Emergent symmetries

Question

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?

**Erratum:** The example of "Kitaev model" I gave above is not correct, please see http://physics.stackexchange.com/questions/65893/why-we-call-the-ground-state-of-kitaev-model-a-spin-liquid for the reason.

**Supplements:** 

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](http://arxiv.org/abs/1409.7820) and [this](http://prb.aps.org/abstract/PRB/v85/i19/e195126) with *emergent $SU(2)$ spin-rotation symmetry*.

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.

**Emergent supersymmetry**, see [this](http://arxiv.org/abs/1407.4497) and [this](http://arxiv.org/abs/1504.05192).

More examples with emergent symmetries are welcome.

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user K-boy

Comments

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.

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user Michael Brown

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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.

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user vik

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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.

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user Yvan Velenik

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@ Yvan Velenik Thanks for your good comments.

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user K-boy

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see also http://arxiv.org/abs/1510.04492

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@ vik, good point, I agree with you. BTW, do you have some concrete example like you said? Thank you very much.

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user K-boy

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@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.

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user vik

Answer 1

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.

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user Luboš Motl

Comments

@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?

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user drake

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@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—

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user drake

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or situations in which an interaction that does not respect the symmetry may be neglected. Therefore the concept of 'approximate symmetry' is more general than 'accidental symmetry'.

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user drake

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Right, I agree and I hope that it's been written above, too. Accidental symmetries are a special case of emergent/appoximate symmetries, a subclass that actually makes sense to be considered.

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user Luboš Motl

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@ drake, physics.stackexchange.com/questions/57717/… is a simple example where the ground state may has a symmetry that the Hamiltonian does not have.

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user K-boy

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And obviously an approximate symmetry might be a good symmetry for all practical purposes. :-) Approximate $\neq$ worthless. Otherwise excellent answer.

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user Michael Brown

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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?

This post imported from StackExchange Physics at 2014-03-09 08:47 (UCT), posted by SE-user Luboš Motl