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Physical review. B, Condensed matter

Incommensurate flux phases on a square lattice

David C. Morse, T. C. Lubensky

Published Nov 1, 1990• 12 citations
Publisher

Abstract

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 1/2${\mathit{KB}}^{2}$ 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.

DOI
10.1103/physrevb.42.7994

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Journal of Physics Condensed Matter•Feb 19, 1990•1 citation
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 the stability of the Fermi-liquid and staggered flux phases in the t-J model for densities off half-filling using a large-N slave-boson approach. We examine the stability of the Fermi-liquid phase with respect to flux-density-wave (FDW) and bond-density-wave (BDW), or spin-Peierls, fluctuations at arbitrary wave vector as a function of doping \ensuremath{\delta} and the ratio \ensuremath{\delta}t/J. For densities near half-filling the Fermi liquid becomes unstable below a critical tc{\mathit{t}}_{\mathit{c}}tc​ towards an FDW state with an incommensurate wave vector k near (\ensuremath{\pi},\ensuremath{\pi}). The critical ratio \ensuremath{\delta}tc{\mathit{t}}_{\mathit{c}}tc​/J for the Fermi-liquid--FDW transition diverges rapidly as \ensuremath{\delta}\ensuremath{\rightarrow}0. For \ensuremath{\delta}>18%, the FDW instability of the Fermi liquid is preempted by a BDW instability at incommensurate k. We examine the stability of the \ensuremath{\surd}2 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}2 staggered flux phase with respect to coupled FDW and BDW fluctuations and find that, upon lowering t/J at small \ensuremath{\delta}, the staggered flux phase remains stable for \ensuremath{\delta}t/J above a nonzero critical ratio and then develops an instability of primarily BDW character. At nonzero \ensuremath{\delta} and still lower \ensuremath{\delta}t/J, the staggered flux phase also develops an instability with respect to long-wavelength FDW fluctuations; tc{\mathit{t}}_{\mathit{c}}tc​/J for this instability is approximately independent of \ensuremath{\delta}. We study the response function for such fluctuations analytically, showing the nontrivial k dependence induced by the presence of a small density of holes in the staggered phase. It is conjectured that this long-wavelength FDW instability is associated with the existence off half-filling of a commensurate flux phase with lower energy than the staggered flux phase.
We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH.
We clarify the ground-state phase diagram of the half-filled square-lattice Hubbard model with Rashba spin-orbit coupling (SOC) characterized by the spin-split energy bands due to broken inversion symmetry. Although the Rashba metals and insulating magnets have been studied well, the intermediate interaction strength of the system remained elusive due to the lack of appropriate theoretical tools to unbiasedly describe the large-scale magnetic structures. We complementarily apply four different methods (sine-square deformed mean-field theory, random phase approximation, Luttinger-Tisza method, and density matrix embedding theory) and succeed in capturing the incommensurate spin-density-wave (SDW) phases with very long spatial periods which were previously overlooked. The transition to the SDW phases from the metallic phase is driven by an unprecedented instability that nests the two parts of the Fermi surface carrying opposite spins. For large SOC, the spiral, stipe, and vortex phases are obtained, when the four Dirac points exist near the Fermi level and their whole linear dispersions nest by a wavelength $\ensuremath{\pi}$, opening a band gap. These two types of transition provide Fermiology that distinguishes the antisymmetric SOC systems, generating a variety of magnetic phases that start from the relatively weak correlation regime and continue to the strongly interacting limit.
Spatially modulated magnetic phases are investigated within the mean field theory for an itinerant electron model, i.e. the Hubbard model on a two-dimensional square lattice. By numerically diagonalizing the Hamiltonian for finite-size systems under a periodic boundary condition, we examine relative stability and physical properties of several possible magnetic states. When the electron fillings are nearly half-full, the diagonally or vertically modulated spin density wave (SDW) state is stabilized over the uniform antiferromagnetic state and a crossover from the vertical to the diagonal states appears. The diagonal or vertical stripe state is characterized by the presence of the midgap band due to the soliton lattice formation inside the main SDW gap, being an insulator. The wave length λ SDW is linearly proportional to the excess carrier concentration. Excess carriers are accommodated in the form of the soliton lattice, forming a charge density wave whose wave length is λ SDW /2.
Stability conditions for the ordered state of the magnetic fluxes induced by rotating electrons on a square lattice are analysed in the mean-field approximation for the extended Hubbard model. It is shown that the flux phase can appear if the exchange energy V1 is sufficiently large. The order parameter, magnetic flux Phi enclosed in the unit cell, and critical temperature of the flux phase are monotonically increasing functions of V1. It is also shown that the flux and the antiferromagnetic phase can coexist.