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Physical review. B, Condensed matter

Systematic study of generalized flux phases

Franco Nori, Benoît Douçot, R. Rammal

Published Oct 1, 1991• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.44.7637

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FILLING LANDAU LEVELS: FERMI SEA GROUND STATE ENERGY, COMPETING INTERACTIONS AND MARGINAL DISPERSIONS IN GENERALIZED FLUX PHASES

Benoît Douçot, Franco Nori, R. Rammal

ABSTRACT: We review recent studies on the energetics of fermions confined to a two dimensional square lattice, and the relations of these results to mean-field approaches to the t−J model. Our goal has been to compute the kinetic energy of the Fermi sea of the spinless fermions for any value of the (1) fermion concentration, (2) magnetic flux, and (3) frustration. For the unfrustrated case, we confirm that the ground state energy, χ(Φ), is a minimum for Φ=π(1−δ), which corresponds to one flux quantum per spinless fermion. We then proceed to do a systematic study of frustration effects, coming from longer range couplings, which modify the picture obtained for the unfrustrated case. The frustrating influence of the kinetic energy of the holes (e.g., by breaking magnetic bonds and suppressing the long-range order present in the undoped systems) is the main focus of this work. We find that, in general, E(Φ) always exhibits cusp-like minima which position moves linearly as a function of the fermion density x. Frustration can induce a competition between different local minima. By first considering the local minima for one particle only, we can understand most of the qualitative features of E(Φ). These local minima occur at simple rational fractions of Φ0, and when the flux slightly deviates from these values a one-particle Landau level structure develops. It is precisely such a spectrum that generates a family of cusps that “move away” from the original flux value as x is increased. Every cusp corresponds to an integer number of filled Landau levels, and the minimum energy cusp corresponds to the one level case. Furthermore, we use perturbation theory, valid for low fermion density x, in order to analyze quantitatively the behavior of the cusp-like energy minima; which originate from the Landau level structure when the flux is close to a rational value. If the flux is slightly away from a given rational value [Formula: see text] each of the q subbands generates a secondary Landau level structure. We have derived a t2−t3 phase diagram indicating regions of similar behavior (i.e., adiabatic continuations can be performed with each region, preserving the E(Φ) structure) and the boundaries between them. We have studied several points belonging to those boundaries and found that anomalous behavior, (e.g., cancelation of the k2 term in the dispersion relation) induced by frustration, can occur.

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The energy of 2D Bloch electrons in a magnetic field is studied as a function of the filling fraction ν and the magnetic flux Φ. Using a new semi-classical quantization method the total energy E(Φ, ν) is calculated and shown to have an absolute minimum which corresponds to one flux quantum per particle. This optimal flux phenomenon is shown to occur under large conditions and for different lattices. An explicit cusp-like behavior of E νs. Φ at fixed ν is found both for the absolute and for the relative minima of E. Furthermore, the ground state energy is shown to be a smooth function of ν. The implications of our results for the stabilization of Anyons and the flux states are discussed.

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Incommensurate flux phases on a square lattice

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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.

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Renormalized mean-field theory of flux phases

Didier Poilblanc

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.

Physical review. B, Condensed matter•Mar 1, 1990•12 citations

Fermionic Sherrington-Kirkpatrick models with Hubbard interaction: Magnetism and electronic structure

R. Oppermann, D. Sherrington

Models with range-free frustrated Ising spin interaction and additional Hubbard interaction are treated exactly by means of the discrete time slicing method of Grassmann-field theory. Critical and tricritical points, spin and charge correlations, and the fermion propagator are derived as a function of temperature and chemical potential $\ensuremath{\mu},$ of the Hubbard coupling U and the spin-glass energy J. U is allowed to be either repulsive $(U>0)$ or attractive $(U<0).$ Cuts through the multidimensional phase diagram are obtained. Analytical and numerical evaluations take important replica symmetry-breaking (RSB) effects into account. Results for the ordered phase are given at least in a one-step approximation (1RSB), and for $T=0$ we report the first two-, three-, and four-step calculations (4RSB) for fermionic spin glasses. The use of exact relations and invariances under RSB together with 2RSB calculations for all fillings and 4RSB solutions for half filling allow to model exact solutions by interpolation. For $T=0,$ our numerical results provide strong evidence that the exact spin-glass pseudogap obeys $\ensuremath{\rho}{(E)=c}_{1}|E\ensuremath{-}{E}_{F}|$ for energies close to the Fermi level with ${c}_{1}\ensuremath{\approx}0.13.$ Rapid convergence of ${\ensuremath{\rho}}^{\ensuremath{'}}{(E}_{F})$ under increasing orders of RSB is observed and ${\ensuremath{\rho}}^{\ensuremath{''}}(E)$ is evaluated to estimate subleading powers. Over a wide range of the pseudogap and after a small transient regime $\ensuremath{\rho}(E)$ regains a linear shape with a larger slope and a small S-like perturbation. The leading term resembles the Efros-Shklovskii Coulomb pseudogap of two-dimensional localized disordered fermionic systems. Beyond half filling we obtain a $\ensuremath{\nu}\ensuremath{-}1\ensuremath{\sim}(\ensuremath{\mu}\ensuremath{-}{U)}^{2},$ $\ensuremath{\mu}>~U$ dependence of the fermion filling factor $\ensuremath{\nu}.$ We find a half filling transition between a phase for $U>\ensuremath{\mu},$ where the Fermi level lies inside the Hubbard gap, into a phase where $\ensuremath{\mu}(>U)$ is located at the center of the upper spin-glass (SG) pseudogap (SG gap). For $\ensuremath{\mu}>U$ the Hubbard gap combines with the lower one of two SG gaps (phase I), while for $\ensuremath{\mu}<U$ it joins the sole SG gap which exists in this half-filling regime (phase II). Shoulders of the combined gaps are shaped by RSB due to spin-glass order. We predict scaling behavior at the half-filling transition which becomes continuous due to $\ensuremath{\infty}\mathrm{RSB}.$ Implications of the half-filling transition between the deeper insulating phase II and phase I for the eventual delocalization by additional hopping processes in itinerant model extensions are discussed. Possible metal-insulator transition scenarios are described. Generalizations to random Hubbard coupling and alloy models as well as frustrated magnetic interactions with ferro- or antiferromagnetic components are discussed.

arXiv:cond-mat/0207734•Physical review. B, Condensed matter•Jun 27, 2003•10 citations
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}=\ensuremathΦ0{\mathrm{\ensuremath{\Phi}}}_{0}\ensuremathΦ0​/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.
ABSTRACT: We review recent studies on the energetics of fermions confined to a two dimensional square lattice, and the relations of these results to mean-field approaches to the t−J model. Our goal has been to compute the kinetic energy of the Fermi sea of the spinless fermions for any value of the (1) fermion concentration, (2) magnetic flux, and (3) frustration. For the unfrustrated case, we confirm that the ground state energy, χ(Φ), is a minimum for Φ=π(1−δ), which corresponds to one flux quantum per spinless fermion. We then proceed to do a systematic study of frustration effects, coming from longer range couplings, which modify the picture obtained for the unfrustrated case. The frustrating influence of the kinetic energy of the holes (e.g., by breaking magnetic bonds and suppressing the long-range order present in the undoped systems) is the main focus of this work. We find that, in general, E(Φ) always exhibits cusp-like minima which position moves linearly as a function of the fermion density x. Frustration can induce a competition between different local minima. By first considering the local minima for one particle only, we can understand most of the qualitative features of E(Φ). These local minima occur at simple rational fractions of Φ0, and when the flux slightly deviates from these values a one-particle Landau level structure develops. It is precisely such a spectrum that generates a family of cusps that “move away” from the original flux value as x is increased. Every cusp corresponds to an integer number of filled Landau levels, and the minimum energy cusp corresponds to the one level case. Furthermore, we use perturbation theory, valid for low fermion density x, in order to analyze quantitatively the behavior of the cusp-like energy minima; which originate from the Landau level structure when the flux is close to a rational value. If the flux is slightly away from a given rational value [Formula: see text] each of the q subbands generates a secondary Landau level structure. We have derived a t2−t3 phase diagram indicating regions of similar behavior (i.e., adiabatic continuations can be performed with each region, preserving the E(Φ) structure) and the boundaries between them. We have studied several points belonging to those boundaries and found that anomalous behavior, (e.g., cancelation of the k2 term in the dispersion relation) induced by frustration, can occur.
The energy of 2D Bloch electrons in a magnetic field is studied as a function of the filling fraction ν and the magnetic flux Φ. Using a new semi-classical quantization method the total energy E(Φ, ν) is calculated and shown to have an absolute minimum which corresponds to one flux quantum per particle. This optimal flux phenomenon is shown to occur under large conditions and for different lattices. An explicit cusp-like behavior of E νs. Φ at fixed ν is found both for the absolute and for the relative minima of E. Furthermore, the ground state energy is shown to be a smooth function of ν. The implications of our results for the stabilization of Anyons and the flux states are discussed.
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.
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.
Models with range-free frustrated Ising spin interaction and additional Hubbard interaction are treated exactly by means of the discrete time slicing method of Grassmann-field theory. Critical and tricritical points, spin and charge correlations, and the fermion propagator are derived as a function of temperature and chemical potential $\ensuremath{\mu},$ of the Hubbard coupling U and the spin-glass energy J. U is allowed to be either repulsive (U>0)(U>0)(U>0) or attractive (U<0).(U<0).(U<0). Cuts through the multidimensional phase diagram are obtained. Analytical and numerical evaluations take important replica symmetry-breaking (RSB) effects into account. Results for the ordered phase are given at least in a one-step approximation (1RSB), and for T=0T=0T=0 we report the first two-, three-, and four-step calculations (4RSB) for fermionic spin glasses. The use of exact relations and invariances under RSB together with 2RSB calculations for all fillings and 4RSB solutions for half filling allow to model exact solutions by interpolation. For T=0,T=0,T=0, our numerical results provide strong evidence that the exact spin-glass pseudogap obeys $\ensuremath{\rho}{(E)=c}_{1}|E\ensuremath{-}{E}_{F}|$ for energies close to the Fermi level with ${c}_{1}\ensuremath{\approx}0.13.$ Rapid convergence of ${\ensuremath{\rho}}^{\ensuremath{'}}{(E}_{F})$ under increasing orders of RSB is observed and ${\ensuremath{\rho}}^{\ensuremath{''}}(E)$ is evaluated to estimate subleading powers. Over a wide range of the pseudogap and after a small transient regime $\ensuremath{\rho}(E)$ regains a linear shape with a larger slope and a small S-like perturbation. The leading term resembles the Efros-Shklovskii Coulomb pseudogap of two-dimensional localized disordered fermionic systems. Beyond half filling we obtain a $\ensuremath{\nu}\ensuremath{-}1\ensuremath{\sim}(\ensuremath{\mu}\ensuremath{-}{U)}^{2},$ $\ensuremath{\mu}>~U$ dependence of the fermion filling factor $\ensuremath{\nu}.$ We find a half filling transition between a phase for $U>\ensuremath{\mu},$ where the Fermi level lies inside the Hubbard gap, into a phase where $\ensuremath{\mu}(>U)$ is located at the center of the upper spin-glass (SG) pseudogap (SG gap). For $\ensuremath{\mu}>U$ the Hubbard gap combines with the lower one of two SG gaps (phase I), while for $\ensuremath{\mu}<U$ it joins the sole SG gap which exists in this half-filling regime (phase II). Shoulders of the combined gaps are shaped by RSB due to spin-glass order. We predict scaling behavior at the half-filling transition which becomes continuous due to $\ensuremath{\infty}\mathrm{RSB}.$ Implications of the half-filling transition between the deeper insulating phase II and phase I for the eventual delocalization by additional hopping processes in itinerant model extensions are discussed. Possible metal-insulator transition scenarios are described. Generalizations to random Hubbard coupling and alloy models as well as frustrated magnetic interactions with ferro- or antiferromagnetic components are discussed.