Compact discrete breathers on flat-band networks

Linear wave equations on flat-band networks host compact localized eigenstates (CLS). Nonlinear wave equations on translationally invariant flat-band networks can host compact discrete breathers-time-periodic and spatially compact localized solutions. Such solutions can appear as one-parameter famil...

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Дата:	2018
Автори:	Danieli, C., Maluckov, A., Flach, S.
Формат:	Стаття
Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2018
Назва видання:	Физика низких температур
Теми:	Динамика нелинейных упругих сред
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/176198
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Цитувати:	Compact discrete breathers on flat-band networks / C. Danieli, A. Maluckov, S. Flach // Физика низких температур. — 2018. — Т. 44, № 7. — С. 865-876. — Бібліогр.: 43 назв. — англ.

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spelling	irk-123456789-1761982021-02-05T01:30:24Z Compact discrete breathers on flat-band networks Danieli, C. Maluckov, A. Flach, S. Динамика нелинейных упругих сред Linear wave equations on flat-band networks host compact localized eigenstates (CLS). Nonlinear wave equations on translationally invariant flat-band networks can host compact discrete breathers-time-periodic and spatially compact localized solutions. Such solutions can appear as one-parameter families of continued linear compact eigenstates, or as discrete sets on families of non-compact discrete breathers, or even on purely dispersive networks with fine-tuned nonlinear dispersion. In all cases, their existence relies on destructive interference. We use CLS amplitude distribution properties and orthogonality conditions to derive existence criteria and stability properties for compact discrete breathers as continued CLS. 2018 Article Compact discrete breathers on flat-band networks / C. Danieli, A. Maluckov, S. Flach // Физика низких температур. — 2018. — Т. 44, № 7. — С. 865-876. — Бібліогр.: 43 назв. — англ. 0132-6414 PACS: 71.10.–w, 71.10.Fd http://dspace.nbuv.gov.ua/handle/123456789/176198 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Динамика нелинейных упругих сред Динамика нелинейных упругих сред
spellingShingle	Динамика нелинейных упругих сред Динамика нелинейных упругих сред Danieli, C. Maluckov, A. Flach, S. Compact discrete breathers on flat-band networks Физика низких температур
description	Linear wave equations on flat-band networks host compact localized eigenstates (CLS). Nonlinear wave equations on translationally invariant flat-band networks can host compact discrete breathers-time-periodic and spatially compact localized solutions. Such solutions can appear as one-parameter families of continued linear compact eigenstates, or as discrete sets on families of non-compact discrete breathers, or even on purely dispersive networks with fine-tuned nonlinear dispersion. In all cases, their existence relies on destructive interference. We use CLS amplitude distribution properties and orthogonality conditions to derive existence criteria and stability properties for compact discrete breathers as continued CLS.
format	Article
author	Danieli, C. Maluckov, A. Flach, S.
author_facet	Danieli, C. Maluckov, A. Flach, S.
author_sort	Danieli, C.
title	Compact discrete breathers on flat-band networks
title_short	Compact discrete breathers on flat-band networks
title_full	Compact discrete breathers on flat-band networks
title_fullStr	Compact discrete breathers on flat-band networks
title_full_unstemmed	Compact discrete breathers on flat-band networks
title_sort	compact discrete breathers on flat-band networks
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2018
topic_facet	Динамика нелинейных упругих сред
url	http://dspace.nbuv.gov.ua/handle/123456789/176198
citation_txt	Compact discrete breathers on flat-band networks / C. Danieli, A. Maluckov, S. Flach // Физика низких температур. — 2018. — Т. 44, № 7. — С. 865-876. — Бібліогр.: 43 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT danielic compactdiscretebreathersonflatbandnetworks AT maluckova compactdiscretebreathersonflatbandnetworks AT flachs compactdiscretebreathersonflatbandnetworks
first_indexed	2025-07-15T13:52:34Z
last_indexed	2025-07-15T13:52:34Z
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fulltext	Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7, pp. 865–876 Compact discrete breathers on flat-band networks C. Danieli1, A. Maluckov1,2, and S. Flach1,3 1Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon, Korea 2Vinca Institute for Nuclear Sciences, University of Belgrade, Serbia 3New Zealand Institute for Advanced Study, Massey University, Auckland, New Zealand E-mail: sergejflach@googlemail.com Received March 1, 2018, published online May 28, 2018 Linear wave equations on flat-band networks host compact localized eigenstates (CLS). Nonlinear wave equations on translationally invariant flat-band networks can host compact discrete breathers-time-periodic and spatially compact localized solutions. Such solutions can appear as one-parameter families of continued linear compact eigenstates, or as discrete sets on families of non-compact discrete breathers, or even on purely disper- sive networks with fine-tuned nonlinear dispersion. In all cases, their existence relies on destructive interference. We use CLS amplitude distribution properties and orthogonality conditions to derive existence criteria and sta- bility properties for compact discrete breathers as continued CLS. PACS: 71.10.–w Theories and models of many-electron systems; 71.10.Fd Lattice fermion models (Hubbard model, etc.). Keywords: compact localized eigenstates, discrete breathers, flat band. Introduction In recent years, flat-band tight binding networks gained interest in the fields of ultra cold atomic gases, condensed matter and photonics, among others [1]. One of the essen- tial features of the corresponding eigenvalue problem of these linear wave equations is the presence of eigenstates which are strictly compact in space. These modes are coined compact localized states (CLS), and their existence is due to destructive interference which suppresses the dis- persion along the network. The CLS introduce macroscop- ic degeneracy in the energy spectrum of the network, which results in one (or more) momentum-independent (or dispersionless) bands in the spectrum, hence called flat bands. The CLS can be found irrespective to the dimension- ality of the network. CLSs can be classified according to the number U of unit cells they occupy. Class = 1U CLSs form an orthogonal basis of the flat-band Hilbert space, since the compact states do not overlap. Moreover, the flat band can be freely tuned to be gapped away from dispersive bands, or to resonate with them. Class 2U ≥ CLSs instead typically form a non-orthogonal basis, and the flat band is gapped away (or at most touching) from dispersive bands. Introduced by Sutherland [2] and Lieb [3] in the 1980's, and then generalized by Mielke and Tasaki in the 1990's [4,5], flat-band lattices and their perturbations provide an ideal test-bed to explore and study unconventional localization and innovative states of matter [6–8]. The effects of differ- ent types of perturbations have been studied in several ex- amples of flat-band networks [9,10], as well as the effects of disorder and nonlinearity and interaction between them [11]. Further studies focused on non-Hermitian flat-band net- works [12], topological flat Wannier–Stark bands [13], Bloch oscillations [14], Fano resonances [15], fractional charge transport [16] and the existence of nontrivial super- fluid weights [17]. Chiral flat-band networks revealed that CLS and their macroscopic degeneracy can be protected under any perturbation which does not lift the bi- partiteness of the network [18]. The engineering of CLS has been longly attempted [19,20], and it has been recently solved for = 1U lattices [21] and for the = 2U CLS in a two-band problem [22]. Experimentally, compact localized states have been realized using ultra cold atoms [23], pho- tonic waveguides networks [24–26], exciton-polariton condensates [27,28] and superconducting wires [29,30] (for a recent survey on the state of the art, see [1]. © C. Danieli, A. Maluckov, and S. Flach, 2018 C. Danieli, A. Maluckov, and S. Flach Nonlinear translationally invariant lattices admit a class of time-periodic solutions localized in real space (typically exponentially), called discrete breathers [31,32]. The pre- cise decay in the tails depends on the band structure of small amplitude linearized wave equations. For analytic band structures (usually due to short range, e.g., exponen- tially or faster decaying, connectivities on the lattice), the discrete breather tails decay exponentially. For non- analytic band structures (usually due to long range, e.g., algebraically decaying, connectivities on the lattice), the tails decay algebraically as well. In the absence of linear dispersion, but presence of nonlinear dispersion, tails de- cay superexponentially. For short range connectivities, but with acoustic parts in the band structure, and with broken space parity, the ac parts of the discrete breather tails decay exponentially, while the dc part (static lattice deformation) will decay algebraically [31,32]. A natural question then arises whether discrete breath- ers can have strictly zero tails, and turn into compact exci- tations. For instance, traveling solitary waves with compact support have been found in the frame of spatially continu- ous partial differential equations by Rosenau and Hyman in the Korteweg–de Vries model [33]. In discrete systems, spatially compact time-periodic solutions have been found by Page in a purely anharmonic one-dimensional Fermi– Pasta–Ulam-like chain in the limit of non-analytic com- pact (box) interaction potential [34]. Moreover, Kevrekidis and Konotop reported on compact solutions in translationally invariant one-dimensional lattices in the presence of non-local nonlinear terms [35]. In this work, we consider flat-band networks as the underlying support for compact time-periodic excitations. The existence of compact discrete breathers in nonline- ar flat-band networks was observed in [36,37]. Further- more, the coexistence between nonlinear terms and spin- orbit coupling has been discussed in the framework of ul- tra-cold atoms in a diamond chain [38]. Perchikov and Gendelman studied compact time-periodic solutions in a one-dimensional nonlinear mechanical cross-stitch net- work [39]. In this case the above mentioned destructive interference translates into several time-dependent forces acting on masses in the mechanical network in such a way that the sum of all forces vanishes, leading to a compactification of the vibrational excitation. In this work, we present a necessary and sufficient con- dition for the existence and continuation of time-periodic and compact in space solutions (herewith called compact discrete breather) on flat-band networks with local nonlin- earity. The existence and continuation condition applies irrespective of the dimensionality of the lattice and the class U of linear CLS. Then, we discuss the linear stabil- ity of compact discrete breathers. For orthogonal CLSs in = 1U networks, the only source of instability are reso- nances with extended states. For class 2U ≥ networks instead, the non-orthogonality between linear CLSs induc- es additional potential local instabilities due to CLS-CLS interaction. Resonances with dispersive states lead to ra- diation and potential complete annihilation. Resonances with neighboring CLSs in general simply yield local insta- bilities which do not annihilate the excitation. The study of the nonlinear stability has been performed numerically, and standard techniques of perturbation theory have been applied to substantiate the numerical findings. The present work is structured as follows: in Sec. 1 we will present the flat-band networks; then in Sec. 2 we introduce the nonlin- ear terms in flat-band model equations, and discuss the continuation criteria of linear CLS to compact discrete breathers. Next, in Sec. 3 we present the linear stability analysis of the compact discrete breathers, which will then be discussed numerically in Sec. 4. 1. Flat-band networks For simplicity we will operate in one spatial dimension. Results in general take over to higher dimensions. We will comment on particular cases where caution is to be execut- ed. The linear time-dependent model equation of the flat- band networks can be presented in a form † 0 1 1 11=n n n ni H H H+ +ψ ψ + ψ + ψ . (1) For all n∈ , n νψ ∈ is a time-dependent complex vector of ν components, each one representing one site of the network. The set of ν sites is called the unit-cell. The ma- trix 0H defines the geometry of the unit-cell, while the matrixes † 1 1,H H define the hopping between nearest- neighboring ones. This model equation can be easily gen- eralized to longer range hopping, as well as higher dimen- sional networks. The phase-amplitude ansatz ( ) =n tψ e iEt nA −= leads to the associated eigenvalue problem † 0 1 1 11= .n n n nEA H A H A H A+ ++ + (2) Then, the Bloch solution = eiqn n qA ϕ of Eq. (2) defined for the wave-vector q gives rise to the Bloch Hamiltonian of the lattices † 0 1 1= ( ) [ e e ]iq iq q q qE H q H H H−ϕ ϕ ≡ + + ϕ . (3) Equation (3) yields the band structure =1= ( )i iE E qν∪ of the problem. We consider lattices which exhibit at least one band independent from the wave vector q, which we call disperionless (or flat) band FBE . The eigenmodes as- sociated to a flat-band are typically compact localized states, and the number U of unit-cells occupied by one CLS is the flat band class. These states can be written in the time-dependent form solutions of Eq. (1): 0 1 , ,0 =0 ( ) = e ,tFB U iE n n l n n l l t − − +   ψ δ     ∑v (4) where the sum indicates the spatial component of the CLSs. The real vectors lv are defined as the following: 866 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 Compact discrete breathers on flat-band networks , , =1 =l l j l j j j a A ν ∑ ev , (5) where the vectors =1{ }j j νe form the canonical basis of 1= , ,ν ν〈 〉e e ; , {0, 1}l ja ∈ ± denotes the sites with non- zero amplitude, and the real numbers ,l jA ∈ defines the amplitudes in the sites with non-zero ,i ja . In the next sec- tion, we will introduce local nonlinear terms to Eq. (1), and we will discuss continuation criteria for the CLS introduced in Eq. (4) as compact solution of the nonlinear regime. 2. Nonlinear flat-band networks and continuation of compact localized states Let us consider the model equation of the flat-band network Eq. (1) in presence of local nonlinear terms † 0 1 1 11= ( ) ,n n n n n ni H H H+ +ψ ψ + ψ + ψ + γ ψ ψ (6) where the matrix ( )nψ 2 =1 ( ) \| \|j n n j j j ν ψ ≡ ψ ⊗∑ e e (7) contains the terms 2\| \|i nψ along the diagonal. We seek for time-periodic solutions of the nonlinear system Eq. (6) 1 , ,0 0 =0 ( ) = e U i t n n l n n l l C t − − Ω +   δ     ∑v (8) with frequency Ω , which are continuation of the CLSs Eqs. (4), (5) that exist in the linear regime = 0γ . We consider the compact solution (8) defined with the profile in space of the linear CLS in Eqs. (4), (5) and fre- quency Ω , and check under which conditions these are solutions of the nonlinear equation (6). At first, let us ob- serve that for all sites where a CLS is zero ( , = 0l ja and outside the range of U cells in Eq. (4)), Eq. (6) is solved. For = 1, ,l U and = 1, ,j ν where a CLS has non-zero amplitude , 0l ja ≠ , Eq. (6) reduces to 3 , , ,= .l j l j l jFB A E A AΩ + γ (9) If for all ,l j such that , 0l ja ≠ , ,l jA A≡ (all sites have same amplitude in absolute value), Eq. (9) turns into 2 = .FBEA Ω− γ (10) If instead there exist non-zero ˆ, ˆ,l j l jA A≠ , Eq. (9) yields different frequencies Ω , which breaks the condition of continuation of CLS as a periodic orbit with compact sup- port. Let us introduce the following definition: Definition: let , 0 ( )n n tψ be a CLS of class U of a flat- band network with ν sites per unit-cell. We call , 0 ( )n n tψ a homogeneous CLS if , ,for all 0l j l ja A A≠ ⇒ ≡ (11) and we call , 0 ( )n n tψ a heterogeneous CLS otherwise. From the above consideration in Eqs. (9), (10), we can obtain the following continuation criteria in the following lemma: Lemma: in a nonlinear flat-band network Eq. (6), a compact state , 0 ( )n n tψ of the linear lattice = 0γ with en- ergy FBE can be continued as a periodic orbit with com- pact support , 0 ( )n nC t with frequency 2= FBE AΩ + γ if and only if it is homogeneous. This lemma states a necessary and sufficient condition for linear CLSs to be continued as time-periodic solutions of the nonlinear regime with compact support. Indeed, ho- mogeneous CLSs in presence of this local nonlinearity do not break the destructive interference, preserving therefore the compactness in space. Heterogeneous CLSs instead in presence of nonlinearity break the destructive interference, loosing therefore the compactness in space. We call the continued homogeneous CLS solutions compact discrete breathers. Their spatial profile is identical to the CLS one, and their frequency is given by 2= , .FBE g g AΩ + ≡ γ (12) In the next section, we will discuss the linear stability of compact discrete breathers. 3. Linear stability analysis Herewith, we consider a perturbation ( )n tε of a com- pact discrete breather , 0 ( )n nC t Eq. (8) solution of the non- linear flat-band model (6) , 0 ( ) = ( ) ( ) .n n n nt C t tψ + ε (13) By linearizing Eq. (6) around one compact discrete breath- er , 0 ( )n nC t , and defining 2g A≡ γ , we obtain † 0 1 1 11=n n n ni H H H+ +ε ε + ε + ε + ( ) 1 2 * , 0 =0 2 e , U i t l n n n n l l g − − Ω ++ Γ ε + ε δ∑ (14) where 1 =0{ }U l l −Γ are the projector operators of nψ over a compact discrete breather , 0 ( )n nC t , =1 =l l j j j j a ν Γ ⊗∑ e e . (15) The resulting dynamical model Eq. (14) for the perturba- tion term nε consists of equations with time-dependent coefficients that occur at sites where the compact discrete breather , 0 ( )n nC t has non-zero amplitudes. The aim of this section is to analytically prove the existence of regions of instability in the parameter space ( , )gΩ ∈ ×  for the compact discrete breather. In order to achieve this, we first express Eq. (14) in the Bloch representation. Than, we com- pute the condition for resonance determining the Floquet Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 867 C. Danieli, A. Maluckov, and S. Flach matrix at = 0g . At last, we obtain the regions of instability around the resonances via the strained coefficient method, focusing on the = 1U and = 2U cases. 3.1. Bloch states representation Let us consider the Bloch representation of Eq. (14) us- ing the following transformation: 1= eiqn n q qN ε φ∑ . (16) This leads to the Bloch equation, ˆ ˆˆ= ( )q qi H qφ φ + ( ) 1 ˆ 2 * =0 e 2e e e , U iql iql i t iql l q q q l g N − − − Ω −  + Γ φ + φ     ∑ ∑ (17) where † 0 1 1( ) e eiq iqH q H H H−≡ + + is the Bloch matrix. The ( )H q matrix admits ν eigenvectors i qv and ν eigenval- ues i qλ . We assume that one flat band 1 =q FBEλ exists with corresponding eigenvector qw of the Bloch matrix. Then we define the expansion of q̂φ in the Bloch eigenbasis ˆ ˆ ˆ ˆ ˆ =2 = .i i q q q q q i f d ν φ +∑w v (18) The resulting equations on the expansion coefficients q̂f of the flat band reads (see Appendix A) ˆ ˆ=q FB qif E f + ( ) 1 ˆ 2 * * =0 e 2e e e U iql iql i t iql q q l q q q l g f f N − − − Ω −+ + Γ +  ∑ ∑ w w ( ) 1 ˆ 2 * * =0 =2 e 2e e e U iql iql i i t iql i i q q l q q l i d d − ν − − Ω −  + + Γ     ∑ ∑ v w (19) while the equation of the coefficients q̂d of the j th disper- sive band reads ˆ ˆ ˆ=j j j q q qd dλ + ( ) 1 ˆ 2 * * =0 e 2e e e U iql iql i t iql j q q l q q q l g f f N − − − Ω −+ + Γ ⋅ +  ∑ ∑ w v ( ) 1 ˆ 2 * * =0 =2 e 2e e e . U iql iql i i t iql i i j q q l q q l i d d − ν − − Ω −  + + Γ ⋅     ∑ ∑ v v (20) Equations (19) and (20) describe the time-dynamics of the flat-band states qf with dispersive states i qd due to the linearized term of Eq. (14). For class = 1U , these equa- tions are decoupled, while for class > 1U they are coupled. In the next subsection, we neglect the terms following from the nonlinearity (set = 0g ), and obtain the resonance condition by computing the Floquet matrix of the system Eqs. (19), (20). 3.2. Floquet matrix For = 0g , we calculate the Floquet matrix A for Eqs. (19), (20) (also called period advancing matrix). For = ( , )q qf dϕ and = /T π Ω , it follows that * * ( ) ( ) e 0 = with . ( ) ( ) 0 e i T i T t T t A A t T t − λ λ  ϕ + ϕ     ≡    ϕ + ϕ           (21) The eigenvalues of the Floquet matrix A will be degener- ate on the unit circle if and only if 2cos ( ) = 1Tλ , which means 2cos ( ) = 1 = , , = , . T T m m m m λ ⇔ λ π ∈ ⇔ λ Ω ∈   (22) Concerning Eqs. (19), (20), Eq. (22) implies that for m∈ = , = , = 2, , .j FB qE m m jΩ λ Ω ν (23) It follows that from the values of the frequency Ω of a compact discrete breather , 0 ( )n nC t contained in Eq. (23), regions of instability (Arnol'd tongues) in the parameter space ( , )gΩ are expected. In order to obtain an approxima- tion of these regions, we apply a standard technique of perturbation theory called strained method coefficient. 3.3. Arnol'd tongues In the following, we estimate the regions of instability in the parameter space ( , )gΩ of Eqs. (19) and (20), separating between the class = 1U case (where the dispersive states are decoupled from the flat-band ones) and the class = 2U case (where dispersive and flat-band states are coupled). 3.3.1. Class U = 1 In the case of class = 1U flat-band network, it holds that 0 =q qΓ w w and Eqs. (19), (20) reduce to ( )2 * ˆ ˆ= 2e e e ,iql i t iql q FB q q q q gif E f f f N − Ω −+ +∑ 2 * * 0ˆ ˆ ˆ =2 = 2 e .j j j i i t i i j q q q qq q q i q gid d d d N ν − Ω   λ + + Γ ⋅     ∑ ∑ v v (24) 868 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 Compact discrete breathers on flat-band networks Without loss of generality, we refer to a two bands prob- lem = 2ν . The equations of the dispersive band compo- nent qd of Eq. (24) read ( )2 * ˆ ˆ ˆ= 2 e .i t q q q q q q gid d d d N − Ωλ + +∑ (25) The strained coefficient method consists in expanding in powers of g both the time-dependent component qd as well as the frequency term q̂λ around one of the resonant frequencies in Eq. (23). Then, we determine the expansion coefficients so that the resulting expansion is periodic. This will define transition curves between stability and instabil- ity regions in the parameter space ( , )gΩ (for further de- tails, see [40]). The expansion of q̂d and q̂λ reads ˆ( ) ˆ ˆ =0 =1 = , =qk l q q lk k l d g u m g +∞ +∞ λ Ω+ δ∑ ∑ (26) for q̂ qλ ≠ λ and for all ˆq q≠ . Vanishing the secular terms (terms which give rise to non-periodicities in the expan- sion) demands the following conditions in the expansion coefficient 1δ (see Appendix B for details) 1 3 1= , , N N δ − − = 1,m 1 2= , N δ − 1.m ≠ (27) This implies that a region of instability appears from each dispersive frequency qλ (obtained for = 1m in Eq. (23)), while for /q mλ for 2m ≥ regions of instability are absent. It is important to notice that for N ∞ the coefficients in Eq. (27) converge to zero, implying that in the limit of in- finite chain, the instability regions disappear, in analogy with [41]. In Fig. 1 we can see a representation of the Arnol'd tongue around one frequency qλ . Analogous conclusions follow from the strained coeffi- cient method applied to the flat-band states qf . Here the expansions reads ˆ( ) ˆ =0 =1 = , = .qk l q FB lk k l f g E m g +∞ +∞ Ω+ σ∑ ∑v (28) The zeroing of the secular terms yields to the following coefficients: 1 = 3, 1 ,σ − − = 1m , 1 = 2 ,σ − 1m ≠ . (29) Equation (29) is independent on N due to macroscopic degeneracy of the flat-band states (see Appendix B for details). The strained coefficient method showed the appearance of regions of instability in correspondence of each disper- sive energy qλ of the dispersive band. However, instability regions do not appear for higher order resonances ( /q mλ for 2m ≥ ) (see Fig. 1). Furthermore, we can notice that this region of instability also follows from the Bogoliubov ex- pansion of Eq. (24) (see Appendix C for details). Before to go ahead to numerical studies, we briefly check the previ- ous approach in the case with = 2U . 3.3.2. Class U ≥ 2 In the case = 2U , without loss of generality, we refer to a two band problem = 2ν , using the saw-tooth network as a test-bed. Equations (19), (20) read (see Appendix B for details) ( )( ){ 2 2 * ˆ ˆ 2= 1 e i t q FB q q q q gif E f f f N − Ω+ α − + + α ∑ ( )ˆ 2 e 2e e eiq iq i t iq q qf f− − Ω −+ + + ( )( )}2 1 e 2 eiq i i t i q qd d− − Ω+ + + , (30) ( ){ 2 * ˆ ˆ 2= 2 e i t q q q q q q gid d d d N − Ωλ + + + α ∑ ( )ˆ 2 e 2e e eiq iq i t iq q qd d− − Ω −+ + + ( )( )}2 1 e 2 eiq i t q qf f− Ω+ + + (31) for = 3 2cos qα + . Both expansions Eqs. (26), (28) have to be applied to Eqs. (30), (31). However, in the first order, the additional terms (the second and the third lines of both equations) do not provide the appearance of further regions of instability (see Appendix B for details). These additional polarized terms (terms dependent on the wave number q) indeed provide interactions between dispersive and flat- band states. However, the strained coefficient method does not report additional instability regions in the parameter space ( , )gΩ due to these terms. In the following, we will discuss numerically the linear stability of the compact dis- crete breather solutions of certain examples of class = 1U and class = 2U one-dimensional nonlinear flat-band net- works. Fig. 1. First order approximation of the Arnold's tongues (grey shaded area) at a dispersive energy qλ . Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 869 C. Danieli, A. Maluckov, and S. Flach 4. Numerical results In this section we numerically study the linear stability properties of the compact discrete breather solutions of certain flat-band topologies. We then relate the numerical observations with the analytical results discussed above. Herewith we numerically solve the eigenvalue problem Eq. (14) obtained from the time-evolution Eq. (6) linear- ized around a compact discrete breather Eq. (13). General- ly, we will obtain complex eigenvalues, and the presence of non-zero real part will highlight instability [42]. We will also discuss the nature of the eigenvector associated to unstable eigenvalues (eigenvalues with non-zero real part). Furthermore, we will substantiate the findings by showing simulations of the time evolution of initially perturbed compact discrete breathers. In the following, we will focus on two models: the cross-stitch lattice and the saw-tooth chain. In Appendix D we detail the numerical methods used along the work. 4.1. Cross-stitch lattice The cross-stitch lattice (Fig. 2(a)) is a one-dimensional two-band network, which possesses one flat band. Associ- ated to the flat band, there exists a countable set of class = 1U compact localized states, whose homogeneous pro- file in space is shown by the black dots in Fig. 2(a). The full band structure of the model is = , ( ) = 4cos ( )FBE h E q h q− + (32) which can be visualized in Fig. 2(b), for = 3h . In this mod- el, the relative position between dispersive and flat bands can be tuned using the free parameter h∈, which leads to crossing between the two bands for \| \|< 2h , band touching for \| \|= 2h , and presence of a band gap for \| \|> 2.h The time-dependent equations of the cross-stitch lattice in the presence of onsite nonlinearity read 2 1 1 1 1= \| \| ,n n n n n n n nia a a b b hb a a− + − +− − − − − + γ 2 1 1 1 1= \| \| ,n n n n n n n nib a a b b ha b b− + − +− − − − − + γ (33) where γ is the nonlinearity strength. As we have in Sec. 2, the CLSs of the linear regime can be continued as compact discrete breathers written as Eqs. (4), (5) with frequency 2= FBE AΩ + γ : , ,0 0 1 ( ) = e . 1 i t n n n nt A − Ω  δ −   (34) In order to study the linear stability of this model, we line- arize Eq. (33) around the compact discrete breathers Eq. (34), and we numerically calculate the eigenenergies of the resulting model for different values of 2=g Aγ (ob- tained fixing = 1A ). The outcome of our computations can be phrased in the following way. Consider first a weakly nonlinear compact discrete breather with \| \| 1g  . Due to = 1U the linear CLS states are all degenerate but span an orthonormal eigenvec- tor basis of the flat-band Hilbert subspace. Therefore, the degeneracy is harmless, and continuing one CLS into the nonlinear regime will not lead to any resonant interactions with neighboring CLSs. Therefore, a compact discrete breather whose frequency Ω is not in resonance with the dispersive part of the linear spectrum ( )E q is linearly stable. However, if a compact discrete breather is tuned into reso- nance with the dispersive part of the linear spectrum, it will become linearly unstable due to the resonance with ex- tended dispersive states. If we tune the nonlinearity to a finite strength, non-perturbative effects will lead to addi- tional instability windows for compact discrete breathers. In Fig. 3, we show the time evolution of perturbed com- pact discrete breathers. We choose the amplitude of the compact breather to initially be A = 1, and then we introduce an initial uniform random perturbation with maximum am- plitude 10–3 along the whole chain of N = 50 unit-cells. In Figs. 3(a) and (b) we show the time evolution of the \| ( ) \|na t component for h = 3, g = 5 and h = 1, g = 1, respectively. Plot (a) has been obtained for h = 3 and g = 5, when the fre- quency = 3 5 = 8Ω + of the compact discrete breather is located outside the dispersive band [–7,1]. The numerical Fig. 2. (a) Profile of the cross-stitch lattice. (b) Band structure for h = 3. Fig. 3. (Color online) Cross-stitch: (a) and (b): Time evolution of the components an(t) of an initially perturbed compact breathers. (c) and (d): Time evolution of the participation number P. Plots correspond to: h = 3, g = 5 (a), (c); h = 1, g = 1 (b), (d). 870 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 Compact discrete breathers on flat-band networks simulation of the time-evolution shows stability of the com- pact breather. Instead, plot (b) has been obtained for h = 1, and g = 1, when the frequency = 1 1 = 2Ω + of the compact breather is in resonance with the dispersive band [–5,3]. The breather will start to radiate and reduce its amplitude, how- ever the resonance condition will not be destroyed down to the linear level since the linear flat band is resonating with the dispersive one. Thus the compact discrete breather is unstable and will be completely destroyed during its per- turbed evolution. The stable and the unstable behavior of these two cases are confirmed in Figs. 3(c) and (d), where we show the time evolution of the participation number ( )4 4= 1/ \| \| \| \|n nP a b+∑ . The participation number takes values between unity (obtained for a single site excitation) to the system size (obtained for uniformly excited states), [1, ]P N∈ , and it estimates the number of non-negligibly excited sites. Indeed, in plot (c) which corresponds to the stable compact discrete breather shown in Fig. 3(a), the participation number P fluctuates around 1.5, confirming that only few sites are excited. In plot (d), which corre- sponds to the unstable compact discrete breather shown in Fig. 3(b), the participation number P fluctuates around 40, confirming the loss of compactness and the instability of the compact breather. 4.2. Saw-tooth The saw-tooth lattice [Fig. 4(a)] is a one-dimensional two-band network with one flat band. Associated to the flat band, there exists a countable set of class = 2U compact localized states, whose homogeneous profile in space is shown by the black dots in Fig. 4(a). We recall that in this network, every CLS is non-orthogonal with its two nearest neighbors. The full band structure of the model is = 1, ( ) = 2 2cos ( )FBE E q q− − (35) which can be observed in Fig. 4(b). Differently from the cross-stitch case, the spectral bands of this model cannot be tuned by certain free parameter, and the network pos- sesses a band gap between the dispersive and the flat band. The time-dependent equations (6) of the saw-tooth in presence of onsite nonlinearity read 2 1= \| \| ,n n n n nia b b a a+− − + γ 2 1 1 1= \| \| .n n n n n n n nib b b b a a b b− + −− − − − − + γ (36) The CLSs of the linear regime can be continued as compact discrete breathers written Eqs. (4), (5) with fre- quency 2= FBE AΩ + γ written as , , 1 ,0 0 0 1 1 ( ) = e . 0 1 i t n n n n n nt A − Ω −      δ + δ    −      (37) Comparing to the = 1U case of the cross-stitch lattice, the new feature is the non-orthogonality of neighboring CLSs at the linear limit. While the flat band is gapped away from the dispersive band, at weak nonlinearities we can ex- pect a resonant interaction between neighboring CLSs, which may, or may not, lead to model dependent linear local instability. It turns out that this instability indeed takes place for the saw-tooth chain. There exists a narrow region of in- stability for 0.1 < < 0g− . Therefore, the fact that the linear flat-band network is of class = 2U makes compact discrete breathers unstable even in the presence of a band gap. How- ever, this instability is local, and therefore might not lead to a destruction of the perturbed compact discrete breather, since there is no way to radiate the excitation to infinity. In Fig. 5 we show the na components (a) and the nb compo- nents (b) of the unstable eigenvector with pure real eigen- value 5= 2.987 10EV −⋅ obtained for = 0.001g − . The ei- genvector is exponentially localized. Let us discuss the time evolution of slightly perturbed compact discrete breathers, where a perturbation of order 310− is equidistributed along all the = 50N unit-cells. In Figs. 6(a) and (b), we show the time evolution of the \| ( ) \|na t component for = 1.5g − and = 0.007g − , respectively, while in Figs. 6(c) and (d) we show the time evolution of the participation number P . In the left column, Figs. 6(a) and (c) correspond to the time evolution of a compact discrete breather for = 1.5g − . In this case, the compact discrete breather is unstable, since its frequency = 1 1.5 = 0.5Ω − − is in resonance with the dispersive band [ 4,0]− . This instabil- ity is also depicted by the participation number P . In the right column, Figs. 6(b) and (d), we plot the time evolution of a compact discrete breather for = 0.007g − . In this case, the pure real eigenvalues and the exponentially localized eigenvector yield an oscillatory behavior in time of the com- pact discrete breather, which is depicted also by the partici- pation number P . Fig. 4. (a) Profile of the saw-tooth lattice. (b) Band structure. Fig. 5. Saw-tooth: na component (a) and nb component (b) of the unstable eigenvector for 3= 10g −− and real eigenvalue =EV 52.987 10−= ⋅ . Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 871 C. Danieli, A. Maluckov, and S. Flach 5. Conclusions In this work, we have discussed the properties of com- pact discrete breathers in some flat-band networks. Linear flat-band networks possess compact localized states. In or- der to continue them into the nonlinear regime to become compact discrete breathers, a homogeneity condition on the amplitude distribution of CLS has to be satisfied, which is known to be present for a number of flat-band networks. The nonlinear compact discrete breathers will then persist as compact states, albeit with tuned modified frequencies. If these frequencies are in resonance with dispersive branches of the linear flat-band network, then the discrete breather will turn linearly unstable, which may lead to a complete destruction of the perturbed breather by dissolving it into dispersive states. If the CLSs form an orthonormal set at the linear limit, no further instabilities are expected in the weakly nonlinear regime. So all it is needed to have a stable compact discrete breather at the weakly nonlinear limit, is to tune the flat-band energy out of resonance with the disper- sive bands. However, there exist flat-band networks for which the CLSs are not orthogonal. In these cases, the flat band is gapped away from the dispersive spectrum, and resonances with the dispersive spectrum are avoided in the weakly non- linear regime. But the overlap with nearest neighbor CLS states can lead to a local instability in the weakly nonlinear regime. We indeed observe that this is the case for the saw- tooth chain. Remarkably the instability does not lead to a complete destruction of the breather, and instead yields a local oscillation of the excitation. The class of heterogeneous CLSs cannot be continued as compact discrete breathers. However, as discussed in [36], flat-band networks that admit heterogeneous CLSs in pres- ence of local nonlinearity admit families of exponentially localized discrete breathers. Additional fine-tuning of pa- rameters and functions can lead to a compactification for a countable set of discrete breathers. Acknowledgments The authors acknowledge financial support from IBS (Project Code No. IBS-R024-D1). A.M. acknowledges support from the Ministry of Education and Science of Serbia (Project III45010). Appendix A: Bloch states representation Let us consider Eq. (6) linearized around one compact discrete breather , 0 ( )n nC t , for 2g A≡ γ † 0 1 1 11=n n n ni H H H+ +ε ε + ε + ε + 1 2 * , 0 =0 (2 e ) , U i t l n n n n l l g − − Ω ++ Γ ε + ε δ∑ (A1) where 1 =0{ }U l l −Γ are the projector operators of the vector nψ over a compact discrete breather , 0 ( )n nC t : , =1 = .l l j j j j a ν Γ ⊗∑ e e (A2) The expansion Eq. (16) in Bloch states 1= eiqn n q qN ε φ∑ (A3) maps Eq. (A1) to 1 1e = ( ) eiqn iqn q q q q i H q tN N ∂ φ φ + ∂ ∑ ∑ ( ) 1 2 * , 0 =0 2e e e , U iqn i t iqn l q q n n l q l g N − − Ω − +   + Γ φ + φ δ     ∑ ∑ (A4) where † 0 1 1( ) e eiq iqH q H H H−≡ + + is the Bloch matrix. This matrix has = 1, ,i ν eigenvalues i qλ and eigenvectors i qv , where 1 =q FBEλ and =i q qv w . Let us multiply Eq. (A4) by ˆ1 e iqn N − and sum over the lattice =1 N n∑ . This yields to ˆ ˆ( ) ( ) =1 =1 1 1e = ( ) e N N i q q n i q q n q q q n q n i H q t N N − −   ∂ φ φ +    ∂        ∑ ∑ ∑ ∑ ( 1 ˆ( )( )0 =0 2e U i q q n l l q q l g N − − + + Γ φ +  ∑ ∑ )ˆ( )( )2 0 , 0 e e .i q q n li t q n n l − + +− Ω + + φ δ  (A5) Fig. 6. (Color online) Saw-tooth. Time evolution of the ampli- tudes of perturbed compact breathers (component \| ( ) \|na t ) and the participation number. Plots correspond to: = 1.5g − (a), (c) 3= 7 10g −− ⋅ (b), (d). 872 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 Compact discrete breathers on flat-band networks Without loss of generality, we choose 0 = 0n . The relation ,0 1 e =iqn q nN δ∑ (A6) yields to Eq. (17) in Sec. 3.1: ˆ ˆˆ= ( )q qi H qφ φ + ( ) 1 ˆ 2 =0 e 2e e e . U iql iql i t iql l q q q l g N − − − Ω −  + Γ φ + φ     ∑ ∑ (A7) Let us now expand q̂φ in the Bloch eigenbasis ˆ ˆ ˆ ˆ ˆ =2 = ,i i q q q q q i f d ν φ +∑w v (A8) where ˆ, i q qf d ∈ are time-dependent complex numbers. Equation (A7) becomes ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ =2 =2 =i i i i i q q FB q qq q q q q i i i f d E f d t ν ν ∂ + + λ +  ∂   ∑ ∑w v w v 1 ˆ =0 =2 e 2e U iql iql i i l q q q q q l i g f d N − ν −     + Γ + +      ∑ ∑ ∑w v 2 * * =2 e e .i t iql i i q q q q i f d ν − Ω −   + +     ∑w v (A9) Next, we regroup Eq. (A9) in terms of q̂f and ˆ i qd and we multiply it by * qw . By the orthogonality of the eigen- vectors qw and i qv of the Bloch matrix ( )H q , we obtain Eq. (19) for the flat-band component qf ˆ ˆ=q FB qif E f + ( ) 1 ˆ 2 * * =0 e 2e e e U iql iql i t iql q q l q q q l g f f N − − − Ω −+ + Γ ⋅ +  ∑ ∑ w w ( ) 1 ˆ 2 * * =0 =2 e 2e e e . U iql iql i i t iql i i q q l q q l i d d − ν − − Ω −  + + Γ ⋅     ∑ ∑ v w (A10) Analogously, we obtain Eq. (20) for the dispersive bands component ˆ i qd by multiplying Eq. (A9) by j qv and using the orthogonality of the eigenvectors of the Bloch matrix ( )H q : ˆ ˆ ˆ=j j j q q qd dλ + ( ) 1 ˆ 2 * =0 e 2e e e U iql iql i t iql j q q l q q q l g f f N − − − Ω −+ + Γ ⋅ +  ∑ ∑ w v ( ) 1 ˆ 2 * * =0 =2 e 2e e e . U iql iql i i t iql i i j q q l q q l i d d − ν − − Ω −  + + Γ ⋅     ∑ ∑ v v (A11) Appendix B: Strained coefficient method Let us consider Eq. (24) in Sec. 3.3.1 for a class = 1U flat-band network ( )2 * ˆ ˆ= 2e e e ,iql i t iql q FB q q q q gif E f f f N − Ω −+ +∑ 2 * * 0ˆ ˆ ˆ =2 = 2 e ,j j j i i t i i j q q q qq q q i q gid d d d N ν − Ω   λ + + Γ ⋅     ∑ ∑ v v (B1) and the expansion of q̂d and q̂λ in Eq. (26) ˆ( ) ˆ =0 = ,qk q k k d g u +∞ ∑ ˆ =1 = .l q l l m g +∞ λ Ω+ δ∑ (B2) This expansion yields to ˆ ˆ ˆ( ) ( ) ( ) =0 =0 =0 =1 =q q qk k k l lk k k k k k l i g u m g u g u t +∞ +∞ +∞ +∞ +∂ Ω + δ + ∂ ∑ ∑ ∑∑ ( )( ) ( )1 2 =0 1 2 e .q qk i t k k k q g u u N +∞ + − Ω   + +     ∑ ∑ (B3) Next, we equate the coefficients of each power of g to zero. From 0g we get ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) 0 0 0 0= ( ) = e .q q q qim tiu m u u t a− ΩΩ ⇒ (B4) Without loss of generality, we can assume all the initial conditions to be equal ˆ( ) 00 qa a≡ . For 1g in Eq. (B3) we get ( )ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )2 11 1 0 0 0 1= 2 eq q q q qi t q iu m u u u u N − ΩΩ + δ + +∑ (B5) which, by Eq. (B4), reads ˆ ˆ( ) ( ) 1 1=q qiu m uΩ + (2 ) 1 0 0 1 ( 2)e eim t i m tN a a N − Ω − Ω −+ δ + + + (2 ) * 0 0 ˆ 2e e . i t i tq q q q a a − λ − Ω−λ ≠  + +    ∑ (B6) Using the general solution 0 ( ) ( ) 0 ( ) = e e ( ) t t r a s ds t a s ds y t b r dr ∫ ∫ + ∫ (B7) of the first-order differential equation ( ) = ( ) ( ) ( ),y t a t y t b t+ for 1m ≠ it follows that Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 873 C. Danieli, A. Maluckov, and S. Flach ( )ˆ( ) 1 1 01 1( ) = e 2 eq im t im tu t a N a t N − Ω − Ω+ δ + + ( ) * 20 e e 2 (1 ) i t im tia m − Ω − Ω+ − + Ω − ( )0 ˆ 2 e eqi t im t qq q i a m − λ − Ω ≠  + − + Ω −λ ∑ ( ) * (2 )0 e e . (2 ) qi t im t q ia m − Ω−λ − Ω  + −  Ω − −λ  (B8) To kill the secular term we need to set 1 = 2 / Nδ − in the second term of the right-hand inside of Eq. (B8). The solu- tion (B7) in the case = 1m reads ( )ˆ( ) * 1 1 0 01 1( ) = e 2 eq i t i tu t a N a a t N − Ω − Ω + δ + + +  ( )0 ˆ 2 e eqi t i t qq q i a − λ − Ω ≠  + − + Ω −λ ∑ ( ) * (2 )0 e e .qi t i t q ia − Ω−λ − Ω  + −  Ω −λ  (B9) To kill the secular term we need to set * 1 0 0( 2) = 0N a aδ + + * 0 01 0 = 2 = 2 eia N a θ⇔ δ − − − + 1 3 1= , N N  ⇔ δ − −    (B10) since 00 0= eia I θ , and the coefficients { }i iδ in Eq. (B2) are real numbers. The very same procedure discussed above for the components q̂d of the dispersive states can be re- peated for the component q̂f of the flat band. Analogously to the expansion (B2), we perform an expansion for the flat-band component q̂f in Eq. (B1): ˆ( ) ˆ =0 = ,qk q k k f g +∞ ∑ v =1 = l FB l l E m g +∞ Ω+ σ∑ . (B11) This ultimately lead to the expansion coefficients 1 1 = 3, 1, = 1, = 2, 1. m m σ − − σ − ≠ (B12) The system size N is absent due to the macroscopic de- generacy of the flat-band states. Therefore, in Eq. (B6) all terms of the sum ˆq q≠∑ have the same time-dependent term e FBiE t− . To kill the secular term in Eq. (B8) for the flat-band component qf for 1m ≠ , the following condition has to be satisfied 1 0 0 1 2= = 2. q b b N σ − −∑ (B13) In Eq. (B9) for the flat-band component qf for = 1m , in order to kill the secular term we need to set * 01 0 0 0 1 1= 2 = 2 ei q q b b b N θ    σ − + − +    ∑ ∑ 1 = { 3, 1}.⇔ σ − − (B14) For flat-band networks of larger class 2U ≥ , both expansions (B2), (B11) have to be applied to Eqs. (19), (20). For the saw- tooth case, these equations reduces to Eqs. (30), (31) here recalled ( )( ){ 2 2 * ˆ ˆ 2= 1 e i t q FB q q q q gif E f f f N − Ω+ α − + + α ∑ ( )ˆ 2 e 2e e eiq iq i t iq q qf f− − Ω −+ + + ( )( )}2 1 e 2 e ,iq i i t i q qd d− − Ω+ + + (B15) ( ){ 2 * ˆ ˆ ˆ 2= 2 e i t q q q q q q gid d d d N − Ωλ + + + α ∑ ( )ˆ 2 e 2e e eiq iq i t iq q qd d− − Ω −+ + + ( )( )}2 1 e 2 e ,iq i t q qf f− Ω+ + + (B16) since in the saw-tooth is a two band problem = 2ν with the following Bloch vectors ,q qw v and projector operators 0 1,Γ Γ 0 1 1 0 01= , = , 0 11 e 1 01 1 e= , = , 0 11 q iq iq q − −    Γ    α +       + Γ    α    w v (B17) where = 3 2cos qα + . Eqs. (B15), (B16) expanded via Eqs. (B2), (B11) lead to additional time-periodic terms dependent on the wave number q (called polarized terms). These terms therefore do not influence the zeroing condi- tion of the secular term presented above for class = 1U flat-band networks. Appendix C: Bogoliubov expansion Let us consider Eq. (24) for the dispersive component i qd for one of the dispersive band i considered for one component only 2 = 2 ei i i i i t i q q q q q gid d d d N − Ω λ + +   . (C1) Let us simplify the notation, by dropping the i . We now apply the Bogoliubov expansion to Eq. (C1) (2 )= e e .i t i t q q qd a b− ω − Ω−ω+ (C2) 874 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 Compact discrete breathers on flat-band networks This yields to * (2 )e (2 ) ei t i t q qa b− ω − Ω−ωω + Ω−ω = * (2 )= e ei t i t q q q qa b− ω − Ω−ωλ + λ + ( ) ( )* * (2 )2 e 2 ei t i t q q q q g a b b a N − ω − Ω−ω + + + +   (C3) which can be decoupled in two equations: ( )= 2 ,q q q q q ga a a b N ω λ + + ( ) ( )* * * *2 = 2 .q q q q q gb b b a N Ω−ω λ + + (C4) In matrix form, Eq. (C4) reads = 2 q q q q a a b b     λ ε ω       −ε Ω −λ       (C5) for = 2 /q g Nλ λ + and = /g Nε . 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USA (2003). ___________________________ 876 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 https://doi.org/10.1103/PhysRevA.96.063838 https://doi.org/10.1103/PhysRevB.94.144302 https://doi.org/10.1103/PhysRevE.96.052208 https://doi.org/10.1016/S0167-2789(98)00077-3 Introduction 1. Flat-band networks 2. Nonlinear flat-band networks and continuation of compact localized states 3. Linear stability analysis 3.1. Bloch states representation 3.2. Floquet matrix 3.3. Arnol'd tongues 3.3.1. Class U = 1 3.3.2. Class U ≥ 2 4. Numerical results 4.1. Cross-stitch lattice 4.2. Saw-tooth 5. Conclusions Acknowledgments Appendix A: Bloch states representation Appendix B: Strained coefficient method Appendix C: Bogoliubov expansion Appendix D: Numerical methods

Compact discrete breathers on flat-band networks

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