Flat-band physics in the spin-1/2 sawtooth chain

We consider the strongly anisotropic spin-1/2 XXZ model on the sawtooth-chain lattice with ferromagnetic longitudinal interaction Jzz = ΔJ and aniferromagnetic transversal interaction Jxx = Jyy = J > 0. At Δ = −1∕2 the lowest one-magnon excitation band is dispersionless (flat) leading to a massively degenerate set of ground states. Interestingly, this model admits a three-coloring representation of the ground-state manifold [H.J. Changlani et al., Phys. Rev. Lett. 120, 117202 (2018)]. We characterize this ground-state manifold and elaborate the low-temperature thermodynamics of the system. We illustrate the manifestation of the flat-band physics of the anisotropic model by comparison with two isotropic flat-band Heisenberg sawtooth chains. Our analytical consideration is complemented by exact diagonalization and finite-temperature Lanczos method calculations.


Introduction
Frustrated quantum Heisenberg spin systems are of great interest nowadays. Exact calculations and rigorous statements, although scarce, are obviously important for this field. One source of such results stems from the flat-band antiferromagnets, i.e., the models with a dispersionless (flat) one-magnon band [1]. The flat one-magnon band leads to localized multi-magnon states which dominate the low-temperature physics in antiferromagnetic flat-band models close to the saturation field. Their contribution to the partition function can be exactly calculated by visualizing the localized multi-magnon states as hardcore-object configurations on a corresponding auxiliary lattice. Then the hard-core description allows to use classical statistical mechanics to describe frustrated quantum spin models. This approach has been successfully used for a wide class of frustrated quantum antiferromagnets supporting flat bands [2][3][4][5][6][7] including the kagome antiferromagnet in two dimensions and the pyrochlore antiferromagnet in three dimensions. We mention that a similar description of flat-band states can be developed for the Hubbard model [1,[8][9][10][11][12]. A popular one-dimensional example of a flat-band antiferromagnet is the Heisenberg sawtooth chain with the special relation between antiferromagnetic exchange interaction along the basal line J 1 > 0 and along the zig-zag path J 2 > 0, of J 2 /J 1 = 2, that a e-mail: derzhko@icmp.lviv.ua was widely used as a playground for localized-magnon physics at low temperatures around the saturation field h sat = 4J 1 , see, e.g., [2][3][4][5][13][14][15].
Later on it was found that flat-band physics and corresponding localized multi-magnon states can appear in frustrated magnets also at zero magnetic field in case that ferro-and antiferromagnetic interactions compete [16]. Again, the sawtooth chain is a prominent example, however, with ferromagnetic bonds J 2 < 0 along the zig-zag path and antiferromagnetic bonds J 1 > 0 along the basal line [16][17][18][19][20][21][22]. Here the flat-band physics is realized at a critical point J 2 /J 1 = −2, where the ferromagnetic ground state gives way for a ferrimagnetic one. It is worth mentioning that the ferro-antiferromagnetic sawtooth chain is an appropriate model to describe the recently synthesized compound Fe 10 Gd 10 [23] and is also relevant for Cs 2 LiTi 3 F 12 that hosts ferro-antiferromagnetic sawtooth chains as magnetic subsystems [24].
Very recently, using the three-coloring description Changlani et al. [25,26] have noticed that the groundstate manifold of the spin-1/2 XXZ sawtooth chain with antiferromagnetic bonds J 1 = J 2 > 0 and with a negative zz anisotropy parameter ∆ = −1/2 (denoted as XXZ0 model) exhibits also a huge degeneracy. As already noticed before (but not investigated) in reference [19], the XXZ0 sawtooth chain also belongs to the class of flat-band systems hosting localized multi-magnon states in zero magnetic field. The three-coloring description of spin systems is a general and promising approach to study frustrated magnets [25][26][27][28][29]. To illustrate the relation between the three-coloring and the flat-band localizedmagnon description by the example of the sawtooth spin chain is one of the aims of the present study.
In the present paper, we examine the spin-1/2 XXZ0 sawtooth-chain model [25] focusing on the specific flatband features, i.e., localized-magnon properties. We also compare this model with the two isotropic Heisenberg sawtooth-chain flat-band cases which were mentioned above and studied previously [2][3][4][5]13,14,16,[19][20][21][22]. To be specific, in what follows we consider the spin-1/2 XXZ Hamiltonian on the sawtooth-chain lattice of N sites (see Fig. 1), where we have N = (N − 1)/2 for open boundary conditions and N = N/2 for periodic boundary conditions. In what follows we choose the following flat-band parameter sets: where model 1 corresponds to the XXZ0 model mentioned above. It is convenient to set J 1 = 1 for model 1 [25], but J 1 = 1/2 for models 2 [16] and 3.

Constituents for many-body physics
We begin with the illustration of some key elements relevant for the localized-magnon picture and for the three-coloring representation. First of all we note that the spin Hamiltonian H commutes with total S z = N i=1 s z i that allows us to consider the eigenstates of the Hamiltonian in each of N + 1 subspaces of S z = N/2, N/2 − 1, . . . , −N/2 separately. Clearly, the fully polarized ferromagnetic state |0 is the only eigenstate of H in the subspace with S z = N/2 with the energy E 0 and it can be considered as the magnon vacuum state. It is straightforward to get the eigenstates and eigenvalues in the subspace with S z = N/2 − 1 (one-magnon states), see below.
Since the sawtooth chain is a one-dimensional array of corner-sharing triangles, see Figure 1, its Hamiltonian can be written as a sum over Hamiltonians of each triangle, (the Zeeman term is omitted). Below we also discuss the eigenstates and eigenvalues of the spin Hamiltonian H as they explain the three-coloring representation for the sawtooth-chain spin model 1 and provide a route to construct the eigenstates of the sawtooth-chain spin models 1 and 2.
It is worth mentioning that the triangles in model 1 have the full C 3 symmetry before they are connected and that models 2 and 3 are isotropic in spin space.

Flat bands and localized magnons
Imposing periodic boundary conditions, s α N +1 = s α 1 (N is even), we find straightforwardly the energies of the onemagnon excitations above the fully polarized ferromagnetic state |0 (magnon vacuum) for all three spin models. All of them exhibit flat bands. For the models at hand we have for model 1, for model 2, and for model 3 (recall that J 1 = 1/2 for models 2 and 3), where E 0 is the energy of the ferromagnetic state. Here, as usually, k acquires N/2 values within the region between −π and π. Note that in the cases 1 and 2 the (lowestenergy) flat-band excitations have zero energy. Furthermore, one can construct the flat-band states as localized states where a magnon is located on three adjacent sites of the lattice (magnon trap), see Figure 1, where a trap built by sites 4, 5, 6 is brownly highlighted. We have for model 1, for model 2, and for model 3, see Figure 1. Note that with periodic boundary conditions 0 ≡ N , i.e., s − 0 ≡ s − N . The local nature of the one-magnon ground states (6) allows to construct many-magnon ground states, see Section 3.

Spin model on a triangle
Here we provide some formulas which we need in the following sections. Eight eigenstates of the triangle Hamiltonian H (2) with different S z = 3/2, 1/2, −1/2, −3/2 for the model 1 may be written in the form where the states |1 , |2 χ , |3 χ , |5 χ , |6 χ , and |8 are the ground states of the triangle with the eigenvalue −3/8 and the states |4 and |7 are excited states with the eigenvalue 9/8. Here The eigenstates |2 χ , |3 χ , |5 χ , and |6 χ , are also the eigenstates of the chirality operator of the triangle and thus their form is important for constructing a threecoloring representation for the ground-state manifold of model 1 [25]. However, to get a better relation to the investigations of references [16,19] we may introduce other linear combinations of these states, namely, We use these eigenstates in Section 3 while constructing many-magnon ground states.  So far only for the models 2 and 3 the construction of localized-magnon states was described in the literature, see, e.g., reference [16] for model 2 and references [2,3,13] for model 3, but not for model 1. Therefore, we sketch now the construction rules of localized-magnon states for model 1 in this section. In addition, we will briefly discuss the relation of the localized-magnon states to the promising three-coloring picture developed in references [25,26].
We begin with a short outline of the three-coloring representation for the ground-state manifold of the spin-1/2 XXZ0 sawtooth chain (i.e., model 1 in our notation). The starting point is the definition of three single-spin coloring states where ω = exp(2πi/3), see equation (11). The states in equation (13) describe spin-1/2 coherent states [36], associated with three particular directions, each separated by 120 degrees, in the plane perpendicular to the z-axis. These states are represented by the colors red, blue, green, respectively. A multi-spin state on a lattice is constructed by putting a single-spin coloring state at each lattice site. The state can be graphically represented as a threecoloring of the lattice (i.e., no two vertices connected by a bond have the same color). Obviously, the two threecoloring states |r 1 b 2 g 3 and |r 1 g 2 b 3 on a triangle are superpositions of states given in equation (10) with the ground-state energy −3/8, where states with different S z are mixed. In other words, the states |r 1 b 2 g 3 and |r 1 g 2 b 3 belong to the ground-state manifold. The three-coloring can be straightforwardly extended to a thermodynamically large lattice. The total number of three-colorings for the open sawtooth chain of N = 2N + 1 sites grows as 2 N , where N = (N − 1)/2 is the number of triangles in the open sawtooth chain. Moreover, constructing resonating color loops, one can single out a localized magnon state [26], see also below for an example. However, by contrast to the flat-band localized-magnon description, the utilization of the three-coloring picture to determine properties of corresponding frustrated spin models, such as model 1, is much less elaborated, i.e., it is still a task to be addressed in the future. One difficulty is certainly the mixing of states with different S z that requires a subsequent projection onto the S z -subspaces to restore this symmetry of Hamiltonian (1). In what follows, we therefore mainly exploit the localized-magnon picture for the one-dimensional sawtooth-chain spin model.
First we mention that in the subspace S z = N/2 − 1 the localized-magnon (or flat-band) states introduced in Section 2 are exact eigenstates. For the sawtooth chain model 1 of N = 2N + 1 sites with open boundary conditions there are two classes of localized one-magnon states, namely, "boundary" states such as and "bulk" states such as (15) where the numbers at the up-and down-arrows correspond to the numbering in Figure 1, top; | . . . ↑ . . . in equation (15) means the state with all the spins 1, 5, 6, . . . up. Both states belong to the ground-state manifold. (For an explicit proof we refer to Appendix A.) Note here that the localized boundary ground states exist also for model 2 1 but not for model 3.
Let us also give an example how to get a localized magnon state from the three-coloring representation. We have where P S z stands for the projector onto the subspace with the specific S z and the numbers 1, . . . , 5 correspond to those given in the first line of Figure 1.
In summary, the ground-state degeneracy for the open sawtooth chain 1 of N = 2N + 1 sites in the subspace S z = N/2 − 1 is N + 1, because all the localized-magnon states are linearly independent [37]. A corresponding consideration holds for the open sawtooth-chain model 2, i.e., the ground-state degeneracy is also N + 1. On the other hand, for the open sawtooth-chain model 3 the degeneracy in the subspace S z = N/2 − 1 is lower and equals N − 1, because the localized boundary states are missing.
We pass to the subspace S z = N/2 − k with k = 2 down spins. Because of the localized nature of the onemagnon excitations, independent localized two-magnon eigenstates can be constructed satisfying the hard-dimer rule (see lines 1 and 2 in Fig. 2), i.e., two localized onemagnon states are not allowed to be in touch. There are C k N −k+2 , k = 2, such states for the open sawtooth 1 The analogue of the state (14) for the model 2 is Line 1: two independent localized magnons can be pictorially represented as a spatial configuration of two hard dimers. A hard dimer extends over two lattice constants. Overlapping of two dimers is forbidden, i.e., two neighboring sites of the auxiliary linear chain cannot be occupied by independent localized magnons. Line 2: corresponding position of the localized magnons (filled brown circles). Lines 3 and 4: two different two-magnon complexes of overlapping localized magnons corresponding to equations (17) (line 3) (18) (line 4).
is the binomial coefficient. This construction rule was first found for model 3 and can be extended to more than two magnons (so called independent localized multi-magnon states) leading finally to a huge ground-state degeneracy of model 3 at the saturation field h sat that grows exponentially with system size N , cf. [2,3,13]. Obviously, the above illustrated construction of independent localized multi-magnon states also holds for models 1 and 2 [16,19]; the (natural) number of the magnons k for these open chains varies in the region 2 ≤ k ≤ (N + 2)/2. However, there are two important differences to model 3: (i) all these multi-magnon states are degenerate at zero field h = 0 [cf. Eqs. (3)-(5)] and (ii) in addition to the independent localized-magnon states also specifically overlapping localized magnons are ground states [16]. Thus, the ground-state degeneracy of models 1 and 2 is much larger than for model 3.
To illustrate overlapping localized magnons, we consider a localized two-magnon complex at the boundary defined as [16,19] where in equation (17) we have defined the symbol c = −1/2 instead of using the concrete value −1/2 to avoid cumbersome expressions in a number of formulas given below, see line 3 in Figure 2 for a pictorial representation of the state (17). In Appendix A we check that this state is among the ground states with S z = N/2 − 2. Line 1: three independent localized magnons can be pictorially represented as a spatial configuration of three hard dimers. Line 2: localized magnon and localized two-magnon complex. Line 3: localized three-magnon complex of one-bracket type, see equation (20). Lines 4, 5, and 6: localized three-magnon complexes of two-bracket type, see equation (22).
A two-magnon complex away from the boundary is given by the formula (18) (c = −1/2), see line 4 in Figure 2. Again, in Appendix A we check that this state is among the ground states with S z = N/2 − 2.
It is easy to count the ground states in the sector S z = N/2 − 2. We have C 2 N independent localized two-magnon states and N + 1 localized states built by localized twomagnon complexes, in total We have checked equation (19) by exact diagonalization of open sawtooth chains 1 of up to N = 39 sites providing evidence for the completeness of the constructed ground states in the subspace S z = N/2 − 2. Moreover, these states are linearly independent [37]. Now we turn to the subspace S z = N/2 − k with k = 3 spins down. Again, because of the local nature of the independent localized-magnon states and the localized two-magnon complexes, we can construct a number of ground states in the sector S z = N/2 − 3 placing such states sufficiently far from each other. This way we construct C 3 N −1 independent localized magnon states and 2C 2 N −1 states consisting of a localized magnon and a localized two-magnon complex, see lines 1 and 2 in Figure 3 for N = 11. More ground states in this subspace are the localized three-magnon complexes, where we have two types, which we will denote as one-bracket type and two-bracket type, see equations (20)- (22) below. These localized threemagnon complexes are sketched in Figure 3 for N = 11, see line 3 for the one-bracket type and lines 4, 5, and 6 for the two-bracket type.
(Note that the last term in the first and the last lines of equation (22) , it is written for similarity to the other lines, where this kind of terms are relevant.) Again, some more detailed calculations checking that the above presented states are ground states with S z = N/2 − 3 are transferred to Appendix A.
In sum, the number of ground states in the sector S z = N/2 − 3 is As previously, we have confirmed the analytical expression (23) by exact diagonalization for models 1 with open boundary conditions of up to N = 39 sites. Moreover, these states are linearly independent.
The construction of the linearly independent ground states for k ≥ 4 follows the same lines as explained above, although it becomes more tedious. We find that model 1 with open boundary conditions is identical to model 2 [16]. After all, the degeneracy of the ground-state manifold of the open sawtooth-chain model 1 with N = 2N + 1 sites in the subspaces S z = N/2 − k for k = 0, 1, 2, . . . , N is given by cf. equations (19) and (23). Then the total degeneracy of the ground-state manifold of the open sawtooth-chain model 1 with N = 2N + 1 sites is As already found for the sectors with k = 2 and 3, the general formula (24) and of course also (25) match perfectly with corresponding exact-diagonalization data.
We add here the known information on the degeneracy of the ground-state manifold in the subspaces S z = N/2 − k for k = 0, 1, 2, . . . , N of the models 2 and 3 with N = 2N + 1 sites and open boundary conditions. For model 2 equation (24) holds [16] and for model 3 the flat-band states exist only in the subspaces S z = N/2 − k, k = 0, 1, 2, . . . , N /2, and the degeneracy is g N (S z = N/2 − k) = C k N −k , i.e., it is smaller, because only independent localized multi-magnon states exist, but no additional complexes [2,3,5].
Let us now briefly discuss the ground-state degeneracy of periodic sawtooth chains of N = 2N sites. For model 3 independent localized multi-magnon ground states exist in the subspace S z = N/2 − k with k = 0, 1, 2, . . . , N /2. Their degeneracy is G 3,5]. For model 2, the ground states in the subspace S z = N/2 − k with k = 0, 1, 2, . . . , N were found in references [16,19]; their degeneracy is For periodic chains, the model 1 exhibits more ground states than the model 2 if k ≥ 3: The total degeneracy of the ground-state manifold of the periodic sawtooth-chain model 1 with N = 2N sites then is cf. equation (25). We confirmed the numbers given in equations (26)-(28) by exact diagonalization for the periodic sawtooth-chain model 1 of up to N = 32 sites (see also Tab. 1). Apparently, the ground-state degeneracies depend on the imposed boundary conditions for finite chains, but in the thermodynamic limit N → ∞ the boundary conditions become irrelevant. Therefore, it is sufficient to consider for the analytical calculations of low-temperature thermodynamic quantities (see the next section) the simpler case of open boundary conditions. However, for the numerical techniques used in Section 5 to study finite systems, periodic boundary conditions are more appropriate, because more symmetries can be used, i.e., longer chains are feasible.

Low-temperature thermodynamics of model 1
From previous investigations of models 2 and 3 it is known that the huge manifold of localized flat-band ground states may dominate the low-temperature thermodynamics [2][3][4][5]14,16]. We may expect that this statement is valid also for model 1. Following this argument, in this section we derive analytical formulas for the field and temperature dependences of magnetization, entropy, specific heat, and susceptibility for model 1 taking into account flat-band states only. We compare this low-temperature approach with numerical data taking into account all eigenstates.
We consider the influence of a magnetic field h on the manifold of the localized ground states of the open sawtooth-chain model 1 with N = 2N + 1 sites. For h = 0 only the single fully polarized ferromagnetic state remains the ground state with energy E 0 (h) = E 0 − hN/2, and all the other localized flat-band states become excited states. The contribution of all these states to the partition function is determined by their degeneracy g N (S z ) given in equation (24) and their Zeeman energy E(h, S z ) = E 0 − hS z : As mentioned already above, this part of the partition function is identical for models 1 and 2 (but not for model 3, where no complexes of overlapping localized magnons exist) and may dominate the low-temperature physics. It yields thermodynamic quantities which depend on x = h/T only. Clearly, the full partition functions of models 1 and 2 are different because of different excited non-flat-band states which come into play at nonzero temperatures, and we can reveal these differences by an exactdiagonalization analysis of finite chains, see Section 5. We note that (how it should be) for h = 0, T → 0 the partition function reproduces the total ground-state degeneracy (25), i.e., Z fbs (x = 0, N ) = W(N ) exp(−E 0 /T ).
The residual ground-state entropy is given by s = ln[W(N )]/N , i.e., we get in the thermodynamic limit for the residual entropy per spin s = ln 2/2 ≈ 0.346 574. As mentioned above, for N → ∞ the boundary conditions become irrelevant: Because only the exponential term in equations (25) and (28) is essential, we get the same value for open and periodic chains. Obviously, s = s(T = 0) is already half of the maximum entropy for T → ∞. Note that the residual entropy for model 3 at the saturation field is smaller, s = (1/2) ln[(1 + √ 5)/2] ≈ 0.240 606 [2,3]. 2 Interestingly, although the residual entropy following from equation (25) resembles that of flat-band systems of the so-called monomer universality class [3,5], the universal magneto-thermodynamics for both systems is different: For monomer flat-band systems the partition function is as in equation (29), however, with F k (x, N ) = exp(kx), see reference [5].
A nonzero residual ground-state entropy leads to efficient magnetic cooling [5,38,39]. Importantly, for models 1 and 2 the residual entropy is present at zero field, i.e., it is relevant for cooling by varying the field around zero [40,41], which is obviously an advantage from the practical point of view compared to model 3.
From the Helmholtz free energy obtained from the partition function (29) by F fbs (T, h, N ) = −T ln Z fbs (T, h, N ) we get thermodynamic quantities such as magnetization, susceptibility, entropy or specific heat. The magnetization M = −∂F/∂h and the susceptibility X = ∂M/∂h are given by the formulas 2 This result follows from the formula for the total number of ground states for the open sawtooth chain 3 of N = 2N + 1 sites at h = hsat, which implies the following recurrence relation: (to prove it, one has to use the identity Evidently, we have arrived at a Fibonacci sequence. In the limit N → ∞, the ratio W(N − 2)/W(N ) tends to ϕ = ( √ 5 − 1)/2, i.e., in the and T X fbs (x, N ) respectively. The entropy S = −∂F/∂T and the specific heat C = T ∂S/∂T are given by the formulas and respectively. Clearly, the magnetization is an odd function of x, whereas the susceptibility, the entropy, and the specific heat are even functions of x. Below we consider the thermodynamic quantities per site to be denoted by small letters, e.g., m = M/N , etc. The contribution of the flat-band states to the partition function is identical for models 1 and 2 in the thermodynamic limit as well as for finite open sawtooth chains. This contribution was discussed in detail in reference [16]. In particular, it was shown that the magnetization m fbs (x) calculated in such a reduced basis depends essentially on N and in the thermodynamic limit N → ∞ it tends to 1/4 at x = h/T → 0 in contradiction to the theorem that the magnetization should vanish in vanishing field at T > 0 for one-dimensional systems. However, for finite chains the "reduced-set" magnetization given by equation (30) may give a good estimate, see Figure 3 and discussion after equation (31) in reference [16].
To illustrate the contribution of the localized flatband states to the low-temperature thermodynamics in a magnetic field, we compare in Figure 4 thermodynamic quantities as they follow from equation (30) It is obvious that for larger magnetic fields (i.e., large values of x in Fig. 4) the localized-magnon states cover the thermodynamics not only at very low temperatures.
In Figure 5 we show the specific heat in dependence on x = h/T at the same three temperatures as in Figure 4 for  As already mentioned above, the universal behavior is identical for models 1 and 2 (but not for model 3). Leaving the range of validity of formulas (30)-(33) which display the contributions of flat-band states only, the physics of the full spin model is determined more and more also by the non-flat-band states. This issue is studied in the next section by large-scale numerical calculations of the full spin models 1, 2, and 3.

Numerical investigations of finite systems
Now we consider models 1 and 2 at zero field and model 3 at the saturation field, i.e., all excitations are non-flatband states. We present full exact diagonalization (ED) and finite-temperature Lanczos (FTL) data for the considered sawtooth-chain models of finite size N . We focus here on periodic chains allowing to access larger system-sizes N by exploiting the translational symmetry not present in open chains. Using ED we can study up to N = 20 and using FTL we will provide data up to N = 32.
We begin with the excitation gaps ∆ (i) (S z ) (i = 1 corresponds to model 1 and i = 2 corresponds to model 2), considered in each sector of S z separately, see Table 1 for N = 20 (and also Tab. I in Ref. [16] for model 2 with N = 16, 20, 24, 28 and k = 1, . . . , 6). For both models the gaps ∆ (i) (S z = N/2 − k) are rather small if k > 1. However, for model 2 the gap ∆ (2) becomes virtually zero for k > 4, but not for model 1. This means that the contribution of low-lying excited states to the partition function enters for model 2 at lower temperatures than for model 1, compare the specific heat in the low-temperature region shown in Figure 7 and discussed below.
First, we pass to the density of states, see Figure 6. Although the models 1 (J 1 = 1, blue curves) and 2 (J 1 = 1/2, magenta curves) do not have identical energy spectra, a similarity between both models is evident. On the other hand, the density of states of model 3 (J 1 = 1/2, red Table 1. Ground-state degeneracies G (i) (S z ) and excitation gaps ∆ (i) (S z ) ≡ E (i) 1 (S z ) − E 0 for the periodic sawtooth-chain model 1 (J 1 = 1, i = 1) and model 2 (J 1 = 1/2, i = 2) both of N = 20 sites (N = 10) in different subspaces S z . E (i) 1 (S z ) is the energy of the lowest excitation in the subspace S z and E 0 = −3.75 is the ground-state energy (which is identical for models 1 and 2). The results in the third and fifth columns coincide with the predictions according to equations (27) and (26). For (finite) periodic sawtooth chains of N = 20 sites, the total number of ground states for model 1 is 4445 and for model 2 it is 3545. curves in Fig. 6) is completely different. A striking feature of the density of states of models 1 and 2 is the collection of about 6% of the states in the low-energy region below E − E 0 0.6, where this region is separated by a quasi-gap from the high-energy region E − E 0 0.6. This feature together with the huge ground-state degeneracy is responsible for the unconventional low-temperature physics of models 1 and 2.
We have to comment on the height of the blue and magenta peaks at E − E 0 = 0 in Figure 6. As it has been explained above, the ground-state degeneracy for the periodic model 1 (4445) is larger than for the periodic model 2 (3545), see Table 1. That would imply that the blue peak at E − E 0 = 0 is higher than the magenta one. However, the gaps for model 2 are much smaller than for model 1 and within the first histogram bar between E 0 and E 0 + ∆E with ∆E = 0.02 or ∆E = 0.002 not only the ground states but also excited states are collected. According to the above discussion of the gaps, there are a lot of excited states in the first ∆E interval for model 2 but much less for model 1.
Finally, in Figure 7 we show the temperature dependence (logarithmic temperature scale) of the specific heat for all three models (models 1 and 2 at zero field and model 3 at the saturation field). The specific heat of model 3 is characterized by single pronounced maximum followed by an exponential decay of c(T ) as T → 0 leading to a virtually vanishing specific heat below T ∼ 0.06. By contrast, the very specific low-energy density of states of models 1 and 2 with much smaller energy gaps and the quasigap at about E = E 0 + 0.06 leads to a distinct separation of temperature scales in the temperature dependence of the specific heat which is characterized by a pronounced   low-temperature profile of c(T ) with two additional maxima below the typical main maximum. The difference in the details of the low-energy spectrum of the two models results in a deviation of the c(T ) curves of both models at low temperatures starting at about T = 0.3. For model 1 the finite-size effects are negligible down to T ∼ 0.01, whereas model 2 exhibits practically no finite-size effects in the temperature region shown in Figure 7. This difference can be attributed to the different sizes of the excitation gaps, cf. Table 1.
To demonstrate the relation of the separation of temperature scales in the c(T ) profile to the very specific structure of the density of states of the models 1 and 2 we show in Figure 8 the specific heat c(T ) for periodic chains of N = 20 sites using the full spectrum (i.e., numerically exact data) together with the approximate data for c(T ) which are calculated using a restricted set of energies E < E 0 + 0.06, i.e., only the low-energy spectrum below the quasi-gap is taken into account. This comparison reveals that indeed the unconventional features in c(T ) below the main maximum are entirely covered by the energy levels below the quasi-gap.

Conclusions
To summarize, in the present paper we have examined three spin-1/2 sawtooth-chain models which for a special choice of parameters exhibit flat-band physics. While the universal flat-band behavior of the two models introduced in references [25] and [16] is identical, it is different from the universal flat-band behavior of the model introduced in references [5,13]: In the latter case flat-band many-magnon localized states are independent localized magnons only, whereas in the former two cases the flat-band ground-state manifold is larger containing in addition specifically overlapping localized magnons (many-magnon complexes). Flat-band states dominate the low-temperature thermodynamics around zero field (models 1 and 2) or around the saturation field (model 3). The localized nature of flat-band states allows the complete analysis of the massively degenerate ground-state manifold, and, as a result, to find analytical expressions for the low-temperature thermodynamics in presence of a magnetic field. These analytical expressions are complemented by numerical calculations of thermodynamic quantities of finite sawtooth spin chains.
The spin-1/2 XXZ0 sawtooth-chain model admits another promising route to examine its properties using the three-coloring representation [25,26]. Interesting and still open problems are: how to present the localized many-magnon complex states within the three-coloring picture? Is it possible to use the three-coloring representation for constructing thermodynamics? In the future, it might be interesting to consider small perturbations to the basic flat-band sawtooth-chain models (1). One extension is to introduce into Hamiltonian (1) the interaction between neighboring apical sites J 1 , ∆ 1 to see traces of the flat-band-states manifolds.

Author contribution statement
All the authors were involved in the preparation of the manuscript. All the authors have read and approved the final manuscript.
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Appendix A: Some analytical calculations supplementary to Section 3 Here we present some detailed analytical calculations mentioned but not presented in the main text.
First we check that in the subspace S z = N/2 − 1 the boundary and bulk localized-magnon states defined in equations (14) and (15) are exact ground states for model 1 of N = 2N + 1 sites with open boundary conditions, where N is the number of triangles in this chain. We begin with the boundary state