First-Order Character and Observable Signatures of Topological Quantum Phase Transitions

Topological quantum phase transitions are characterized by changes in global topological invariants. These invariants classify many-body systems beyond the conventional paradigm of local order parameters describing spontaneous symmetry breaking. For noninteracting electrons, it is well understood that such transitions are continuous and always accompanied by a gap closing in the energy spectrum, given that the symmetries protecting the topological phase are maintained. Here, we demonstrate that a sufficiently strong electron-electron interaction can fundamentally change the situation: we discover a topological quantum phase transition of first-order character in the genuine thermodynamic sense that occurs without a gap closing. Our theoretical study reveals the existence of a quantum critical endpoint associated with an orbital instability on the transition line between a 2D topological insulator and a trivial band insulator. Remarkably, this phenomenon entails unambiguous signatures related to the orbital occupations that can be detected experimentally.

Figure 1

(a) $T=0$ phase diagram in the $1/U$ vs $1/M$ plane for $J=U/4$ and $\lambda =0.3$. Besides delimiting the different phases—Mott, topological, and band insulator (MI, QSHI, and BI, respectively, in the main text)—the color quantifies the many-body character, as measured by the value of $\mathrm{\Xi }=2[\mathrm{\Sigma }(0)-\mathrm{\Sigma }(\infty )]$. The orange squares and the dotted line mark the continuous BI-QSHI transition for small $U$. The blue diamonds and the thick solid line mark the first-order transition between the same phases for large $U$. The two lines are connected by a quantum critical point. White circles and a dashed line denote the boundary of the MI. (b),(c) Total energy $⟨\mathcal{H}⟩$ as a function of $M$, for two values of $U$. Vertical arrows mark the transition. The red and blue curves in (c) denote the solutions coming from the QSHI and from the BI, respectively. The branches with higher energy correspond to metastable solutions that can be followed beyond the transition point and give rise to hysteresis. (d) Effective band splitting parameter $M_{\mathrm{eff}}$ near the quantum critical point showing the critical behavior of the transition.

Figure 2

Evolution of the poles $P(\mathbf{k})$ (positive part) of the single-particle Green’s function near the $\mathrm{\Gamma }$ point across the topological transition for $U=5.5$ (left panel) and $U=7.0$ (right panel). At weak interaction (left panel) the transition occurs with a closure of the band gap with the formation of a semimetallic state (Dirac cone). At strong interaction (right panel) the transition takes place without closure of the spectral gap. Here we follow the QSHI solution in its whole existence region.

Figure 3

First-order character of the topological quantum phase transition in the strongly correlated regime. (a) Local orbital polarization $⟨T_z⟩$, (b) orbital compressibility $\kappa =\partial ⟨T_z⟩/\partial M$ and (c) local spin susceptibility ${\chi }_{\mathrm{loc}}$ as a function $M$. (d) Temperature dependence of the jump in the orbital occupation of the first (higher-lying) orbital.