Measuring topology in a laser-coupled honeycomb lattice: from Chern insulators to topological semi-metals

In the model introduced in [16], cold fermionic atoms are trapped in a honeycomb structure formed by two intertwined triangular optical lattices, whose inequivalent sites are labeled by A and B respectively (see figures 1(a)–(c)). In the tight-binding regime—applicable for sufficiently deep optical potentials V_A,B(x)—atoms are only allowed to hop between neighboring sites of the two triangular sublattices, which correspond to NNNs of the honeycomb lattice (denoted 〈〈n_τ,m_τ〉〉, with τ = A,B). The second-quantized Hamiltonian takes the form

$$\begin{equation} \hat H_{\rm NNN}= - t_{\rm A} \sum_{\langle \langle n_{\rm A}, m_{\rm A} \rangle \rangle} \hat a^{\dagger}_{n_{\rm A}} \hat a_{m_{\rm A}} - t_{\mathrm{B}} \sum_{\langle \langle n_{\mathrm{B}}, m_{\mathrm{B}} \rangle \rangle} \hat b^{\dagger}_{n_{\mathrm{B}}} \hat b_{m_{\mathrm{B}}}, \end{equation} \tag{ 1 }$$

where $\hat a_{m_{\mathrm { A}}}$ ($\hat b_{m_{\mathrm { B}}}$) is the field operator for annihilation of an atom at the lattice site r_mA (r_mB) associated with the A (B) sublattice, and where t_A,B are the tunneling amplitudes. Furthermore, the two sublattices are coupled through laser-assisted tunneling, where hopping is induced between neighboring sites of the honeycomb lattice by a laser coupling the two internal states associated with each sublattice. This corresponds to tunneling processes linking NNs sites of the honeycomb lattice, denoted as 〈m_A,m_B〉, which are described by the Hamiltonian

$$\begin{equation} \hat H_{\rm NN}= - t \sum_{\langle m_{\rm A}, m_{\mathrm{B}} \rangle} \left( \mathrm{e}^{\mathrm{i} \phi (m_{\rm A}, m_{\mathrm{B}})} \hat a^{\dagger}_{m_{\rm A}} \hat b_{m_{\mathrm{B}}} + \mathrm{h.\,c.} \right). \end{equation} \noindent \tag{ 2 }$$

Here, the phases ϕ(m_A,m_B) generated by the laser fields are the analogues of the Peierls phases familiar from condensed matter physics [22, 23], with r_mA and r_mB specifying the nearest neighboring sites of the hexagonal lattice. Following the approach of Jaksch and Zoller [12], these phases can be expressed in terms of the momentum p transferred by the laser-assisted tunneling as

$$\begin{equation} \phi (m_{\rm A}, m_{\mathrm{B}})= {\boldsymbol{p}} \cdot ({\boldsymbol{r}}_{m_{\rm A}} + {\boldsymbol{r}}_{m_{\mathrm{B}}})/2 = - \phi (m_{\mathrm{B}},m_{\rm A}), \end{equation} \noindent \tag{ 3 }$$

so that the phases have opposite signs for r_A → r_B and r_B → r_A hoppings (see figure 2(a) and [11, 13]). Finally, the model also features an on-site staggered potential, described by

$$\begin{equation} \hat H_{\rm stag}= - \varepsilon \sum_{m} \left( \hat a^{\dagger}_{m_{\rm A}} \hat a_{m_{\rm A}} - \hat b^{\dagger}_{m_{\mathrm{B}}} \hat b_{m_{\mathrm{B}}}\right), \end{equation} \tag{ 4 }$$

which explicitly breaks the inversion symmetry of the honeycomb lattice [3]. The total Hamiltonian, given by

$$\begin{equation} \hat H_{\rm tot}=\hat H_{\rm NN}+\hat H_{\rm NNN}+\hat H_{\rm stag} \end{equation} \tag{ 5 }$$

is characterized by the hopping amplitudes (t, t_A and t_B), the momentum transfer p = (p_x,p_y), as well as the mismatch energy ε.

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To eliminate the explicit spatial dependence of our Hamiltonian (5), we perform the unitary transformation

$$\begin{equation} \begin{array}{@{}ll@{}} \displaystyle \hat a^{\dagger}_{m_{\rm A}} \rightarrow \tilde a^{\dagger}_{m_{\rm A}}=\hat a^{\dagger}_{m_{\rm A}} \exp(\mathrm{i} {\boldsymbol{p}} \cdot {\boldsymbol{r}}_{m_{\rm A}} /2 ), \\ \displaystyle \hat b^{\dagger}_{m_{\mathrm{B}}} \rightarrow \tilde b^{\dagger}_{m_{\mathrm{B}}}=\hat b^{\dagger}_{m_{\mathrm{B}}} \exp(-\mathrm{ i} {\boldsymbol{p}} \cdot {\boldsymbol{r}}_{m_{\mathrm{B}}} /2 ), \end{array} \end{equation} \noindent \tag{ 6 }$$

giving a transformed Hamiltonian

$$\begin{eqnarray} &&\fl \hat H_{\rm tot} = - t \sum_{\langle n_{\rm A}, m_{\mathrm{B}} \rangle} \left( \tilde a^{\dagger}_{n_{\rm A}} \tilde b_{m_{\mathrm{B}}} + \tilde b^{\dagger}_{m_{\mathrm{B}}} \tilde a_{n_{\rm A}} \right)- t_{\rm A} \sum_{\langle \langle n_{\rm A}, m_{\rm A} \rangle \rangle} \mathrm{e}^{\mathrm{i} \tilde \phi (n_{\rm A}, m_{\rm A})} \tilde a^{\dagger}_{n_{\rm A}} \tilde a_{m_{\rm A}} - t_{\mathrm{B}} \sum_{\langle \langle n_{\mathrm{B}}, m_{\mathrm{B}} \rangle \rangle} \mathrm{e}^{\mathrm{i} \tilde \phi (n_{\mathrm{B}}, m_{\mathrm{B}})} \tilde b^{\dagger}_{n_{\mathrm{B}}} \tilde b_{m_{\mathrm{B}}} \nonumber\\ &&- \varepsilon \sum_{m} \left( \tilde a^{\dagger}_{m_{\rm A}} \tilde a_{m_{\rm A}} - \tilde b^{\dagger}_{m_{\mathrm{B}}} \tilde b_{m_{\mathrm{B}}} \right ), \end{eqnarray} \tag{ 7 }$$

with new Peierls phases given by

$$\begin{equation} \begin{array}{@{}ll@{}} \displaystyle \tilde \phi (n_{\rm A}, m_{\rm A})= {\boldsymbol{p}} \cdot ({\boldsymbol{r}}_{m_{\rm A}} - {\boldsymbol{r}}_{n_{\rm A}})/2,\\ \displaystyle \tilde \phi (n_{\mathrm{B}}, m_{\mathrm{B}})= {\boldsymbol{p}} \cdot ({\boldsymbol{r}}_{n_{\mathrm{B}}} - {\boldsymbol{r}}_{m_{\mathrm{B}}})/2. \end{array} \end{equation} \noindent \tag{ 8 }$$

The transformed Hamiltonian (7), featuring complex hopping terms along the links connecting NNN sites, is similar to the Haldane model [3], but with important differences highlighted in section 2.4.

Since r_nA,B − r_mA,B = δ_μ − δ_ν = ± a_λ are the primitive lattice vectors of the honeycomb lattice (see figure 1), where μ,ν,λ = 1,2,3, the phases $\tilde {\phi }$ in equation (8) no longer depend on the spatial coordinates. Therefore, the Hamiltonian (7) is invariant under discrete translations, $[\hat H_{\mathrm { tot}}, \mathcal {T}_{1,2}]=0$ where $\mathcal {T}_{1,2} \psi ({\boldsymbol {r}})=\psi ({\boldsymbol {r}}+{\boldsymbol {a}}_{1,2})$, allowing us to invoke Bloch's theorem and reduce the analysis to a unit cell formed by two inequivalent sites A and B. In momentum space, the Hamiltonian takes the form of a 2 × 2 matrix,

$$\begin{equation} H({\boldsymbol{k}})= - \left(\begin{array}{@{}cc@{}} \varepsilon + 2 t_{\rm A} f ({\boldsymbol{k}} - {\boldsymbol{p}} / 2) & t g ({\boldsymbol{k}}) \\ t g^{*} ({\boldsymbol{k}}) & - \varepsilon + 2 t_{\mathrm{B}} f ({\boldsymbol{k}} + {\boldsymbol{p}} / 2) \end{array}\right) , \end{equation} \tag{ 9 }$$

where

$$\begin{equation} f ({\boldsymbol{k}})=\sum_{\nu=1}^{3}\cos \bigl ( {\boldsymbol{k}} \cdot {\boldsymbol{a}}_{\nu} \bigr), \quad g ({\boldsymbol{k}})=\sum_{\nu=1}^{3} \exp (-\mathrm{i} {\boldsymbol{k}} \cdot {\boldsymbol{\delta}}_{\nu}), \end{equation} \tag{ 10 }$$

and k = (k_x,k_y) belongs to the first Brillouin zone (FBZ) of the system. We rewrite this Hamiltonian in the standard form

$$\begin{equation} H({\boldsymbol{k}})= \epsilon ({\boldsymbol{k}}) \hat{1}+ {\boldsymbol{d}} ({\boldsymbol{k}}) \cdot \hat{\boldsymbol{\sigma}}, \end{equation} \tag{ 11 }$$

with

$$\begin{equation} \epsilon ({\boldsymbol{k}})= - t_{\rm A} f ({\boldsymbol{k}} - {\boldsymbol{p}} / 2) - t_{\mathrm{B}} f ({\boldsymbol{k}} + {\boldsymbol{p}} / 2), \end{equation} \tag{ 12 }$$

where $\boldsymbol {\hat \sigma }$ is the vector of Pauli matrices, and d(k) has real-valued Cartesian components defined by

$$\begin{equation} \begin{array}{@{}ll@{}} \displaystyle d_x ({\boldsymbol{k}})-\mathrm{i}d_y ({\boldsymbol{k}})= - t g({\boldsymbol{k}}), \\ d_z ({\boldsymbol{k}})= - \varepsilon - t_{\rm A} f ({\boldsymbol{k}} - {\boldsymbol{p}} / 2) \displaystyle + t_{\mathrm{B}} f ({\boldsymbol{k}} + {\boldsymbol{p}} / 2). \end{array} \end{equation} \tag{ 13 }$$

The eigen-energies of the Hamiltonian (11) are E_±(k) = (k) ± d(k), where we introduced the 'coupling strength' d(k) = |d(k)|.

Our Hamiltonian (11)–(13) differs from the expression derived in [16], where different Peierls phases were used⁸. Both models are exactly equivalent when t_B = 0, where the system describes a semi-metal (see sections 3 and 4).

We now briefly describe how the energy spectrum changes with the parameters (t_A,t_B,ε) of the Hamiltonian (11)–(13), in the absence of momentum transfer p = 0. When t_A,B = ε = 0, the band structure is that of graphene [19, 20], namely, the spectrum is given by

$$\begin{equation} E_{\pm} ({\boldsymbol{k}})= \pm \vert t g({\boldsymbol{k}})\vert , \quad t_{\mathrm{A,B}}=\varepsilon=0 . \end{equation} \tag{ 14 }$$

The two bands touch at zero energy for particular points K_± (the so-called Dirac points), where g(K_±) = 0, and around which the spectrum is quasi-linear with momentum, E_±(k) ≈ ± v_F|k|. We will still use the term 'Dirac' points to denote K_±, even if the gap is open (in the vicinity of these points the excitations describe massive Dirac fermions). For ε ≠ 0, a bulk gap Δ∝ε opens at the Dirac points, where the gap width is defined as Δ = min(E₊) − max(E₋).⁹ This is not a necessary condition to open a gap, as the NNN couplings t_A,t_B are also able to do so. For t_A,B ≠ 0 and ε = 0, the spectrum is now

$$\begin{equation} E_{\pm} ({\boldsymbol{k}}; {\boldsymbol{p}}=0,\varepsilon=0)=D_{+}({\boldsymbol{k}}) \pm \sqrt{ \vert t ~g({\boldsymbol{k}})\vert]^2 +[ D_{-}({\boldsymbol{k}}) ]^2 }, \end{equation} \tag{ 15 }$$

with D_±(k) = −(t_A ± t_B) f(k). Next we note that |g(k)|² = 3 + 2f(k), showing that a gap Δ∝|t_A − t_B| opens at the Dirac points due to NNN couplings. For finite momentum transfer p ≠ 0, the energy spectrum

$$\begin{equation} E_{\pm} ({\boldsymbol{k}})= \epsilon ({\boldsymbol{k}}; {\boldsymbol{p}} , t_{\mathrm{A,B}}) \pm \sqrt{ \left[ tg({\boldsymbol{k}})\right]^2 + [d_z ({\boldsymbol{k}}; {\boldsymbol{p}}, \varepsilon , t_{\mathrm{A,B}}) ]^2 } \end{equation} \tag{ 16 }$$

leads to more complex spectral structures and phases, to be explored in section 3.

In the following, we study the phases of non-interacting fermions in an optical-lattice setup described by Hamiltonian (11)–(13). Such a system forms a metal (or a semi-metal) when the gap is closed Δ = 0, and an insulator when Δ > 0. In the latter case, we set the Fermi energy E_F in the middle of the bulk gap. This classification in terms of the band structure is not exhaustive, and it must be completed by a description of the topological properties of this band structure. This is examined in the following section 2.2. In addition, the properties of some peculiar semi-metals are also explored in this work (see section 4).

When the two-band spectrum E(k) exhibits an energy gap Δ, one can define a topologically invariant Chern number [24], which encodes the topological order of the system. As shown in [4], the Chern number ν is equal to the transverse Hall conductivity, σ_H = ν in units of the conductivity quantum, provided the Fermi energy is located in the bulk gap. The Chern number is given by the standard TKNN expression [4, 25]

$$\begin{equation} \nu= \frac{\mathrm{i}}{2 \pi} \int_{\mathbb{T}^2} \langle \partial_{k_x} u_{(-)}({\boldsymbol{k}}) \vert \partial_{k_y} u_{(-)} ({\boldsymbol{k}}) \rangle - (k_x \leftrightarrow k_y) {\mathrm{d}}^2 \boldsymbol{k}, \end{equation} \tag{ 17 }$$

$$\begin{equation} \hphantom{\nu}= \frac{1}{2 \pi} \int_{\mathbb{T}^2} {\boldsymbol{1}}_z \cdot (\nabla _{{\boldsymbol{k}}} \times {\boldsymbol{A}} ({\boldsymbol{k}}) ) {\mathrm{d}}^2 {\boldsymbol{k}} , \end{equation} \tag{ 18 }$$

where |u₍₋₎(k)〉 denotes the single-particle eigenstate associated with the lowest bulk band E₋(k). The Berry's connection—or vector potential—A(k) is defined by

$$\begin{equation} {\boldsymbol{A}} ({\boldsymbol{k}}) = \mathrm{i} \langle u_{(-)} \vert {\boldsymbol{\nabla}} _{{\boldsymbol{k}}} \vert u_{(-)} \rangle. \end{equation} \tag{ 19 }$$

This quantity, which defines the parallel transport of the eigenstates over the FBZ [24], also determines the topological order of the system [2]. The integration in equation (18) is taken over the FBZ, a two-torus denoted as $\mathbb {T}^2$, where the contribution due to any singularities of A(k)—to be discussed later on—should be excluded.

It is convenient to parametrize the 'coupling' vector d(k) in terms of the spherical angles θ ≡ θ(k) and ϕ ≡ ϕ(k), defined as

$$\begin{equation} \tan \phi = d_y ({\boldsymbol{k}}) / d_x ({\boldsymbol{k}}) , \quad \cos \;\theta= d_z ({\boldsymbol{k}}) / d ({\boldsymbol{k}}), \end{equation} \tag{ 20 }$$

where $\phi = \pi - \arg g({\boldsymbol {k}})$ for t > 0. In what follows we shall assume that t > 0, without loss of generality. In this representation, the Hamiltonian (11) takes the form

$$\begin{equation} H ({\boldsymbol{k}})= \epsilon ({\boldsymbol{k}}) \hat{1} - d ({\boldsymbol{k}}) \left(\begin{array}{@{}cc@{}} \cos \theta & \mathrm{e}^{-\mathrm{ i} \phi} \sin \theta \\ \mathrm{e}^{\mathrm{i} \phi} \sin \theta & - \cos \theta \end{array}\right). \end{equation} \tag{ 21 }$$

The lowest eigenstate of (21) is given by

$$\begin{equation} \vert u_{(-)} \rangle =\left(\begin{array}{@{}c@{}} -\mathrm{e}^{-\mathrm{i} \phi} \sin (\theta /2) \\ \cos (\theta /2) \end{array}\right), \end{equation} \noindent \tag{ 22 }$$

and from equations (19) to (22), we obtain an explicit expression for the Berry's connection,

$$\begin{equation} {\boldsymbol{A}} ({\boldsymbol{k}})=\frac{1}{2} (1- \cos \theta) {\boldsymbol{\nabla}} _{{\boldsymbol{k}}} \phi. \end{equation} \tag{ 23 }$$

A crucial point to note is that the Berry's connection (23) has singularities at the points in k-space where d_x(k) = d_y(k) = 0 and d_z(k) < 0. The condition d_x(k) = d_y(k) = 0, which coincides with the zeros of the function g(k), is always fulfilled at the special points ${\boldsymbol {K}}_{\pm } = (\frac {2 \pi }{3}, \pm \frac {2 \pi }{3 \sqrt {3}})$, where we have set a = 1. The second condition d_z(K_±) < 0 is only satisfied for certain values of the model parameters (t_A,B, ε, p). In terms of the coupling vector d, the singularity takes place at the 'South pole' where θ = π and ϕ is arbitrary, so that the state |u₍₋₎〉 is multivalued there. Note that this singularity can be removed locally by a gauge transformation, but not globally [26]. Moreover, we find that the phase $\phi = \pi - \arg g({\boldsymbol {k}})$ yields opposite vorticities at the two inequivalent Dirac points,

$$\begin{equation} v_{\pm}=\oint_{\gamma_{\pm}} \boldsymbol{\nabla} \phi ({\boldsymbol{k}}) \cdot {\rm d} {\boldsymbol{k}}= \pm 2 \pi, \end{equation} \tag{ 24 }$$

where γ_± denotes closed loops around the two Dirac points K_±.

If these singularities were absent, the integrand in equation (18) would constitute an exact differential form over the entire FBZ. In this trivial case, Stokes theorem would then ensure that the integral in equation (18) is zero, since this exact two-form is integrated over a closed manifold¹⁰. To account for these singularities, Stokes theorem can be applied to a contour avoiding them [2, 27]. In particular, the Chern number (18) can be written as a sum of integrals performed over the excluded singularities, i.e. by contributions from small circles of infinitesimal radius γ_± around the excluded Dirac points k = K_± at which A(k) is singular,

$$\begin{equation} \nu = - \frac{1}{2 \pi} \sum_{{\boldsymbol{K}} _{-}, {\boldsymbol{K}} _{+}} \oint_{\gamma_{\pm}} {\boldsymbol{A}} ({\boldsymbol{k}}) \cdot {\rm d} {\boldsymbol{k}} . \end{equation} \noindent \tag{ 25 }$$

Using equation (24) and taking into account the fact that cos θ(k) remains well-defined close to K_±, we find the simple expression for the Chern number

$$\begin{eqnarray} \nu &=& \frac{1}{4 \pi} ( \cos [ \theta ({\boldsymbol{K}}_{+}) ] v_{+} ~+~ \cos [ \theta \left({\boldsymbol{K}}_{-}\right) ] v_{-} ) \nonumber \\ &=& \frac{1}{2} \Biggl( \frac{d_z ({\boldsymbol{K}}_{+})}{\vert d_z ({\boldsymbol{K}}_{+}) \vert} - \frac{d_z ({\boldsymbol{K}}_-)}{\vert d_z ({\boldsymbol{K}}_{-}) \vert} \Biggr ), \end{eqnarray} \noindent \tag{ 26 }$$

which only involves the sign of the 'mass' term d_z(k) (13) at the two inequivalent Dirac points K_±. A detailed demonstration of equation (26), which further highlights the role played by the singularities, is presented in appendix A. The important result in equation (26) shows that the Chern number ν can now be directly evaluated, without performing the integration over the FBZ in equation (17). From equations (13) to (26), one can already deduce that non-trivial Chern numbers ν ≠ 0 can only be obtained when d_z(k) has opposite signs at the two inequivalent Dirac points K_±, which can only be achieved for p ≠ 0. In the following section 2.3, we give a physical interpretation in terms of effective magnetic fluxes and time-reversal-symmetry breaking. We also comment on pathological time-reversal symmetric configurations, which necessarily lead to a trivial topological order ν = 0.

To conclude this section, we note that a Chern insulator is also characterized by current-carrying edge states that propagate along the edge of the system. This edge transport is guaranteed by the opening of a non-trivial bulk gap (Δ,ν ≠ 0, see figure 6(b)), and it leads to the quantization of the Hall conductivity via the bulk-edge correspondence [27]. The latter is observed through transport measurements in solid-state experiments. In the cold-atom framework, such measurements are not convenient, as they would require atomic reservoirs coupled to the optical lattice. However, alternative methods, based on Bragg spectroscopy [28, 29], have been proposed to extract and image these topological edge states [30]. We will use the appearance of chiral edge states later in this paper to strengthen the identification of Chern insulators (section 3.2). They are obtained from the spectrum of Hamiltonian (5) in a finite geometry [27], as explained in appendix B.

In this section, we examine the effects of the Raman-induced phases in equation (3) from a less formal point of view, by associating effective 'fluxes' to these Peierls phases. First, one can evaluate the number of magnetic flux quanta penetrating each hexagonal plaquette , which yields (see figure 2(a))

Therefore, in the absence of NNN hopping (i.e. t_A,B = 0), the system has a trivial flux configuration Φ = 0 and remains invariant under time reversal.

Importantly, when NNN hopping terms are introduced (i.e. t_A,B ≠ 0), triangular sub-plaquettes are penetrated by non-zero magnetic fluxes, explicitly breaking time-reversal symmetry and potentially leading to quantum Hall phases [3]. Considering the sub-plaquettes formed by the A–B and A–A hoppings, illustrated in figure 2(a), one finds that

$$\begin{eqnarray} \Phi_1 & = & -\! {\boldsymbol{p}} \cdot {\boldsymbol{a}}_3/4 \pi=(p_2 - p_1)/2 , \nonumber \\ \Phi_2 & = & - \!{\boldsymbol{p}} \cdot {\boldsymbol{a}}_2/4 \pi=-p_2/2,\\ \Phi_3 & = & {\boldsymbol{p}} \cdot {\boldsymbol{a}}_1/4 \pi=p_1/2,\nonumber \end{eqnarray} \noindent \tag{ 27 }$$

where we expressed the recoil momentum p = p₁b₁ + p₂b₂ in terms of the basic reciprocal lattice vectors b_1,2, for which b_j·a_l = 2πδ_jl. The sub-plaquettes formed by the A–B and B–B hoppings have a similar flux structure. Thus, the space-dependent Peierls phases (3) produce a flux configuration characterized by three local fluxes Φ_1,2,3, and which is translationally invariant over the whole lattice (see figure 2(a)). We also note that $\sum _{\alpha }\Phi _{\alpha }=0$, which indicates that the total flux penetrating each hexagonal plaquette remains zero, as found above [3].

The system remains invariant under time reversal when H({Φ_1,2,3}) ≡ H(−{Φ_1,2,3}), where {Φ_1,2,3} represents the flux configuration stemming from a given p. Besides the obvious case p = 0, we find from equation (27), that this occurs:

• if p₁ and p₂ are both integers, i.e. if p is a vector of the reciprocal lattice;
• if one of the components p₁,p₂ or p₂ − p₁ is an even integer, and in particular, if p is collinear with one of the basis vectors b_1,2. For example, when p₁ = 0 (resp. p₂ = 0), one finds Φ₃ = 0 and Φ₁ = −Φ₂ (resp. Φ₂ = 0 and Φ₁ = −Φ₃).

In these pathological 'staggered flux' cases, the system remains invariant under time reversal and therefore topologically trivial (note that the number of magnetic flux quanta Φ_α is only defined modulo 1).

We can verify that these singular time-reversal configurations equally correspond to the condition

$$\begin{equation} d_z ({\boldsymbol{K}}_{+})=d_z ({\boldsymbol{K}}_{-}),\quad \forall t_{\mathrm{A}},\;t_{\mathrm{B}}. \end{equation} \noindent \tag{ 28 }$$

As established in equation (26), the condition (28) naturally leads to a trivial Chern insulator ν = 0 when Δ > 0, as expected for a time-reversal-invariant system exhibiting a gap. One can check that the condition (28) can be simply rewritten in terms of the vector p = p₁b₁ + p₂b₂,

$$\begin{equation} \sin(\pi p_2)-\sin(\pi p_1)+\sin\left[ \pi(p_2-p_1)\right]=0, \end{equation} \noindent \tag{ 29 }$$

whose solutions exactly reproduce the pathological cases listed above. When p satisfies the condition (29), the system is necessarily a trivial insulator or a metal depending on the other parameters. In section 3.1, we explore other values of the momentum recoil p, where trivial or non-trivial phases can be found depending on the specific values of the parameters t_A,t_B,ε.

We conclude this section by comparing the laser-coupled honeycomb lattice (5), with the original Haldane model (see [3] and figure 2(b)). In the latter, the hopping factor t₁ between NN sites of the honeycomb lattice is real, while NNN hoppings t₂ are multiplied by a constant phase factor e^±i2πϕ_H (the sign being determined by the orientation of the path). Thus, in the Haldane model, the three small triangular subplaquettes illustrated in figure 2(b) are all penetrated by the same flux Φ_1,2,3 = ϕ_H, whereas the large central triangular plaquette is penetrated by a flux −3ϕ_H. This leads to a staggered magnetic field configuration, with a vanishing total flux penetrating the hexagonal unit cell . We stress that time-reversal symmetry is necessarily broken in the Haldane model, for any finite value of the phase ϕ_H ≠ 0. This important difference between the two models highlights the richness of the laser-coupled honeycomb lattice (11)–(13), where the flux configuration and the nature of the spectral gaps strongly depend on the orientation of the vector p entering the Peierls phases.

We first investigate the effects of the Raman recoil momentum p. Here, the staggered potential is set to ε = 0. The phase diagrams shown in figure 3 illustrate the appearance of topological phases as a function of the Cartesian components p_x and p_y, for several values of the tunneling rates t_A,B. The areas corresponding to non-trivial topological phases, characterized by the Chern numbers ν = ± 1, are indicated by blue and red colors, respectively. Green areas correspond to the trivial insulating phase ν = 0, and white areas signify the 'undesired' metallic regime (Δ ≈ 0). The size of the bulk gap Δ is simultaneously shown through the color intensity. Panel (a) shows the isotropic case with equal NNN tunneling amplitudes set to t_B = t_A = 0.3t. Here, non-trivial topological phases (ν = ± 1) are generally separated by semi-metallic or metallic phases, and these topological regions depict triangular patterns. Panels (b) and (c) correspond to anisotropic cases where the hopping amplitude t_B is reduced to t_B = 0.2t in panel (b) and set to zero in panel (c). As the anisotropy increases, the metallic regions and trivial insulating phases progressively modify the non-trivial islands.

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In the special case t_B = 0, we find that all the regions that were non-trivial for t_B > 0 reduce to semi-metals: when t_B = 0, no topological insulating phase is found (contrary to what the skyrmion behavior of the vector d would suggest [16], see section 4). We stress that the semi-metallic behavior of the special case t_B = 0 is found for the entire parameter space (i.e. for all t_A, ε and p), and equally happens for the case t_A = 0 and t_B ≠ 0. This subtle effect is highlighted in figure 6(c), presented in section 3.2, where the band structure E = E(k_y) clearly shows the indirect gap closing for the case t_B = 0. This energy spectrum suggests that a small perturbation could open the bulk gap and lead to a Chern insulator. However, in section 3.3, we show that the staggered potential does not open such a non-trivial gap in the case t_B = 0. We therefore conclude that the condition t_A,B ≠ 0 should be satisfied to generate a robust Chern insulator.

The Hamiltonian is a periodic function of p, and the resulting periodicity of phases is conspicuous in the phase diagrams illustrated in figure 3. The central elementary lattice (the 'resized' FBZ) cell is marked by a black hexagon¹¹ in all panels of figure 3. We find that the most convenient non-trivial topological insulating phases (i.e. phases protected by the largest bulk gaps Δ ∼ 2t) are found for p∝(sin Nπ/3,cos Nπ/3), where N is an integer. Therefore, setting p_x = 0 potentially leads to topological phases with large bulk gaps, which is the most interesting situation for an experimental realization (see section 4).

We now explore how the topological phases evolve as the laser recoil momentum p and the tunneling amplitudes t_A,B are modified. As motivated above, we set p_x = 0, and then compute the phase diagrams in the p_y–t_A plane. First of all, we investigate the isotropic case t_A = t_B (the effects of anisotropy will be discussed in section 3.2). The phase diagram presented in figure 4 indicates that in the realistic situation where t_A ≈ t_B, the sizes of the topological gaps Δ are maximum for t_A ≈ t_B ≈ 0.3t, where Δ ≈ 2t for p_y ≈ 4K_y (see also figure 3(a)). Furthermore, this figure indicates that one should generally observe phase transitions between metallic and non-trivial topological phases as p_y is varied. Importantly, we note that the system remains metallic (Δ = 0) when the 'natural' hoppings t_A,B are larger than the Raman-induced hopping t, in particular when t_A ≈ t_B ≈ t. In the following, we show that an anisotropy t_A ≠ t_B, or the inclusion of a staggered potential ε ≠ 0, can turn this metallic phase into a topological one.

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In figure 5, we show the phase diagram in the plane p_y–t_B for a large and fixed value of the tunneling rate t_A = t. This important result shows that when t_A ≈ t, the anisotropy |t_A − t_B| ≠ 0 is necessary to open non-trivial topological gaps. This effect occurs for a relatively large range of the anisotropy, namely for t_B∈]0,t_A], and for specific values of the momentum p_y. For larger anisotropy |t_A − t_B| > t, the topological phases are destroyed and only metallic and trivial insulators survive. Specific phase transitions between semi-metallic and Chern insulating phases, indicated in figure 5 by three successive dots, are further illustrated through the edge-state analysis, in figure 6. The energy spectrum E = E(k_y) shown in this figure is obtained for a finite geometry, revealing the projected bulk bands E_±(k_x,k_y) → E_±(k_y) as well as the edge states dispersions (see [27] and appendix B). In panel 6(b), one indeed observes the presence of topological edge states within the bulk gap, which is the hallmark of a Chern insulator, i.e. through the bulk-edge correspondence [27]. Finally, we note the robustness of the topological edge states within the semi-metal regime Δ = 0, in figures 6(a), (c), a fact which is further analyzed in section 4.

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In this section, we explore the effect of the staggered potential. In figure 7, we show the phase diagram as a function of the staggered potential strength ε and of the recoil momentum component p_y, for several configurations of the tunneling amplitudes t_A,B (we set p_x = 0). First, we show the case t_A = t_B = t in figure 7(a). In this situation, large metallic regions and small non-trivial islands are found in the phase diagram, which can already be anticipated from figure 4 for ε = 0. Interestingly, in the totally symmetric case, where t_A = t_B = t, the topological phases vanish for ε = 0, and they are thus separated along the ε-axis¹² . These results indicate that the staggered potential is necessary to induce topological phases in this situation where t_A = t_B = t. However, for large values of the staggered potential, a trivial phase with ν = 0 is always privileged, in agreement with the general belief that such a staggered potential generically leads to trivial phases [3].

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For t_A = t_B < t, non-trivial Chern insulating phases can be formed both with and without the staggered potential. In figure 7(b), we illustrate the effects of the staggered potential for the optimized values of the tunneling rates t_A = t_B = 0.3t. Here, one observes two topological phases with ν = ± 1, which are separated by a small metallic region. This result highlights that one should generally observe phase transitions between semi-metallic and non-trivial topological phases as p_y is varied. On the other hand, varying the staggered potential to large values always privileges the transition to a trivial phase with ν = 0.

In the extreme case where t_B = 0, shown in figure 7(c), one finds the contour of the phase diagram presented in [16]. However, we stress that the two central regions featured in this diagram do not correspond to Chern insulating phases, as their corresponding bulk gap is closed. This indirect gap closing is further illustrated in figure 6(c). In the next section, we analyze this important point in more detail.

When a spectral gap is opened, Δ > 0, the winding number (30) is exactly equal to the Chern number (18). This potentially gives rise to a Chern insulator, as illustrated in figures 8(a) and (b), where we compare how the energy gap Δ, the winding number w and the ToF measurement w_ToF vary as a function of the recoil momentum p_y. As expected from the topological property of the Chern number, we find that the ranges where Δ > 0 and ν = w = ± 1 lead to the clear plateaus depicted by the observable w_ToF(p_y) ≈ ± 1 (see appendix C for a discussion on finite size effects). We also demonstrate the robustness of these plateaus in figure 9(a), where the quantity w_ToF is computed as a function of the Fermi energy E_F, and where a large plateau ∼ Δ is observed for fixed values of the other parameters. Therefore, the ToF winding number w_ToF shows a robust behavior, and exhibits a clear plateau when the Fermi energy lies in the band gap. In other words, the Chern insulating phase is characterized by a 'quantized' winding number w_ToF, which is protected against finite changes of the parameters through the existence of a topological bulk gap¹⁵.

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Let us stress the important fact that the Chern number ν in equation (18) no longer reflects the quantized Hall conductivity when the bulk gap is closed, in which case the system effectively describes a metallic phase. However, the topological order and skyrmion patterns associated with the winding number w survive even when the bulk gap is closed, as we now explore in the next subsection.

First of all, we find that the ToF winding number w_ToF, given by equation (33), can be robust even when the gap is closed. This effect is illustrated in figures 8(c) and (d), where we compare how the energy gap Δ, the winding number w and the ToF winding number w_ToF vary as a function of the recoil momentum p_y. From figures 8(c) and (d), we find that the winding number w displays non-trivial plateaus w = ± 1, in regions where the bulk energy gap is closed Δ = 0. In figure 8(d), we precisely set the Fermi energy at the gap closing point E_F = −0.25t, which can be determined from the spectra in figures 10(b)–(e), thus completely filling the lowest bulk band. In this specific configuration, the observable winding number w_ToF(p_y) depicts plateaus, and it converges toward the quantized value w_ToF → w as the resolution of the grid is increased (see appendix C). However, as the Fermi energy is tuned away from this ideal value, we find that the plateaus w_ToF(p_y) ∼ ± 1 progressively lose their robustness. This dramatic effect is illustrated in figure 9(b), where w_ToF is computed as a function of the Fermi energy. Here, in contrast with the Chern insulator case shown in figure 9(a), the winding number w_ToF is strongly parameter-dependent: it only reaches w_ToF ≈ w = + 1 at the specific Fermi energy E_F = −0.25t (see figure 9(b)). Consequently, the robust behavior of these topological semi-metals will only be observed if the Fermi energy is precisely tuned at the gap closing point (such as in figure 8(d)). This important fact makes topological semi metals more challenging to detect than Chern insulating phases. We point out that the phase diagrams and skyrmion configurations presented in [16], and reproduced in figures 8(c) and (d), only feature trivial insulators and 'topological semi-metals', since all the computations were performed for the peculiar configuration t_B = 0, whose corresponding spectrum remains gapless in the 'non-trivial' regions (see also figure 10).

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The topological semi-metal is an intriguing phase, and in this new context, an important question arises: if the topological winding number w remains stable for Δ = 0 (see figure 8(d)), under which conditions does the value of this quantity change? We address this question by analyzing the energy spectrum E(k_y) together with the skyrmion pattern depicted by the vector n(k) in figure 10. Here, the parameters are the same as in figure 8(d) and p_y is varied between K_y and 8K_y, where transitions between $w=0 \leftrightarrow w=+1$, but also between $w=+1 \leftrightarrow w=-1$, are expected. From figure 10, we find that the winding number w, and its corresponding skyrmion pattern, remain extremely stable as long as the energy bands do not touch at the Dirac points. When a direct gap closing occurs at the Dirac point K_D, where K_D denotes K₊ and/or K₋, we observe a topological phase transition signaled by a change in the winding number w (see figures 10(b) and (d)). In particular, we find that this direct gap closing is accompanied with the cancellation d_z(K_D) = 0 at the band touching point K_D. In fact, the gap-closing condition d_z(K_D) = 0 should necessarily be satisfied at the transition between different values of the winding number w, as can be deduced from the equivalence (32) and from the simple expression (26).

The topological phase transitions are clearly visible in the skyrmion patterns of figure 10, where a non-trivial winding number w = ± 1 emerges when the vector n(k) has covered the whole Bloch sphere once. We remind the reader that this full coverage of the Bloch sphere is achieved when the vector field n(k) reaches the north (n = + 1_z) and south (n = −1_z) poles at the two inequivalent Dirac points K_±. In figure 10, we observe a radical change in the behavior of n(K_±) as p_y is varied. For example, for p_y = K_y (figure 10(a)), the vector field n(k) visits the north pole twice (i.e. at K₊ and K₋) but never the south pole (w = 0), while for p_y = 4K_y (figures 10(c)) the vector visits the entire Bloch sphere once (w = 1). Between these two topologically different configurations, the energy bands E_± touch at k = K₋ for p_y = 2K_y (see figure 10(b)), a singular situation where the gapless phase is generally equivalent to a standard semi-metal [19, 20]. We note that transitions $w=0 \leftrightarrow w =\pm 1$ require a single band-touching point (see figure 10(b)), while transitions $w=+1 \leftrightarrow w =- 1$ involve two band-touching points (see figure 10(d)). Therefore, in agreement with the equivalence (32), we observe that the topological phase transitions between different topological semi metals are of the same nature as the transitions between different Chern insulators, in the sense that both phenomena occur through direct gap closing (driven here by the control parameter p_y).

In summary, we conclude that the laser-coupled honeycomb lattice and the ToF method of [16] offers the possibility to explore the topological order of topological semi-metals, which survive in the absence of a band gap. However, we remind the reader that this detection scheme relies on the evaluation of the winding number w_ToF, through a ToF measurement of the vector field n(k), which only converges toward the quantized value w for a complete filling of the lowest energy band E₋(k). Thus, the experimental detection of topological semi-metals would constitute a subtle task, in the sense that the Fermi energy should be finely tuned in order to maximize the filling of the lowest band (figure 8(d)). Let us stress that the winding number can only take three possible values, w = 0,±1. Therefore, an experimental plateau w_ToF(p_y) ∼ ± 1, stemming from a slightly incomplete filling of the band E₋(k) and from finite size effects, would already provide an acceptable witness of non-trivial topological order.

We end this section by observing that the transitions between topologically different semi-metals are driven by the laser recoil momentum p, and therefore, this interesting effect cannot be captured by the original Haldane model.

When the Berry's connection is regular over the entire FBZ, the Berry's curvature $\mathcal {F}={\mathrm { d}} \mathcal {A}$ is an exact differential form. In this case, the Chern number (A.1) is given by the integral over a closed manifold (i.e. the two-torus $\mathbb {T}^2$) of an exact differential form

$$\begin{equation} \nu= \frac{1}{2 \pi} \int_{\mathbb{T}^2} \mathcal{F} = \frac{1}{2 \pi} \int_{\mathbb{T}^2} {\rm d} \mathcal{A} = 0 , \end{equation} \tag{ A.5 }$$

which is trivial from Stokes theorem. In particular, when d_z(K_±) > 0 at the two inequivalent Dirac points, the Berry's connection (A.4) remains regular over the entire FBZ and the Chern number necessarily vanishes.

Now suppose that the Berry's connection $\mathcal {A}$ in equation (A.4) is singular at one Dirac point, say K_D = K₊. This situation occurs when d_z(K₊) < 0 and d_z(K₋) > 0, which we now consider to be the case in this paragraph. In the presence of such a singularity, the Berry's curvature $\mathcal {F}$ is no longer an exact differential form, and Stokes theorem cannot be applied globally over the torus. In order to compute the integral (A.1), we partition the FBZ into two complementary regions, R_I and R_II, whose common boundary ∂R is chosen to be a loop γ₊ encircling K₊. Here, we define the region R_II as the one that contains the Dirac point K₊ at which the singularity takes place (see figure A.1(b)). Then, we define specific gauges within each region [2, 26, 27],

$$\begin{equation} \vert u_{(-)} \rangle_{\rm I} =\left(\begin{array}{@{}cc@{}} -\mathrm{e}^{-\mathrm{i} \phi} \sin (\theta /2) \\ \cos (\theta /2)\end{array}\right), \quad \vert u_{(-)} \rangle_{\rm II} =\left(\begin{array}{@{}ll@{}} - \sin (\theta /2) \\ \mathrm{e}^{\mathrm{i} \phi} \cos (\theta /2) \end{array}\right). \end{equation} \noindent \tag{ A.6 }$$

In this patchwork configuration, the Berry's connection $\mathcal {A}=\{ \mathcal {A}_{\mathrm { I}} , \mathcal {A}_{\mathrm {II}} \}$ is a locally-defined quantity, which is now given by

$$\begin{equation} \boldsymbol{A}_{\rm I} ({\boldsymbol{k}})=\frac{1}{2} (1- \cos \theta) {\boldsymbol{\nabla}}_{{\boldsymbol{k}}} \phi , \end{equation} \tag{ A.7 }$$

$$\begin{equation} \boldsymbol{A}_{\rm II} ({\boldsymbol{k}})= - \frac{1}{2} (1 + \cos \theta) {\boldsymbol{\nabla}}_{{\boldsymbol{k}}} \phi, \end{equation} \tag{ A.8 }$$

inside the regions R_I and R_II, respectively. We note that the gauge structures of the two individual regions are connected at the frontier ∂R = γ₊ through the gauge transformation

$$\begin{equation} \vert u_{(-)} \rangle_{\rm II}= \mathrm{e}^{\mathrm{i} \phi({\boldsymbol{k}})} \vert u_{(-)} \rangle_{\rm I}, \end{equation} \tag{ A.9 }$$

$$\begin{equation} \boldsymbol{A}_{\rm II} ({\boldsymbol{k}})={\boldsymbol{A}}_{\rm I} ({\boldsymbol{k}}) - {\boldsymbol{\nabla}}_{{\boldsymbol{k}}} \phi. \end{equation} \tag{ A.10 }$$

Furthermore, we note that the Berry's connection A_II(k) is now regular at the Dirac point K₊, where d_z(K₊) < 0. Therefore, the locally-defined Berry's connection $\mathcal {A}=\{ \mathcal {A}_{\mathrm { I}} , \mathcal {A}_{\mathrm {II}} \}$ is regular over the entire FBZ, and the integral (A.1) can now be computed by applying Stokes theorem to the two different regions [2, 27]

$$\begin{eqnarray} \nu&=&\frac{1}{2 \pi} \int_{\mathbb{T}^2} \mathcal{F} = \frac{1}{2 \pi} \int_{R_{\rm I}} {\rm d} \mathcal{A}_{\rm I} + \frac{1}{2 \pi} \int_{R_{\rm II}} {\rm d} \mathcal{A}_{\rm II} \nonumber \\ &=&\frac{1}{2 \pi} \oint_{\partial R}( \boldsymbol{A}_{\rm II} - \boldsymbol{A}_{\rm I}) \cdot {\rm d} \boldsymbol{k} \nonumber \\ &=& - \frac{1}{2 \pi} \oint_{\gamma_{+}} \boldsymbol{\nabla}_{{\boldsymbol{k}}} \phi ({\boldsymbol{k}}) \cdot {\rm d} {\boldsymbol{k}}= - v_{+} /2 \pi= - 1. \end{eqnarray}\noindent \tag{ A.11 }$$

Therefore, when the singularity only takes place at the Dirac point K₊, the Chern number is non-trivial and its value is directly related to the vorticity v₊ associated with this Dirac point [27].

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In the opposite situation, where the singularity only takes place at the other Dirac point K_D = K₋, namely when d_z(K₊) > 0 and d_z(K₋) < 0, a similar calculation (with ∂R = γ₋) yields

$$\begin{equation} \nu = - v_- /2\pi = + 1. \end{equation} \tag{ A.12 }$$

When the Berry's connection (A.4) is singular at both Dirac points, namely when d_z(K_±) < 0, the Chern number (A.1) is necessarily trivial. Indeed, the gauge transformation

$$\begin{equation} \vert u_{(-)} \rangle \rightarrow \vert \tilde{u}_{(-)} \rangle = \mathrm{e}^{\mathrm{i} \phi({\boldsymbol{k}})} \vert u_{(-)} \rangle \end{equation} \noindent \tag{ A.13 }$$

simultaneously removes the singularities at both Dirac points. In this case, Stokes theorem can be applied globally over the entire FBZ, leading to a zero Chern number (see the case with no singularity).

From the results presented above, we conclude that the Chern number ν characterizing the topological order of our system can only take non-trivial values ν = ± 1 when the Berry's connection (A.4) features a unique singularity inside the FBZ. Therefore, this non-trivial regime is reached when the function d_z(k) has opposite signs at the two inequivalent Dirac points K_±. Then, from equations (A.11) and (A.12), we finally obtain the result

$$\begin{equation} \nu=\frac{1}{2} ( {\rm sign} (d_z ({\boldsymbol{K}}_{+}) ) - {\rm sign} (d_z ({\boldsymbol{K}}_{-})) ), \end{equation} \noindent \tag{ A.14 }$$

already announced in equation (26).