Anatomy of Topological Surface States:
Exact Solutions from Destructive Interference on Frustrated Lattices
The hallmark of topological phases is their robust boundary signature whose intriguing properties—such as the one-way transport on the chiral edge of a Chern insulator and the sudden disappearance of surface states forming open Fermi arcs on the surfaces of Weyl semimetals—are impossible to realize on the surface alone. Yet, despite the glaring simplicity of noninteracting topological bulk Hamiltonians and their concomitant energy spectrum, the detailed study of the corresponding surface states has essentially been restricted to numerical simulation. In this work, however, we show that exact analytical solutions of both topological and trivial surface states can be obtained for generic tight-binding models on a large class of geometrically frustrated lattices in any dimension without the need for fine-tuning of hopping amplitudes. Our solutions derive from local constraints tantamount to destructive interference between neighboring layer lattices perpendicular to the surface and provide microscopic insights into the structure of the surface states that enable analytical calculation of many desired properties including correlation functions, surface dispersion, Berry curvature, and the system size dependent gap closing, which necessarily occurs when the spatial localization switches surface. This further provides a deepened understanding of the bulk-boundary correspondence. We illustrate our general findings on a large number of examples in two and three spatial dimensions. Notably, we derive exact chiral Chern insulator edge states on the spin-orbit-coupled kagome lattice, and Fermi arcs relevant for recently synthesized slabs of pyrochlore-based EuIrO and NdIrO, which realize an all-in-all-out spin configuration, as well as for spin-ice-like two-in-two-out and one-in-three-out configurations, which are both relevant for PrIrO. Remarkably, each of the pyrochlore examples exhibit clearly resolved Fermi arcs although only the one-in-three-out configuration features bulk Weyl nodes in realistic parameter regimes. Our approach generalizes to symmetry protected phases, e.g. quantum spin Hall systems and Dirac semimetals with time-reversal symmetry, and can furthermore signal the absence of topological surface states, which we illustrate for a class of models akin to the trivial surface of Hourglass materials KHg where the exact solutions apply but, independently of Hamiltonian details, yield eigenstates delocalized over the entire sample.
pacs:71.10.Fd, 73.21.Ac, 73.20.At, 03.65.Vf
The experimental discovery of the quantum Hall effect in 1980 [klitzingdordapepper, ] decisively put topological phases in the limelight, and especially during the past decade the interplay between theoretical ideas and experimental advances has led to spectacular developments with intriguing prospects for future technological applications hasankane (); qizhang (); Weylreview (); topocomp (). Most early work focused on topological insulators [hasankane, ; qizhang, ], the most basic of which are simple two-dimensional lattice generalizations of the quantum Hall states, namely Chern insulators [haldane, ; tknn, ; hofstadter, ; changzhangfengshenzhang, ; jotzumesserdesbuquoislebratuehlingergreifesslinger, ], while quantum spin Hall insulators stem from two time-reversed copies thereof [kanemele, ; kanemele2, ]. Weyl semimetals, experimentally realized in 2015 [xubelopolskialidoustetal, ; lvwengwantmiaoetal, ; luwangyeranfujoannopoulossoljacic, ; hasanreview, ], are paradigmatic examples of a gapless topological phase existing in three dimensions [volovik, ; murakami, ; wanturnerbishwanathsavrasov, ; burkovbalents, ], whose time-reversal invariant cousins, the Dirac semimetals, were unraveled in 2014 [liuzhouwangwengprabhakaran, ; liujianzhouwangzhangweng, ].
What makes these topological phases so intriguing is their robust and novel boundary states. Despite their central importance, and the simplicity of their bulk description, explicit solutions for the boundary states of topological phases are only known in a very limited number of special cases kitaev (); liuqizhang (); maokuramotoimurayamakage (); shenshanlu (); koenigbuhmannmolenkamphughesliuqizhang (); mongshivamoggi (); zhouluchushenniu (); ojanen (); aklt (). While powerful transfer matrix methods, which in some special cases allow analytical progress, have been developed transfer1 (); transfer2 (); transfer3 (), there is a glaring absence of generic analytical solutions that do not require fine-tuning, that are valid in any dimension, in the entire surface Brillouin zone, at finite size, and without the need for approximations.
In this work, we devise a general strategy for finding exact surface state solutions for trivial as well as for topological phases in any dimension, notably including Chern insulators and Weyl semimetals as well as their time-reversal invariant counter parts in quantum spin Hall insulators and Dirac semimetals. Rather than stemming from fine-tuning of hopping amplitudes our method is rooted in the underlying lattice structure. The lattices we consider can be seen as composed by -dimensional layers of different variety, referred to as and lattices, that are stacked on top of each other in an alternating fashion such that the full -dimensional lattice is geometrically frustrated (Fig. 1). Prominent examples of this type are kagome lattices in and pyrochlore in .
Frustrated lattices are usually studied in the context of magnetism and lead to rich physics while being notoriously difficult to understand even at a qualitative level balents10 (). In glaring contrast, we find that frustration greatly simplifies the study of surface states of both trivial and topological variety. Under very general conditions we find exact eigenstates of the form
where is the -dimensional quasi-momentum parallel to the surface, is a Bloch state of the th lattice layer in a system composed of such layers and intermediate lattice layers, and is a simple function determined by the local connectivity between neighboring layers and the Bloch states of the individual layers. A salient feature of (1) is the vanishing amplitudes on the lattice layers, which is directly related to how the exact solutions are found: assuming vanishing amplitudes on the lattices puts constraints on and provides a bootstrapping procedure uniquely leading to (1). The existence of these solutions hinges only on the counting of local constraints in combination with locality and translation invariance, and as such is insensitive to Hamiltonian details. In this context we stress that the local constraints are not a feature of the Hamiltonian but rather an emergent exact property of the eigenstates in Eq. (1), which is, however, not fulfilled for any other eigenstate. It is also noticeable that the exact solutions are for the full tight-binding model and thus extend in the full -dimensional surface Brillouin zone and thereby also describe the attachment to bulk bands as the states switch surface. Moreover, the solutions remain exact at any finite size, i.e., for any number of layers, .
Our approach is akin to the construction of flat band models arising due to local constraints on ’line graphs’ such as kagome and pyrochlore lattices (see e.g. Ref. bergmanwubalents, ). In contrast to our setup, these models require precise fine-tuning of the hopping amplitudes, typically allowing real and strictly nearest-neighbor hopping only. A second key difference is that the flat bands studied earlier are -dimensional bulk bands while our solutions provide a -dimensional manifold corresponding to the surface Brillouin zone. A similarity is, however, that band touchings necessarily occur in both setups.
Expanding on the seminal work by Mielke mielke (), a large body of work, including effects of interactions and disorder on line graphs, has accumulated during the past 25 years. In this context, valuable insights have been obtained for antiferromagnetic Heisenberg models on frustrated lattices with a flat band corresponding to a localized magnons schulenburg (); schnack (); zengelser (); asakawasuzuki () as well as on flat-band Hubbard models derzhkorichtermaksymenko (); gulacsikampfvollhardt (). Alongside the extensive literature on theory (see also Refs. mielke, ; tasaki, ; mielketasaki, ; flachleykambodyfeltmatthies, ; bodyfeltleykamdanieliyu, ; huberaltman, ; goldmanurbanbercioux, ), intriguing recent experiments babouxjacqimbiodigalopin (); mukherjeespracklenchoudhurygoldman () have underscored the value of these works.
It is conceivable that a similar progress on topological (and trivial) surface states can be spurred by the present work. Indeed, earlier work by two of us exploring Eq. (1) in the special case of -oriented slabs of the pyrochlore lattice trescherbergholtz (); bergholtzliutreschermoessnerudugawa () has already borne fruit: for thin slabs this provided a natural platform for nearly flat bands with higher Chern numbers trescherbergholtz () and led to the subsequent discovery of an entire zoo of novel fractional Chern insulators qualitatively different from their quantum Hall relatives ChernN (); ChernN2 (); bergholtzliutreschermoessnerudugawa (). For thicker slabs, we discovered that Fermi arcs can in fact persist without Weyl nodes in the bulk—and that when Weyl nodes do occur their dispersion is generally both anisotropic and tilted bergholtzliutreschermoessnerudugawa (); titledweylcones (). In particular, the tilting can easily be so strong that the Weyl cones become “over-tilted” forming a compensated metal where the Weyl point is a singular point connecting two Fermi pockets bergholtzliutreschermoessnerudugawa (). These systems were later popularly coined type-II Weyl semimetals saluyanovgreschwangwutroyerdaibernevig () and subsequently experimentally identified in a growing list of intriguing materials weyl2exp (); weyl2exp2 (); weyl2exp3 (); weyl2exp4 (); weyl2exp5 (); weyl2expWTe2_1 (); weyl2expWTe2_2 (); weyl2expMoxW1?xTe2 (). The phenomenon of Fermi arcs without the presence of Weyl nodes has been corroborated by recent experimental findings [brunotamaiwucucchi, ; xuautesmattlvyao, ].
Building on our previous work, we here explore the connection between frustration and surface topology in much more detail and generality whereby we derive a number of generic results regarding correlation functions, surface dispersion, Berry curvature, energy gaps, and the bulk-boundary correspondence. While we mostly focus on two- and three-dimensional examples our results apply mutatis mutandis to any dimension. We also refine the earlier analysis of pyrochlore slabs to relate more directly to experiments on pyrochlore iridates. In particular, this makes contact to beautiful recent experimental progress in growing (thin) single-crystal slabs of the pyrochlore iridates EuIrO fujita2015 () and NdIrO gallagher2016 ()—two materials that both exhibit an all-in-all-out spin ordering and that we conclude are likely to have Fermi-arc-like surface states without possessing Weyl nodes in the bulk. For the yet to be grown slabs of PrIrO, we, however, find that Weyl nodes exist depending on the particular spin ordering goswami2016 (), which is either spin-ice-like two-in-two-out and one-in-three-out, while Fermi arcs exist in either configuration.
This work is structured as follows. In Sec. II, we introduce a generic recipe for constructing our models and present the exact solutions and consequences thereof in general terms. In Sec. III, we illustrate the effectiveness of our recipe with a number of examples. In Sec. III.1, we introduce a one-dimensional chain, and analyze the exact expressions for its end modes. In Sec. III.2, we focus on two dimensions and derive exact edge state solutions on the kagome lattice including the chiral edge states occurring when the system is a Chern insulator. We also discuss the connection between the local lattice structure and topology. In Secs. III.3-III.5, we investigate the surface states of three-dimensional lattice models, most saliently obtaining exact solutions for the Fermi arcs of Weyl semimetals. Throughout this exposition, we comment on the relevance of our solutions for naturally existing, synthesized and artificial materials. We conclude with a discussion in Sec. IV.
Ii Setup and general considerations
In this section, we introduce five conditions—illustrated in Fig. 1 and detailed below—, which, when they are fulfilled, allow us to find a -dimensional manifold of exact wave functions and energies corresponding to the topological surface theory of a given -dimensional topological phase. After detailing the general setup, we describe a number of results that can be derived directly from the exact surface state solutions.
ii.1 Lattice structure
We study -dimensional models with periodic boundary conditions in dimensions while they are left open in the remaining dimension giving the possibility of surface state solutions. More precisely, we consider models that can be decomposed in terms of alternating layers of two different -dimensional periodic lattices; a lattice with degrees of freedom ( lattice) and a lattice with degrees of freedom ( lattice) such that the surfaces of the material are formed by lattices as shown schematically in Fig. 1(a), which we refer to as condition (i). In this work, for the sake of transparency, we mostly consider examples in which the degrees of freedom equal the number of sites in the unit lattice cells, there is one available state per site. Note, however, that our results can readily be generalized to include more degrees of freedom which is necessary for instance for time-reversal symmetric models including onsite spin degrees of freedom.
The key assumption is that the lattices are only connected to each other via the intermediate lattices, and hence that direct hopping between different lattices is prohibited as shown in Fig. 1(b). We refer to this as condition (ii). This is a realistic scenario, because it is unlikely that the orbitals of electrons sitting on different lattices will overlap. Upon solving the Schrödinger equation, we find precisely exact solutions to the wave function, which have total-zero weight on the intermediate-lattice sites if the layers are connected such that the full model is geometrically frustrated, condition (iii), as shown schematically in Fig. 1(c). This is due to geometric frustration, which allows the hoppings from the lattices to the intermediate lattice to interfere out. We refer to this interference as the local hopping constraint, and wave functions obeying this constraint can always be found when the lattice satisfies conditions (ii) and (iii). This shows that the connection of the lattices via the intermediate lattices is essential for our problem. Hopping within the and within the lattices is allowed and will be elaborated upon in the next section.
We distinguish two types of connectivity in this stacking construction. In the first, the intermediate sites on the lattice are connected to sites in the minimal unit cells on both neighboring lattices in a symmetric way. In the second case, the intermediate sites on the lattice are connected differently to the sites in the minimal unit cells of the lattice below than to the sites in the unit cells in the lattice above, as shown in the bottom panel of Fig. 1(d). In either case, there is a natural constraint—zero total hopping amplitude to the lattice sites—that leads to a bootstrapping procedure and exact eigenstates that can be expressed entirely in terms of the Bloch eigenstates of the lattice layers. In the symmetric case, the solutions are rather mundane with , while the latter case, where the local constraint necessarily connects multiple minimal unit cells, gives more interesting solutions for including those that correspond to topological surface states. The latter situation is referred to as condition (iv). Note that in this discussion we have assumed that the coupling strength between the intermediate lattice and the degrees of freedom in the unit cell of the lattice above and equals the coupling strength between the intermediate lattice and the degrees of freedom in the unit cell of the lattice below. We refer to this as the isotropic case. If this coupling were anisotropic, we can find nontrivial solutions for , which depend on both the crystal momentum and the strength of the various perpendicular hopping parameters, regardless of whether the local constraint involves multiple unit cells. If we now include pertinent terms in the Hamiltonian such that the system indeed supports a topological phase, as shown in Fig. 1(e), we find that our wave function solutions describe a topological boundary state. This final condition is referred to as condition (v).
We can thus list five conditions that need to be fulfilled to find topological boundary states.
The lattices are formed by alternating -dimensional lattices, referred to as and lattices, which have periodic boundary conditions. There are open boundary conditions in the direction of stacking and the outermost layers are lattices [Fig. 1(a)];
The lattices are only coupled to each other via intermediate lattices and cannot directly communicate [Fig. 1(b)];
The and lattices are connected in a geometrically frustrated fashion meaning that there are several inequivalent ways of hopping from the neighboring lattices to the single orbital in the lattice unit cell leading, together with (ii), to the emergence of a local constraint obeyed by the exact solutions in Eq. (1) [Fig. 1(c)]. (In the presence of pertinent symmetries, this can be generalized to several orbitals in the lattice unit cell.);
There is no way of choosing a minimal unit cell such that the local constraint obeyed by the exact solutions in Eq. (1) takes place within a single unit cell on both of the involved lattices [Fig. 1(d)]. Alternatively, this condition can be satisfied if the coupling between the and lattices is anisotropic;
The bulk Hamiltonian supports the pertinent topological phase [Fig. 1(e)].
Exact wave-function solutions corresponding to -dimensional bands can be found whenever conditions (i)-(iii) are fulfilled. Fulfilling condition (iv), the exact solution generically yields exponentially localized boundary states, and whenever the bulk supports a given topological phase, condition (v), the exact solution corresponds to its surface theory.
ii.2 Generic tight-binding models
We consider tight-binding models describing noninteracting identical particles on the lattices described above. For the sake of clarity we set , i.e. we consider intermediate lattices with a single degree of freedom per unit cell. The Hamiltonian describing a system with stacked lattices is written directly in momentum space and reads with , the annihilation operator of an electron in the full lattice, and an -dimensional matrix given by
where is the -dimensional Hamiltonian for the lattice, is the -dimensional Hamiltonian for the intermediate lattice, and is an matrix connecting the lattice to the intermediate lattice. In general, this connecting Hamiltonian can be written as
where is the hopping amplitude from site in the unit cell of the lattice to the intermediate lattice, and is a -dependent phase derived from the local lattice structure. Note that all hopping amplitudes are allowed to be complex, i.e. allowing for spin-orbit coupling as well as commensurate magnetic fields incorporated via Peierls substitution.
ii.3 Exact eigenstates
Using the Hamiltonian in Eq. (2), we find a subset of solutions to the Schrödinger equation, , corresponding to the number of degrees of freedom in the lattice, given by
where are the eigenvalues of the lattice Hamiltonian and the eigenstates with components , thereof, is the -dimensional momentum,
ensures normalization,111As , the normalization factor is smooth approaching . labels the lattice layer, and is a prefactor given by
where are the components of in the th lattice on sublattice site . Using that the weight of the wave function on the intermediate site of the lattice is zero, can also be expressed in terms of the components of :
where is given in Eq. (3). From this equation, we can formalize condition (iv): this condition is fulfilled when , i.e. , and broken when , i.e. . Note that explicitly calculating , one can still find when . This is due to the explicit form of and closely related to topology in the model, condition (v), which is further discussed towards the end of this section. We want to emphasize that the exact solution is completely independent of the Hamiltonian on the intermediate lattice. It should, however, be mentioned that the remaining -solutions to the Schrödinger equation, which are only numerically accessible, do depend on this Hamiltonian and are subject to deformation by changing the perpendicular hopping strength.
Inspecting the solution in Eq. (5), we notice three properties. First, the solution has zero weight on the intermediate sites, which means that the Hamiltonian for the intermediate lattice is completely irrelevant to the solution and as such can generally include arbitrary terms. Second, only the connectivity of the lattices via the intermediate lattices encoded by is relevant. The coupling between lattices may differ in strength effectively yielding strongly or weakly coupled layers. Third, we can now understand why satisfying condition (iv) leads to boundary states. If , the weight of the wave function on layer ,
is the same for all , and the wave function is equally localized on each lattice. However, if , the eigenstate will localize to one of the boundaries. If , decreases with increasing and the wave function is strongly localized on the first layer corresponding to the surface on one side. When , increases with increasing and the state is localized on the last layer corresponding to the surface on the other side. Therefore, when , we have exponentially localized boundary states with a localization length, and the solution in Eq. (5) thus corresponds to the solution for the boundary state. Now, if also has a -dependent structure the boundary state can switch surfaces, which for a three-dimensional material means a constant energy contour represents a Fermi arc. When discussing the exponential surface localization we will make use of the localization length .
Satisfying condition (iv) means we have found a suitable geometry for the lattices to find boundary states. However, when one plugs the solution to the eigenfunctions into Eq. (8), we may still find for systems deep in the topologically trivial regime. In the trivial regime it is also possible to find unprotected, weakly-localized boundary states for which has a nontrivial structure. Therefore, we need to impose a fifth condition, condition (v), that introduces nontrivial topology in the models such that we for instance obtain a Chern insulator and Weyl semimetal, which are examples of two- and three-dimensional models, respectively. One needs to minimally break time-reversal symmetry to find a Chern insulator, such that in two dimensions, the Hamiltonian should include at least one such term. To obtain a Weyl semimetal, one could either break inversion or time-reversal symmetry. In the cases studied in this work, we break the latter symmetry by turning the lattice into a Chern insulator. In section II.5, we present an argument to understand why this leads to an eigenstate that switches surfaces.
It is worth emphasizing that the exact wave function solution directly enables the computation of correlation functions within the surface bands which are otherwise only numerically or approximately tractable. For example, the expectation value of any operator , which acts the same within each layer, reads . This expression can readily be extended to more complicated, layer dependent, operators.
Diagonalizing the Hamiltonian in Eq. (2) leads to the band spectrum of the full system with bands, which are divided into -bulk parts separated by band gaps. When conditions (i) - (v) are met, boundary states are present, which can be identified in the bulk spectrum as bands crossing a gap and connecting two bulk parts.
It should be noted that, while the exact solutions remain unchanged, all other eigenstates change while deforming the coupling to the lattices. For instance, as we will demonstrate in Section III, increasing the inter-layer coupling strength drives a transition between the quasi-two-dimensional layered Chern insulators phase and a truly three-dimensional Weyl semimetal regime.
ii.4 Attachment of bulk bands and surface switching
It is a generic property of our exact solutions that in part of the surface Brillouin zone they attach to bulk bands in the limit of many stacked layers. This happens precisely at those points in the -dimensional (surface) Brillouin zone where the boundary state connects to the bulk where , i.e. where the penetration depth diverges. Below we will provide a variational argument that shows that the gap—either from below or from above—vanishes as for large . In contrast, whenever the penetration is finite there is a gap to neighboring bands also in the limit .
There are two types of penetration depth divergences and concomitant bulk band attachments. First, at it immediately follows that independently of model details. Second, many models feature -dimensional families of such points, typically along high symmetry paths cutting through the surface Brillouin zone. The existence of these motifs depend on details of the lattice geometry and the tight-binding Hamiltonian. In particular, the boundary states may switch surfaces at , i.e. as . Notably, -dimensional families of such points necessarily exist for topologically nontrivial models although they can also occur in topologically trivial models.
It is intuitively plausible that the energy gap between the boundary states and the bulk bands at the points should disappear. We can explicitly demonstrate that this is indeed the case by introducing a class of ansatz wave functions describing bulk state with nearly the same energy as the exact solution . In this case, the exact solution to the low-energy boundary state can be written as
We can now make an ansatz for a class of states expected to be close in energy,
which has to be orthogonal to the exact solution in Eq. (10), such that we find
This leads to
However, the trial wave function in Eq. (11) is not an eigenfunction of the Hamiltonian because it has zero weight on the intermediate site of the lattice, which it should not have.
Therefore, the trial wave function can be made an eigenstate by mixing in the state for the intermediate site , which results in altering the entries given in , such that we can create an approximate eigenstate, which becomes exact in the limit of large N:
Therefore, we find that the energy difference between the two states behaves as
for large . Therefore, the energy difference between the exact solution and our variational bulk state disappears as at for large . In fact, there will be many such states as signaled by the family of states constructed (varying ). Note, however, that depending on details, in particular the strength of the inter-layer tunneling , the variational state may be lower or higher in energy. For weak , the variational state is always lower in energy than the exact solution. However, for stronger , this can change in the Brillouin zone, as is strikingly manifested in the case of Weyl points at which these energies are equal and a sign change of takes place.
ii.5 Berry curvature and surface state topology
The appearance of boundary states is closely related to nontrivial topology in the bulk of a material. For Chern insulators and Weyl semimetals, this bulk topology is manifested by a nonzero Chern number. The Chern number for an isolated band of a two-dimensional periodic lattice is computed by integrating the Berry curvature over the Brillouin zone:
where is the Berry curvature given by with the Berry connection. The total Chern number of any system has to be zero, i.e. . While correlation functions are easily calculated, the derivatives entering the Berry curvature complicate analytical calculations thereof. Alternatively, this problem can be seen from the fact that a (derivative-free) expression of the Berry curvature involves all energy eigenstates of the model—not just the solvable surface bands. Nevertheless, a number of instructive results can be derived.
For (quasi-)three-dimensional models we use the solution to the surface state in Eq. (5) such that one can write the Berry curvature of a system with two-dimensional lattices as
being the Berry curvature of a single layer and
We emphasize that contains the full -dependence of the Berry curvature in Eq. (18). In the limit , Eq. (19) reduces to , thus for large the Berry curvature exhibits a peak scaling with at those parts in the Brillouin zone where the exact solution is not a surface state but is completely delocalized over all layers. In contrast, whenever , the Berry curvature saturates as a function of consistent with the exponential localization of the wave functions to the surface layers.
The Chern number of the solvable bands in a system with lattices can be found upon integrating the Berry curvature in Eq. (18) over the Brillouin zone as shown in Eq. (17). When everywhere, the derivative over the second part of Eq. (18) yields zero such that the total Chern number equals that of one layer . However, in the generic situation when , we observe in our examples that the total Chern number grows with as
which means that the Chern number that is associated with each lattice is absorbed into the surface state.
Furthermore, it can be shown that a finite Chern number on the lattice implies Fermi arcs by considering a generic two-band model on the lattice whose Hamiltonian reads
where are the Pauli matrices, , and
projects on to the upper (+) and lower (-) bands, respectively, with . Using the projector, the Chern number can be written in terms of as
The Chern number can thus be interpreted as the number of times wraps the unit sphere. Provided the general structure of one can show that, for a very generic class of models, this implies that renders both the numerator and denominator of given in Eq. (8) to vanish at different whenever the single layer Chern number is finite. Thus has zeros and infinities when the lattice is a Chern insulator implying surface switching and the existence of Fermi arcs in the sense that the surface state is entirely localized at the top and bottom layer at different points in the surface Brillouin zone. An explicit example of this is provided in section III.3.
iii.1 One dimension: end modes on a chain
As a simple warmup, we start by studying a one-dimensional system, which has two degrees of freedom in the lattice and one degree of freedom in the intermediate lattice shown in Fig. 2. This chain readily satisfies conditions (i)-(iii), and the absence of a surface momentum parameter means that condition (iv) can only be satisfied if the and intermediate lattices are coupled in an anisotropic fashion.
In its most generic form, the Hamiltonian for each lattice reads , where is the annihilation operator of an electron in the lattice and can be written in the Dirac form
with the energy eigenvalues
and eigenstates . Due to the lack of a momentum parameter, the eigenvalues and the amplitudes of the eigenstates are constants. The Hamiltonian for the one-dimensional chain is given in Eqs. (2) and (3) with the phases , and we set the intermediate lattice Hamiltonian and , and with . Using Eq. (8), we find
which is a function of . When and , and according to Eq. (9) the wave function in Eq. (5) has equal weight on each lattice . However, when and/or , we find and there are end modes on the chain. These modes either reside at the same end, e.g. at and yields hence both end modes are exponentially localized around the lattice —or they live at opposite ends, e.g. when the Hamiltonian for the lattice in Eq. (24) reads and we require the perpendicular hopping parameters to satisfy and yielding and such that the end modes and appear at and , respectively. The exactly obtained end modes of the chain thus switch ends as a function of the perpendicular hopping parameters .
iii.2 Two dimensions: Chern and quantum spin Hall insulators on kagome and related lattices
In this section, we study two-dimensional lattice models by stacking (periodic) chains containing two sites in the lattice unit cell (red and blue in Fig. 3), which are connected via an intermediate chain with a single site per unit cell (green). The two types of chains are stacked in an alternating fashion in such a way that the lattice geometry is frustrated and a local hopping constraint is naturally realized, such that conditions (i)-(iii) are fulfilled. The four different stacking possibilities are shown in Fig. 3, where the lattices in Figs. 3 and 3 are related to the lattices in Figs. 3 and 3, respectively, via sublattice exchange in every other two-site chain.
The lattice for the models in Figs. 3 and 3 are the two-site periodic chains in red and blue, and the intermediate lattice is the one-site periodic chain in green. For the lattices in Figs. 3 and 3, however, one has to consider the composite of three chains as the lattice indicated by the black brackets in the figures. The intermediate lattice is the one-site green chain in between. We treat all four models in detail and find a subset of solutions to the Schrödinger equation for each of them. First, we will show that when condition (iv) is not fulfilled, as is the case for the lattices in Figs. 3, 3 and 3, the system remains topologically trivial, signaled by an absence of edge states and vanishing topological invariants, regardless of the microscopic Hamiltonian (as long as it is local in the sense of condition (ii)). Then, by considering models living on the lattice displayed in Fig. 3, which does fulfill condition (iv), we illuminate the relevance of condition (v). Strikingly, we find that whenever the system has a bulk band characterized by a unit Chern number our exact solutions describe the chiral edge states of the model.
Let us now proceed to demonstrate what is advertised in the preceding paragraph by considering a generic description of translation invariant tight-binding models on the aforementioned lattices. The Hamiltonian for each chain with two sublattices in red and blue is , where is the annihilation operator of an electron in the lattice and
The corresponding energy eigenvalues are given by
and are the eigenstates. In all four cases, the Hamiltonian for the full models is given in Eqs. (2) and (3), and we set and . The latter can be interpreted as a gauge choice, and does not impede our general approach: one can always choose the perpendicular hopping parameter to be real by suitably redefining to account for the “flux” through each closed path of the lattice [redderuhrig, ].
We first focus on the model in Fig. 3. The lattice Hamiltonian is given by Eq. (27), and the pertinent phases are and such that by using Eq. (8) we immediately find that and condition (iv) is thus not fulfilled. The weight of the wave function on each individual chain given in Eq. (9) is thus equal for each chain , which labels the lattice, meaning that the state is fully delocalized. We thus expect to find a topologically trivial system. Indeed, regardless of any details of for the lattice Hamiltonian, the Chern number remains zero.
Next, we look at the system in Fig. 3. The suitably redefined lattice Hamiltonian now accounts for the five sites in the unit cell, hence we can find exact expressions for five edge state bands. The lattice Hamiltonian is given by
and the concomitant phase factors are , and and . Before analyzing the five solutions to the Schrödinger equation for the full system, we first take a closer look at the solution for the lattice Hamiltonian . We observe that we can interpret the lattice as existing out of two sub- lattices, the two-site chains in red and blue, and a sub-intermediate lattice, the green sites. We thus find solutions of the following form, which look similar to the solution in Eq. (5):
where is the normalization factor, are the eigenstates of the two-site chain, and , which can be straightforwardly derived. The corresponding eigenvalues are given in Eq. (28). Plugging this and the phases into Eq. (8) yields such that we retrieve a system in which the wave functions are completely delocalized. For the remaining three solutions, we find with with the energy and again , independent of the Hamiltonian used for the chain. Again, regardless of the hopping terms included in the Hamiltonian, the system stays in a topologically trivial phase, as expected by the absence of edge states stemming from the breaking of condition (iv).
Now, we turn to the lattice model in Fig. 3, which shows slightly more complex behavior. The lattice Hamiltonian again includes five sites in the unit cell and reads
These two wave functions are thus equally localized to each lattice present in the full model. However, we notice that there may occur some localization inside the lattice as is not trivially equal to , which is shown in Figs. 4(a) and 4(b).
For the remaining three solutions to the wave function, which read with , we find a nontrivial as shown in Fig. 5. However, one can see in the corresponding energy spectrum in Fig. 5 that the localized states are not topologically protected, which is supported by the retrieval of a zero-Chern number for these parameters.
Finally, we turn to the kagome lattice in Fig. 3, which shows significantly richer and more complex behavior. This lattice is also of special interest as it occurs naturally in many materials and can also be engineered in cold atom systems [joguzmanthomashosur, ; zhangchenmazhou, ]. The lattice Hamiltonian is given by Eq. (27), and the phases are and , such that corresponds to the expression in Eq. (30), and constraint (iv) is fulfilled. This allows us to review condition (v). To specify the lattice Hamiltonian in Eq. (27), we use the following:
where , , and are nearest-neighbor hopping parameters, and is a staggering potential, and . We emphasize that details of the lattice Hamiltonian are irrelevant as long as it is local and translation invariant, and we have introduced nearest-neighbor hopping terms only to be able to review the localization of the state in a transparent fashion. We find that only when at least and/or as is shown in Fig. 4. We first review the situation where and , in which case both sublattice and inversion symmetry are broken but time-reversal symmetry is preserved. We thus expect to be unable to find a Chern-insulator phase, which is indeed what is revealed in Figs. 4(a) and 4(c). The case in which and is more interesting. Now inversion symmetry is preserved but sublattice and time-reversal symmetry are broken such that we are able to find a Chern insulator as is shown in Figs. 4(b) and 4(d). We see that the energy spectrum has two bulk gaps with a band crossing each of these gaps. These two bands (in orange) correspond to our exact wave function solution in Eq. (5) and their inverse localization length reveals that the right mover localizes to the chain and the left-mover to the chain , which is in accordance with the chirality of the edge states in a Chern insulator. This localization is corroborated by the weight of the wave function on each lattice given in Eq. (9) shown in Fig. 6 for a system of five layers. The Chern number for the fully periodic kagome lattice is governed by the values of , and we find for half-filling that the Chern number for . This leads to another interesting observation, namely, that when the system is no longer a Chern insulator, the exact solution still localizes, which is reflected by in Fig. 4(b) still having a structure for , . We have thus found a Chern insulator on the kagome lattice with chiral-edge states whose exact wave function solution is given in Eq. (5) when both time-reversal and sublattice symmetry are broken simultaneously. The breaking of the first is a minimal requirement to find a Chern insulator, whereas the breaking of the second is inherent to our specific choice of lattice as preserving sublattice symmetry would yield such that one trivially finds . Note that a quantum spin Hall insulator can simply be retrieved by taking two time-reversed copies of the Chern insulator, which introduces spin degree of freedom. In that case, the system supports four helical-edge states, whose wave function is described by our exact solution.
iii.3 Three dimensions, first example: Dirac and Weyl semimetals from stacked checkerboard models
We now turn to three-dimensional models built from stacking two-dimensional layers in a frustrated fashion. In the first example, the lattice is a checkerboard lattice, which has two sites in the unit cell, and the intermediary lattice is a square lattice with one site in the unit cell. By stacking them we obtain the lattice shown in Fig. 7, which is a three-dimensional cousin of the kagome model in Fig. 3. The two-dimensional surface Brillouin zone is shown as an inset in Fig. 8. Conditions (i) - (iv) are fulfilled and we have a Weyl phase when condition (v) is fulfilled in which case we expect that the exact solution corresponds to (a family of) Fermi arcs. In particular, this model is well suited for exploring the connection between the topology of the two-dimensional layers and the band switching properties of the surface bands.
The Hamiltonian for the individual checkerboard lattice is given by , where