In this paper, we propose a unified formalism, using Green's functions, to integrate out the electrons in an insulator under uniform electromagnetic fields. We derive a perturbative formula for the Green's function in the presence of uniform magnetic or electric fields. Applying the formula, we derive the formula for the polarization, the orbital magnetization, and the orbital magnetopolarizability, without assuming time reversal symmetry. Specifically, we realize that the terms linear in the electric field can only be expressed in terms of the Green's functions in one extra dimension. This observation directly leads to the result that the coefficient of the θ-term in any dimensions is given by a Wess-Zumino-Witten like term, integrated in the extended space whose boundary is the original physical Brillouin zone, with the group element replaced by the Green's function. This generalizes an earlier result for the case of time reversal invariance. 1

In this brief report, we show that in a 1D insulating system with periodic boundary conditions, the coefficient of the θ-term in the effective theory is not only determined by the topological index, i α∈occ u kα | ∂ ∂k |u kα dk. Specifically, the relative position between the electronic orbitals and the ions also alters the coefficient, as one would expect when one identifies (−eθ/2π) as the polarization. This resolves a paradox when we apply our previous result to the Su-Shreiffer-Heeger model where the two ground states related by a lattice translation have θ differed by π. We also show that the static dielectric screening is the same with or without boundaries, contrary to what we have commented in our previous paper.

We numerically study the quantum oscillations in superconducting vortex-mixed states with d-wave pairing. We show that in the parameter range of an underdoped cuprate superconductor, the commonly held assumption that the period is given by the underlying Fermi-surface area using the Onsager relation becomes invalid. Using this result, we conclude that the interpretation of the recent experimental data on YBCO as a signal of an underlying Fermi surface with four hole pockets created by a ͑ , ͒ folding cannot be ruled out. DOI: 10.1103/PhysRevB.79.180510 PACS number͑s͒: 74.25.Jb Recent experiments have shown quantum oscillations in underdoped YBCO samples in strong magnetic field ϳ45 T. [1][2][3][4][5] This has been interpreted as a signal of the underlying Fermi surface ͑FS͒. The period of the oscillation is large and implies that the FS has been reconstructed, probably, by some translational symmetry-breaking order in the ground state.6-9 However, for both materials that have been studied, the simplest construction, a ͑ , ͒ folding forming four hole pockets in the original Brillouin zone ͑BZ͒, would imply a pocket too small comparing to the nominal doping by about 25%. 1,3 This partly motivated a number of workers to interpret the data in terms of more complicated reconstruction such as incommensurate spin-density wave ͑SDW͒. 5,7Furthermore, the measured Hall effect is negative 10 and this led LeBoeuf et al. to propose that the quantum oscillations originate from electron pockets. Whether the negative Hall effect is due to flux flow 11 is currently a subject under debate. We believe that in these experiments, the samples are still in a vortex-mixed state. One evidence is the measured torque hysteresis, 5 which implies that vortices exist at least up to 45 T. Furthermore, the commonly quoted core size of 20 Å ͓see, for example, the extrapolation based on scanning tunneling microscopy ͑STM͒ measurement 12 as well as Nernst measurements 13 ͔ lead to an estimate of H c2 of 100 T. Therefore, to interpret the oscillations data, one has to understand the quantum oscillations in the mixed state.Up to now, all discussions assume that quantum oscillations in the mixed state maintain the same frequency as in the normal state and are given by the Onsager relation, which relates the frequency to the Fermi-surface area. This is true of all experiments performed up to date where it is possible to scan the magnetic field across H c2 . 14,15 However, there is no clear argument why the frequency should remain the same and previous theories predict a small shift. 16,17 We note that all previous experiments have been done on conventional s-wave low T c superconductors, with the possible exception of the organics which may be d-wave, and the high-T c cuprates may be in quite a different parameter regime. For example, the coherence length 0 is very short, on the order of 4 or 5 lattice constants, and is the consequence of a large energy gap ⌬ 0 . The number of Landau levels N is about 10 in the high-T c experiments, as opposed t...

In this paper, we investigate in general how thermodynamic quantities such as the polarization, magnetization and the magneto-electric tensor are affected by the boundaries. We show that when the calculation with periodic boundary conditions does not involve a Berry's phase, the quantity in question is determined unambiguously by the bulk, even in the presence of gapless surface states. When the calculation involves a Berry's phase, the bulk can only determine the quantity up to some quantized value, given that (i) there are no gapless surface states, (ii) the surfaces do not break the symmetries preserved by the bulk, and (iii) the system is kept at charge neutrality. If any of the above conditions is violated, the quantity is then determined entirely by the details at the boundary. Due to the strong dependence on the boundary, this kind of thermodynamic quantity, such as the isotropic magneto-electric coefficient, cannot be measured in the bulk without careful control at the boundary. One thus cannot distinguish between a topological insulator and a trivial insulator in three dimensions by any local measurement in the bulk.

In this paper we address two questions concerning the effective action of a topological insulator in one and three dimensional space without boundaries, such as a torus. The first is whether a uniform θ-term with θ = π is generated for a strong topological insulator. The second is whether such a term has observable consequences in the bulk. The answers to both questions are positive, but the observability in three dimension vanishes for infinite system size.