$2d\,$ Fu–Kane–Mele invariant as Wess–Zumino action of the sewing matrix

Abstract

We show that the Fu–Kane–Mele invariant of the 2d time-reversal invariant crystalline insulators is equal to the properly normalized Wess–Zumino action of the so-called sewing-matrix field defined on the Brillouin torus. Applied to 3d, the result permits a direct proof of the known relation between the strong Fu–Kane–Mele invariant and the Chern–Simons action of the non-Abelian Berry connection on the bundle of valence states.

Appendix

We establish here (in the reversed order) the properties of the WZ amplitude $\,\exp [\mathrm {i}S_\mathrm{WZ}(w)]\,$ claimed in Remark in Sect. 2.

First, for a smooth family of fields $\,w_t(k)\,$ defined on $\,\mathbb T^2\,$ such that $\,w_t\circ \vartheta =-w_t^\mathrm{T}\,$ for all $\,t\,$ (with $\,\vartheta \,$ induced by $\,k\mapsto -k$)  the well known formula for the derivative of the WZ action, easily following from the definition (2.10), gives:

$$\begin{aligned} \frac{_d}{^{dt}}S_\mathrm{WZ}(w(t))= & {} \frac{_1}{^{4\pi }}\int _{\mathbb T^2}\hbox {tr}\big ( (w^{-1}_t\partial _tw_t)(w_t^{-1}dw_t)^2\big )\nonumber \\= & {} \frac{_1}{^{4\pi }}\int _{\mathbb T^2}\vartheta ^* \hbox {tr}\big ((w^{-1}_t\partial _tw_t)(w_t^{-1}dw_t)^2\big )\nonumber \\= & {} \frac{_1}{^{4\pi }}\int _{\mathbb T^2} \hbox {tr}\big (((w^\mathrm{T}_t)^{-1}\partial _tw_t^\mathrm{T})((w_t^\mathrm{T})^{-1}dw_t^\mathrm{T})^2\big )\nonumber \\= & {} \frac{_1}{^{4\pi }}\int _{\mathbb T^2} \hbox {tr}\big (((w^\mathrm{T}_t)^{-1}\partial _tw_t^\mathrm{T})((w_t^\mathrm{T})^{-1}dw_t^\mathrm{T})^2\big )^\mathrm{T}\nonumber \\= & {} -\frac{_1}{^{4\pi }}\int _{\mathbb T^2} \hbox {tr}\big (((dw_t)w_t^{-1})^2(\partial _tw_t)w_t^{-1}\big )\nonumber \\= & {} -\frac{_1}{^{4\pi }}\int _{\mathbb T^2} \hbox {tr}\big ((w_t^{-1}\partial _tw_t)(w_t^{-1}dw_t)^2\big )\,=\,0, \end{aligned}$$

(7.1)

implying the invariance of the WZ amplitude $\,\exp [\mathrm {i}S_\mathrm{WZ}(w)]\,$ under smooth deformations of $\,w\,$ preserving the symmetry (2.5).

Next, let us consider a change of the trivialization of the valence bundle

$$\begin{aligned} e_i(k)=\sum \limits _{i'}U_{i'i}(k)\,e'_{i'}(k) \end{aligned}$$

(7.2)

with unitary $\,U(k)$.  For the sewing matrices, this gives

$$\begin{aligned} w_{ij}(k)= & {} \big \langle e_i(-k)|\theta e_j(k)\big \rangle = \sum \limits _{i',j'}\overline{U_{i'i}(-k)\,U_{j'j}(k)} \,\big \langle e'_{i'}(-k)|\theta e'_{j'}(k)\big \rangle \nonumber \\= & {} \sum \limits _{i',j'}U^{-1}_{\ ii'}(-k)\,U^{-1}_{\ jj'}(k)\,w'_{i'j'}(k)\quad \end{aligned}$$

(7.3)

so that $\,w(k)=U^{-1}(-k)w'(k)(U^{-1}(k))^\mathrm{T}\,$ or

$$\begin{aligned} w'(k)=U(-k)w(k)U(k)^\mathrm{T}. \end{aligned}$$

(7.4)

The map $\,\mathbb T^2\ni k\mapsto U(k)\in U(n)\,$ can be smoothly contracted to the one

$$\begin{aligned} \mathbb T^2\ni k\mapsto U_{n_1,n_2}(k)=\mathrm{diag}[\mathrm{e}^{\mathrm {i}(n_1k_1+n_2k_2)},1, \dots ,1], \end{aligned}$$

(7.5)

where $\,n_1,n_2\in \mathbb Z\,$ are the winding numbers of $\,\det {U(k)}\,$ along the basic cycles of $\,\mathbb T^2$.  By the previous argument, it is enough to check the relation $\,\exp [\mathrm {i}S_\mathrm{WZ}(w')] =\exp [\mathrm {i}S_\mathrm{WZ}(w)]\,$ for $\,U(k)=U_{n_1,n_2}(k)$.  Besides, by an $\,SL(2,\mathbb Z)\,$ change of variables $\,k$,  one may achieve that $\,n_1=0$.  Let $\,\mathcal{D}\,$ be the unit disc in $\,\mathbb C$.  Then $\,\mathcal{B}=\mathcal{D}\times S^1\,$ is a 3-manifold with the boundary $\,S^1\times S^1\cong \mathbb T^2$.  Since $\,\det {w(k)}\,$ has no windings, there exists a smooth contraction

$$\begin{aligned}{}[0,1]\times \mathbb T^2\ni (r,k)\,\longmapsto \, W(r,k)\in U(n) \end{aligned}$$

(7.6)

such that $\,W(1,k)=w(k)\,$ and $\,W(r,k)=I\,$ for $\,r\,$ close to zero. We shall identify $\,W\,$ with a smooth map defined on $\,\mathcal{B}\,$ by setting

$$\begin{aligned} W(r\mathrm{e}^{\mathrm {i}k_1},\mathrm{e}^{\mathrm {i}k_2})=W(r,k). \end{aligned}$$

(7.7)

Consider two other smooth maps $\,V_{1,2}:\mathcal{B}\rightarrow U(n)\,$ given by

$$\begin{aligned} V_1(r\mathrm{e}^{\mathrm {i}k_1},\mathrm{e}^{\mathrm {i}k_2})=\mathrm{diag}[\mathrm{e}^{-\mathrm {i}n_2k_2},1,\dots ,1], \qquad V_2(r\mathrm{e}^{\mathrm {i}k_1},\mathrm{e}^{\mathrm {i}k_2})=\mathrm{diag}[\mathrm{e}^{\mathrm {i}n_2k_2},1,\dots ,1].\nonumber \\ \end{aligned}$$

(7.8)

The product map $\,W'=V_1WV_2:\mathcal{B}\rightarrow U(n)\,$ is a smooth extension of $\,w'(k)=U_{0,n_2}(-k)w(k)U_{0,n_2}(k)^\mathrm{T}\,$ to the interior of $\,\mathcal{B}\,$ so that, by Witten’s prescription,

$$\begin{aligned} S_\mathrm{WZ}(w')&=\int _\mathcal{B}(W')^*H=\int _\mathcal{B}(V_1WV_2)^*H\nonumber \\&=\int _\mathcal{B}\Big (V_1^*H+ W^*H+V_2^*H+\frac{_1}{^{4\pi }}\,d\,\hbox {tr}\big ((V_1^{-1}dV_1)WV_2d(WV_2)^{-1}\nonumber \\&\quad + (W^{-1}dW)V_2dV_2^{-1}\big )\Big ),\quad \end{aligned}$$

(7.9)

where we applied twice the formula

$$\begin{aligned} (W_1W_2)^*H=W_1^*H+W_2^*H+\frac{_1}{^{4\pi }}\,d\,\hbox {tr}\big ((W_1^{-1}dW_1)W_2dW_2^{-1} \big ) \end{aligned}$$

(7.10)

holding for two $\,U(n)$-valued maps $\,W_{1,2}\,$ on the same domain.  Since $\,V_1^*H=0=V_2^*H\,$ for dimensional reasons,  we infer that

$$\begin{aligned}&S_\mathrm{WZ}(w')-S_{WZ}(w)\nonumber \\&\quad =\,\frac{_1}{^{4\pi }}\int _{\partial \mathcal{B}}\hbox {tr}\Big ((V_1^{-1}dV_1)W(V_2dV_2^{-1})W^{-1}\nonumber \\&\qquad -(V_1^{-1}dV_1)(dW)W^{-1}+(W^{-1}dW)V_2dV_2^{-1}\Big )\nonumber \\&\quad =\,\frac{_1}{^{4\pi }}\int _{\mathbb T^2}\hbox {tr}\Big (\mathrm{diag}[-\mathrm {i}n_2dk_2,0,\dots ,0] \,\,w\,\,\mathrm{diag}[-\mathrm {i}n_2dk_2,0,\dots ,0]\,w^{-1}\nonumber \\&\qquad -\mathrm{diag}[-\mathrm {i}n_2dk_2,0,\dots ,0]\,(dw)w^{-1} +(w^{-1}dw)\,\mathrm{diag}[-\mathrm {i}n_2dk_2,0,\dots ,0]\Big ).\nonumber \\ \end{aligned}$$

(7.11)

Changing the variables $\,k\mapsto -k\,$ in the integral on the right hand side and using the symmetry (2.5) of $\,w\,$ together with the invariance of trace under the transposition,  one shows that the integral in question is equal to its negative, hence it vanishes.

Finally, note that $\,\widetilde{W}(r,k)=-W(r,-k)^\mathrm{T}\,$ is also a contraction of $\,w(k)\,$, and as $\,W$,  it may be regarded as defined on $\,\mathcal{B}$.  Then

$$\begin{aligned} S_\mathrm{WZ}(w)=\int _\mathcal{B}\tilde{W}^*H=\int _\mathcal{B}(-W^\mathrm{T})^*H=-\int _\mathcal{B}W^*H=-S_\mathrm{WZ}(w) \end{aligned}$$

(7.12)

modulo $\,2\pi$,  implying  that $\,\exp [\mathrm {i}S_\mathrm{WZ}(w)] =(\exp [\mathrm {i}S_\mathrm{WZ}(w)])^{-1}=\pm 1$.