Quantum spin compass models in 2D electronic topological metasurfaces

We consider a metasurface consisting of a square lattice of cylindrical antidots in a two-dimensional topological insulator (2DTI). Each antidot supports a degenerate Kramer's pair of eigenstates formed by the helical topological edge states. We show that the on-site Coulomb repulsion leads to the onset of the Mott insulator phase in the system in a certain range of experimentally relevant parameters. Intrinsic strong spin-orbit coupling characteristic for the 2DTI supports a rich class of the emerging low-energy spin Hamiltonians which can be emulated in the considered system, which makes it an appealing solid state platform for quantum simulations of strongly correlated electron systems.

We consider a metasurface consisting of a square lattice of cylindrical antidots in a two-dimensional topological insulator (2DTI). Each antidot supports a degenerate Kramer's pair of eigenstates formed by the helical topological edge states. We show that the on-site Coulomb repulsion leads to the onset of the Mott insulator phase in the system in a certain range of experimentally relevant parameters. Intrinsic strong spin-orbit coupling characteristic for the 2DTI supports a rich class of the emerging low-energy spin Hamiltonians which can be emulated in the considered system, which makes it an appealing solid state platform for quantum simulations of strongly correlated electron systems.
Spin lattice models are ubiquitous in theoretical physics. Besides their natural applications for the description of the behavior of magnetic systems, a variety of the condensed matter problems, related to hightemperature superconductivity [1], thin superfluid films [2], quantum Hall bilayers [3], and nonlinear optical lattices [4] allow mapping into spin lattice Hamiltonians. An interesting subclass of such models is represented by Compass models (CM), for which the characteristic feature is direction-dependent spin-spin interaction [5]. The first model of this type was introduced back in the 1982 [6] to describe the interplay between Jahn-Teller effect and magnetization dynamics. Since then CMs have been applied for modelling of emergent phenomena in variety of strongly correlated systems such as hightemperature superconductors [7,8], vacancy centers networks [9], colossal magnetoresistance manganites [10], and materials supporting spin-liquid phases [11]. One of the most prominent examples is Kitaev model [12] employed extensively in the rapidly developing field of topological quantum computation. For any spin model, it is highly desirable to find a material platform which allows flexible control over its effective parameters [13]. While it has been argued that certain quantum CMs can be emulated with use of cold atoms in optical lattices [14,15], corresponding solid state platforms are still yet to appear. Here, we demonstrate that a metasurface formed by a square lattice of antidots in two-dimensional topological insulators (2DTI) is an attrective alternative for this.
2DTIs are materials that have both a 2D bulk energy gap (like ordinary insulators) and 1D gapless conducting edges [16][17][18], protected by time reversal symmetry and characterized by the spin-momentum locking, which means that at a specific boundary the direction of the propagation of an edge state is uniquely defined by electrons spin projection. Naturally, for the closed boundaries, the energy of an edge state is quantized, and for the case of rotational symmetry the corresponding eigenstates are characterized by specific projections of the orbital and spin angular momentum. Such topological resonators have been actively studied recently in topological photonic systems [19][20][21], but can be realized as well for electrons, example being an antidot (ring shape aperture) in 2DTI. The electronic spectrum of such a system was obtained in Refs. [22,23] and associated quantum impurity models have been considered in a number of follow up works [24,25].
In the current paper, we show that a square lattice of antidots in 2DTI emulates a quantum spin compass model with additional spin-orbit interaction of the Dzyaloshinskii-Moriya type. The parameters of the model can be flexibly tuned by change of the geometry of the lattice (antidot size and inter-dot distance), which can be routinely achieved within the state of the art fabrication techniques. The proposed system can thus serve as a solid state quantum simulator of a wide class of quantum compass models with possible applications ranging from emulation of correlated electron materials to topological quantum computation. The geometry of the system we consider is shown schematically in Fig. 1.
We take an example of CdTe/HgTe/CdTe quantum well [18] since this is currently the most common of 2DTI where topologically protected edge states has been observed experimentally [26], but other material platforms are also possible.
The model Hamiltonian of a CdTe/HgTe/CdTe quantum well is represented by 4 by 4 block-diagonal matrix, which consists of the blocks related to each other by time-reversal symmetry operation [18], H = arXiv:2106.08606v2 [cond-mat.mes-hall] 17 Jun 2021 The parameters entering into this expression are determined by the geometry of the QW. Further, we take A = 375 meV nm, B = −1.12 eV nm 2 , D = 00, C = 0, M = −10 meV [22]. The parameter M (Dirac mass), defined by a thickness of a QW, is of special importance, as only the case M < 0 corresponds to the topologically non-trivial regime. The eigenvalues and eigenvectors of an individual axially symmetric antidot can be found with use of the following ansatz for a 4-spinor: where θ is an angular coordinate, and quantum number m = ±1/2, ±3/2, ±5/2, ... gives z-component of the total angular momentum j z commuting with the Hamiltonian. Using this substitution we get the radial part of a wavefunction in terms of the Macdonald functions K m±1/2 . Applying the Dirichlet boundary condition at the antidot edge χ 1,2 (r)| r=a = 0 we get a secular equation defining the eigenergies and eigenfunctions. The spectrum of a single antidot as a function of an antidot radius a is shown in Fig. 2. As one can see, the spectrum is symmetric with respect to the gap centre (zero energy). For small radii there exist only two bound states corresponding m = ±1/2 with energies approaching the gap edges as one decreases the radius. At larger radii the states corresponding to larger |m| appear. In what follows we consider the radius a = 15 nm, which corresponds to the case of a single pair of the bound states. It should be stressed, that each of the eigenenergies corresponds to the Kramer's doublet, representing a mix-ture of purely orbital degenerate states [27] and degenerate spin states. We introduce the pseudospin index σ = (↑ , ↓) to label the partners of the doublet. Their wavefunctions ψ and ψ are related to each other by the time reversal operator T : ψ = T ψ = iσ y ψ * .
In Figs. 3(a,b) we plot the probability distribution functions and corresponding spin density profile for the m = 1/2 eigenstate for a = 15 nm. It can be seen that the wavefunction is localized at the scale of several a. Moreover the spin distribution of the state is highly nonisotropic which is a consequence of the spin-orbit coupling. In the situation, when apertures are placed reasonably close to each other, the wavefunctions of the states localized at each of them overlap, and electrons can thus tunnel between antidots. If one neglects the doublet corresponding to higher energy (upper line in Fig. 2), the system can be described in terms of a tight binding Hamiltonian of the Hubbard type, which for a square lattice of antidots reads: where c iσ is the annihilation operator for the state with specific pseudospin projection localized at site i of the lattice. The first term corresponds to the tunneling between the sites, while the last term describes on-site Coulomb repulsion.
Since we account for nearest neighbour hopping only, there exist only two inequivalent tunneling matrices t σσ i,i+x , t σσ i,i+ŷ corresponding to hoppings along orthogonal lattice translation vectors, which read: where R is the distance between the antidot centers, is the bound state energy, andê = [x, y]. Substituting the expression for ψ ↑ from Eq. (2) and recalling that ψ ↓ = T ψ ↑ , we notice that t ↑↑ i,i+ê = t ↓↓ i,i+ê and t ↑↓ i,i+ê = −(t ↓↑ i,i+ê ) * . Moreover, since χ 1 , χ 2 are real functions, the diagonal elements t σσ are real. Also, the absolute value of tunnelling amplitudes should depend only on the distance between antidots and not on the orientation of the antidot pair. These general considerations allow to parametrize the tunnelling matrix aŝ where t and α are real numbers which depend on the antidot radius and intersite distance, and phase φê depends on the hopping direction. Numerical integration shows that φ x = 0 and φ y = π/2. The dependence of the tunneling amplitude t and phase α versus intersite distance R for a = 15 nm is shown in Fig. 4(a). As expected the tunnelling amplitude decays exponentially with the inersite distance. Interestingly, the angle α reaches the value of π/4, which corresponds to the case of the equivalency of spin conservative and spin flip tunnelings, at some finite value of R. The Coulomb interaction energy can be estimated as: where ρ ↑↓ (r) = ψ † ↑↓ (r)ψ ↑↓ (r) is an electron density for the corresponding pseudospin projection, and V is the interaction potential which in principle should include both static and dynamical screening. We will neglect the latter, since edge states lie in the band gap of the bulk material where there is vanishing density of the free electrons. As to the static screening, it was considered in Ref. [28] where it has been shown that the the effect of the image potential can be neglected for moderate dielectric contrasts and simple expression for the potential V = e 2 /(ε|r 1 − r 2 |), where ε ≈ 10 is the static dielectric constant of HgTe can be safely used.
In Fig. 4(b) we plot the dependence of U on the antidot radius a. Its non-monotonic behaviour can be attributed to the fact, that for small antidots the eigenfunctions are weakly localized in the radial direction, because of the approaching of the corresponding energies to the band gap edge. At the same time, for large antidot wavefunction becomes delocalized along its periphery of the radius ≈ 2a. The interplay between these two effects defines the radius at which U becomes maximal. For the considered parameters U = U max ≈ 4 meV is achieved at a ≈ 13 nm.
Let us consider the situation, when the Fermi energy in the system is tuned in such a way, that one has exactly one electron per each dot (regime of half filling). It is well known, that in this situation the tunneling between the neighbouring sites can be completely blocked by the interaction. This regime corresponds to the so called spin-orbit coupled Mott insulator and is achieved when U > 4t [29], which in our case corresponds to the distances between the antidots R > 10a.
Our geometry resembles some previously proposed ones, where Mott insulators were realized with arrays of semiconductor quantum dots [30][31][32]. However, there is one crucial difference, namely strong spin-orbit coupling inherent for the topological insulators. It gives an additional twist to our model, which enables to emulate much wider class of the accessible low-energy Hamiltonians.
By employing the standard Schrieffer-Wolff transformation and excluding the states with two electrons sitting on the same sites, we can map the low-energy sector of the original Hamiltonian 3 into the following spin lattice model: where exchange constant J = 4t 2 /U > 0, a, b = 1, 2, 3 denote components of the pseudospin operator,x,ŷ are basis vectors along corresponding axes and R x (2α), R y (2β) are SO(3) rotation matrices around x and y axes, respectively. The states with pseudospins S z = ±1/2 correspond to the occupations of the partners of the Kramers doublet, other states being their linear combinations. For the case of α = π/4 which is reached for the intersite distance ≈ 10a, we can rewrite the Hamiltonian as where the first term corresponds to the so-called 90 o spincompass model and the second term to the conventional Dzyaloshinskii-Moriya interaction (DMI). The Hamiltonian in Eq. (8) is essentially a quantum spin compass model with added DMI interaction. Pure spin compass models are usually characterized by the highly degenerate ground states, which sometimes allows for the dimensionality reduction and even the exact solution such as in the case of the Kitaev model, corresponding to the honeycomb lattice [5]. While for the case of the square lattice, the DMI interaction is likely to lift the ground state degeneracy, it will be instructive to consider the structures with geometrical frustration, such as Lieb or Kagome lattices, and to explore the interplay between the geometrical frustration and strong spin-orbit coupling. Moreover, inclusion of the edge states characterized by larger value of m opens the access to the multi-band Hubbard model and to the multi-band Mott insulators with strong spin-orbit interaction.
In the classical limit, the Hamiltonian (8) is characterized by the spiral wave ground states [33]. At the same time, quantum fluctuations can substantially modify the ground state properties of the system. Specifically, in [34] the one-dimensional analog of Hamiltonian (8) was analyzed, where the spins are aligned along x axis. It has been shown using exact diagonalization and renormalization group methods, that the spiral long range order characterized by the order parameter is destroyed and only local order is preserved. To probe the onset of different collective phases experimentally, one can resort to the measurement of the zerobias conductance [32], which can be performed for different filling factors, controlled by the gate voltage. To spot quantum phase transitions, one would need cryogenic temperatures T satisfying the condition T ≈ 0.01t ≈ 100mK in order to preserve the correlations from being washed away by thermal fluctuations [35,36]. This temperature range can be routinely achieved in the modern dilution cryostats.
In conclusion, we have proposed an experimentally viable quantum simulator of a spin-orbit coupled Mott insulator based on array of antidots in the two dimensional topological insulator. If antidot size is sufficiently small, the low-energy behaviour of the system can be described by the quantum spin compass model with Dzyaloshinskii-Moriya interaction. The alternative lattice geometries supporting the geometrical frustration would give access to even richer class of the available spin models, which could be of particular interest in the domains of quantum simulation and quantum computation.