Chern-Simons Terms andnField in Haldane's Model for the Quantum Hall Effect without Landau Levels

Abstract

It is shown that the nonzero quantized value of the Hall conductivity in Haldane's model [Phys. Rev. Lett. 61, 20151 appears because the electron Green's function is a topologically nontrivial function of momentum and frequency. If the n field is introduced in this model then the same topological invariant, defined in the momentum-frequency space, gives a nonzero coefficient for the Hopf-ChernSimons term of the n field. A Chern-Simons-like term, which mixes the electromagnetic and n fields and determines the electric charge of the Skyrmion, can also have a nonzero coefficient in the model. PACS numbers: 75.10.Lp, 05.30.Fk, 11.15.KcIn a recent paper, 1 Haldane considered a (2+1)-dimensional model in which fermions on a hexagonal lattice interact with a periodic external magnetic field. The field is distributed so that the total magnetic flux through a unit cell equals zero. In this situation there is no Landau quantization or degeneracy. Plane waves, characterized by quasimomenta k\ and k 2, are eigenfunctions of the model. Nevertheless, the quantum Hall effect takes place in this model.Volovik has suggested 2,3 an analog of the quantum Hall effect for a thin film of 3 He-,4 in the absence of an external magnetic field. He has found 3 a general expression for the Hall conductivity which I briefly rederive here for consistency.Let us consider at 7=0 a system of (2+l)D noninteracting fermions in a uniform or periodic potential with the action So-Tr [-4\ ¥ Hk)G-{ (k)yr(k).(1) J {2K) 5Here k = (co,&],A;2), the trace is taken over spin and other discrete indices of electrons, and G(k) is the matrix Green's function. I use natural units, so in what follows charge is measured in units of e, spin in units of ft, and Hall conductivity in units of e 2 /h. Now let us switch on a weak external electromagnetic field A^ir), 11 =0,1,2, r = (*,*,>>), which varies slowly in space-time. In first order of A n this adds the term x a G -'(k +P /2) ( ok Mto the action. Let us now integrate out fermions to find the effective action for the electromagnetic field. The lowest-order term in powers and gradients of A^ is the well-known Chern-Simons term:The structure of this term is fixed by gauge invariance. We can substitute y/(r) -exp[/0(r)]^'(r) in (1) and (2) and integrate out y/(r). In this case we shall obtain the action (3) with A M substituted by A^ -b6lbr^ This action will be the same as (3) only if Ohe tensor in this expression is the antisymmetric one e^v X and the coefficient C\ does not depend on r. As the parameters of the model (hopping integrals, chemical potential, etc.) can slowly vary in space-time, the latter condition means that the coefficient C\ in (3) is a topological invariant of the model parameters. 4

Frequency and temperature dependence of the anomalous ac Hall conductivity in a chiralpx+ipysuperconductor with impurities

“…The anomalous Hall effect was studied for this model earlier in Ref. [79], and the anomalous Nernst effect was discussed in Ref. [80].…”

“…For non-superconducting systems, the current operator can be expressed in terms of the derivatives ∂Ĥ/∂p of the HamiltonianĤ with respect to the electron quasimomenta p. For a topologically non-trivial band structure, the off-diagonal matrix elements of the current operator between different bands produce a non-zero Berry curvature, and the anomalous Hall conductivity can be expressed completely in terms of the Berry curvature [46,47,[77][78][79][80].…”

Marginality of bulk-edge correspondence for single-valley Hamiltonians

“…The Hall conductivity ͑in units of e 2 / h͒ associated to a fully gapped model is generically given by 20,21 …”

Time-Reversal Symmetry Breaking by a (d+id) Density-Wave State in Underdoped Cuprate Superconductors

“…The Hamiltonian (4) with t ′ = 0 and µ = 0 was studied in Ref. [13], where it was found that this system exhibits the intrinsic quantum Hall effect without an external magnetic field. This confirms that the time-reversal symmetry is spontaneously broken in the (d + id) density-wave state.…”