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Help Privacy Terms. Elliptic problems with nonlinearities indefinite in sign S Alama, G Tarantello journal of functional analysis 1 , , Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems R Michalek, G Tarantello Journal of differential equations 72 1 , , In Eq. We note that for the TI surface with a Zeeman field 23 , a similar Chern—Simons Lagrangian arises as a result of the parity anomaly, where the coefficient in front of the Chern—Simons term is exactly the Chern number C. This suggests that the ground state carries a magnetization associated with it.
In the following, we take this into account and write the off-diagonal term in Eq. We obtain. The total net magnetization M is shown by the arrow for each case. The effective model Eq. Our further studies on the wrapping effect shows that the exciton gap remains very robust and therefore one can neglect the wrapping effect for the following analysis.
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To facilitate the study of excitonic instability, we transform H I onto the chiral basis. Then, considering only interband forward scattering channel, the low-energy effective vectorial interaction component can be derived as see Methods for details ,. The polar excitonic phase is topologically trivial with zero chiral Chern number. Therefore, even though the ground state energies of the two states are close, the mass term of the polar exciton insulator is negligible in the mean-field theory.
Hence, the resultant mean-field Hamiltonian exactly matches with Eq. The results from mean-field calculations. In the above calculations, we temporarily focused on the p -wave interaction H eff 1 without considering the s -wave component. In order to be more general, we assume in the following two independent couplings V p and V s for the p -wave and s -wave interaction respectively, and treat both terms on equal footing.
The former and latter are stabilized for large V s and V p respectively. Recalling the interband optical absorption process shown by Fig. The corresponding microscopic theory can be constructed by quantization of the vector potential A r , i. The coupling of the external field to the surface electrons is described within the formalism of the minimal coupling, which leads to the electron—photon interaction,.
This verifies the spin-flip mechanism illustrated by Fig. This procedure leads to the effective action S int describing the renormalization from photons:. After a unitary transformation to the chiral basis, Eq. The interband optical processes automatically select the p -wave channel due to the spin-flip scatterings Fig. According to the mean-field study above and the calculated phase diagram in Fig. The resultant state displays the predicted fractional charges associated with vortex excitations, the spontaneous in-plane magnetization and thus the anisotropic transport of excitons on the TI surface.
Excitonic insulators require strong interactions since the topological surface states become unstable only when the interaction is as large as the ultraviolet cutoff band gap as shown in Fig.
Therefore, we argue that topological Kondo insulators TKIs are good candidates for excitonic instability on TI surface. Indeed, magnetism and hysteresis in magnetotransport have been observed in SmB 6 Whether these experiments indeed observe an excitonic insulator require future work.
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We would like to note that our theory is not related with the recent works on excitonic contributions to quantum oscillations 53 or the intervalley excitonic instabilities 54 in SmB 6. In addition to the short-range interaction, we also considered a long-range Coulomb interaction on the TI surface and make generalization to finite temperature It is found that the critical interaction V c for BCS exciton condensation is significantly reduced for long-range interaction.
At finite temperature, taking into account the thermal fluctuation, the strict long-range order is absent, the Kosterliz—Thouless transition temperature T KT can be calculated by evaluating the phase stiffness of the exciton order based on the mean-field calculations Experimentally, the driven TI surface offers a rare platform for realization of new optical devices and facilitates the study of possible collective instabilities of Dirac electrons.
Experiments measuring bulk metallic TI materials have obtained the lifetime of the excited states to be around a few picoseconds 57 — 59 , which is significantly larger than than that in graphene. These indicate promising observation of a transient or quasi-equilibrium excitonic insulator 59 , Besides, the surface has a reduced dielectric constant which can be tuned even smaller by doping or gating 62 , and this possibly makes the long-range interaction more effective.
We provided a concrete example and discussed its possible realization in strongly correlated TI surface. Interestingly enough, similar to the TIs due to band inversion, the predicted topological behaviors, i. This suggests that realizations of topological p -wave excitonic insulators are likely in many other physical systems.
For example, the spectrum under study can be continuously deformed from the Dirac cone to a quadratic band crossing point QBCP This indicates that the instability is also possible and may be more feasible in interacting 2D electron gas with spin-momentum locking and QBCP. In this sense, the present results are general and may provide new physical settings for the realization of quantum anomalies in condensed matter system, and may stimulate different directions to search for more exotic topological state of matters.
From Eq. Then, a Fourier transformation yields,. After transformation to the polar coordinate, we arrive at the zero-energy BdG-like equations in real space. The action in the functional representation is then cast into:. The above action formally describes a 2D Dirac fermion with a mass v F k 2. With further including the vortex degrees of freedom in the order parameter, the Lagrangian can be written as,. The behavior of vortex degrees of freedom can be extracted from the above Lagrangian after integrating out the Grassmann fields.
To the second order expansion, the Chern—Simons action in Eq. Let us denote the four k vectors in the Hubbard interaction Eq. Then we make a unitary transformation into the chiral basis. Only those scattering terms that involve two bands can open a gap in the Dirac state, while the scatterings within one band only renormalize the Fermi velocity but do not contribute to any instabilities. Moreover, the translation invariant mean-field treatment to the second and fifth term contributes to a self-energy that only renormalizes the chemical potential.
Both H eff 1 and H eff 2 , p -wave and s -wave type respectively, are reduced from the original Hubbard interaction, which are the least irrelevant channels that can possibly gap out the Dirac cone. Authors are grateful to A. Burkov, T. Sedrakyan, C. Ting, Q. Wang, L. Hao, and L. Shao for fruitful discussions. This work was supported by Program under Grant No. Journal peer review information : Nature Communications thanks Ping Zhang and the other anonymous reviewer s for their contribution to the peer review of this work.
Peer reviewer reports are available. Onur Erten, Email: ude. Baigeng Wang, Email: nc. Supplementary Information accompanies this paper at National Center for Biotechnology Information , U. Nat Commun.
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Published online Jan Xing 1. Author information Article notes Copyright and License information Disclaimer. Corresponding author. Received Jul 2; Accepted Dec Abstract Excitonic insulators are insulating states formed by the coherent condensation of electron and hole pairs into BCS-like states. Open in a separate window. Discussion Excitonic insulators require strong interactions since the topological surface states become unstable only when the interaction is as large as the ultraviolet cutoff band gap as shown in Fig.
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Mean-field analysis of the Hubbard interaction Let us denote the four k vectors in the Hubbard interaction Eq. Supplementary information Peer Review File K, pdf. Acknowledgements Authors are grateful to A. Author contributions R.
Data availability The data that support the findings of this study are available from the corresponding author upon reasonable request. Notes Competing interests The authors declare no competing interests. Footnotes Journal peer review information : Nature Communications thanks Ping Zhang and the other anonymous reviewer s for their contribution to the peer review of this work. Contributor Information Onur Erten, Email: ude.
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Bubbling Solutions for the SU(3) Chern‐Simons Model on a Torus
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