Fig. 7: Experimental probes of the trimer quantum spin liquid (TQSL).

Two experimental probes that together conclusively characterize a state as a TQSL are the measurements of Z- and X-loop operators. a The main experimental probe that distinguishes between trivial and TQSL phases entails checking the two U(1) conservation laws³⁹ relating the electric field (arrows) flux with enclosed charges (number of different sublattice sites) along an arbitrary closed loop (example shown in blue). b Two Z-loop operators evaluated for the loop shown in blue in (a) for the 10 × 6 shape 1 cluster from Fig. 4a. The two U(1) conservation laws are violated in the trivial phase, but are approximately satisfied in both the TQSL and VBS (valence bond solid) phases. The second transition between the TQSL and the VBS present on this cluster is not distinguishable from Z-loop measurements alone, as both phases are of trimer character. c Expectation value of ∣Z∣, evaluated by sampling from DMRG ground state of 32 × 4 cylinder and averaging over the bulk of the system (discarding four rows of atom on each end of the system). The violations of the U(1) conservation laws are high in the trivial and columnar phase and low in the TQSL phase (denoted by dashed green lines extracted from energy susceptibility peaks in Fig. 3). Sampling errors are smaller than the marker size. d The X-loop experimental probe can be explicitly constructed for a given breathing move on the irregular honeycomb lattice of trimer domain walls shown here (see Supplementary Note 1). The X-loop operator is a product of σ_x operators needed to expand the loop. e An example of a simple breathing move is shown with configurations before and after the move differing only within the yellow shell around the perimeter of the loop, a general feature of any breathing move. The moves needed to expand the honeycomb, in this case, lead to the many-body X-loop operator that is a product of 24 individual honeycomb σ_x operators.