# How to judge whether a symmetry will be spontaneously broken while only given a Hamiltonian preserving this symmety

+ 2 like - 0 dislike
295 views

As asked in the title, is Hamiltonian containing enough information to judge the existence of spontaneously symmetry breaking?

Any examples?

This post imported from StackExchange Physics at 2014-04-24 02:31 (UCT), posted by SE-user Laurent

+ 2 like - 0 dislike

The Hamiltonian of a theory describes its dynamics. Symmetry is "spontaneously broken" when a certain state of the quantum theory doesn't have the same symmetry as the Hamiltonian (dynamics). The standard example is a theory with a quartic potential. In field theory, say we have a potential $V (\phi) \sim \lambda{(\phi^2 - a^2)}^2$ in the lagrangian/hamiltonian, for some real valued scalar field $\phi$. This potential has minima at $\phi_{\pm}=\pm a$.

The theory (Hamiltonian) has a $\phi \rightarrow - \phi$ symmetry, but note that each of those vacua don't have the same symmetry i.e. vacuum states are not invariant under the symmetry. The solutions $\phi_{\pm}$ transform into each other under the transformation. So this is an example where the symmetry of the theory is broken by the vacuum state, "spontaneously" (roughly by itself, as the theory settles into the vacuum state).

So, to find whether a symmetry will be spontaneously broken (given a Hamiltonian), you have to check the symmetry of the states (typically vacua) of the theory and compare them to the symmetry of the Hamiltonian.

This post imported from StackExchange Physics at 2014-04-24 02:31 (UCT), posted by SE-user Siva
answered Mar 15, 2013 by (720 points)
+ 1 like - 0 dislike

I'm not sure what OP has in mind, but consider e.g. the inverted harmonic oscillator $$H~=~\frac{p^2}{2m}-\frac{1}{2}kq^2,$$ which has spontaneously broken $\mathbb{Z}_2$ symmetry $(q,p) \to (-q,-p)$. The stable positions are $q=\pm \infty.$

This post imported from StackExchange Physics at 2014-04-24 02:31 (UCT), posted by SE-user Qmechanic
answered Feb 12, 2013 by (3,110 points)
+ 1 like - 0 dislike

In general, you should look to solve the potential for a function that gives a minimum. Does this minimum of the potential respect the symmetry? If not, you are very likely to find an anomaly. The classic example of this is the minimum of the higgs potential not respecting the gauge invariance of the SU(2) x U(1) fields coupled to it, but there are simpler examples, like the one @Qmechanic points out.

This post imported from StackExchange Physics at 2014-04-24 02:31 (UCT), posted by SE-user Jerry Schirmer
answered Feb 13, 2013 by (130 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar\varnothing$sicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.