Hofstadter Butterfly and Broken-Symmetry Quantum Hall States in  -Type Organic Dirac Fermion Systems

The electronic state under magnetic fields in the  -type organic Dirac fermion systems,  -(ET) 2 I 3 and  -(BETS) 2 I 3 , has been studied to clarify the spatial order in the quantum Hall state. The four-band tight-binding model with Peierls phase factors was employed, and the generated Hofstadter butterfly and its Chern numbers confirmed the validity of the Dirac fermion picture in these materials. The four-component envelope function of the N = 0 Landau level with valley degeneracy was investigated. It was found that the two degenerate valley states have different weights on A and A' molecules connected by inversion. This valley-site correspondence is also recognized for the N = 0 spin-split Landau levels under the Zeeman effect and the spin-orbit interaction. The spontaneous valley symmetry breaking in the N = 0 Landau levels due to the exchange interaction results in the  = ±1 quantum Hall states accompanied by the spatial modulation of charge and spin densities at A and A' sites in a unit cell.


Introduction: Broken-Symmetry Quantum Hall States
Layered organic conductors -(BEDT-TTF)2I3 and -(BEDT-TSeF)2I3, which are often abbreviated as -(ET)2I3 and -(BETS)2I3, are considered to be twodimensional (2D) Dirac fermion (DF) systems under high pressure [1][2][3].On their conducting layers, ET or BETS molecules form an anisotropic triangular lattice called the -type configuration where the unit cell contains four molecules.Four 2D -bands are constructed from the HOMO of these molecules.Under appropriate pressure, the third and fourth bands have no overlap and touch at two points, k0 and −k0, in the Brillouin zone forming two tilted type-I Dirac cones (valleys).The Fermi level is stoichiometrically located at the Dirac point without any carrier doping.The emergence of 2D DF state with type-I Dirac cones under pressure has been experimentally confirmed for both real systems by the -Berry phase of Shubnikov-de Haas oscillation using doped samples with contact charge [4,5].
At ambient pressure, -(ET)2I3 undergoes a metal-insulator transition at 135K due to charge ordering caused by electron correlation.On the other hand, -(BETS)2I3 undergoes a metal-insulator crossover at around 50K, which is believed to be a result of the small topological insulator gap due to relatively strong spin-orbit interaction (SOI) [3,6].Appropriate high pressure can suppress the insulating behavior of both systems, resulting in the metallic 2D DF state.Although their band parameters differ slightly, the qualitative feature of their DF state is expected to be similar, except for the effect of SOI.
Recently, the  =1 quantum Hall (QH) plateau was observed in the Hall resistance of -(BETS)2I3 under high pressure [7].This is the first observation of the QH effect in a bulk single crystal of organic conductors, which is caused by the finite natural electron doping.Moreover, it is the  =1 QH effect of a 2D DF system with type-I Dirac cones having -Berry phase.In general, the 2D DF system in which the Landau levels (LLs) have four-fold spin and valley degeneracy shows the QH effect with Hall plateau In the conventional 2D electron systems with multiple valleys, such as the nchannel inversion layer of Si MOSFET, the QH effect due to valley splitting has also been observed [8].However, their valley splitting does not originate from the spontaneous degeneracy breaking, but from the inter-valley coupling under the confinement potential, although the splitting is enhanced by the exchange interaction [9].
In graphene, which is the typical 2D massless DF system on the honeycomb lattice, the broken-symmetry QH states have been studied experimentally and theoretically [10][11][12][13][14][15][16].Graphene's LLs are believed to have a four-fold spin and valley degeneracy due to the small Zeeman splitting compared to the LL spacing.Several possible states have been theoretically discussed, particularly for the  = 0 QH states, which result from the degeneracy breaking of the N = 0 LL.These include the spinpolarized (ferromagnetic) states, the canted antiferromagnetic state (CAF), the bond order (intervalley coherent) state with Kekule distortion (BO), the charge density wave state (CDW) [11], and the CAF-BO coexistence state [14,15].Additionally, the STM technique has visually observed the spatial modulation pattern of charge density in these broken-symmetry states [16].Graphene has two carbon sites (A and B) in a unit cell, and two -bands touching at two Brillouin zone corners (K and K' points), forming two Dirac cones (valleys).The N = 0 LL at the K(K') valley consists only of the wave function at the B(A) site.This valley-site correspondence causes the spatial charge and/or spin modulation in a unit cell in broken-symmetry QH states.
The similar spatial modulation is also expected in the  = 1 QH state in the type organic DF system with a different lattice structure and relatively large Zeeman splitting [17].The key to this problem is the presence and aspect of the valley-site correspondence in the N = 0 LL of the -type organic conductor.

Tight-Binding Model with Peierls Phase and Spin-Orbit Interaction
In this paper, we discuss the real-space charge/spin order pattern in the brokensymmetry QH states of the -type organic DF system.Unlike graphene, each conducting layer of the -type organic conductors has four molecular sites (A, A', B, and C) in a unit cell on the anisotropic triangular lattice, as shown in Fig. 3 (b).To clarify the spatial structure of the wave functions under magnetic fields, we use the 2D four-band tight-binding model for the -type organic conductors [1,2] instead of the two-band k-linear effective model, i.e., the tilted Weyl model [18,19].Additionally, we take into account the SOI, which is considered significant in -(BETS)2I3 [3].
In a 2D electron system with in-plane potential modulation, a moving electron experiences an effective magnetic field perpendicular to the plane.Therefore, the SOI is dependent only on the normal spin component z s (Ising-type SOI) [20].Thus, the Hamiltonian of the system is decoupled into two spin sectors as ) . In the -type organic DF system, ( , ) Hs k is a 44  matrix including the dimensionless SOI strength  at zero magnetic field [21].
The bases of the matrix are the Bloch sums constructed from HOMOs of molecules A, A', B, and C.
Under the normal magnetic field B = (0, 0, B), transfer integrals in the Hamiltonian obtain the Peierls phase factors.These factors extend the original unit cell to the magnetic unit cell.We choose the Landau gauge A = (0, Bx, 0) and consider the magnetic field that satisfies the following relation.
Here, p' and q are coprime integers and 4' pp = .becomes the following 44 qq  matrix. ( Here, the diagonal blocks are given by the following 44  matrices.(1 ) The Zeeman energy zB sB  is also added, where B is the Bohr magneton and Lande's g-factor is assumed to be g = 2.The SOI parameter  is introduced to preserve time reversal symmetry at zero field according to Ref. [21].The off-diagonal blocks of ( , ) z Hs k are given by the following 44  matrices. ( ( , ) ( , )  Here, k is a wave number in the magnetic Brillouin zone ( / and ky is related to the center coordinate X0 by , where / l eB = is the magnetic length.X0 indicates the location of the real-space orbital motion.Hofstadter levels lose k-dispersion at large q where the magnetic Brillouin zone becomes much smaller than the original Brillouin zone.Therefore, we calculate the Hofstadter levels only for k = 0 by choosing a sufficiently large value of q [22].In addition, we assume infinitesimal  to differentiate valley states by introducing small valley splitting.

Validity of Dirac Fermion Picture in -Type Organic Conductors
First, we consider only the orbital effect, i.e., the spinless case in which the SOI and Zeeman effect are ignored. .These features are characteristic of the 2D massless DF system in the spinless case.Importantly, they are obtained directly from the four-band tight-binding model without any approximation using the k-linear Weyl model.Therefore, it has been confirmed that the -type conductor behaves like the massless DF system between the van Hove singularities even when the multiband effect is included.The similar confirmation has also been made for graphene [22].Outside the van Hove singularities, each LL shows double splitting, reflecting the disappearance of the valley degeneracy, as seen in Fig. 2(b).

Valley-Site Correspondence in the N = 0 Landau Level
To investigate the possible real-space pattern of electron density at the brokensymmetry QH state, we the LL wave function at each molecular site for each valley.The wave function of each LL state is obtained by the above diagonalization of ( , 0) H k .The values of the envelope function at A, A', B, and C sites in the magnetic unit cell are obtained from the eigenvector for each Hofstadter level.For sufficiently low magnetic one LL for each valley is formed from  .We can see that the envelope functions correspond to the semiclassical cyclotron orbits surrounding two valleys in k-space.This reflects the two-fold valley degeneracy of the LLs.Note the asymmetry of the envelope functions of A and A' sites between −k0-and +k0-valleys.In particular, in the N = 0 LL at the Dirac point energy, the probability density of A (A') site is larger in the +k0-valley (−k0-valley) than in the −k0-valley (+k0valley).Namely, in the N = 0 LL, the +k0-valley is rich in A, while the −k0-valley is rich in A'.This is nothing but the valley-site correspondence in the -type organic DF system, although it is not perfect, unlike graphene.On the other hand, the probability density of B and C sites is symmetric between −k0 and +k0-valleys.
In -type systems, A and A' sites are connected by the inversion operation and are equivalent under inversion symmetry.However, once the inversion symmetry is broken, the N = 0 LL splits into the −k0 and +k0-valley levels because A and A' sites are no longer symmetric.If the valley-degenerate N = 0 LL is occupied almost half, the degeneracy could be spontaneously broken due to exchange interaction.In the spinless case, this generates the broken-symmetry  = 0 QH state where either A or A' site is charge-rich in a unit cell.

Effect of Spin-Orbit Interaction
Next, we will discuss the spinful case in which we take into consideration account the SOI and Zeeman effect.In -type organic conductors, the Zeeman energy zB sB  cannot be ignored relative to the LL spacing because the group velocity is much smaller than in graphene.Therefore, within the one-body effect -type organic conductors exhibit a significant spin splitting caused by the Zeeman effect, which is further modified by the SOI.However, we will ignore the Zeeman effect when   This spatial modulation may be detected experimentally using STM or NMR.
In graphene, the STM measurement visually observed the real-space modulation pattern of the broken-symmetry QH states, including the BO and CDW states [16].If the DF state can be realized in the -type organic conductor in a vacuum, the spatial modulation of charge density can be observed by STM in the QH states.The use of 13 C-NMR in the QH states may be a more effective method for detecting spatial spin modulation because four molecular sites give different NMR peaks.At zero magnetic field, in fact, previous site-resolved NMR measurements have successfully detected the ferrimagnetic spin polarization in -(ET)2I3 [29].

Conclusions
In summary, we studied the broken-symmetry QH states in -type organic 2D with  = ±2, ±6, ±10, ... .Although the Zeeman effect breaks the spin degeneracy, the  = 1 QH effect is expected only when both spin and valley degeneracy are broken in the N = 0 LL.The valley degeneracy could be broken in the bulk crystal by the electron-electron interaction, more precisely by the exchange interaction.In the spin-split N = 0 LL with valley degeneracy, the Pauli principle prevents two electrons from occupying the same valley at a given center coordinate.Therefore, the repulsive interaction between two electrons belonging to the same valley is effectively weakened.This leads to the spontaneous breaking of the valley degeneracy.One valley is selectively occupied, leading to filling-dependent valley splitting.The  = 1 QH effect is expected when the N = 0 LL with one spin is about half-filled and spontaneously splits into two valley levels with the Fermi level in between.
quantum.b and a are the lattice constants in the xand y-direction, respectively.The factor 4 on the right side reflects the fact that the crystal unit cell (eight triangular plaquettes) is four times larger than the unit cell of the molecular triangular lattice (two plaquettes), which determines the Peierls phase factors.The magnetic unit cell ( denoted by j (= 1, ..., q) aligned in the x-direction.In the magnetic field, the bases of the tight-binding Hamiltonian ( , ) z Hs k are the Bloch sum constructed from four types of molecules in a magnetic unit cell.Thus( , )   z Hs k
we call the Hofstadter levels, are obtained for a wave number k and a spin sz.

Figure 1 (( / ) 8  2 Chs 2 (
Fig. 1(b).The detailed spectra are shown in Fig. 2 on an enlarged scale.As shown in Thus, for a fixed X0 and a single valley, the probability density at each site is obtained as the sum of the square norm of the envelope functions of the degenerated p Hofstadter levels.

Figure 3 (
Figure 3(a) shows the probability density of the site-dependent envelope discussing the wave functions to clarify the contribution of the SOI, because the Zeeman effect only shifts the electron energy by zB sB  without changing the wave functions.

Figure 4 (
Figure 4(a) shows the Hofstadter butterfly for the -type system with finite

Figure 4 ( 1 zs 6 .
Figure 4(b) shows the probability density of site-dependent envelope functions DF systems using the four-band tight-binding model instead of the two-band linearized model.The Hofstadter butterfly and its Chern numbers confirmed the conventional DF picture.We investigated the four-component envelope function of spin-split N = 0 LLs with valley degeneracy, considering SOI.We found that the two valley states have different weights on A and A' molecules connected by inversion.The exchange interaction spontaneously breaks the valley degeneracy, leading to a spatially modulated  =±1 QH state with different charge and spin densities on A and A' molecules.