Topological field theory of time-reversal invariant insulators

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 are classified by the second Chern number. These topological properties are the natural generalizations of the time reversal-breaking quantum Hall insulator in $2+1$ dimensions. The TRI quantum spin Hall insulator in $2+1$ dimensions and the topological insulator in $3+1$ dimensions 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 measurable phenomena, the most striking of which is the topological magnetoelectric effect, where an electric field generates a topological contribution to the magnetization in the same direction, with a universal constant of proportionality quantized in odd multiples of the fine-structure constant $\alpha =e^2∕\hslash c$. 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.

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Figure 1

(Color online) Illustration of Berry’s phase curvature in a two-band model. Berry’s phase ${\oint }_C\mathbf{A}\cdot \mathbf{d}\mathbf{r}$ around a path $C$ in the BZ is half of the solid angle subtended by the image path $d\left(C\right)$ on the sphere $S_2$.

Figure 2

(Color online) (a) Illustration of a square lattice with cylindrical geometry and the chiral edge states on the boundary. The definitions of $x$ and $y$ axes are also shown. (b) One-dimensional energy spectrum of the model in Eq. (12) with $m=-1.5$. The red solid and black dashed lines stand for the left and right moving edge states, respectively, while all other blue lines are bulk energy levels. (c) Illustration of the edge states evolution for $k_y=0\to 2\pi$. The arrow shows the motion of end states in the space of center-of-mass position versus energy. (d) Polarization of the one-dimensional system versus $k_y$ (see text).

Figure 3

(Color online) Illustration of the $\mathbf{d}(k,\theta )$ vector for the 1D Dirac model (30). The horizontal blue solid circle shows the orbit of $\mathbf{d}\left(k\right)$ vector in the 3D space for $k∊[0,2\pi )$ with $\theta$ fixed. The red dashed circle shows the track of the blue solid circle under the variation in $\theta$. The cone shows the solid angle $\Omega \left(\theta \right)$ surrounded by the $\mathbf{d}\left(k\right)$ curve, which is $4\pi$ times the polarization $P\left(\theta \right)$.

Figure 4

(Color online) Illustration of the interpolation between two particle-hole symmetric Hamiltonians $h_1\left(k\right)$ and $h_2\left(k\right)$.

Figure 5

(Color online) (a) Schematic energy spectrum of a parametrized Hamiltonian $h_{mn}\left(\theta \right)$ with open boundary conditions. The red dashed (blue solid) 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 dashed line) and ${\mu }_2$ (blue dotted line). The areas below the curves $\rho \left({\mu }_1\right)$ and $\rho \left({\mu }_2\right)$ are $+e∕2$ and $-e∕2$, respectively, which show the half charge confined on the boundary.

Figure 6

(Color online) Illustration of the $2\times 2$ single-site Hamiltonians. Each point on the sphere represents a unit vector $\widehat{\mathbf{d}}=\vec{d}∕\mid d\mid$, and the north and south poles correspond to the particle-hole symmetric Hamiltonians $h_{1,2}=\pm {\sigma }^3$, respectively. The blue path shows an interpolation between $h_1$ and $h_2$ satisfying constraint (46), which always encloses a solid angle $\Omega =2\pi$.

Figure 7

The Feynman diagram that contributes to topological term (52). The loop is a fermion propagator and the wavy lines are external legs corresponding to the gauge field.

Figure 8

(Color online) Illustration showing that a band insulator with arbitrary band structure ${\epsilon }_i\left(k\right)$ can be continuously deformed to a flat band model with the same eigenstates. Since no level crossing occurs at the Fermi level, the two Hamiltonians are topologically equivalent.

Figure 9

(Color online) Three-dimensional energy spectrum of the surface states for the parameters (a) $c=1$, $m=-3$ and (b) $c=1$, $m=-1$. The Dirac points where energy gap vanishes are marked with deepest red color. For $m=-3$ there is one Dirac point at $\Gamma$ point, while for $m=-1$ there are three of them at $X$ points.

Figure 10

(Color online) Illustration of the surface Landau-level spectrum given by Eq. (70). Each level in the figure is $N_{xy}$-fold degenerate. The solid circles are the occupied states of the zeroth Landau level and the red open square shows the states that are filled when the gauge vector potential $A_z$ is shifted adiabatically from 0 to $2\pi ∕L_z$.

Figure 11

(Color online) (a) Illustration of the Hall effect induced by a spatial gradient of $P_3$. The colors represent different values of $P_3$, which decrease from 0 at the bottom to $1∕2$ on top. The blue dashed arrow shows the direction of a uniform electric field $E_x$ and the red solid arrows show the Hall current density induced, given by the formula $j_y=({\partial }_z{P}_3∕2\pi )E_x$. (b) Energy spectrum of the $(3+1)$-dimensional lattice Dirac model (73) in a magnetic field $B_z$ toward the $z$ direction. The boundary conditions are periodic in the $x$ and $y$ directions and open on the $z$ direction. The red dashed and black solid curves show the surface states on the top and bottom surfaces, respectively, each of which is a $B_z{L}_x{L}_y∕2\pi$-fold degenerate Landau Level. The parameters of model (73) are chosen to be $m=-3$, $c=1$.

Figure 12

(Color online) Illustration of the nodal points in the surface state energy spectrum of $(4+1)$-dimensional and $(3+1)$-dimensional insulators. (a) The nodal points in the $(k_x,k_y,\theta )$ BZ for a $z=\mathrm{const}$. surface of the $(4+1)$-dimensional system. The red hollow (blue filled) points stand for nodal points at time-reversal symmetric (asymmetric) wave vectors. The dashed lines are guides for the eyes. There are two pairs of asymmetric nodal points and three symmetric points in this example, which correspond to a bulk second Chern number $C_2=7$. (b) The nodal points in the $(k_x,k_y)$ BZ for a $z=\mathrm{const}$ surface of the $(3+1)$-dimensional system. According to the dimensional reduction procedure (see text) the 2D surface energy spectrum is given by the $\theta =0$ slice of the 3D surface spectrum in (a). Since $\theta =\pi$ corresponds to a vacuum Hamiltonian $h_0$, no nodal points exist in that plane. Consequently, the number of nodal points in the 2D BZ (five in this example) has the same parity as $C_2$ in the $(4+1)$-dimensional system.

Figure 13

(Color online) (a) Domain wall configuration $\theta \left(z\right)$ (blue solid line) and corresponding $\sin \theta \left(z\right)$ (red dashed line) as defined by Eq. (94) with $\xi =20$. (b) Dispersion relation of the two-dimensional bands trapped on the domain wall for the TRI Hamiltonian $h_0$. (c) The masses of surface Dirac fermion (blue solid line) and the half-quantized Hall conductance carried by the domain wall (red dashed line) in the Hamiltonian $h\left(\lambda \right)=h_0+\lambda h_1$ as a function of $\lambda$. The arrows on the left and right sides show schematically the rotation of angle $\theta$ across the domain wall for $\lambda <0$ and $\lambda >0$, respectively.

Figure 14

(Color online) Illustration of the QH effect in ferromagnet-topological insulator heterostructure. [(a) and (b)] The electric field $E_x$ (with direction into the paper) and the induced Hall currents $j_t$ and $j_b$ for parallel and antiparallel magnetizations, respectively. In (b) the Hall currents on the two surfaces are opposite and form a circulating current. (c) A top-down view of the device for Hall measurement. The gray regions are leads that contact to the surface of the topological insulator. (d) For the case of parallel magnetization, the chiral edge states are trapped on the side surfaces of the topological insulator. These carry the quantized Hall current.

Figure 15

(Color online) (a) Illustration of the magnetization induced by an electric field in a cylindrical geometry. The magnetization of the FM layer points outward from the side surface of the TI, and a circulating current is induced by the electric field. (b) Illustration of the charge polarization induced by a perpendicular magnetic field in the same geometry. “⊕” and “⊖” indicate the positive and negative charges induced by the magnetic field on the top and bottom surfaces, respectively. (c) Illustration of a topological insulator with fully gapped surface states induced by a hedgehog magnetization configuration. $P_3$ of the topological insulator is well defined since the integral along any two paths $L$ and $L^{\prime }$ gives the same $\Delta P_3=P_3\left(\vec{x}\right)-P_3\left({\vec{x}}_0\right)$. (d) Illustration of a topological insulator with parallel magnetization on the surface. A 1D domain wall between 2D surface regions with different Hall conductances (the upper and lower hemisphere as shown by red and blue, respectively) is induced, which carries chiral edge states as shown by ⊗ and ☉. In this case the $P_3$ of the topological insulator cannot be determined since two different paths $L$ and $L^{\prime }$ lead to different $\Delta P_3$.

Figure 16

(Color online) (a) Illustration of the Faraday rotation ${\theta }_{\mathrm{topo}}$ on the interface between a ferromagnet (or equivalently, a paramagnet in magnetic field) and a topological insulator. (b) Illustration of the total Faraday rotation angle $\theta$ as a function of magnetic field $B$ in the sandwich structure as shown in the inset. The zero-field extrapolation of $\theta \left(B\right)$ determines the topological term ${\theta }_{\mathrm{topo}}$.

Figure 17

(Color online) (a) Illustration of the charge pumping effect in the $(2+1)$-dimensional lattice Dirac model with spatial dependent $\theta =\theta \left(y\right)$ and time dependent $\phi =\phi \left(t\right)$. This effect is the dimensional reduction in the domain-wall QH effect in the $(3+1)$-dimensional system as shown in (b), in which ${\partial }_t\phi$ plays the role of electric field $E_z$. (c) Illustration of the half charge trapped at the crossing point of the $\theta$ and $\phi$ domain walls. This effect is the dimensional reduction in the charge trapped on the domain wall by a magnetic field, as shown in (d). $\phi \left(x\right)$ corresponds to the gauge vector potential $A_z$ as shown by red arrows lying in the top surface, the curvature of which leads to a magnetic field $B_y$ in the yellow shaded region with net flux $2\pi$. Due to the half quantized Hall conductance of the $\theta$ domain wall, such a magnetic flux induces a half charge.

Figure 18

(Color online) (a) Illustration of an interface between the vacuum (VAC) with Hamiltonian $h_0\left(\mathbf{k}\right)$ and a QSH insulator (QSH) with Hamiltonian $h_1\left(\mathbf{k}\right)$. An interpolation $h(\mathbf{k},\theta ),\theta ∊[0,2\pi ]$ can be defined between $h_0$ and $h_1$. On the left (right) half of the interface marked by red (blue), $\theta$ has a domain wall from 0 to $-\pi \left(\pi \right)$. A fractional charge $Q=e∕2$ is trapped on the domain wall between the left and right half of the interface, which can be calculated by an integration of $\Omega$ along the loop $L$ (see text). (b) Physical realization of the domain wall between two interfaces in (a) by an antiphase domain wall of magnetic field or magnetization. The red curve shows schematically the charge-density distribution.

Figure 19

(Color online) Illustration of the dimensional reduction from the surface of (a) a $(3+1)$-dimensional topological insulator to the edge of (b) a $(2+1)$-dimensional QSH insulator. The red hollow and blue filled circles in (a) are the positions of gapless $(2+1)$-dimensional Dirac cones in the surface BZ. The blue horizontal line at $k_y=\pi$ defines the edge theory of a QSH insulator, as shown in (b).

Figure 20

(Color online) Illustration of quantized charge pumping on the boundary of a QSH insulator induced by a rotating magnetic field. During the time period $t∊[0,T]$ the magnetic field is rotated from $B$ to $-B$, and a half charge $Q=\int dt{\int }_{-L}^{L}dy{j}_x=e∕2$ is pumped along $x$ direction. $L$ is taken as a cutoff with $j_x$ vanishingly small for $\mid y\mid >L$.

Figure 21

(Color online) The family tree of topological insulators. The black points on the “trunk” (i.e., zeroth generation) stand for the fundamental topological insulators in odd space-time dimensions characterized by a nontrivial Chern number in momentum space. The blue and red markers show the descendants of the $(4n-1)$-dimensional and $(4n+1)$-dimensional insulators, respectively. Physical effects associated with some of the prominent topological insulators are indicated in parenthesis. A $Z_2$ classification is defined for each blue hollow circle (square) under (pseudo-) particle-hole symmetry $C$ $\left(\tilde{C}\right)$, and for each red filled circle (square) under (pseudo-) time-reversal symmetry $T$ $\left(\tilde{T}\right)$. The definitions of $C$, $\tilde{C}$, $T$, and $\tilde{T}$ are given in Sec. 6B. For all the physically realizable systems with spatial dimensions $d⩽3$, the names of topological insulators are labeled, where ${\mathrm{CI}}_n$ $\left({\mathrm{TI}}_n\right)$ stands for a particle-hole symmetric (TRI) topological insulator in $n+1$ dimension. The black circles and squares stand for other topological phenomena obtained from dimensional reduction, which are also described by the phase space Chern-Simons theories but do not correspond to $Z_2$ topological insulators. The phase space Chern-Simons theory ${\mathrm{CS}}_{2n+1}^t$ [as defined in Eq. (134)] corresponding to each topological phenomenon is also specified on the figure.