Magnetism of the $s=1/2$ $J_1$-$J_2$ square-kagome lattice antiferromagnet

The spin-$1/2$ Heisenberg antiferromagnet on the square-kagome (SK) lattice has attracted growing attention as a model system of highly frustrated quantum magnetism. A further motivation for theoretical studies comes from the recent discovery of SK spin-liquid compounds. The SK antiferromagnet exhibits two non-equivalent nearest-neighbor bonds $J_1$ and $J_2$. One may expect that in SK compounds $J_1$ and $J_2$ are of different strength. We present a numerical study of finite systems by means of the finite-temperature Lanczos method. We discuss the temperature dependence of the specific heat $C(T)$, the entropy $S(T)$, and of the susceptibility $X(T)$ of the $J_1$-$J_2$ SK Heisenberg antiferromagnet varying $J_2/J_1$ in the range $0 \le J_2/J_1 \le 4$. We also discuss the zero-field ground state of the model. We find indications for a magnetically disordered singlet ground state for $0 \le J_2/J_1 \lesssim 1.65$. Beyond $J_2/J_1 \sim 1.65$ the singlet ground state gives way for a ferrimagnetic ground state. In the region $0.77 \lesssim J_2/J_1 \lesssim 1.65$ the low-temperature thermodynamics is dominated by a finite singlet-triplet gap filled with low-lying singlet excitations leading to an exponentially activated low-temperature behavior of $X(T)$. On the other hand, the low-lying singlets yield an extra maximum or a shoulder-like profile below the main maximum in the $C(T)$ curve. For $J_2/J_1 \lesssim 0.7$ the low-temperature thermodynamics is characterized by a large fraction of $N/3$ weakly coupled spins leading to a sizable amount of entropy at very low temperatures. In an applied magnetic field the magnetization process features plateaus and jumps in a wide range of $J_2/J_1$.


I. INTRODUCTION
Highly frustrated quantum antiferromagnets on twodimensional lattices have attracted an enormous attention over more than three decades, see, e.g., [1][2][3][4][5]. "Now in the early 2020s, quantum magnetism is a mature field showing no signs of senescence. To the contrary, there is a tremendous amount of activity studying exotic magnetic phenomena especially with strong quantum fluctuations." [6] Over many years the kagome antiferromagnet (KHAF) has been the holy grail in this field. Quite recently the square-kagome antiferromagnet, the 'little brother' of the kagome antiferromagnet, has received more appreciation because several magnetic compounds with square-kagome lattice structure have been found which do not exhibit magnetic order down to very low temperatures [7][8][9][10]. The square-kagome lattice (sometimes also called shuriken or squagome lattice) [11][12][13][14] is a two-dimensional tiling built of squares and corner-sharing triangles. The classical ground state of the square-kagome Heisenberg antiferromagnet (SKHAF) is highly degenerated (classical spin liquid). There are Starting in 2013 the interest in the spin-1/2 SKHAF has been growing as a model system exhibiting a nonmagnetic quantum ground state, magnetization plateaus, flat-band physics near the saturation field and quantum scars [14,[17][18][19][20][21][22][23][24][25][26][27][28][29]. All these papers were focused on zerotemperature properties. Only, in the early paper [16] specific-heat data calculated by a simple renormalization group approach were reported. The thermodynamics of the balanced spin-1/2 SKHAF, i.e., J 1 = J 2 = J, has been studied quite recently in Ref. [30] using the finitetemperature Lanczos method (FTLM). At zero magnetic field we find that the KHAF and SKHAF exhibit a striking similarity of the temperature profile of C(T ), X(T ) and S(T ) down to very low temperature T . Thus, for X(T ) and S(T ) an almost perfect coincidence for both models was observed. For the specific heat there is a perfect agreement of the C(T ) data down to T /J = 0.3. A characteristic feature common in both models is the existence of low-energy singlet excitations filling the magnetic spin gap [13,[31][32][33]. These lowenergy singlets yield a low-temperature shoulder below arXiv:2212.10838v2 [cond-mat.str-el] 21 Jun 2023 the major maximum in the C(T ) profile [30,34]. We mention that such a shoulder has been observed in a recent experiment on the kagome quantum antiferromagnet YCu 3 (OH) 6 Br 2 [Br x (OH) 1−x ] [35]. The subtle details of the singlet excitations depending on the shape and the size N of the finite lattices lead to deviations between the behavior of C(T ) for both models at very low T .
Bearing in mind the recent experimental studies on square-kagome quantum antiferromagnets [7][8][9][10] and the non-equivalence of the nearest-neighbor bonds J 1 and J 2 we may expect that for the modeling of square-kagome compounds it is natural to consider a spin model with J 1 ̸ = J 2 . Moreover, the J 1 -J 2 model is interesting in its own right as highly frustrated model that allows to tune the competition between the bonds.
So far only a few papers exist which study the zerotemperature properties of the J 1 -J 2 model [18,20,[23][24][25] where in Ref. [23] the focus is on the magnetization process of the J 1 -J 2 model with only slight deviations from the balanced model, i.e., the difference between J 1 and J 2 is small. In our paper we will fill the gap of missing nonzero-temperature studies and present FTLM data for the magnetization M , the Wilson ratio P , the specific heat C, the entropy S and the uniform magnetic susceptibility X of the J 1 -J 2 SKHAF. In addition, we will analyze the ground state of the finite lattices used for the FTLM studies which allows to get a relation between ground-state and finite-temperature properties of the investigated systems.
The corresponding Heisenberg Hamiltonian augmented with a Zeeman term is given by where s 2 i = s(s + 1) = 3/4. The J 1 bonds represent the nearest-neighbor exchange connecting A sites on the squares, whereas the J 2 bonds represent the nearestneighbor exchange connecting A with B sites on the triangles, see the left inset in Fig. 1. In what follows we set J 1 = 1.
The paper is organized as follow. In Section II we introduce our numerical scheme. In Section III we present our results where in Section III A we briefly discuss the ground-state properties as well as the excitation gaps of the model which may be relevant for the interpretation of the low-temperature thermodynamics. The results for the temperature dependence of the Wilson ratio P (T ), the specific heat C(T ), the entropy S(T ) as well as the susceptibility X(T ) at zero magnetic field are presented and discussed in Section III B. Finally, in Section III C we discuss the magnetization process in an applied magnetic field. In the last Section IV we summarize our findings. In two appendices we show the finite lattices considered in our paper (App. A) and provide some additional figures to illustrate finite-size effects (App. B).

II. CALCULATION SCHEME
The magnetic system under consideration is modeled by the spin-1/2 Heisenberg Hamiltonian given in Eq. (1). We use the conservation of the z-component of the total spin S z = i s z i as well as lattice symmetries, i.e., the Hilbert space splits into subspaces characterized by the eigenvalues of S z (magnetic quantum number M ) and of the symmetry operator, see, e.g., Refs. [36,37]. To calculate the ground state we perform Lanczos exact diagonalization in the sector M = 0. For that we use Jörg Schulenburg's publicly available package spinpack [38,39].
For the FTLM scheme we also exploit the package spinpack as well as the conservation of S z and the symmetries to decompose the Hilbert space into much smaller subspaces. The FTLM is meanwhile a well established and accurate approach to calculate thermodynamic quantities of frustrated quantum spin systems [30,34,[40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55]. We do not present a detailed description of the method, rather we will provide the basics of the FTLM for convenience. Within the FTLM the sum over an orthonormal basis in the partition function is replaced by a much smaller sum over R random vectors: where the | ν ⟩ label random vectors for each symmetryrelated orthogonal subspace H(γ) of the Hilbert space with γ labeling the respective symmetry. In Eq. (2), the exponential of the Hamiltonian has been replaced by its spectral representation in a Krylov space spanned by the N L Lanczos vectors starting from the respective random vector | ν ⟩, where | n(ν) ⟩ is the n-th eigenvector of H in this Krylov space. For more information we refer the interested reader to the reviews [45,48] and to our recent FTLM papers of the KHAF [34] and SKHAF [30]. A detailed discussion of the accuracy of the FTLM can be found in Refs. [34] and [53]. Based on this knowledge, we chose the number of random vectors R along the lines of our previous study [34]. The absence of magnetic long-range order for the balanced s = 1/2 SKHAF (J 1 = J 2 ) was established by previous studies [13,18,20,25,28,29]. The nature of the ground state is still under debate, candidates are a pinwheel valence-bond-crystal ground state [18,28], a loopsix valence-bond state [20,29] or a topological nematic spin liquid [25]. The ground-state phase diagram of the J 1 -J 2 model was studied in Ref. [25] using a Schwingerboson mean field theory as well as in Refs. [18,20] using a resonating valence-bond approach.
Here we present Lanczos exact diagonalization data for N = 30 and N = 36. Note that a brief discussion of the ground state for N = 24 and N = 30 was already given in Refs. [20,24]. Our ground-state data will be useful to compare with the Schwinger-boson data [25] as well as for the interpretation of the low-temperature thermodynamics.
To get an impression on possible ground state magnetic order we first consider an order parameter introduced in Ref. [2] that measures the total strength of the overall spin-spin correlations without any assumptions on possible magnetic order with a related ordering vector Q. It is defined as Numerical ground-state data for m + are depicted in Fig. 1, main panel. It is obvious that in a wide parameter range 0 ≤ J 2 ≲ 1.65 the order parameter m + is approximately of the same small size as for the balanced model (J 2 = 1) which is known to be in a non-magnetic singlet ground state. Thus we may argue that there is no magnetic ground-state state order for J 2 ≲ 1.65. The steep increase of m + beyond J 2 ≈ 1.65 is related to a transition from a singlet ground state to a ferrimagnetic ground state with non-zero magnetization M . The jumps in the m + (J 2 ) curve visible for 1.65 ≲ J 2 ≲ 1.83 are related to a stepwise increase of M up to M 1/3 = M sat /3. The ground state with M 1/3 present for J 2 ≳ 1.83 is a ferrimagnetic up-up-down (UUD) state, i.e., ⟨s z i∈A ⟩ and ⟨s z i∈B ⟩ are antiparallel. To give an example, for N = 36, J 2 = 2, we have ⟨s z i∈A ⟩ = 0.39779 and ⟨s z i∈B ⟩ = −0.29558. We mention that for the classical model the transition to the UUD state takes place at J 2 = 2, i.e., the order-bydisorder mechanism [56,57] leads to a shift of the transition to the collinear UUD state to smaller values of J 2 . Bearing in mind the Schwinger-boson mean-field study of the ground state reporting 5 ground state phases [25] it is worth to have a closer look on the details of the m + (J 2 ) profile. Indeed, there are small discontinuous changes in where the values at about 0.85 and 1.33 are close to transition points reported in [25]. We also mention that below J 2 ≈ 0.77 the spins on B-sites become weakly coupled to the neighboring A-site spins, whereas the nearestneighbor correlations on the J 1 bonds asymptotically approach the value of the square-plaquette singlet ground state (see the right inset in Fig. 1), i.e., the system enters a plaquette ground-state phase at low values of J 2 .
For the low-temperature thermodynamics the spin gap (singlet-triplet gap) ∆ t as well as the singlet-singlet gap ∆ s are relevant. Corresponding data are shown for N = 30, N = 36 and N = 42 in Fig. 2. Our data provide evidence that there is a finite spin gap ∆ t in the region between J 2 ≈ 0.77 (J 2 ≈ 0.74) and J 2 ≈ 1.65 (J 2 ≈ 1.65) for N = 36 (N = 30). We notice only a small finite-size dependence of the spin gap away from J 2 = 1, whereas in the vicinity of J 2 = 1 it shrinks with increasing N . However, it is known that ∆ t remains finite at J 2 = 1 for N → ∞ [29]. The vanishing of the spin gap at J 2 ≈ 0.77 coincides with the above reported value at which a small discontinuous change in m + occurs, whereas the closing of the spin gap at J 2 ≈ 1.65 is related to the emergence of a ferrimagnetic ground state. Thus, m + as well as ∆ t yield indications for ground-state phase transitions between a gapped and a gapless phase. A similar behavior was found in Ref. [25], where, however, the region of the gapped phase is 0.84 ≤ J 2 ≤ 1.27. While ∆ t determines the low-temperature behavior of the susceptibility X, the existence of low-lying singlet excitations within the spin gap, i.e., ∆ s < ∆ t , is crucial for the low-temperature behavior of the specific heat C. From Fig. 2 it is obvious that in the whole region with a finite spin gap we have ∆ s < ∆ t . As for the balanced model J 1 = J 2 = 1 there are a number of singlets within the spin gap. The details of their energy distribution will determine the temperature profile of C at very low T .  Here A and B label the two non-equivalent sites and J1 and J2 label the two non-equivalent nearest-neighbor bonds. Right inset: Nearest-neighbor spin-spin correlation for N = 36: ⟨si · sj⟩NN,A−A = (⟨s0 · s1⟩ + ⟨s0 · s3⟩/2 and ⟨si · sj⟩NN,A−B = (⟨s0 · s4⟩ + ⟨s1 · s4⟩/2, see Fig. 12   P , the specific heat C, the entropy S, and the uniform susceptibility X.
The modified Wilson ratio is defined as [58,59] It is a measure of the ratio of the density of magnetic excitations with M > 0 and the density of all excitations including singlet excitations with M = 0. As shown for the KHAF [58,59] and for the balanced SKHAF [30] a vanishing P as temperature T → 0 is a hallmark of a quantum spin-liquid ground state with dominating singlet excitations at low T . In contrast, for quantum spin models with semi-classical magnetic ground-state order, such as the square-lattice Heisenberg antiferromagnet, the Wilson ratio diverges according to a power-law [58,59]. We show the modified Wilson ratio in Fig. 3. For J 2 = 0.8, 0.9, 1.0, 1.1, 1.2, , 1.3, 1.4, 1.5 singlet excitations are noticeably below the first triplet excitation. As a result there is an obvious downturn of P as T → 0. Also the upturn of P as T → 0 for J 2 = 1.7 and 1.8 (ferrimagnetic ground state) is evident. More subtle is the situation for J 2 < 0.8, where the plaquette groundstate phase emerges. Here the low-lying spectrum is dominated by the weakly coupled spins on the B sites. which leads to a maximum in P at low temperatures, see Fig. 3(b). This behavior can be understood by considering the ground state in the limit of decoupled B spins, i.e., for J 2 = 0. In this limit we get a size independent Wilson ratio P 0 = lim T →0 P = π 2 /(3 ln 2) = 4.74628. Obviously, the height of the low-temperature maximum in P approaches P 0 as decreasing J 2 . At very low T the Wilson ratio approaches a constant value of about P ≈ 2. (Note, however, that our FTLM is not appropriate to get accurate data precisely at T = 0, because in the limit of very weakly coupled B spins very tiny energy differences appear in the low-energy spectrum.) As reported in Ref. [58] this behavior corresponds to a gapless spin liquid; in particular, for the one-dimensional s = 1/2 Heisenberg antiferromagnet (Bethe chain) P 0 is exactly 2 [58,60].  Let us now discuss the specific heat C(T ), the entropy S(T ) and the uniform susceptibility X(T ). We use a logarithmic temperature scale which makes the lowtemperature features transparent, see Figs. 4, 6, and 7. In panels (a) we show data for J 2 ≥ 1 and in panels (b) for J 2 ≤ 1. In all these figures we also show the corresponding data for the balanced model [30] which may serve as benchmark data. The typical main maximum is related to the magnitudes of J 1 and J 2 . Its position T max and its height C max /N exhibit a quite regular behavior, see Fig. 5 (a) and (b).
From Fig. 5 it is also evident that T max and C max /N are equal for N = 30 and 36 for all values of J 2 , i.e., the main maximum in C(T )/N is not affected by finite-size effects, see also shown in Fig. 4 the temperature profile exhibits a lowtemperature maximum below the main maximum that indicates an extra-low energy scale. Though, we show in Let us finally mention that for some values of J 2 there is even some additional structure at very low T ≲ 0.02 which most likely can be attributed to finite-size effects.
In highly frustrated quantum magnets we may have a high density of states at low excitation energies [30,34,61]. To shed light on the density of low-lying eigenstates we present the entropy S(T )/N in Fig. 6. We observe, that already at T ∼ 0.2 about 50% of the maximum entropy S(T → ∞) = N ln 2 is acquired. Note that for the unfrustrated square-lattice Heisenberg antiferromagnetthe corresponding value at T ∼ 0.2 is only about 10%, cf. Ref. [34]. Moreover, there is a change in the curvature or even a plateau-like feature in the S(T ) profile below this temperature. In particular, for J 2 ≲ 0.7 we see such a plateau at S/(N ln 2) ∼ 0.1 which can be attributed to a high density of states caused by the weakly coupled B spins in this parameter region. For some values of J 2 (e.g. for J 2 = 2.0 and 1.2) there is a finite value of S(T = 0)/N due a degeneracy of the ground state. However, S(T = 0)/N will become zero as N → ∞. For more information on finite-size effects, see Fig. 16 in Appendix B, where data for N = 30, N = 36 and N = 42 are compared.
Next we turn to the zero-field susceptibility X displayed in Fig. 7. For J 2 values where we have a finite singlet-triplet gap ∆ t , see Fig. 2, X exhibits an exponentially activated low-temperature behavior and there is a maximum in X(T ). Its position T max and its height X max /N exhibit a quite regular behavior, and the finitesize effects in T max and X max /N are small, see Fig. 8.
(For more information on finite-size effects, see Fig. 17 in Appendix B, where data for N = 30, N = 36, and N = 42 are compared.) Around J 2 = 1 the position T max is largest, although, it is still at a pretty low temperature compared to T max = 0.935 for the square-lattice Heisenberg antiferromagnet [62,63], which demonstrates the crucial role of frustration also for the susceptibility. It is also obvious that T max is directly related to the spin gap ∆ T , compare Fig. 8(a)  ing J 2 towards the transition to the ferrimagnetic ground state naturally leads to a diminishing of T max and an increase of X max /N . At J 2 ∼ 1.65 we get T max = 0 and X max /N → ∞.
A similar behavior can be observed for decreasing J 2 towards J 2 = 0. Again the singlet-triplet gap ∆ t becomes smaller and it is effectively zero below J 2 ∼ 0.77 (plaquette ground-state phase), i.e., T max approaches zero. However, here the weakly coupled spins on the B sites lead to extremely low-lying magnetic and non-magnetic excitations. In fact, we find that for the finite systems considered here the ground state is still a non-magnetic singlet but magnetic excitations dominate the X(T )/N profile down to very low T . Thus, the susceptibility indeed vanishes at T = 0, but X(T )/N approaches zero only at T ∼ 10 −4 , 10 −5 , 10 −9 for J 2 = 0.7, 0.5, 0.1, respectively. The magnetization process of strongly frustrated quantum magnets exhibits a number of interesting features, such as plateaus and jumps [64]. Previous studies for the balanced model [13,14,29,30,65] report on wide plateaus at 1/3 and 2/3 of the saturation magnetization M sat . Moreover, there is the typical macroscopic jump to saturation due to the presence of independent localized multi-magnon ground states stemming from a flat one-magnon band [15,[66][67][68].
Let us first present the zero-temperature magnetization curve for selected values of J 2 , see Fig. 9. Both plateaus as well as the jump to saturation known from the balanced model are present for all values J 2 ≤ 1, whereas for J 2 > 1 the jump and the related preceding 2/3 plateau are missing. For J 2 ≤ 1 both plateau states are nonclassical valence-bond states, cf. Refs. [13,29,30,65], where the upper plateau state is the exactly known magnon-crystal product state, i.e., spins on the B sites are fully polarized and the A-spins on a square occupy the lowest triplet eigenstate of the square plaquette with S z plaqu = 1, for an illustration of this state, see, e.g., Fig. 2a in Ref. [69]. For J 2 ≲ 0.7 the transition between the two plateaus becomes steplike. The jump to saturation as well as the magnon-crystal product state are related to the flat one-magnon band which is the lowest one for J 2 ≤ 1. In contrast, for J 2 > 1, the flat onemagnon band is not the lowest one, and, therefore the flat-band related features are not present in the magnetization curve. However, the flat-band related localized multi-magnon states including the magnon crystal are still eigenstates living now as quantum scar states somewhere in the middle of the spectrum [26].
The valence-bond state of the lower plateau is not exactly known but it is approximately described by a product state with fully polarized spins on the B sites and a singlet state of the A-spins on a square, see the inset in Fig. 9, where the spin-spin correlations ⟨s i · s j ⟩ N N,A−A and ⟨s i · s j ⟩ N N,A−B in the 1/3 plateau state are shown. On the other hand, for J 2 > 1 the 1/3 plateau state is semi-classical, namely it is the ferrimagnetic UUD state, cf. Sec. III A and see the inset in Fig. 9.
Using the entire set of calculated magnetization curves for N = 36 (which includes altogether 22 J 2 values in the region 0 ≤ J 2 ≤ 2) we can construct the J 2 -B phase diagram shown in Fig. 10. The saturation magnetization (uppermost line) is given by gµ B B sat = 2 + J 2 for J 2 ≤ 1 and gµ B B sat = 3J 2 for J 2 ≥ 1, and it is size-independent. But also for the other phase boundaries derived from numerical data, the finite-size effects are very small, see Fig. 13 in appendix B.
For elevated temperatures the experimental detection of plateaus may become intricate, because often there is a fast melting of plateaus and jumps, i.e., they are smeared out already at pretty low T , see, e.g., Refs. [65,70]. Therefore, to detect plateaus and jumps in experiments the differential susceptibility X(T, B) = dM (T, B)/d(gµ B B) as a function of B measured at various T is more suitable, cf., e.g., Ref. [71]. Magnetization plateaus show up as pronounced minima in X(B), however, requiring sufficiently low temperatures. On the other hand, a jump of the magnetization leads to a high peak in X(B) at low T .
We present the influence of the temperature on the magnetization curve M (B) and on the differential susceptibility X(B) in Fig. 11 for selected values of J 2 . We observe, that the melting process is most rapid for J 2 ∼ 1, whereas for small and large J 2 the plateaus and the jumps are still well visible at T = 0.2. We notice that the oscillations present for J 2 ≥ 1 at T = 0.05 (green curves) above the 1/3 plateau are finite-size effects.

IV. SUMMARY AND CONCLUSIONS
In our study we performed numerical calculations of thermodynamic quantities such as the magnetization M (T ), the specific heat C(T ), the entropy S(T ) and the susceptibility X(T ) for the J 1 -J 2 spin-half square-kagome Heisenberg antiferromagnet (SKHAF) by using the finitetemperature Lanczos method (FTLM) applied to finite lattices of N = 30, N = 36 and N = 42 sites. Since the SKHAF exhibits two non-equivalent nearest-neighbor bonds, the extension of previous studies [16,30], which were restricted to J 1 = J 2 , on the generalized model with J 1 ̸ = J 2 is natural with respect to experimental realization of the SKHAF, see Refs. [7][8][9][10]. Moreover, the generalized model may serve as a model allowing to tune the competition of antiferromagnetic bonds in a highly frustrated spin system.
The exact-diagonalization data for the ground state indicate magnetic disorder in a wide range of J 2 /J 1 ratios. Only for J 2 ≳ 1.65J 1 the ground state features ferrimagnetic order. In the region 0.77 ≲ J 2 /J 1 ≤ 1.65 the low-temperature thermodynamics is determined by a finite singlet-triplet gap with low-lying singlets within this gap. Therefore, the susceptibility decays exponentially to zero as temperature T → 0, while the specific heat exhibits an extra maximum at low T related to the singlets. For smaller values of J 2 /J 1 ≲ 0.7 the ground state becomes a plaquette ground state with weakly coupled spins on B sites which become asymptotically decoupled as J 2 /J 1 → 0. As a result, the entropy acquires already a large amount at very low temperatures.
In non-zero magnetic field B we find well pronounced   plateaus at 1/3 and 2/3 of the saturation magnetization and a jump from the 2/3 plateau to saturation in the whole region 0 ≤ J 2 /J 1 ≤ 1, whereas for J 2 /J 1 > 1 only the 1/3 plateau is present. At at low and moderate temperature the plateaus are reflected as minima in the differential susceptibility X(B) = dM (B)/d(gµ B B) and a jump is seen as a peak in X(B). Bearing in mind the numerous studies of the lowenergy physics of the related kagome Heisenberg antiferromagnet we argue that our work may also stimulate other studies using alternative techniques, such as tensor network methods, DMRG, numerical linked cluster expansion or Green's function techniques [29,[72][73][74][75][76][77].  Here we provide the finite lattices studied in our paper, see Here we present data for the widths of the 1/3 and 2/3 plateaus (Fig. 13) and show the specific heat for large values of J 2 (Fig. 14). Moreover, we provide additional information on finite-size effects for the specific heat C(T ) (Fig. 15), the entropy S(T ) (Fig. 16) and the susceptibility X(T ) (Fig. 17) by comparing data for N = 30, N = 36, N = 42. there is a noticeable difference around the maximum which is, however, at pretty low T .