staggered flux or a normal Fermi liquid.

DOI
10.1103/physrevb.41.4502

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We present a mean-field theory of the t-J model on a square lattice that becomes exact in a particular large-N limit. A detailed study of the free-energy manifold is carried out for the case of filling fraction \ensuremath{\nu}=(1/4 as a function of t/J. Over the manifold examined, the ground state is either an insulating crystal of an N-hole bound state (for small t/J) or a normal Fermi liquid (for large t/J). For (1/4\ensuremath{\le}\ensuremath{\nu}\ensuremath{\le} 1) / 2 , and t=0+{0}^{+}0+, states with the localized holes, and nonzero spin correlations restricted to clusters of 4 or smaller, are global free-energy minima; in particular they are lower in energy than the state with uniform flux \ensuremath{\Phi}=\ensuremath{\nu}, with staggered flux or a normal Fermi liquid.
We investigate certain generalized flux phases arising in a mean-field approach to the t-J model. First, we establish that the energy of noninteracting electrons moving in a uniform magnetic field has an absolute minimum as a function of the flux at exactly one flux quantum per particle. Using this result, we show that if the hard-core nature of the hole bosons is taken into account, then the slave-boson mean-field approximation for the t-J Hamiltonian allows for a solution where both the spinons and the holons experience an average flux of one flux quantum per particle. This enables them to achieve the lowest possible energy within the manifold of spatially uniform flux states. In the case of the continuum model, this is possible only for certain fractional fillings and we speculate that the system may react to this frustration effect by phase separation.
We investigate ground state properties of the half-filled staggered-flux Hubbard model on a square lattice. Energy gaps to charge and spin excitations and magnetic as well as dimer orders are calculated as a function of interaction strength U/t by means of a constrained-path quantum Monte Carlo method. It is found that the system is a semimetal at U/t≲5.6 and a Mott insulator, with long-range antiferromagnetic order, at U/t≳6.6. In the range 5.6≲U/t≲6.6, the ground state is a correlated insulator where both magnetic and dimer orders are absent. Furthermore, spin excitation in the intermediate phase appears to be gapless, and the measured spin-spin correlation function exhibits power-law decaying behavior. The data suggest that the nonmagnetic ground state is a possible candidate for the putative algebraic spin liquid.
Fermion mean-field solutions for the Heisenberg and the t-J models have been obtained by considering particle-hole correlations and carrying out the mean-field-theory calculations in momentum space. At half-filling the solutions have mixed (s\ifmmode\pm\else\textpm\fi{}d)-wave-like symmetry and have a lower ground-state energy than the s (uniform phase) and the d phases. On doping, the solutions with mixed s and d symmetry still have lower ground-state energy than the uniform phase. In the presence of doping, the density of states at the Fermi level in the mixed phases is depressed from its value in the uniform phase and is in good agreement with the experimental data.
We investigate the phase diagram of the half-filled SU(N)\mathrm{SU}(N)SU(N) Hubbard-Heisenberg model with hopping ttt, exchange JJJ, and Hubbard UUU, on a two-dimensional square lattice. In the large-NNN limit, and as a function of decreasing values of t∕Jt∕Jt∕J, the model shows a transition from a ddd-density wave state to a spin dimerized insulator. A similar behavior is observed at N=6N=6N=6 whereas at N=2N=2N=2 a spin density wave insulating ground state is stabilized. The N=4N=4N=4 model, has a ddd-density wave ground state at large values of t∕Jt∕Jt∕J which as a function of decreasing values of t∕Jt∕Jt∕J becomes unstable to an insulating state with no apparent lattice and spin broken symmetries. In this state, the staggered spin-spin correlations decay as a power law, resulting in gapless spin excitations at $\stackrel{P\vec}{q}=(\ensuremath{\pi},\ensuremath{\pi})$. Furthermore, low lying spin modes with small spectral weight are apparent around the wave vectors $\stackrel{P\vec}{q}=(0,\ensuremath{\pi})$ and $\stackrel{P\vec}{q}=(\ensuremath{\pi},0)$. This gapless spin liquid state is equally found in the SU(4) Heisenberg $(U∕t\ensuremath{\rightarrow}\ensuremath{\infty})$ model in the self-adjoint antisymmetric representation. An interpretation of this state in terms of a $\ensuremath{\pi}$-flux phase is offered. Our results stem from projective (T=0)(T=0)(T=0) quantum Monte Carlo simulations on lattice sizes ranging up to $24\ifmmode\times\else\texttimes\fi{}24$.