  Although hydrogen is the simplest of atoms, it does not form the simplest of
solids or liquids. Quantum effects in these phases are considerable (a
consequence of the light proton mass) and they have a demonstrable and often
puzzling influence on many physical properties, including spatial order. To
date, the structure of dense hydrogen remains experimentally elusive. Recent
studies of the melting curve of hydrogen indicate that at high (but
experimentally accessible) pressures, compressed hydrogen will adopt a liquid
state, even at low temperatures. In reaching this phase, hydrogen is also
projected to pass through an insulator-to-metal transition. This raises the
possibility of new state of matter: a near ground-state liquid metal, and its
ordered states in the quantum domain. Ordered quantum fluids are traditionally
categorized as superconductors or superfluids; these respective systems feature
dissipationless electrical currents or mass flow. Here we report an analysis
based on topological arguments of the projected phase of liquid metallic
hydrogen, finding that it may represent a new type of ordered quantum fluid.
Specifically, we show that liquid metallic hydrogen cannot be categorized
exclusively as a superconductor or superfluid. We predict that, in the presence
of a magnetic field, liquid metallic hydrogen will exhibit several phase
transitions to ordered states, ranging from superconductors to superfluids.
