Topological insulators with SU(2) Landau levels
We construct continuum models of 3D and 4D topological insulators by coupling spin- fermions to an SU(2) background gauge field, which is equivalent to a spatially dependent spin-orbit coupling. Higher dimensional generalizations of flat Landau levels are obtained in the Landau-like gauge. The 2D helical Dirac modes with opposite helicities and 3D Weyl modes with opposite chiralities are spatially separated along the third and fourth dimensions, respectively. Stable 2D helical Fermi surfaces and 3D chiral Fermi surfaces appear on open boundaries, respectively. The charge pumping in 4D Landau level systems shows quantized 4D quantum Hall effect.
Time-reversal (TR) invariant topological insulators (TIs) have become a major research focus in condensed matter physics(1); (2); (3). Different from the 2D quantum Hall (QH) and quantum anomalous Hall systems which are topologically characterized by the first Chern number (4); (5); (6); (7); (8), time-reversal invariant TIs are characterized by the second Chern number in 4D (9); (10) and the index in 2D and 3D(11); (12); (13); (10); (14); (15); (16). Various 2D and 3D TIs are predicted theoretically and identified experimentally exhibiting stable gapless 1D helical edge and 2D surface modes against TR invariant perturbations (13); (17); (18); (19); (20); (21). Topological states have also been extended to systems with particle-hole symmetry and superconductors (22); (23); (24).
Most current studies of 2D and 3D TIs focus on Bloch-wave bands in lattice systems. The nontrivial band topology arises from spin-orbit (SO) coupling induced band inversions(13). However, Landau levels (LL) are essential in the study of QH effects because their elegant analytical properties enable construction of fractional states. Generalizing LLs to high dimensions gives rise to TIs with explicit wave functions in the continuum, which would facilitate the study of the exotic fractional TIs. Efforts along this line were pioneered by Zhang and Hu (9). They constructed LLs on the compact sphere by coupling fermions with the SU(2) monopole gauge potential, and various further developments appeared (25); (26); (27); (28); (29). Two-dimensional TIs based on TR invariant LLs have also been investigated (12). Two of the authors generalized LLs of Schrödinger fermions to high-dimensional flat space (30) by combining the isotropic harmonic potential and SO coupling. LLs have also been generalized to high dimensional Dirac fermions and parity-breaking systems (31); (32).
In all the above works, angular momentum is explicitly conserved; thus, they can be considered as LLs in the symmetric-like gauge. In 2D, LL wave functions in the Landau gauge are particularly intuitive: they are 1D chiral plane wave modes spatially separated along the transverse direction. The QH effect is just the 1D chiral anomaly in which the chiral current generated by the electric field becomes the transverse charge current. In this Letter, we develop high dimensional LLs with flat spectra as spatially separated helical Dirac or chiral Weyl fermion modes, i.e., the SU(2) Landau-like gauge. They are 3D and 4D TIs defined in the continuum possessing stable gapless boundary modes. To our knowledge, these are the simplest TI Hamiltonians constructed so far. Recently, there have been considerable interests of 2D topological band structures with approximate flat spectra. (33); (34); (35). Our Hamiltonians defined in the continuum possess exact flat energy spectra in 3D and 4D, and are independent of the band inversion mechanism. For the 4D case, they exhibit the 4D quantum Hall effect (9); (10), which is a quantized nonlinear electromagnetic response related to the spatially separated (3+1)D chiral anomaly. Our methods can be easily generalized to arbitrary dimensions and also to Dirac fermions.
The 3D case.— We begin with the 3D TR invariant LL Hamiltonian for a spin- fermion as
which couples the 1D harmonic potential in the direction and the 2D Rashba SO coupling through a -dependent SO coupling strength. possess translation symmetry in the plane, TR and parity symmetries. Eq. (1) can be reformulated in the form of an SU(2) background gauge potential as , where , and takes the Landau-like gauge as and . In Ref. [(30)], a symmetric-like gauge with is used, which explicitly preserves the 3D rotational symmetry. However, the SU(2) vector potentials and are not gauge equivalent. As shown below, the physical quantities of Eq. (1), such as density of states (DOS), are not 3D rotationally symmetric. Nevertheless, we will see below that these two Hamiltonians give rise to the same forms of helical Dirac surface modes, and thus they belong to the same topological class. A related Hamiltonian is also employed for studying electromagnetic properties in superconductors with cylindrical geometry (36).
Equation (1) can be decomposed into a set of 1D harmonic oscillators along the axis exhibiting flat spectra, a key feature of LLs. We define a characteristic SO length scale . Each of the reduced 1D harmonic oscillator Hamiltonian is associated with a 2D helical plane wave state as , where and ; the helicity operator is defined as . The -th LL eigenstates are solved as
where ; are eigenstates of the helicity satisfying with helicity eigenvalues ; are the eigenstates of the -th harmonic levels with the central positions located at , and . The energy spectra of the -th LL is , independent of and .
In the 2D LL case, spatial coordinates and are noncommutative if projected to a given LL, say, the lowest LL (LLL). The LLL wave functions in the Landau gauge are 1D plane waves along the direction whose central positions linearly depend on as . These 1D modes with opposite chiralities are spatially separated along the direction. Consequently, the plane can be viewed as the 2D phase space of a 1D system, in which plays the role of . The momentum cutoff of the bulk states is determined by the system size along the direction as . Right and left moving edge modes appear along the upper and lower boundaries perpendicular to the axis, respectively. These chiral edge modes cannot exist in purely 1D systems such as quantum wires.
Similarly, the 3D LL wave functions in Eq. (2) are spatially separated 2D helical plane waves along the axis. As shown in Fig. 1 (A), for states with opposite helicity eigenvalues, their central positions are shifted in opposite directions. Let us perform the LLL projection. Among the LLL states with good quantum numbers , it easy to check the following matrix elements . Therefore, we have where are arbitrary linear superpositions of in the LLL. This proves that
where is the LLL projection operator. Since the LLL states span the complete basis for the plane waves in the plane, the projections of and in the LLL remain themselves. As a result, we obtain the following commutation relations after the LLL projection as
Interestingly, the 3D LL states can be viewed as states in the 4D phase space of a 2D system (with and coordinates) augmented by the helicity structure, in which plays the role of the magnitude of the 2D momentum and the sign of corresponds to . In fact, the momentum cutoff of the bulk states is exactly determined by the spatial size along the direction as
in which . Applying the open boundary condition along the direction, and periodic boundary conditions in the and directions with spatial sizes and respectively, we can easily count the total number of states to be . The dependence of may seem puzzling for a 3D system, but expressing in terms of Eq. (5), we find , which is the conventional state counting of a 2D system expressed in terms of the 4D phase space volume. It means that an effectively 4D density of states are squeezed into a 3D real space.
The topological property of this 3D LL system manifests through its helical surface spectra. Let us consider the upper boundary located at for . For simplicity, we only consider the LLL as an example. If , the positive helicity states with are confined at the boundary. The 1D harmonic potential associated with and is truncated at , and thus the surface spectra acquire dispersion. If we neglect the zero-point energy, the surface mode dispersion is approximated by with . If the chemical potential lies above the LLL, it cuts the spectra at surface states with a Fermi wavevector . The Fermi velocity is . The surface Hamiltonian is approximated as with an electronlike Fermi surface with . In another case, if the upper boundary is located at , then the negative helicity states with , and all the positive helicity states are pushed to the boundary as surface modes. Depending on the value of , we can have a holelike Fermi surface with , a Dirac Fermi point, or an electronlike Fermi surface with . Similarly, any other LL gives rise to a branch of gapless helical surface modes, and each filled bulk LL contributes one helical Fermi surface. For the lower boundary, the analysis is parallel to the above. Each filled LL gives rise to an electronlike helical Fermi surface with , or holelike with . According to the standard Z classification, this system is topologically nontrivial if the Fermi energy cuts an odd number of Landau levels. So far, we have assumed the harmonic frequency and SO coupling frequency to be equal in Eq. (1). As explained in the Supplemental Material, although the equality of these two frequencies is essential for the spectra flatness, the topology does not require this equality.
The 4D case The above procedure can be straightforwardly generalized to any higher dimension. For example, the 4D LL Hamiltonian is denoted as
where and refer to the coordinate and momentum of the 4th dimension, respectively, and is the 3-momentum in the space. Eq. 6 can be represented as , where the SU(2) vector potential takes the Landau-like gauge with for and . Equation 6 preserves the translational and rotational symmetries in the space and TR symmetry. Similar to the 3D case, the 4D LL spectra and wave functions are solved by reducing Eq. (6) into a set of 1D harmonic oscillators along the axis as where and . The LL wave functions are
where, the central positions ; are eigenstates of 3D helicity with eigenvalues . Inside each LL, the spectra are flat with respect to and . This realizes the spatial separation of the 3D Weyl fermion modes as shown in Fig. 1 (B). Similarly to the 3D case, after the LLL projection, we have the relation , and then the noncommutative relations among coordinates are
for . The 4D LLs can be viewed as states of the 6D phase space of a 3D system: increasing the width along the direction corresponds to increasing the bulk momentum cutoff . If the LLL is fully filled, the total number of states is given by . Reexpressing we find , which is the conventional state counting of a 3D system expressed in terms of the 6D phase space volume. With an open boundary imposed along the direction, 3D helical Weyl fermion modes appear on the boundary.
Now let us consider the generalized 4D quantum Hall effects (10) as the nonlinear electromagnetic response of ()D LL system to the external electric and magnetic (EM) fields, with in the space. Without loss of generality, we choose the EM fields as and , which are minimally coupled to the spin- fermion,
where . Here is the magnetic vector potential in the Landau gauge with and . We define , where is assumed.
The field further reorganizes the chiral plane wave states inside the -th 4D LL into a series of 2D magnetic LLs in the plane. For the moment, let us set . The eigenvalues remain the same as before without splitting, while the eigen-wave-functions are changed. We introduce a magnetic LL index in the plane. For the case of , the eigen-wave-functions are spin polarized as
where and is the zero mode channel of the operator with the eigenvalue of . The central positions of the direction harmonic oscillators are . For , the eigenmodes of come in pairs as
where coefficients , , and the eigenvalues are . The corresponding eigen-wave-functions are where the central positions .
For the solutions of Eq. (10) with the same 4D LL index , the 2D magnetic LLs with index are singled out. The central positions of states in this branch are linear with , and thus run across the entire axis, while those of other branches with only lie in one half of the space as shown in Fig. 2. After turning on , is accelerated with time as , and thus the central positions moves along the axis. Only the branch of the magnetic LL states contribute to the charge pumping which results in an electric current along the direction. Within the time interval , the number of states with each filled 4D LL passing a cross section perpendicular to the axis is , which results in the electric current density . If the number of fully filled 4D LLs is , the total current density along the axis is
where is the fine-structure constant. This quantized non-linear electromagnetic response is in agreement with results from the effective theory (10) as the 4D generalization of the QH effect. If we impose open boundary conditions perpendicular to the direction, the above charge pump process corresponds to the chiral anomalies of Weyl fermions with opposite chiralities on the two 3D boundaries, respectively. Since they are spatially separated, the chiral current corresponds to the electric current along the direction.
Although the DOS of our 4D LL systems is different from the 4D TIs in the lattice system with translational symmetry (10), their electromagnetic responses obey the same Eq. (12). It is because that only the spin-polarized branch of LLs, , is responsible for the charge pumping. For this branch, field quantizes the motion in the plane so that the and coordinates play the role of and , respectively. Consequently, the system can be viewed as the 4D phase space of coordinates and , and scales uniformly as with the conventional thermodynamic limit of a 4D system.
The 3D LL Hamiltonian Eq. (1) can be realized in strained semiconductors by generalizing the method in Ref. (13) from 2D to 3D. For semiconductors with the zinc-blende structure, the components of the strain tensor generate SO coupling as
where m/s and m/s for GaAs and InSb, respectively (37). The -dependent Rashba SO coupling is induced by the strain component , which can be generated by applying the pressure along the  direction without inducing and (38); (13). Furthermore, if the pressure linearly varies along the axis, such that where is the strain gradient, then we arrive at the SO coupling in Eq. (1) with the relation . A strain gradient of 1% over the length of m leads to the LL gap eV in GaAs. It corresponds to the temperature of mK, which is accessible within the current low temperature technique. The strain gradient at a similar order has been achieved experimentally (39); (13). The harmonic potential in Eq. (1) is equivalent to a standard parabolic quantum well along the axis (40), which is constructed by an alternating growth of GaAs and AlGaAs layers. The harmonic frequency can be controlled by varying the thickness of different layers.
In conclusion, we have generalized LLs to 3D and 4D in the Landau-like gauge by coupling spatially dependent SO couplings with harmonic potentials. This method can be generalized to arbitrary dimensions by replacing the Pauli-matrices with the matrices in corresponding dimensions. These high dimensional LLs exhibit spatial separation of helical or chiral fermion modes with opposite helicities, which give rises to gapless helical or chiral boundary modes. The 4D LLs give rise to the quantized nonlinear electromagnetic responses as a spatially separated (3+1)D chiral anomaly. Many interesting open problems are left for future studies. For example, in the Supplemental Material, we present a preliminary effort on constructing of Laughlin wave functions in 4D LLs with spin polarizations. The generalizations of LLs to 3D and 4D Dirac fermions are also given there. A full study of the interaction effects on the high dimensional fractional topological states based on LLs will be investigated in future publications.
Y.L. and C.W. thank J. Hirsch, N.P. Ong, L. Sham, Z. Wang and Y.S. Wu for helpful discussions. Y.L. and C.W. are supported by the NSF DMR-1105945 and AFOSR FA9550-11-1-0067(YIP), and S.-C. Z. is supported by the NSF under Grant No. DMR-1305677. C.W. also acknowledges the support from the NSF of China under Grant No. 11328403. Y.L. thanks the Inamori Fellowship and the support at the Princeton Center for Theoretical Science.
Appendix A Supplemental Material
In this supplemental material, we cover the classic equations of motion of the 3D Landau levels, further discussion of the topology class, the construction of Laughlin-type wavefunction, and the 3D Landau levels for relativistic fermions.
Appendix B Classical equations of motion
The classical equations of motion for (Eq. (1) in the main text) are derived as
where and are spin operators in the Heisenberg representation defined as and .
If we choose the initial condition of spin such that and , then is conserved and lies in the plane. The motion is reduced to a coplanar cyclotron one in the vertical plane perpendicular to . In other words, the orbital angular momentum and spin are locked, a feature from SO coupling.
Appendix C Discussions on the mismatch of the trap and spin-orbit frequencies
In principle there should be two different frequencies in the 3D LL Hamiltonian Eq. 1 in the main text, the harmonic frequency denoted as below and the spin-orbit frequency . Below, we will explain that our analysis does not depend on the exact equality between and , and thus is generic. This equality is only important to maintain the spectra flatness which the usual 3D TIs do not possess.
Let us move back to a familiar example of the usual 2D LLs of quantum Hall systems in the Landau gauge. In a more general form, the Hamiltonian can also be represented in terms of two independent frequencies as
where and . The case of corresponds to the usual 2D LL Hamiltonian
In other words, . needs to be larger than , i.e. , to ensure the spectra bounded from below. If , the spectra are exactly flat. If , the eigenstates are still plane-wave modes along with the good quantum number whose central positions are shifted according to . The feature of the spatially separated chiral modes does not change. As a result, the topology remains the same as before, but their spectra become dispersive. If is close to 1, the dispersion is very slow. In this case, we can view this potential difference as a soft external potential imposing on the bulk Hamiltonian. The edge spectra of this quantum Hall system remains chiral, only the Fermi velocities are modified. This situation is met in mesoscopic quantum Hall systems. The topology of such a system is still characterized by the number of chiral edge modes.
Coming back to our case of 3D LL Hamiltonian Eq. (1) in the main text, the situation is very similar. As long as the time-reversal symmetry is preserved, and the energy scale of perturbations is much smaller than the Landau level gap, the topology is maintained. The concept of helicity due to spin-orbit coupling replaces chirality in the usual 2D Landau levels. When and do not match, the Hamiltonian is expressed as
which gives rise to the following dispersive spectra as
However, the eigenstates remains spatially separated helical 2D plane wave modes with good quantum numbers of , whose central positions are shifted according to . As a result, the topology remains the same as before. In other words, the mismatch between and is equivalent to add the bulk Hamiltonian Eq. (1) with an external potential
In the case of close to 1, this soft external potential imposes a finite sample size along the axis with the width where . Inside this region, is smaller than the LL gap, and the LL states are bulk states. The corresponding 2D wavevectors for these bulk states satisfy . Outside this width , the LL states can be viewed as boundary modes with a fixed helicity. Again each branch of LL that cut the Fermi surface will contribute a helical surface Fermi surface, and thus the topology does not change, only the Fermi velocities of surface modes are affected.
Appendix D Delocalized helical modes on the side boundary
In the main text, we have solved the helical surface states of the 3D LL Hamiltonian of Eq. (1) on the plane. Each LL below the chemical potential contributes one helical Fermi surface. We further check the side surface below. According to the theorem proved in Ref. [(44)], as long as one surface exhibits a helical Fermi surface, all surfaces are topologically equivalent to odd numbers of channels of gapless delocalized helical modes. This theorem also applies to the side surface in our case in which the translational symmetry is absent along the direction. Following the reasoning in Ref. [(44)], let us consider two orthogonal surfaces of the plane (top) and plane (side) which intersect at the line of axis. Without loss of generality, we only consider the gap between the lowest and the second LLs. The translational symmetry along the axis is still maintained in spite of the open boundaries, and is conserved across the boundary between and surfaces. For a given energy lying inside this gap, it cuts surface states on the plane with one helical iso-energy ring in momentum space centering around the origin . Let us focus on the channel of modes with . There are a pair of modes on the surface, which cannot be scattered into each other at the intersection edge between and surfaces due to TR symmetry. Each of them has to continue on the surface. According to the theorem in Ref. [(44)], the oddness of the number of TR pairs does not change on the surface and cannot be localized even though the axis is not translational invariant. With varying the energy across the gap, we arrive at a branch of delocalized helical modes on the side surface of the plane. On the other hand, for the case of , the modes in the surface with the energy are not TR partners, and thus they can be scattered into each other at the intersection edge. There is no guaranty that they need to continue on the side surface of plane, or, the modes on the plane with are generally gapped and may be localized. In summary, on the side surface, although we cannot define Dirac cones due to the lack of translational symmetry, there do exist an odd number of helical delocalized modes.
Appendix E Laughlin-type wave functions
These high dimensional LLs may provide a convenient platform for the further study of high dimensional interacting fractional topological states. The construction of the Laughlin-type wave functions for interacting fermions is difficult for these SO coupled high dimensional LL systems. Nevertheless, for the 4D case in the magnetic field, the LLL states with both and are spin-polarized, and their total DOS is finite as . Even though they are degenerate with other LLL states with , they are favored by repulsive interactions if they are partially filled.
The Laughlin wavefunction in the Landau gauge for the 2D LLs has been constructed in Ref. (41). We generalize it to the 4D case, and define and , then LL states are represented as up to a Gaussian factor . The Laughlin-type wavefunction can be constructed as
where is an odd number; represents the Slater determinant for the fully filled single particle states and and . The study of the topological properties of Eq. 21 will be deferred to a later publication.
Appendix F High dimensional LLs of Dirac electrons
The LL quantization based on 2D Dirac electrons have attracted a great deal of research attention since the discovery of graphene. Here we generalize LLs to the 3D and 4D relativisitc fermions, which are the square root problems to their Schrödinger versions in Eq. 1 and Eq. 6 For the 3D case, we have
Its square exhibits a diagonal block form with a super-symmetric structure as
where is just the 3D LL Hamiltonian given in Eq. 1 in the main text, and is given as
The eigenvalues of Eq. (22) are . The eigen-wave-functions are constructed based on the eigenstates of as
where the eigenstate of takes the form as
The 4D LL Hamiltonian for Dirac Hamiltonian can be constructed as
where is the phonon annihilation operator in the direction. The eigenvalues are still , and the eigenstates are
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