Phases and phase transitions of an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> chain on metallic and

Motivated by recent scanning tunneling microscopy experiments on chains of Co adatoms on Cu surfaces, we investigate the physics of a spin-$3/2$ Heisenberg chain with single-ion anisotropy ($D$) on metallic and semi-metallic surfaces. In the strong Kondo coupling ($J_k$) limit, a perturbative analysis maps the system onto a Haldane spin-1 chain with single-ion anisotropy, ferromagnetically coupled to the metallic surface. This Haldane state, arising from underscreening of the $S=3/2$ chain, is stable against small $D$ and characterized by topological edge modes. The nature of the $D$-driven transitions out of this state depends on the environment. Coupling to a metal (semi-metal) is a relevant (irrelevant) perturbation at the decoupled fixed point between the spin-1 chain and the two-dimensional electron gas. In the large positive $D$ limit, the system maps onto an anisotropic spin-$1/2$ Kondo system. For large negative $D$, in the Ising phase, spins are frozen. For small $J_k$, the nature of the metallic phase dominates. On a two-dimensional semi-metal, the Kondo coupling is irrelevant at the decoupled fixed point ($J_k= 0$), leading to a Kondo breakdown phase at weak coupling, irrespective of $D$. In contrast, on a two-dimensional metal, the resulting dissipative Ohmic bath is a marginally relevant perturbation, inducing antiferromagnetic ordering along the chain. In this case, $D$ drives a spin-flop transition between Ising and XY ordered phases. At $D=0$, we observe continuous transitions between the Kondo breakdown or dissipation-induced long-range ordered phases and the underscreened Haldane phase. These phase diagrams are supported by scaling arguments and sign-free auxiliary-field quantum Monte Carlo simulations.