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I was wondering what is the difference between a spin-fluid Heisenberg Magnet and spin-glass Heisenberg Magnet. As far as I understand in a spin-glass the spins are randomly oriented compared to a ferromagnet material but I don't quite understand what we mean by spin-fluid phase. And also in the article I am reading its mentioned that spin-fluid phase is found to be generically "gapless", but I am not sure what we mean by phase being gapless and how is it related to the spin phase of the material?

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Let me answer your question in 3 steps:

Spin glass

A spin glass describes a frustrated magnetic system in analogy to classical (structural) glass formers. Frustration means that the system can not easily find configurations that satisfy orientation to neighboring spins. A typical example is anti-ferromagnetic interactions on a triangular lattice. Another example has the standard Hamiltonian

$$H = -\sum\limits_{i,j}J_{ij}\mathbf{s}_i\mathbf{s}_j$$

but using randomly distributed $J_{ij}$ in contrast to $J_{ij} = \pm1$ in the Ising model. So it is more precise to say that the interaction parameters $J_{ij}$ are randomly distributed, instead of saying that the spins are randomly oriented.

Spin liquid

Above the glass transition temperature, the energy is high enough for the system to reach its equilibrium state on a given timescale (the glass transition is a dynamic phenomenon, so it depends on a timescale, typically set by the cooling rate). At this temperature the system is ergodic and called a spin-fluid in analogy to the supercooled liquid phase of structural glass formers. Below the glass transition, the system is no longer able to reach its equilibrium state, is non-ergodic and displays aging. This is called the spin-glass phase, again in analogy to structural glass formers.

Gapless phase

The term "gapless" refers to the energy spectrum and means that there is not a minimum excitation energy. It means that you can find pairs of spins that are effectively decoupled from the rest of the system and which interact only weakly with each other. Thus, you need only a small amount of energy to exchange their spins.

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