The Mott transition as a topological phase transition

We show that the Mott metal-insulator transition in the standard one-band Hubbard model can be understood as a topological phase transition. Our approach is inspired by the observation that the midgap pole in the self-energy of a Mott insulator resembles the spectral pole of the localized surface state in a topological insulator. We use numerical renormalization-group–dynamical mean-field theory to solve the infinite-dimensional Hubbard model, and represent the resulting local self-energy in terms of the boundary Green's function of an auxiliary tight-binding chain without interactions. The auxiliary system is of generalized Su-Schrieffer-Heeger model type; the Mott transition corresponds to a dissociation of domain walls.

Figure 1

Mapping from the Hubbard model (left) to a fully noninteracting system (right) in which physical degrees of freedom ($\circ$) couple to auxiliary tight-binding chains ($\square$).

Figure 2

Lattice self-energy at $T=0$ obtained from NRG-DMFT [panels (a) and (d)] and corresponding $t_n$ of the auxiliary chain [panels (b) and (e)]. Left panels show results for the MI ($U/t=9$, $D=4$): the hard gap in $\mathrm{Im}\mathrm{\Sigma }\left(\omega \right)$ and the Mott pole at $\omega =0$ produce an SSH-type chain in the topological phase, hosting an exponentially localized boundary zero mode, panel (c). Right panels show the metallic FL ($U/t=3$, $D=3$): the low-energy ${\omega }^2$ psuedogap in $\mathrm{Im}\mathrm{\Sigma }\left(\omega \right)$ produces a generalized SSH chain with $1/n$ decay, in the trivial phase.

Figure 3

Modified SSH model with two poles at $\pm {\omega }_p$ inside a gap of width $2\delta$ [spectral function illustrated in panel (a)]. Chain parameters $t_n$ presented in panels (c) and (d) for ${\omega }_p/D={10}^{-2}$ and ${10}^{-4}$ with common $\delta /D=0.2$, showing a domain wall at $n_{\mathrm{dw}}$. States localized at the boundary and the domain wall hybridize and gap out to give exact eigenstates with energies $\pm {\omega }_p$ [panels (e) and (f)]. The domain wall position [panel (b), points] follows ${\omega }_p\sim Dexp(-n_{\mathrm{dw}}\delta /D)$ (lines).

Figure 4

Self-energy $-\mathrm{Im}\mathrm{\Sigma }\left(\omega \right)$ from NRG-DMFT (a),(c) and corresponding auxiliary chain parameters $t_n$ (b),(d) close to the Mott transition in the FL phase at $T=0$. Top panels for $U/t=5.82$; lower panels for $U/t=5.86$ (both with $D=3$). Self-energy peaks of finite width centered on $\pm {\omega }_p$ produce a generalized SSH chain with periodic domain wall structure. Low-energy ${\omega }^2$ behavior of the self-energy manifests as long-distance ${(-1)}^n/n$ behavior in the chains. As the transition is approached, the self-energy peaks sharpen into poles and ${\omega }_p\to 0$; correspondingly, the location of the first domain wall moves out, and the beating period increases, leaving a single boundary-localized topological state in the MI.

Figure 5

Auxiliary chain parameters $t_n$ of the toy model Eq. (6), with parameters $\beta =3$, $d=15$, $\varphi =0.1$, and $\lambda =30$ [panel (a)]. The resulting ${\mathrm{\Delta }}_0\left(\omega \right)$ [panel (b)] is in good agreement with the true lattice self-energy of the Hubbard model for $U/t=5.6$, $D=3$ [panel (c)].