Topological Field Theory of Time-Reversal Invariant Insulators

TL;DR

The paper identifies a fundamental TRI topological insulator in 4+1 dimensions, described by a second-Chern-number–driven Chern-Simons theory, and demonstrates that lower-dimensional TRI insulators (3+1D and 2+1D) arise via dimensional reduction, inheriting a Z2 classification. It derives axion electrodynamics for 3+1D insulators and phase-space Chern-Simons descriptions for 2+1D descendants, predicting measurable responses such as the topological magnetoelectric effect and half-quantized surface phenomena. A unifying framework of phase-space Chern-Simons theories is developed, linking Chern numbers in higher dimensions to their lower-dimensional counterparts and providing a systematic classification across dimensions, with edge theories and pumping phenomena explained by domain-wall physics. The work unifies topological insulators through a generating functional CS_{2n+1}(λ) and clarifies the boundary-state stability and the Bott-periodic pattern of Z2 invariants, offering experimental routes (e.g., Faraday/Kerr rotation, magnetization-induced Hall effects) to observe these topological effects. Overall, it provides a comprehensive topological-field-theory foundation for TRI and QSH insulators across dimensions and a practical bridge to observable electromagnetic responses.

Abstract

We show that the fundamental time reversal invariant (TRI) insulator exists in 4+1 dimensions, where the effective field theory is described by the 4+1 dimensional Chern-Simons theory and the topological properties of the electronic structure is classified by the second Chern number. These topological properties are the natural generalizations of the time reversal breaking (TRB) quantum Hall insulator in 2+1 dimensions. The TRI quantum spin Hall insulator in 2+1 dimensions and the topological insulator in 3+1 dimension can be obtained as descendants from the fundamental TRI insulator in 4+1 dimensions through a dimensional reduction procedure. The effective topological field theory, and the $Z_2$ topological classification for the TRI insulators in 2+1 and 3+1 dimensions are naturally obtained from this procedure. All physically measurable topological response functions of the TRI insulators are completely described by the effective topological field theory. Our effective topological field theory predicts a number of novel and measurable phenomena, the most striking of which is the topological magneto-electric effect, where an electric field generates a magnetic field in the same direction, with an universal constant of proportionality quantized in odd multiples of the fine structure constant $α=e^2/\hbar c$. Finally, we present a general classification of all topological insulators in various dimensions, and describe them in terms of a unified topological Chern-Simons field theory in phase space.

Figures (21)

• Figure 1: Illustration of the Berry's phase curvature in a two-band model. The Berry's phase $\oint_C {\bf A\cdot dr}$ around a path $C$ in the BZ is half of the solid angle subtended by the image path $d(C)$ on the sphere $S_2$.
• Figure 2: (a) Illustration of a square lattice with cylindrical geometry and the chiral edge states on the boundary. The definition of $x$ and $y$ axis are also shown by black arrows. (b) One-d energy spectrum of the model in Eq. (\ref{['DiracH']}) with $m=-1.5$. The red and black line stands for the left and right moving edge states, respectively, while the blue lines are bulk energy levels. (c) Illustration of the edge states evolution for $k_y=0\rightarrow 2\pi$. The arrow shows the motion of end states in the space of center-of-mass position versus energy. (d) Polarization of the one-d system versus $k_y$. (See text)
• Figure 3: Illustration of the ${\bf d}(k,\theta)$ vector for the 1D Dirac model (\ref{['Dirac1d']}). The horizontal blue circle shows the orbit of ${\bf d}(k)$ vector in the 3D space for $k\in[0,2\pi)$ with $\theta$ fixed. The red circle shows the track of the blue circle under the variation of $\theta$. The cone shows the solid angle $\Omega(\theta)$ surrounded by the ${\bf d}(k)$ curve, which is $4\pi$ times the polarization $P(\theta)$.
• Figure 4: Illustration of the interpolation between two particle-hole symmetric Hamiltonians $h_1(k)$ and $h_2(k)$.
• Figure 5: (a) Schematic energy spectrum of a parameterized Hamiltonian $h_{mn}(\theta)$ with open boundary conditions. The red (blue) lines indicate the left (right) end states. The $\theta$ values with zero-energy left edge states are marked by the solid circles. (b) Illustration to show that the open boundary of a $Z_2$ nontrivial insulator is equivalent to a domain wall between $\theta=\pi$ (nontrivial) and $\theta=0$ (trivial vacuum). (c) Illustration of the charge density distribution corresponding to two different chemical potentials $\mu_1$ (red) and $\mu_2$ (blue). The area below the curve $\rho(\mu_1)$ and $\rho(\mu_2)$ is $+e/2$ and $-e/2$, respectively, which shows the half charge confined on the boundary.