Topological Invariant and Quantum Spin Models from Magnetic $\pi$ Fluxes in Correlated Topological Insulators

The adiabatic insertion of a $\pi$ flux into a quantum spin Hall insulator gives rise to localized spin and charge fluxon states. We demonstrate that $\pi$ fluxes can be used in exact quantum Monte Carlo simulations to identify a correlated $Z_2$ topological insulator using the example of the Kane-Mele-Hubbard model. In the presence of repulsive interactions, a $\pi$ flux gives rise to a Kramers doublet of spin-fluxon states with a Curie-law signature in the magnetic susceptibility. Electronic correlations also provide a bosonic mode of magnetic excitons with tunable energy that act as exchange particles and mediate a dynamical interaction of adjustable range and strength between spin fluxons. $\pi$ fluxes can therefore be used to build models of interacting spins. This idea is applied to a three-spin ring and to one-dimensional spin chains. Because of the freedom to create almost arbitrary spin lattices, correlated topological insulators with $\pi$ fluxes represent a novel kind of quantum simulator, potentially useful for numerical simulations and experiments.

Popular Summary

As exotic states of matter, topological insulators have, for a number of years now, commanded the fundamental interest, and fueled the scientific creativity, of a broad swath of condensed matter physicists. But, identifying these remarkable states of matter unambiguously has been fiendishly difficult. One interesting approach—a powerful one if viable—relies on the answer to the following question: Is it possible—and if yes, how—to detect unambiguously a topological insulator from its bulk properties, as opposed to from its surface properties? In this theoretical paper, we answer this question in the context of 2-dimensional (2D) topological insulators with substantial electronic interactions (thus termed “correlated topological insulators”). Our approach exploits a unique response of these systems to localized magnetic defects engineered through injections of magnetic fluxes.

In a 2D topological insulator, insertion of a magnetic flux of the size of half a flux quantum—a $\pi$ flux—gives rise to two types of states localized around the $\pi$ flux: a so-called Kramers pair of spin fluxons carrying spin $\pm 1/2$ and a pair of charge fluxons carrying charge $\pm e$. The existence of these states and their quantum numbers are intimately tied to the properties of the topological insulator, in particular, its topological invariant. Previous work on $\pi$ fluxes was limited to 2D topological insulators in which electronic interactions can be safely ignored. We have shown, with a paradigmatic model for correlated topological insulators, that electronic interactions remove charge fluxons from the low-energy excitation spectrum of the systems but leave spin fluxons detectable via a Curie-law-type contribution to the bulk magnetic susceptibility—a property routinely measured in condensed matter physics labs. Combining these types of theoretical predictions, which are enabled by the powerful quantum Monte Carlo computational methods and easily accessible bulk-measurement techniques, leads to a simple, yet highly effective tool for identifying 2D correlated topological insulators.

Another, but no less significant, usefulness of this approach, which we have also demonstrated, is that the creation of spin fluxons in response to magnetic fluxes opens a new route to artificially designing and simulating quantum spin models within the bulk gap of topological insulators—creative stimuli to new theoretical and experimental efforts.

Figure 1

For a lattice with periodic boundaries, $\pi$ fluxes can be inserted in pairs. Each flux threads a hexagon (highlighted in blue) of the honeycomb lattice, and the pair is connected by a branch cut (blue line). Hopping processes crossing the branch cut acquire a phase $e^{\mathrm{i}\pi }=-1$.

Figure 2

In a quantum spin Hall insulator, a $\pi$ flux gives rise to four states (with charge $q$ and spin $S^z$) localized near the flux, which lie inside the bulk energy gap between the valence and conduction bands (labeled “VB” and “CB” in the figure, respectively) [12, 31]. The states correspond to a Kramers doublet of spin fluxons $|\uparrow ⟩$, $|\downarrow ⟩$ with energy $E^{\uparrow \downarrow }$ and a doublet of charge fluxons $|+⟩$, $|-⟩$ with energies $E^{+}$, $E^{-}$.

Figure 3

Spin susceptibility of the Kane-Mele model ($\lambda /t=0.2$) with two $\pi$ fluxes at the maximal distance on an $18\times 18$ lattice, for different Rashba couplings $\alpha$. At temperatures $k_{B}T\lesssim 0.1t$, each $\pi$ flux contributes $\frac{1}{2k_{B}T}$ to the susceptibility, leading to ${\chi }_s\approx \frac{1}{k_{B}T}$. Also shown is the spin-gap energy scale $0.2{\Delta }_s$ for $T=0$, $\alpha =0$. For $\alpha >0$, the chemical potential is adjusted to retain a half-filled band.

Figure 4

(a) Spin gap ${\Delta }_s(\mathit{q}=0)$ and single-particle gap ${\Delta }_{\mathrm{sp}}(\mathit{q}=\mathit{K})$ in the thermodynamic limit as a function of the Hubbard repulsion $U$, at $T=0$ ($\lambda /t=0.2$, $\alpha =0$). (b) Scaling of $S^{xx}(L/2)$ using the critical exponents of the 3D XY model, $z=1$, $\nu =0.6717(1)$, and $\beta =0.3486(1)$ [52]. The intersection gives the critical point $U_c/t=5.70(3)$. The lattice size is $L\times L$. Error bars are smaller than the symbols.

Figure 5

Integrated dynamical spin-structure factor $S_{\Omega }(i)$ at $T=0$ on a $9\times 9$ lattice. (a) Localized spin fluxons created in the topological-insulator phase at $U/t=4$. (b) Absence of spin fluxons in the magnetic phase at $U/t=6$. (c) Maximum of $S_{\Omega }(i)$, as a function of $U/t$. Here $\lambda /t=0.2$, $\alpha =0$, and $\Omega /t=0.2$.

Figure 6

(a) Spin (${\chi }_s$) and charge (${\chi }_c$) susceptibilities of the Kane-Mele-Hubbard model ($\lambda /t=0.2$, $\alpha =0$) at $U/t=4$. We consider $L\times L$ lattices with one pair of $\pi$ fluxes placed at the maximal distance. At low temperatures, the spin susceptibility reveals a Curie law ${\chi }_s=\frac{2}{k_{B}T}$, whereas the charge susceptibility is suppressed by the charge gap. (b), (c) Spin susceptibility as a function of temperature for different values of $U/t$ ($\lambda /t=0.2$, $\alpha =0$). (a)–(c) show that with increasing $U/t$, the range of the interaction between spin fluxons increases, leading to deviations from the Curie law ${\chi }_s=\frac{2}{k_{B}T}$ at low temperatures. (d) For $U>U_c=5.70(3)t$, ${\chi }_s$ reflects the presence of long-range magnetic order in the bulk. Error bars are smaller than the symbol size. The arrows indicate the energy scale associated with the spin gap.

Figure 7

Spin susceptibility as a function of temperature for two $\pi$ fluxes arranged as shown in the inset ($\lambda /t=0.2$, $\alpha =0$, $9\times 9$ lattice). With increasing $U/t$, the strength of the interaction between spin fluxons increases, as revealed by the shift of the temperature below which deviations from a $\frac{2}{k_{B}T}$ Curie law occur. Statistical errors are smaller than the symbol size. Inset: $S_{\Omega }(i)$ for $U/t=4$, using the same color coding as in Fig. 5a.

Figure 8

Spin susceptibility as a function of temperature for four $\pi$ fluxes arranged as shown in the inset ($\lambda /t=0.2$, $\alpha =0$, $U/t=4$) on $L\times L$ lattices. The data reveal a Curie law $\frac{4}{k_{B}T}$ at intermediate temperatures and $\frac{2}{k_{B}T}$ at low temperatures. Statistical errors are smaller than the symbol size. Inset: $S_{\Omega }(i)$ for $L=15$, using the same color coding as in Fig. 5a.

Figure 9

Integrated local density of states $A_{\Omega }(i)$ ($\Omega =0.2t$; see text) at $T=0$ for the Kane-Mele model ($\lambda /t=0.2$, $\alpha =0$) with a periodic chain of $\pi$ fluxes. We show a part of the $72\times 12$ lattice used and the size of the magnetic unit cell containing two $\pi$ fluxes. The latter has width $a_{\pi }=4a$, where $a\equiv 1$ corresponds to the norm of the lattice vectors of the underlying honeycomb lattice.

Figure 10

Spectrum of eigenvalues of the Kane-Mele model with a periodic chain of $\pi$ fluxes (cf. Fig. 9). Here, $\lambda /t=0.2$, $\alpha =0$, and the honeycomb lattice has dimensions $72\times 12$. Points correspond to eigenvalues and lines to the band structure $ϵ(k)=\pm 2\tilde{t}\sin (2ka)$, with $\tilde{t}\approx 0.126t$ and $a\equiv 1$.

Figure 11

(a) Spin and (b) charge susceptibility of the Kane-Mele-Hubbard model ($\lambda /t=0.2$, $\alpha =0$) at $U/t=4$. We consider $L\times 12$ lattices with $L/2$ $\pi$ fluxes arranged in a periodic one-dimensional chain. The inset in (b) shows the charge susceptibility as a function of inverse temperature on a logarithmic scale. The key in (a) applies to all panels.

Figure 12

Spin susceptibility as a function of temperature. Symbols correspond to quantum Monte Carlo results for the Kane-Mele-Hubbard model ($\lambda /t=0.2$, $\alpha =0$, $U/t=4$) on $L\times 12$ lattices with $L/2$ $\pi$ fluxes arranged in a periodic one-dimensional chain. Lines are quantum Monte Carlo results for the one-dimensional XXZ model with $J^{zz}/|J^{xy}|=-0.1$ and $L/2$ lattice sites (spins).