Coexistence of symmetry-protected topological order and Neel order in the spin-1/2 ladder antiferromagnet C9H18N2CuBr4

Topological phases of matter are beyond the paradigm of Landau's symmetry breaking and have challenged our understanding of condensed matter systems. Here we report a new type of symmetry-protected topological phase of matter in the spin-1/2 coupled two-leg ladder antiferromagnet C9H18N2CuBr4, DLCB for short. In this two-sublattice antiferromagnet with a weak easy-axis anisotropy, we find no evidence of a conventional spin-flop transition in the magnetization with the magnetic field applied parallel to the easy axis at T=0.4 K, well below TN=2.0 K. Moreover, the temperature dependence of the gapped transverse excitations across TN indicates that they are not the conventional S=1 magnons associated with explicit symmetry breaking. Instead, the thermal renormalization of the gap energy shows a remarkable agreement with a calculation for the three-dimensional O(3) nonlinear sigma model. Accordingly, the spin gap in DLCB is not due to the spin anisotropy but to the separation between a spin singlet state and a triplet excited state. Since an antiferromagnetic spin-1/2 ladder systems can be mapped onto the spin-1 chain, the notion of the Haldane gap is proposed to explain the opening of the spin gap in DLCB. Therefore, the ground state of DLCB is best described as a quantum superposition of a Haldane phase and a Neel-ordered phase, which resembles the quantum state of a qubit in quantum computing. Our results indicate the presence of a symmetry-protected topological order coexisting with an antiferromagnetic order in this material.