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What are the distinguishment between (1) spin ice, (2) spin liquid and (3) quantum spin ice, (4) quantum spin liquid?

Apparently the quantum effect for the later (3) and (4) becomes important. But what physical phenomenon of the (1), (2), (3), (4) set their differences?

Spin ice: Possible candidates include that Dy$_2$ Ti$_2$ O$_7$ and Ho$_2$Ti$_2$O$_7$, etc. may harbor a classical spin ice. They potentially satisfy an ‘ice rule’: each tetrahedron of the pyrochlore lattice are precisely two spins that point inward and two that point outward. It may also have magnetic monopole if we have 3-in 1-out tetrahedron or anti-monopole for 1-in 3-out tetrahedron. In classical spin ice systems may be thermally fluctuating loop gas. The loops can be ‘magnetic’ field lines of an artificial magnetic fields.

Quantum spin ice: Possible candidates include Tb$_2$Ti$_2$O$_7$, Yb$_2$Ti$_2$O$_7$, Pr$_2$Zr$_2$O$_7$, etc. In quantum spin ice, the physics is determined by quantum fluctuations of oriented loops. If these loops form a liquid phase where the loop line tension becomes zero, then it becomes a quantum spin liquid. But how this spin liquid supports an emergent gapless photon? The associated magnetic field lines can be tensionless. Magnetic monopoles as the defect (in tetrahedra) can be gapped quasiparticle excitations where these field lines end. But how the gapless photons can emerge from the quantum fluctuation from spin ice?

Quantum spin liquids: Possible candidates include herbertsmithite.

How can quantum spin liquid have emergent photons, but not for spin ice or quantum spin ice?

For example, can one understand from here: If the loops form a spin ice's liquid phase where the loop line tension becomes zero, then it becomes a quantum spin liquid. And how the photon emerges from here?

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When we talk about spin liquid, we typically mean quantum spin liquid. Thanks to @Stephen Powell's comment below, I learned that there is also a thing called "classical spin liquid". As far as I know, the difference among these concepts can be distinguished as follows:

  • Classical or quantum?
    • Classical spin ice/liquid is a finite temperature thermal phase, described by statistical mechanics, as a thermal ensemble (mixed state) of all configurations in the low-energy sector.
    • Quantum spin ice/liquid is a zero-temperature quantum phase (a quantum many-body ground state), described by quantum mechanics, as a coherent superposition (pure state) of all configurations in the low-energy sector.
  • What is the low-energy sector? The low-energy configurations are a subset of spin configurations that satisfy certain local rules. The 2-in-2-out ice rule is an example of the local rules. The local rules are typically imposed by energy penalties, i.e. violating the local rules will lead to excitations out of the low-energy sector.
    • Spin ice: if the local rule is simply the ice rule. Violating the ice rule leads to monopole excitations. Monopole excitations appear in pairs, as the end points of a magnetic field line. So the local rule enforces that there are no monopoles in the low-energy configurations, which means the magnetic field lines can not end. So the low-energy configurations are just all configurations consist of closed loops of magnetic field lines.
    • Spin liquid: if the local rule is more general. In this sense, spin ices can be considered as a subset of spin liquids (both classical and quantum). In many cases, there is no strict and explicit local rules for spin liquid, just frustrated correlations. In some 2D spin liquid, the local rules are imposed as fusion rules of strings and the low-energy configurations are sting-nets. In higher dimensions, local rules may also single out closed membranes or other extended geometric objects in the low-energy sector.

Quantum spin ice is a special class of quantum spin liquid, known as U(1) spin liquid in 3D, whose low-energy fluctuations are described by a U(1) gauge theory (hence the name U(1)). More general spin liquids are not limited to 3D (in fact 2D spin liquids are largely discussed) and the gauge group is not limited to U(1).

Why do gapless photons emerge in quantum spin ice? Because photon = light = electromagnetic wave = closed magnetic field lines moving in the space. The fluctuating loops in the quantum spin ice are mathematically the same as the fluctuating photons in our vacuum.

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  • $\begingroup$ There is certainly such a thing as a classical spin liquid—you need only put the phrase into Google Scholar to see this. But it is true that people often use the phrases "quantum spin liquid" and "spin liquid" interchangeably. See this review article by Leon Balents for some discussion of the classical case. $\endgroup$ – Stephen Powell Nov 13 '16 at 14:45
  • $\begingroup$ Many thanks. How can I see the different emergent photons of U(1) spin liquids/gauge theory and Z2 spin liquids/gauge theory from your picture of fluctuating magnetic field lines of U(1) or Z2 nature? For example, being gapless or gapped/Higgsed? $\endgroup$ – wonderich Nov 14 '16 at 4:29
  • $\begingroup$ @wonderich If the loop is oriented, then the emergent gauge group is U(1). If the loop is unoriented, then the emergent gauge group is Z2. $\endgroup$ – Everett You Nov 14 '16 at 19:07
  • $\begingroup$ How about the loop of Zn gauge theory? Is that Zn oriented or non-oriented in your language? $\endgroup$ – wonderich Nov 15 '16 at 2:15
  • $\begingroup$ @wonderich $Z_n$ gauge theory has oriented strings, and $n$ strings of the same orientation can fuse together to vacuum. That is why when $n=2$, the string becomes effectively not-oriented. $\endgroup$ – Everett You Nov 18 '16 at 14:53

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