Zero Field Hall Effect in (2+1)-dimensional QED

In QED of two space dimensions, a quantum Hall effect occurs in the absence of any magnetic field. We give a simple and transparent explanation. In solid state physics, the Hall conductivity for non-degenerate ground state is expected to be given by an integer, the Chern number. In our field-free situation, however, the conductivity is $\pm 1/2$ in natural units. We fit this half-integral result into the topological setting and give a geometric explanation reconciling the points of view of QFT and solid state physics. For quasi-periodic boundary conditions, we calculate the finite size correction to the Hall conductivity. Applications to graphene and similar materials are discussed

An algebraic approach to minimal models in CFTs

CFTs are naturally defined on Riemann surfaces. The rational ones can be solved using methods from algebraic geometry. One particular feature is the covariance of the partition function under the mapping class group. In genus $g=1$, this yields modular forms, which can be linked to ordinary differential equations of hypergeometric type with algebraic solutions.Comment: 30 pages. Revised and extended version. (The paper was originally part of arXiv:1305.0469, which had been split into the present paper and arXiv:1705.07627.

Rational CFTs on Riemann surfaces

The partition function of rational conformal field theories (CFTs) on Riemann surfaces is expected to satisfy ODEs of Gauss-Manin type. We investigate the case of hyperelliptic surfaces and derive the ODE system for the $(2,5)$ minimal model.Comment: 90 page

The (2,5) minimal model on degenerating genus two surfaces

In the (2, 5) minimal model, the partition function for genus g = 2 Riemann surfaces is expected to be given by a quintuplet of Siegel modular forms that extend the Rogers-Ramanujan functions on the torus. Their expansions around the g = 2 boundary components of the moduli space are obtained in terms of standard modular forms. In the case where a handle of the g = 2 surface is pinched, our method requires knowledge of the 2-point function of the fundamental lowest-weight vector in the non-vacuum representation of the Virasoro algebra, for which we derive a third order ODE

CFTs on Riemann surfaces of genus g ≥ 1: dependence on moduli

The dependence of the Virasoro-N-point function on the moduli of the Riemann surface is investigated. We propose an algebraic geometric approach that applies to any hyperelliptic Riemann surface. Our discussion includes a demonstration of our methods with the case g = 1

Boundary states and edge currents for free fermions

We calculate the ground state current densities for 2+1 dimensional free fermion theories with local, translationally invariant boundary states. Deformations of the bulk wave functions close to the edge and boundary states both may cause edge current divergencies, which have to cancel in realistic systems. This yields restrictions on the parameters of quantum field theories which can arise as low energy limits of solid state systems. Some degree of Lorentz invariance for boosts parallel to the boundary can be recovered, when the cutoff is removed

Spontaneous Edge Currents for the Dirac Equation in Two Space Dimensions

Spontaneous edge currents are known to occur in systems of two space dimensions in a strong magnetic field. The latter creates chirality and determines the direction of the currents. Here we show that an analogous effect occurs in a field-free situation when time reversal symmetry is broken by the mass term of the Dirac equation in two space dimensions. On a half plane, one sees explicitly that the strength of the edge current is proportional to the difference between the chemical potentials at the edge and in the bulk, so that the effect is analogous to the Hall effect, but with an internal potential. The edge conductivity differs from the bulk (Hall) conductivity on the whole plane. This results from the dependence of the edge conductivity on the choice of a selfadjoint extension of the Dirac Hamiltonian. The invariance of the edge conductivity with respect to small perturbations is studied in this example by topological techniques