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In quantum mechanics, a (unitary) symmetry is a unitary operator $U$ which commutes with the hamiltonian $H$ of the system:

$$ [U, H] = 0 $$

For instance, in the uniform Ising model with periodic boundary conditions:

$$ H = -J\sum_i\sigma_i^z\sigma_{i+1}^z $$

there is a parity symmetry $P$ and the translation $T$ symmetry (periodic boundary conditions are assumed):

$$ P = \prod_i \sigma_i^x,\quad T = \prod_k T_{k,k+1} $$

where $T_{k,k+1}$ is the exchange operator of two neighbouring spins. Indeed, this model also has an extensive amount of unitary symmetries, since spin-$1/2$ operators are both hermitian and unitary and are conserved quantities of the system:

$$ [\sigma_i^z,H] = 0,\;\forall i $$

In general, starting from the spectrum of $H$, one can construct "infinite" unitary symmetries of the system, by linear combination of the projectors times a phase:

$$ H|\psi_{\alpha,\nu}\rangle = E_{\alpha,\nu}|\psi_{\alpha,\nu}\rangle \rightarrow \begin{cases} U = \sum_{\alpha,\nu} e^{i\phi_\nu} |\psi_{\alpha,\nu}\rangle\langle\psi_{\alpha,\nu} | \\ [U,H] = 0 \end{cases} $$

where the spectrum of $H$ has been divided into sets labelled by an index $\nu$.

Is there some sense in which it is meaningful to consider only certain symmetries and not others? The only one I can think of is that $U$ is a local unitary operator, i.e. it can be written as the evolution operator a local hermitian operator $V = \sum_i V_i$ (where $V_i$ acts only on some finite region of the system and not all $V_i$ are the identity):

$$ U = e^{-i\sum_i V_i} = \prod_i e^{-i V_i} $$

For instance, this is the case for the parity symmetry and translation symmetry of the Ising chain, but not for $\sigma_i^z$.

There is also the problem of "equivalent" symmetries. For instance, there is a sense in which the two following symmetries are both parity symmetries of the Ising model:

$$ P = \prod_i\sigma_i^x,\quad P' = \prod_i\sigma_i^y $$

since they both flip the spins along the $z$ direction.

I'm asking this question because, for instance, in many-body systems, one is often interested in "perturbations which preserve the symmetry". Of course, one is not interested in perturbations which conserve every symmetry of $H$ (by the definition above), but rather the "physically meaningful" symmetries. Is there some sense which allows one to identify such symmetries and "equivalent" formulations, especially for many-body quantum systems?

Furthermore, it is particularly important in Landau's symmetry breaking paradigm of phase transitions, which says that spontaneous symmetry breaking is always accompanied by a phase transition.

Therefore, another, possibly more precise statement of my question, although somewhat different, is: what are the symmetries whose "spontaneous breaking" lead to a phase transition?

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    $\begingroup$ Does this answer your question? What is *physical meaning*? $\endgroup$
    – hft
    Nov 3, 2022 at 19:11
  • $\begingroup$ No, because I know there is an answer, which is probably related with locality (as mentioned in my answer), that will answer my question. Although the vagueness of my title might not attract the people who know such an answer. There is clear irrelevance to certain symmetries and clear relevance for others, I want to know if there is a general criterion for distinguishing the two. $\endgroup$ Nov 4, 2022 at 8:41
  • $\begingroup$ If you really want to be precise, a possible re-statement is the following: for which symmetries is spontaneous symmetry breaking an important concept? $\endgroup$ Nov 4, 2022 at 8:43
  • $\begingroup$ It is the physical problem addressed that makes certain symmetries relevant - e.g., when one studies a phase transition, one is mostly interested in the symmetry that is broken in this transition; just like there are symmetries that are relevant when looking for eigenstates or when classifying radiative transitions in atoms/molecules (typical textbook stuff.) $\endgroup$
    – Roger V.
    Nov 9, 2022 at 10:22
  • $\begingroup$ I understand that's how it usually works when looking at a specific model. We choose it such that it describes what we are trying to find. But I want to know why it works. If we start with a model, we find infinite symmetries which a priori are all equivalent, how come we are able to say that only certain ones will be relevant (at least within spontaneous symmetry breaking)? Namely, parity symmetry implies spontaneous symmetry breaking in the ground state of the Ising model, could we have known that a priori, without an "educated guess"? $\endgroup$ Nov 9, 2022 at 10:29

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