Rydberg atom arrays have emerged as a powerful platform to simulate a number of exotic quantum ground states and phase transitions. To verify these capabilities numerically, we develop a versatile quantum Monte Carlo sampling technique which operates in the reduced Hilbert space generated by enforcing the constraint of a Rydberg blockade. We use the framework of stochastic series expansion and show that in the restricted space, the configuration space of operator strings can be understood as a hard rod gas in $d+1$ dimensions. We use this mapping to develop cluster algorithms which can be visualized as various non-local movements of rods. We study the efficiency of each of our updates individually and collectively. To elucidate the utility of the algorithm, we show that it can efficiently generate the phase diagram of a Rydberg atom array, to temperatures much smaller than all energy scales involved, on a Kagom\'e link lattice. This is of broad interest as the presence of a $Z_2$ spin liquid has been hypothesized recently.
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Data for all plots available on Zenodo at https://zenodo.org/records/14922067
List of changes
1. Ordering of Fig 1 and 2 switched. 2. Fig.6(a) moved to Appendix. 3. All instances of Monte Carlo sizes have been explicitly labelled as "L=..." . 4. The following sentences have been added after the first paragraph of Sec.4 : "In the phase diagram discussed below, we thus expect to find four phases. At high temperature, we expect a simple paramagnetic phase with maximal entropy and no signature in the order parameters we use to distinguish the other phases. At low temperature and $\Omega\gg\delta$, we expect the quantum paramagnet, which differs from the the high temperature one only in terms of entropy as it should have zero entropy for temperatures below the energy gap. At $\Omega\ll\delta$, we expect the classical spin liquid for $\Omega<T<\delta$, which has a non-zero entropy which is lesser than the maximal possible. In addition to the entropy, we also use a string order parameter to distinguish between the classical spin liquid and the quantum paramagnet as the former shows a non-zero value for strings of significant size whereas the latter does not. The quantum spin liquid is expected to emerge when we lower the temperature to $T\ll\Omega$ starting from the classical spin liquid regime, and this is characterized by zero entropy and a string order parameter behavior similar to that of the classical spin liquid. We do not find evidence of this last phase in the phase diagram which we are able to generate using the QMC simulations."
