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Instances of one-dimensional lattices with FB. (a) Double-comb lattice (the frequency of cavity cn is zero, while the one of cavities an and bn is ωc). (b) Field spectrum of the double-comb lattice showing a FB at energy ω = ωc. (c) Stub lattice (the frequency of all cavities is set to zero). (d) Field spectrum of the stub lattice, showing a zero-energy FB. In (a) and (c) we highlight the primitive unit cell (red dashed line) and possible choices of CLSs. Notice that CLSs are not overlapping in the double-comb lattice in panel (a), but they do overlap in the stub lattice (c).
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Flat bands (FBs) are energy bands with zero group velocity, which in electronic systems were shown to favor strongly correlated phenomena. Indeed, a FB can be spanned with a basis of strictly localized states, the so called "compact localized states" (CLSs), which are yet generally non-orthogonal. Here, we study emergent dipole-dipole interactions...
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... cells, a CLS |ϕ n ⟩ is generally spread over U ≥ 1 cells, where the positive integer U is the so called class of the CLS set [61]. In the special case U = 1, each |ϕ n ⟩ is entirely localized within the nth cell so that the CLSs do not overlap in space and are thus orthogonal, i.e. ⟨ϕ n |ϕ n ′ ⟩ = δ nn ′ (an instance is the double-comb lattice of Fig. 2). For U ≥ 2, instead, CLSs necessarily overlap in space with one another, which remarkably causes them to be generally non-orthogonal. It is easy to convince oneself that for a 1D (generally multipartite) lattice CLSs of class U = 2 are such that each CLS overlaps only its two nearest-neighbour CLSs, being orthogonal to the all the ...
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... order to make the reader familiarize with CLSs and their properties (especially non-orthogonality), we present next some examples of lattices exhibiting FBs [see Fig. 2 and Fig. 3]. In this section, we will only discuss 1D models, meaning that here the cell index n is an integer and the wavevector k a real number (see Appendix A for more details on those ...
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... double-comb lattice is a tripartite lattice, which comprises three sublattices a, b and c [see Fig. 2(a)]. Each cavity c n of the central sublattice is coupled to the nearest-neighbour cavities c n±1 with photon hopping rate J and with rate t to cavities a n and b n (upper and lower sublattices, respectively). Cavities a n and b n have the same bare frequency ω c , while the one of c n is set to zero. The spectrum of B, which is plotted ...
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... n of the central sublattice is coupled to the nearest-neighbour cavities c n±1 with photon hopping rate J and with rate t to cavities a n and b n (upper and lower sublattices, respectively). Cavities a n and b n have the same bare frequency ω c , while the one of c n is set to zero. The spectrum of B, which is plotted in a representative case in Fig. 2(b), features a FB at energy ω FB = ω c and two dispersive bands. The occurrence of such FB is easy to predict since the antisymmetric ...
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... decouples from state |c n ⟩ (hence the rest of the lattice) due to destructive interference and is thereby an eigenstate of the bath Hamiltonian H B with energy ω c . Evidently, there exists one such state for each cell n, explaining the origin of the FB at energy ω FB = ω c . As each |ϕ n ⟩ is strictly localized within a unit cell [see Fig. 2(a)], states {|ϕ n ⟩} form a set of orthogonal CLSs of class U = 1 based on the previous definition. In (a) and (c) we highlight the primitive unit cell (red dashed line) and possible choices of CLSs. Notice that CLSs are not overlapping in the double-comb lattice in panel (a), but they do overlap in the stub lattice ...
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... stub lattice (or 1D Lieb lattice) [56,62,63] is the tripartite lattice sketched in Fig. 2(c), where each cavity b n is coupled to cavities c n±1 with rate J and side-coupled to cavity a n with rate J √ ∆ where ∆ ≥ 0 is a dimensionless parameter. The spectrum of H B is symmetric around ω = 0, at which energy a FB arises (ω FB = 0). The gap separating the FB from each dispersive band is proportional to J √ ∆ [see Fig. 2(c)]. ...
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... sketched in Fig. 2(c), where each cavity b n is coupled to cavities c n±1 with rate J and side-coupled to cavity a n with rate J √ ∆ where ∆ ≥ 0 is a dimensionless parameter. The spectrum of H B is symmetric around ω = 0, at which energy a FB arises (ω FB = 0). The gap separating the FB from each dispersive band is proportional to J √ ∆ [see Fig. 2(c)]. Similarly to the previous lattices and CLSs, the origin of the zero-energy FB can be understood by noting that ...
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... isolated FB, i.e. separated by a finite gap from all the remaining bands [recall Eq. (24) where only the contribution of the FB is retained]. This rules out in particular those lattices where a FB arises on the edge of a dispersive band. In 1D, such band touching happens for example in the stub lattice for ∆ = 0 (see Section IV B 3 and Fig. 2) and in the Kagomè model (see Section A 4 and Fig. 3). Indeed, it turns out that in either case, an atom dispersively coupled to the FB seeds a BS with features analogous to typical BSs close to the band edge of an isolated dispersive band [e.g. as in Fig. 1 (c)-(e)]. This is witnessed by Fig. 3(c) for the Kagomè model, which shows ...
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... interesting consequence of this is that for nearestneighbour CLSs [e.g. in the sawtooth model of Fig. 1(a) or the stub lattice of Fig. 2(c)] an effective spin Hamiltonian arises with strictly nearest-neighbour ...
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... the three-partite stub lattice [Hamiltonian parameters and sublattice indexes are defined in Fig. 2(c)], the Bloch Hamiltonian can be cast in the ...
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... cells, a CLS |ϕ n ⟩ is generally spread over U ≥ 1 cells, where the positive integer U is the so called class of the CLS set [68]. In the special case U = 1, each |ϕ n ⟩ is entirely localized within the nth cell so that the CLSs do not overlap in space and are thus orthogonal, i.e. ⟨ϕ n |ϕ n ′ ⟩ = δ nn ′ (an instance is the double-comb lattice of Fig. 2). For U ≥ 2, instead, CLSs necessarily overlap in space with one another, which remarkably causes them to be generally non-orthogonal. It is easy to convince oneself that for a 1D (generally multipartite) lattice CLSs of class U = 2 are such that each CLS overlaps only its two nearest-neighbour CLSs, being orthogonal to the all the ...
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... order to make the reader familiarize with CLSs and their properties (especially non-orthogonality), we present next some examples of lattices exhibiting FBs [see Fig. 2 and Fig. 3]. In this section, we will only discuss 1D models, meaning that here the cell index n is an integer and the wavevector k a real number (see Appendix A for more details on those ...
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... double-comb lattice is a tripartite lattice, which comprises three sublattices a, b and c [see Fig. 2(a)]. Each cavity c n of the central sublattice is coupled to the nearest-neighbour cavities c n±1 with photon hopping rate J and with rate t to cavities a n and b n (upper and lower sublattices, respectively). Cavities a n and b n have the same bare frequency ω c , while the one of c n is set to zero. The spectrum of B, which is plotted ...
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... n of the central sublattice is coupled to the nearest-neighbour cavities c n±1 with photon hopping rate J and with rate t to cavities a n and b n (upper and lower sublattices, respectively). Cavities a n and b n have the same bare frequency ω c , while the one of c n is set to zero. The spectrum of B, which is plotted in a representative case in Fig. 2(b), features a FB at energy ω FB = ω c and two dispersive bands. The occurrence of such FB is easy to predict since the antisymmetric ...
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... decouples from state |c n ⟩ (hence the rest of the lattice) due to destructive interference and is thereby an eigenstate of the bath Hamiltonian H B with energy ω c . Evidently, there exists one such state for each cell n, explaining the origin of the FB at energy ω FB = ω c . As each |ϕ n ⟩ is strictly localized within a unit cell [see Fig. 2(a)], states {|ϕ n ⟩} form a set of orthogonal CLSs of class U = 1 based on the previous definition. In (a) and (c) we highlight the primitive unit cell (red dashed line) and possible choices of CLSs. Notice that CLSs are not overlapping in the double-comb lattice in panel (a), but they do overlap in the stub lattice ...
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... stub lattice (or 1D Lieb lattice) [63,69,70] is the tripartite lattice sketched in Fig. 2(c), where each cavity b n is coupled to cavities c n±1 with rate J and side-coupled to cavity a n with rate J √ ∆ where ∆ ≥ 0 is a dimensionless parameter. The spectrum of H B is symmetric around ω = 0, at which energy a FB arises (ω FB = 0). The gap separating the FB from each dispersive band is proportional to J √ ∆ [see Fig. 2(c)]. ...
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... sketched in Fig. 2(c), where each cavity b n is coupled to cavities c n±1 with rate J and side-coupled to cavity a n with rate J √ ∆ where ∆ ≥ 0 is a dimensionless parameter. The spectrum of H B is symmetric around ω = 0, at which energy a FB arises (ω FB = 0). The gap separating the FB from each dispersive band is proportional to J √ ∆ [see Fig. 2(c)]. Similarly to the previous lattices and CLSs, the origin of the zero-energy FB can be understood by noting that ...
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... isolated FB, i.e. separated by a finite gap from all the remaining bands [recall Eq. (24) where only the contribution of the FB is retained]. This rules out in particular those lattices where a FB arises on the edge of a dispersive band. In 1D, such band touching happens for example in the stub lattice for ∆ = 0 (see Section IV B 3 and Fig. 2) and in the Kagomè model (see Section A 4 and Fig. 3). Indeed, it turns out that in either case, an atom dispersively coupled to the FB seeds a BS with features analogous to typical BSs close to the band edge of an isolated dispersive band [e.g. as in Fig. 1 (c)-(e)]. This is witnessed by Fig. 3(c) for the Kagomè model, which shows ...
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... interesting consequence of this is that for nearestneighbour CLSs [e.g. in the sawtooth model of Fig. 1(a) or the stub lattice of Fig. 2(c)] an effective spin Hamiltonian arises with strictly nearest-neighbour ...
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Flat bands (FBs) are energy bands with zero group velocity, which in electronic systems were shown to favor strongly correlated phenomena. Indeed, a FB can be spanned with a basis of strictly localized states, the so called compact localized states (CLSs), which are yet generally non-orthogonal. Here, we study emergent dipole-dipole interactions be...
