Broken-Symmetry Quantum Hall States in Organic Dirac Fermion Systems
Osada Group
Recently, the ν = 1 quantum Hall (QH) plateau was observed in the Hall resistance of an organic conductor α-(BETS)2I3, which has been known as a two-dimensional (2D) massless Dirac fermion (DF) system with finite spin-orbit coupling (SOC), under high pressure [1]. In general, the 2D DF systems show the QH effect at the LL fillings ν = ±2, ±6, ±10, ... when their Landau levels (LLs) have four-fold spin and valley degeneracy. The ν = 1 QH effect is expected only when both spin and valley degeneracy are broken in the N = 0 LL. The spontaneous valley symmetry breaking is caused by the exchange interaction. We have studied the spatial order of the broken-symmetry QH states in α-type organic DF systems, α-(ET)2I3 and α-(BETS)2I3.
In order to clarify the broken-symmetry QH state of the α-type organic DF system with four molecular sites (A, A', B, and C) in the unit cell, we have studied its electronic state under magnetic fields using the four-band tight-binding model including Peierls phase factors. Figure 1 shows the magnetic field dependence of the energy levels (Hofstadter butterfly) for the spinless case (no Zeeman effect and no SOC). The Chern number for each gap confirms the validity of the conventional DF picture in these materials.

Fig. 1. (left) Band dispersion of an α-type organic conductor α-(ET)2I3 under pressure at zero magnetic field. (right) Magnetic field dependence of energy levels (Hofstadter butterfly) in an α-type organic conductor α-(ET)2I3 for the spinless case. The energy gap between quantized levels is colored according to the Chern number NCh.
The four-component envelope functions of the N = 0 LL with valley degeneracy were investigated based on the Hofstadter calculation. The site-resolved probability density of the LLs is shown in Fig. 2. When the SOC is considered, it breaks the spin degeneracy of the N = 0 LL and opens a spin-split gap. We can see that the degenerating −k0- and +k0-valley states have different probability weights on the A and A' molecules, which are connected by the inversion operation. This valley-site correspondence is recognized independently of the presence of the Zeeman effect or the SOC. When the spontaneous valley symmetry breaking, which is equivalent to the inversion symmetry breaking in this case, occurs due to the exchange interaction in the N = 0 LLs, their valley splitting leads to the ν = ±1 QH effects. These broken-symmetry QH states are accompanied by the spatial modulation of charge and spin densities at A and A' sites in a unit cell, as shown in the right panel of Fig. 2.

Fig. 2. (left) Probability densities of LLs under the SOC at four molecular sites of α-(ET)2I3 at about 33 T. Zeeman shift is not included. The zero-field gapped dispersion is also shown over them. (right) Schematics of spontaneous valley splitting of the "N = 0↑" and "N = 0↓" LLs and the examples of the charge and spin density patterns of the broken-symmetry ν = ±1 QH states when the SOC is dominant for the spin splitting.
In graphene, which is the typical 2D DF system, the broken-symmetry QH states, especially the ν = 0 QH states have been intensively studied both theoretically and experimentally. Since graphene has the ten times larger group velocity than the α-type organics, the spin splitting is almost negligible compared to the LL spacing, so that we have to consider the SU(4) symmetry breaking of the N = 0 LL. In fact, various QH states, the QH ferromagnet (QHF), the canted antiferromagnet, the charge order, the Kekule distortion, etc. have been proposed as the ν = 0 QH state. On the other hand, in the α-type organic DF system, the spin splitting resulting from the Zeeman effect and the SOC is sufficiently large compared to the LL spacing. The ν = 0 QH state is considered to be the QHF state without spontaneous symmetry breaking (SSB) as previously pointed out [3]. Therefore, only the ν = ±1 QH states are accompanied by the SSB in the organic DF system.
References
- [1] K. Iwata, A. Koshiba, Y. Kawasugi, R. Kato, and N. Tajima, J. Phys. Soc. Jpn. 92, 053701 (2023).
- [2] T. Osada, J. Phys. Soc. Jpn. 93, 034711 (2024).
- [3] T. Osada, J. Phys. Soc. Jpn. 84, 053704 (2015)