Thermodynamics of the spin-half square kagome lattice antiferromagnet

Over the last decade, the interest in the spin-$1/2$ Heisenberg antiferromagnet (HAF) on the square kagome (also called shuriken) lattice has been growing as a model system of quantum magnetism with a quantum paramagnetic ground state, flat-band physics near the saturation field, and quantum scars. A further motivation to study this model comes from the recent discovery of a gapless spin liquid in the square kagome magnet $\mathrm{K}{\mathrm{Cu}}_6\mathrm{Al}\mathrm{Bi}{\mathrm{O}}_4{\left(\mathrm{S}{\mathrm{O}}_4\right)}_5\mathrm{Cl}$ [M. Fujihala et al., Nat. Commun. 11, 3429 (2020)]. Here, we present large-scale numerical investigations of the specific heat $C\left(T\right)$, the entropy $S\left(T\right),$ as well as the susceptibility $\chi \left(T\right)$ by means of the finite-temperature Lanczos method for system sizes of $N=18,24,30,36,42,48$, and $N=54$. We find that the specific heat exhibits a low-temperature shoulder below the major maximum which can be attributed to low-lying singlet excitations filling the singlet-triplet gap, which is significantly larger than the singlet-singlet gap. This observation is further supported by the behavior of the entropy $S\left(T\right)$, where a change in curvature is present just at about $T/J=0.2$, the same temperature where the shoulder in $C$ sets in. For the susceptibility the low-lying singlet excitations are irrelevant, and the singlet-triplet gap leads to an exponentially activated low-temperature behavior. The maximum in $\chi \left(T\right)$ is found at a pretty low temperature $T_{\max }/J=0.146$ (for $N=42$) compared to $T_{\max }/J=0.935$ for the unfrustrated square-lattice HAF signaling the crucial role of frustration also for the susceptibility. We find a striking similarity of our square kagome data with the corresponding ones for the kagome HAF down to very low $T$. The magnetization process featuring plateaus and jumps and the field dependence of the specific heat that exhibits characteristic peculiarities attributed to the existence of a flat one-magnon band are discussed as well.

Figure 1

Main panel: Modified Wilson ratio $P$, cf. Eq. (3), of the spin-$1/2$ SKHAF ($N=42$) compared with respective data of the kagome HAF ($N=42$) and square-lattice HAF ($N=40$). Note that the data for $N=36$ (not shown) practically coincide with the data for $N=42$. Note further that corresponding data for $N=36$ for the kagome and the square-lattice HAF where previously presented in Ref. [47]. Inset: Sketch of the square kagome lattice. Here A and B label the two nonequivalent sites and $J$ and $J^{\prime }$ label the two nonequivalent nearest-neighbor bonds. All calculations are preformed for the balanced case $J^{\prime }=J$.

Figure 2

(a) Specific heat of the SKHAF as function of temperature for various systems sizes (logarithmic temperature scale). Lattice structures for $N=24,30,36,42$ as in Ref. [18], also see the Appendix. The red arrow shows the emergence of the shoulder in $C\left(T\right)/N$. (b) Specific heat of the SKHAF and the kagome HAF ($N=42$) as function of temperature (linear temperature scale).

Figure 3

Histogrammed density of states of the $N=42$ SKHAF (solid curves) and the kagome HAF (dashed curves) as a function of the respective excitation energy $E^{*}$: total density of states, black, total density of states for $\left|M\right|=1$, red, for $\left|M\right|=2$, green, and for $\left|M\right|=3$, blue (curves from left to right). For technical details, see Ref. [50].

Figure 4

(a) Entropy of the SKHAF as function of temperature for various systems sizes (logarithmic temperature scale). (b) Entropy of the SKHAF and the kagome HAF ($N=42$) as function of temperature (linear temperature scale).

Figure 5

(a) Susceptibility of the SKHAF as function of temperature for various systems sizes (logarithmic temperature scale). (b) Susceptibility of the SKHAF and the kagome HAF Heisenberg antiferromagnet ($N=42$) as function of temperature (linear temperature scale).

Figure 6

Magnetization of the SKHAF for various lattice sizes and $T=0$ (main panel) and width of plateaus (inset).

Figure 7

Magnetization of the SKHAF for $N=42$ and $T\ge 0$.

Figure 8

Differential susceptibility $(dM/dB)$ of the SKHAF for various temperatures.

Figure 9

(a) Main: Specific heat of the SKHAF for $N=42$ and selected values of $B$. Inset: $M\left(B\right)$ with arrows indicating the values of $B$ used in the main panel. (b) Finite-size dependence of the specific heat for $B$ values in the middle of the $1/3$ (red lines) and $2/3$ (black lines) plateaus. Note that for $N=48$ and 54 lower sectors of $S_z$ are not taken into account in Eq. (2) with the result that the upper maximum is not correctly evaluated.

Figure 10

Position $T_{\max }$ (solid line) and height $C_{\max }$ (dashed-dotted line) of the upper main maximum in $C\left(T\right)$ in dependence on the magnetic field $B$ shown for the SKHAF of $N=42$ sites, the kagome HAF of $N=42$ sites and the square-lattice HAF of $N=40$ sites.

Figure 11

Finite lattices used for FTLM. The structures 24, 30, 36, 42 are the same as used in Ref. [18].