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Modern Physics Letters B

ORBITAL PARAMAGNETISM IN TWO-DIMENSIONAL LATTICES

Fedor Vasilievich Kusmartsev

Published Apr 10, 1991• 1 citation
Publisher

Abstract

We calculate the ground state energy and the magnetization of spinless fermions on a two-dimensional lattice in an external magnetic field. We prove that the absolute minimum of the energy corresponds to a flux value equal to the filling, i.e. the “commensurate flux phase” state is preferable. The magnetization of these fermions has a paramagnetic character of special orbital type.

DOI
10.1142/s021798499100068x

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•Aug 9, 2019•0 citations
We calculate the ground state energy and the magnetization of spinless fermions on a two-dimensional lattice in an external magnetic field. We prove that the absolute minimum of the energy corresponds to a flux value equal to the filling, i.e. the “commensurate flux phase” state is preferable. The magnetization of these fermions has a paramagnetic character of special orbital type.
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.
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.
Kinetic and magnetic energies are calculated for the two-dimensional Hubbard model using the variational Monte Carlo method and including a fictitious flux phase. In contrast to earlier results for the two-dimensional t-J model, we found all the flux phases studied lie higher in energy than the paramagnetic or the antiferromagnetic states. Near half-filling, the t-J model with no double occupancy on a lattice site is dominated by the magnetic energy, while the Hubbard model with partial double occupancy is dominated by the kinetic energy. When flux phases are introduced into the Hubbard model, the decrease in the magnetic energy is not nearly enough to offset the cost in kinetic energy.
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.