Higgs-Axion conversion and anomalous magnetic phase diagram in TlCuCl$_3$

What is so unique in TlCuCl$_3$ which drives so many unique magnetic properties, such as a massive Higgs mode at the magnetic critical point, long-lived paramagnons, and dimerized antiferromagnetism? To study these properties, we emply here a combination of {\it ab-initio} band structure, tight-binding parameterization, and an effective quantum field theory. Within a density-functional theory (DFT) calculation, we find an unexpected bulk Dirac cone without spin-orbit coupling (SOC). Tracing back to its origin, we identify, for the first time, the presence of a Su-Schrieffer-Heeger (SSH) like dimerized Cu chain lying in the 2D crystal structure. The SSH chain, combined with SOC, stipulates an anisotropic 3D Dirac cone where chiral and helical states are intertwined (namely, 3D SSH model). As a Heisenberg interaction is introduced, we show that the dimerized Cu sublattices of the SSH chain condensate into dimerized spin-singlet magnets. In the magnetic ground state, we also find a naturally occurring topological phase, distinguished by the axion invariant. Finally, to study how the topological axion excitation couples to magnetic excilations, we derive a Chern-Simons-Ginzburg-Landau action from the 3D SSH Hamiltonian. We find that axion term provides an additional mass term to the Higgs mode, and a lifetime to paramagnons, which are independent of the quantum critical physics.