Origin of the intermediate-temperature magnetic specific heat capacity in the spin-liquid candidate ${\mathrm{Ca}}_{10}{\mathrm{Cr}}_7{\mathrm{O}}_{28}$

We present several approximate calculations of the specific heat capacity of the model for ${\mathrm{Ca}}_{10}{\mathrm{Cr}}_7{\mathrm{O}}_{28}$ proposed by Balz et al. [Phys. Rev. B 95, 174414 (2017)], using methods including exact diagonalization, thermal pure quantum states, and high-temperature expansions. In none of these cases are we able to reproduce the magnitude of the zero-field specific heat capacity shown in the intermediate-temperature $(\sim 5–15\mathrm{K})$ experimental data. We discuss possible reasons for the discrepancy, and what it might tell us about the magnetic Hamiltonian for ${\mathrm{Ca}}_{10}{\mathrm{Cr}}_7{\mathrm{O}}_{28}$.

Figure 1

A section from a single bilayer of ${\mathrm{Ca}}_{10}{\mathrm{Cr}}_7{\mathrm{O}}_{28}$. The blue and red spheres denote ${\mathrm{Cr}}^{5+}$ ions from layers 1 and 2, respectively. All distinct couplings in the breathing bilayer kagome model are indicated by their symbol and associated value, as determined by Balz et al. in Ref. [11].

Figure 2

The specific heat capacity of the breathing bilayer kagome model (1) using the strong-triangle approximation, in zero magnetic field (solid black curve) and in a magnetic field of $12\mathrm{T}$ (dotted black curve). Experimental data, previously published in Ref. [11], are also shown for comparison.

Figure 3

The specific heat capacity of the breathing bilayer kagome model (1) calculated using exact diagonalization of a cluster of 24 spins (12 per layer), in zero magnetic field (solid black curve) and in a magnetic field of $12\mathrm{T}$ (dotted black curve). “ED*” denotes that the exact diagonalization was performed with $J_{\mathrm{inter}}=0$. Experimental data, previously published in Ref. [11], are shown for comparison.

Figure 4

The specific heat capacity of the breathing bilayer kagome model (1) calculated using thermal pure quantum states for clusters of 18, 24, and 30 spins, in the absence of a magnetic field. For 18 spins we averaged across 100 runs, for 24 spins we averaged across eight runs, and for 30 spins only a single run was used. In all cases the uncertainty margin at each temperature never exceeds $\sim 1\%$. The experimental zero-field specific heat capacity of ${\mathrm{Ca}}_{10}{\mathrm{Cr}}_7{\mathrm{O}}_{28}$, previously published in Ref. [11], is also shown. For clarity, we have included an inset which enlarges the heat capacity curves at the lowest temperatures.

Figure 5

The specific heat capacity of the breathing bilayer kagome model (1) in the absence of a magnetic field, calculated using the high-temperature expansion to various orders as stated in the legend. The curves labeled “HTE*” were calculated using a software package [23] allowing only four different coupling strengths, so we set $J_{\nabla 1}=J_{\mathrm{\Delta }2}$ in these cases (which is in any case consistent with the quoted uncertainties in these coupling constants). The various Padé curves were also computed using the aforementioned software package; they are based on the tenth-order expansion obtained by the software, labeled here as “HTE* $10\mathrm{th}$ order.” For comparison, we show our 30-spin thermal pure quantum states curve for the model, as well as the experimental zero-field specific heat capacity data previously published in Ref. [11].