Abstract
The quest for exotic quantum states of matter has become one of the most challenging tasks in modern condensed matter communications. Interplay between topology and strong electronelectron interactions leads to lots of fascinating effects since the discovery of the fractional quantum Hall effect. Here, we theoretically study the Rashbatype spinorbit coupling effect on a fractional quantum spin Hall system by means of finite size exact diagonalization. Numerical evidences from the ground degeneracies, states evolutions, entanglement spectra, and static structure factor calculations demonstrate that nontrivial fractional topological TaoThoulesslike quantum state can be realized in the fractional quantum spin Hall effect in a thin torus geometric structure by tuning the strength of spinorbit coupling. Furthermore, the experimental realization of the TaoThoulesslike state as well as its evolution in optical lattices are also proposed. The importance of this prediction provides significant insight into the realization of exotic topological quantum states in optical lattice, and also opens a route for exploring the exotic quantum states in condensed matters in future.
Introduction
Since the discovery of spinHall effect in 2004^{1,2}, the study of spinorbit coupling effects on quantum physics has been triggered great research interests both in condensed matter community and material science in this decade, especially fueled by the realization of the time reversal invariant topological insulators^{3,4} and Weyl semimetals^{5,6}, leading to the most challenging task of the search for exotic quantum states and their realizations in modern condensed matter physics. Learned from the physics of fractional quantum Hall effects^{7}, interplay between topology and strong electronelectron interactions displays lots of fascinating effects. The prominent fractional quantum Hall states occurring at certain unique values of the filling factor , k integer, have been explained by Laughlin as an “incompressible quantum fluid state”^{8}. Subsequently, Tao and Thouless^{9} proposed an alternative ground state with a gap to excitations, which has a chargedensitywave (CDW)like structure and can be connected to the Laughlin’s state by an adiabatic change of the aspect ratio ^{10,11,12,13,14} without undergoing a quantum phase transition. Namely, TaoThouless state is the exact ground state once N_{x} → O(1). In addition, the stripe formation of conventional CDW state in a fractional quantum Hall system has also been argued for systems with even denominator filling factors^{15,16,17,18}. It is crucial to point out that the main difference between conventional CDW and TaoThouless states is that the latter state associates nonvanished fractional quantum Hall conductance, while the former one does not. Importantly, these studies show the richness of quasiparticle phases in the fractional quantum Hall systems.
In recent years, the fractional quantum (spin) Hall effects^{19,20} have been realized theoretically in the fractional Chern insulators^{21,22,23,24} without an external magnetic field, where strongly interacting particles partially fill up the topological flatband with nonzero Chern number^{21,25,26}. The stability of the edge states in fractional quantum spin Hall systems against interactions and disorders was analyzed by Levin and Stern^{20}: A criterion σ_{sH}/e* (σ_{sH} the spin Hall conductance and e* the Abelian quasiparticles of charge), is proposed to determine whether the system is a nontrivial fractional topological insulator or not. Neupert et al.^{27} studied the stability of fermionic fractional topological insulating phase with the filling factor and pointed out that the system favors a fractional quantum spin Hall state for two decoupled spin species, but it should lead to an unstable fractional topological insulating phase according to Levin and Stern’s criterion. In fact, the fractional quantum spin Hall effect can possess fractionalized excitations in the bulk irrespective of the existence of gapless edge modes^{28}. Moreover, by tuning the interspin interaction, the fractional quantum spin Hall state evolves into a conventional CDW stripe phase^{27,28,29,30} through a phase transition signalled by the closing of energy and quasispin excitation gaps^{30}. Furthermore, Bosonic analogues of those fractional topological insulators have also been extensively studied by Repellin et al.^{31}, and were found to be robust to perturbations in the bulk by introducing a spinorbit coupling.
In this Letter, we propose a theoretical realization of fractionalized topological TaoThoulesslike quantum state in a fractional quantum spin Hall system with a thin torus geometric structure by tuning the strength of Rashbatype spinorbit coupling based on the framework of finite size exact diagonalization method. The obtained TaoThoulesslike state has the property of time reversal symmetry, which is a counterpart of TaoThouless state in fractional quantum Hall systems^{9} or in fractional Chern insulators^{32} with time reversal symmetry breaking. Additionally, we also present a discussion of the possible experimental realization and detection of the TaoThoulesslike topological quantum state as well as its evolution in optical lattices.
Results
The dispersion of singleparticle bands
The singleparticle band dispersion of the Hamiltonian (see model Hamiltonian in Methods) on the system with torus and cylinder structures are shown in Fig. 1(b,c), which have a large bulk energy gap with the amplitude of 2t_{1} well separating the two spinmixed flatbands and conduction bands. Introducing the spinorbit coupling the system still keeps the time reversal invariance but the inversion symmetry is broken, the two spin degenerate flatbands will be split [see Fig. 1(d)] except at the time reversal invariant points. It is interesting to point out that there are some helical edge states emerging inside the bulk energy gap and crossing each other at the Γ(k_{x} = 0) point forming the Diraclike dispersion relation protected by time reversal symmetry, similar to the band structure of a topological band insulator^{33}. As the bulk energy gap is much larger than the energy scale of the interactions, we can safely project Hamiltonian onto the states in the lowest two spinmixed flatbands in the exact diagonalization using a torus geometric structure. The repulsive interaction defined in Hamiltonian (1) (see model Hamiltonian in Methods) include a NN term which is parameterized by the coupling V and the dimensionless number λ. Previous studies^{27,28,29,30} have pointed out that the system favors a fractional quantum spin Hall and a conventional CDW stripe phases at small and large values λ of interspin interaction.
The ground state properties of manybody Hamiltonian
The ground state spectra of the effect of the Rashba spinorbit coupling α_{R} (see model Hamiltonian in Methods) on a fractional quantum spin Hall state are displayed in the top row of Fig. 2, where the parameters are chosen as α_{R} = 0 and α_{R} = 0.08 for (a1) and (b1), respectively, and shown that the ground state manifold is defined as a set of lowest states [ninefold degeneracies in (a1) and threefold degeneracies in (b1)] well separated from other excited states by a clear energy gap. Here it should be pointed out that the results [Fig. 2(a1] have been reported in our previous studies (see refs 29 and 30), we still present here to facilitate the following discussion concerned to the state evolution by applying the effect of spinorbit coupling on the fractional quantum spin Hall state. From Fig. 2(a1, we also notice that the energy gap is always significantly larger than the energy splitting of the ground states for various system sizes. Although these states are not exactly degenerate on a finite system, their energy difference should fall off exponentially as the system size increases. In addition, it is interesting to find that for those states with threefold or ninefold degeneracy, if (k_{x}, k_{y}) is the momentum sector for one of the states in the ground states manifold, then the other state should be obtained in the sector (k_{x} + N_{e}, k_{y} + N_{e}) [modulo (N_{x}, N_{y})], similar to that in fractional Chern insulators^{22}. The relationship of the quantum numbers of the ground states manifold steams from the topological nontrivial characteristics^{27,30}, which can be confirmed by the calculations of spectral flow [see the supplementary information (SI) for details].
Entanglement spectra of ground states
We turn to reveal the nature of these states shown in the top row of Fig. 2 by using a powerful tool of particle entanglement spectra (PES)^{23,34,35,36,37}, which provides an independent signature of the excitation structure of system as a fingerprint and remaining their characteristics in the thermodynamic limit. The entanglement energy levels ξ can then be displayed in groups labeled by the total momentum (k_{x}, k_{y}) for the N_{A} particles, and shown in Fig. 2(a2. When the entanglement spectrum is gapped, the number of states below the gap is a signature of a given topological phase, which is tightly related to the number of quasihole excitations, a hallmark of the fractional phase^{23,31,38}. In our previous study^{29}, we have demonstrated the nature of fractional quantum spin Hall state in Fig. 2(a2), which follows the counting rule of fractional quantum spin Hall state^{30,31}. However, in Fig. 2(b2), the number of states below the gap of PES equalling to 168 deviates from the counting rule of fractional quantum spin Hall state, but precisely matches the conventional CDW counting^{32,39}: . Therefore, this result suggests that such state is a conventional CDW or CDWlike state.
To further wellunderstood the behaviors of the state shown in Fig. 2(b1), we adopt the aspect ratio dependent calculation of the PES. By tuning the aspect ratio in a fixed system size from a thin torus to a more two dimensional one , the calculated PES are shown in Fig. 2(a3. As a reference, we compare with Fig. 2(a2, and notice that the PES for the fractional quantum spin Hall state in both cases clearly display the similar gapped structure as well as sharing the same counting numbers below the gap of PES, as expected from our intuition. Moreover, comparing with Fig. 2(b2 as well as (a3), it is surprised that since changing the aspect ratio to a more two dimensional torus the structure of PES in Fig. 2(b3) is similar to that in Fig. 2(a3) rather than that in Fig. 2(b2), and the states below the gap of PES in Fig. 2(b3) is no long matching the conventional CDW counting instead by the one for fractional quantum spin Hall state. Therefore, it suggests that such CDWlike phase obtained in a thin torus geometric structure is indeed the TaoThoulesslike state signalled by connecting to fractional quantum spin Hall state through an adiabatic change of the aspect ratio [see Fig. 2(b2] and evolved from fractional quantum spin Hall state without undergoing a quantum phase transition (see SI for details).
Static structure factor
In addition, we also present the static structure factor (SSF) calculations to further solidify our findings, and shown in Fig. 3(a). It is clearly shown that the TaoThoulesslike state has a typical feature of CDW stripe order displaying the double unidirectional Bragg peaks aligned along the xdirection appeared at momenta q = Q_{1} [=(0, 2)] and Q_{2}[=(0, 4)] in the SSF S_{q} calculation with the Rashbatype spinorbit coupling strength α_{R} = 0.08 in the thin torus geometric structure with . Furthermore, by tuning the aspect ratio γ to a more two dimensional torus geometric structure , shown in Fig. 3(b), the double unidirectional peaks in SSF S_{q} calculation of the TaoThoulesslike state is disappeared and instead by a featureless characteristic. By comparison with the featureless in SSF S_{q} calculation of the fractional quantum spin Hall state^{29}, it suggests the phase, which is sensitive to the shape of lattice geometric structure, is indeed a TaoThoulesslike state. All these obtained results are consistent with expected from the PES calculations and further solidify our findings.
Experimental realizations
We propose an experimental realization of state of TaoThoulesslike in future. Interestingly, it might be to study the spinorbit coupling effect on the fractional quantum spin Hall state and obtain the state evolution. Very recently, a scheme of direct experimental realization of Rashbatype spinorbit coupling^{40} and topological Haldane model^{41} in optical lattices was proposed, which might be helpful to establish the spinorbit coupling effects on the flatbands model with time reversal symmetry or equivalent bilayer flatbands model in experiments^{42}. Considering the interaction strength can be easily tuned in cold atom setups, our work will provide guidance for the experimental realization of the TaoThoulesslike state and its evolution as well as exciting manybody fractional topological phases.
Methods
Model Hamiltonian
We start with a theoretical model Hamiltonian of electrons on a checkerboard lattice^{29,30} shown in Fig. 1(a):where consists of two copies of πflux phase with flatbands as in ref. 26, and is the density operator on the site i with spin σ(=↑, ↓). In the singleparticle part Hamiltonian , we denote as the creation operator for an electron with lattice momentum k and spin σ in the sublattice α = A, B, and we introduce a spinor . Then, the second quantized singleparticle Hamiltonian readswhere the three vectors B_{k} are respectively defined asthe identity and the triplet Pauli matrices τ = (τ_{0}, τ_{1}, τ_{2}, τ_{3}) act on the sublattice index. The parameters t_{1}, t_{2}, and t_{3} represent the nearest neighbor (NN) hopping, nextnearestneighboring (NNN) hopping, and nextnext nearest neighbor (thirdNN) hopping amplitudes, respectively. In addition, the second term in Hamiltonian (1) describes the Rashbatype spinorbit coupling and has the form^{43,44}:where α_{R} represents the strength of the Rashbatype spinorbit coupling, δ_{x} and δ_{y} are the unit vectors along the and directions shown in Fig. 1(a).
Manybody exact diagonalization
We exactly diagonalize the manybody Hamiltonian (1) projected to the lowest two flatbands for a finite system with N_{x} × N_{y} unit cell (total number of sites N_{s} = 2 × N_{x} × N_{y}) shown in Fig. 1(a). We denote the number of fermions as N_{e}, and filling factor as . Because of the periodic boundary condition implementing translational symmetries, we diagonalize the system Hamiltonian in each total momentum sector with (k_{x}, k_{y}) as integer quantum numbers. Without loss of generality, we set the t_{1} as an energy unit and the interaction V = 1, λ = 0, and the filling factor throughout this paper. Similar results for filling states can also be easily obtained when the NNN repulsion is included (see the SI for details).
Entanglement spectra
We partition the system in the way as described in ref. 23 and divide the N_{e} particles into two subsystems of N_{A} and N_{B} particles, and trace out the degrees of freedom carried by the N_{B} particles. The eigenvalues e^{−ξ} of the resulting reduced density matrix ρ_{A} = Tr_{B}ρ, where is defined in a dfold [ninefold in Fig. 2(a1) and threefold in Fig. 2(b1)] degenerate state Ψ_{i}〉.
Static structure factor
The static structure factor is defined^{30,45,46,47} asthe indicates the density at site j projected onto the lowest two flatbands of singleparticle Hamiltonian, and the wave function Ψ〉 is incoherent summation over the degenerate ground states.
Additional Information
How to cite this article: Liu, C.R. et al. Realizing TaoThoulesslike state in fractional quantum spin Hall effect. Sci. Rep. 6, 33472; doi: 10.1038/srep33472 (2016).
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Acknowledgements
We thank D. N. Sheng, Y. S. Wu, and S. Q. Shen for helpful discussions. This work was supported by the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB04040300), the National Natural Science Foundation of China (Grant Nos. 11274069, 11404359, and 11474064), and the State Key Programs of China (Grant No. 2012CB921604). W.L. also gratefully acknowledges the financial Sponsored by Shanghai YangFan Program (Grant No. 14YF1407100) and Youth Innovation Promotion Association of the Chinese Academy of Sciences.
Author information
Author notes
 ChenRong Liu
 & YaoWu Guo
These authors contributed equally to this work.
Affiliations
Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China
 ChenRong Liu
 , YaoWu Guo
 & Yan Chen
State Key Laboratory of Functional Materials for Informatics and Shanghai Center for Superconductivity, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China
 ZhuoJun Li
 & Wei Li
CAS Center for Excellence in Superconducting Electronics, Shanghai 200050, China
 ZhuoJun Li
 & Wei Li
Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China
 Yan Chen
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Contributions
W.L. and C.R.L. performed the theoretical calculation; W.L. and Y.W.G. wrote the supplemental information; W.L., Z.J.L. and Y.C. discussed the theoretical results; W.L. wrote the paper; W.L. and Y.C. revised the paper.
Competing interests
The authors declare no competing financial interests.
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