# Emergent symmetries

As we know, spontaneous symmetry breaking(SSB) is a very important concept in physics. Loosely speaking, zero temprature SSB says that the Hamiltonian of a quantum system has some symmetry, but the ground state breaks the symmetry.

But what about the opposite case of SSB? The ground state of a quantum system possesses some kind of symmetry while the Hamiltonian does not have this symmetry. For example, the exactly solvable Kitaev-type model Hamiltonians explicitly break the spin rotational symmetry, but the ground states are spin liquids, which possess the spin rotational symmetry.

I wonder whether this opposite case of SSB plays an important role like SSB in physics?

Note:

Examples with exact emergent symmetries:

A simple example with exact emergent $SU(2)$ spin-rotation symmetry can be found here A simple model that exhibits emergent symmetry?

Another example with exact emergent $U(1)$ symmetry is presented in the Supplemental Material of this paper, where it is appeared on page 2 under Eq.(A7).

Examples with approximate emergent symmetries:

A chiral spin-liquid phase with emergent $SU(2)$ spin-rotation symmetry in the framework of Schwinger-fermion mean-field study of the strongly correlated Kane-Mele-Hubbard model.

The example with approximate emergent lattice 3-fold rotation symmetry is the existence of Ferromagnetic(FM) ground state in the Kitaev-Heisenberg model, where the model Hamiltonian explicitly breaks the lattice 3-fold rotation symmetry but the FM phase possesses the lattice 3-fold rotation symmetry.

Another example with emergent chiral symmetry was proposed by X.G.Wen in his paper, as seen on page 18, title C.

A third example with emergent time-reversal symmetry can be found here.

An example with an emergent global topological U(1) symmetry is presented here.

More examples with emergent symmetries are welcome.

Erratum: The example of "Kitaev model" I gave in my 2nd paragraph is not correct, please see Why we call the ground state of Kitaev model a Spin Liquid? for the reason.

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Changed the title cause I believe what you are talking about would be commonly refered to as "emergent symmetries." For example there have been proposals for emergent Lorentz symmetry - but I've never understood how the models work. –  Michael Brown Mar 1 '13 at 8:30
I think, the phrase "emergent symmetry" is also applicable in the sense that although the full theory may lack certain symmetries, its low energy counterpart (effective theory with a smaller UV cut-off) that describes excitations about its ground state may possess these additional symmetries. –  vik Sep 17 '13 at 8:29
@K-boy: Free electron gas is an example. The full Lagrangian is $L = \omega - \left(\frac{|\vec{K}|^2}{2m} - E_f\right),$ where $E_f$ is the Fermi energy. Now if we look into excitations with energy $E < \Lambda \ll E_f$ then we can write $\vec{K} = \vec{K}_f + \vec{k}$ with $|\vec{k}| \ll |\vec{K}_f|$, where $\vec{K}_f$ is Fermi momentum. The effective theory is written in terms of $\vec{k}$ as $L_{eff} = \omega - \vec{v}_f.\vec{k}$, where $v_f$ is Fermi velocity. This is a CFT. –  vik Sep 19 '13 at 2:52
Calling it a CFT, although correct, is probably an overkill, but the low energy theory is Lorentz invariant while the high energy theory is not. –  vik Sep 19 '13 at 20:14
One should also mention the notion of "enhancement of symmetry" (just google for it). One example: when $q$ is large enough, there is a range of temperatures at which the 2d $q$-state clock model (a discrete spin system invariant under a discrete subgroup of $SO(2)$) has a massless phase, that is, it behaves at large scales like a low-temperature 2d XY model. Everything occurs as if the symmetry group is enhanced to the full $SO(2)$. Another important example is the roughening transition. –  Yvan Velenik Oct 18 '13 at 17:33

A key difference between spontaneously broken symmetries and "emergent symmetries" is that emergent symmetries are never exact while spontaneously broken symmetries are backed by exact maths although the ground state isn't invariant. In most cases, the "emergent symmetries" only emerge if some parameters are fine-tuned, and even if it is so, they are only valid within some approximation scheme. In a generic situation, one has no reason to assume that a symmetry will "emerge" if it is not present fundamentally.

When there is a reason to expect such a thing, we use special names that are linked to the reason. In particular, the most solid example of an "emergent symmetry" – and a phrase that is actually being used by actual competent researchers, unlike "emergent symmetries" – is the "accidental symmetry".

http://en.wikipedia.org/wiki/Accidental_symmetry

It is a symmetry such as the lepton number and baryon number that is very well, approximately conserved because the terms in the equations (or action) that would violate it exist but because of a limited choice of renormalizable terms, all such terms may be shown to be high-dimension operators i.e. non-renormalizable. So their effects are negligible at low energies even though the lepton number and baryon numbers are almost certainly violated at higher energies, by the evaporating black holes or earlier than that.

In the Standard Model, the lepton number and the baryon number are conserved at the level of the renormalizable Lagrangians simply because one can't build renormalizable, gauge-invariant, Lorentz-invariant operators out of the given fields for gauge bosons, leptons, and quarks (and the Higgs field).

Your examples of Kitaev-style models and rotational symmetry are a bit less consequential. One may say that the ground state of a physical system is "rotationally invariant". But if the whole theory isn't rotationally invariant, the invariance of the ground state is pretty much a vacuous fact and its very validity is a matter of conventions (especially about a way how the symmetry-breaking theory is embedded into a larger theory that is symmetry-preserving). One won't be able to organize the spectrum into any representations of the symmetry group because it is not a genuine symmetry commuting with the Hamiltonian. Cubic crystals behave as rotationally symmetric materials in some aspects, but they see preferred directions in many other aspects.

There isn't any reason for an emergent or accidental Lorentz symmetry. This case is even much worse than the case of the "emergent rotational symmetry". In all known examples, a huge amount of fine-tuning – potentially fine-tuning of infinitely many parameters – is needed for a fundamentally Lorentz-breaking theory to reproduce Lorentz-invariant results, even at low energies. One must realize that the "maximum speed" of all the particle species including all of their possible bound states must be tuned to the same value called $c$. For each particle species, it's at least one additional tuning. There's no reason why all these fine-tunings should conspire and work properly so no viable theory in physics can make such assumptions about "emergent symmetries".

There's no name used by experts for "emergent Lorentz symmetry" etc. because the phenomenon envisioned in this name can't physically occur. The OP made it sound that this is just a formality and one only needs to learn the "right name". But physics isn't about terminology. The first question is whether such a hypothetical mechanism occurs in Nature and the answer is essentially No. So there's nothing to invent names for.

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"There's no name used by experts for "emergent Lorentz symmetry" etc. because the phenomenon envisioned in this name can't physically occur." I'm probably inclined to agree with you about the physicality of these things - though to be frank I've never been motivated to spend time on the models so I don't really know anything - but I've heard experts use the phrase in serious discussions. :) Anyway, if you think the name is problematic feel free to change it - it was my choice not the OP's. –  Michael Brown Mar 1 '13 at 13:11
And obviously an approximate symmetry might be a good symmetry for all practical purposes. :-) Approximate $\neq$ worthless. Otherwise excellent answer. –  Michael Brown Mar 1 '13 at 13:13
Right, Michael, after all, the most general "emergent" symmetry is nothing else than an approximate symmetry. An approximate symmetry isn't really there but it emerges for some reasons that aren't really well-described. The reasons may be that some parameters are tuned to the nearly symmetric values but some explicit breaking is included, too. But there's no invariant way to distinguish this situation from other cases in which one may observe approximate symmetries. I agree that approx. symmetries are useful - they're vague and somewhat ill-defined, too. How strong the violation may be? –  Luboš Motl Mar 1 '13 at 13:44
@K-boy I don't know anything about spin glasses. Are you saying that the vacuum state may have more symmetries than the generator of n-point functions (partition function $Z[j]$)? How is that possible? –  drake Mar 1 '13 at 19:13
@LubošMotl I agree that "the most general "emergent" symmetry is nothing else than an approximate symmetry", 'emergent' and 'approximate' are synonymous in this context. However, with your definition of accidental symmetry—which is the most common in the context of QFT—, 'accidental' is not synonymous of 'emergent' or 'approximate'. An 'accidental symmetry' is an 'approximate symmetry' in the low-energy regime. But there are also 'approximate symmetries' in other regimes such as those in the high-energy regime—when masses may be neglected— –  drake Mar 1 '13 at 19:39