condensed matter - What is spontaneous symmetry breaking in QUANTUM systems?

Most descriptions of spontaneous symmetry breaking, even for spontaneous symmetry breaking in quantum systems, actually only give a classical picture. According to the classical picture, spontaneous symmetry breaking can only happen for non-linear systems. Classical linear systems, such as Harmonic oscillators, can never have spontaneous symmetry breaking. (Here "linear" means that the equation of motion is linear.)

But the real QUANTUM systems are always linear since the Schrodinger equation is alway linear. So how can a linear quantum system have spontaneous symmetry breaking? Do we have a simple intuitive understanding for spontaneous symmetry breaking WITHIN QUANTUM mechanics? (without using the classical picture, such a Mexican hat -- the logo of physics.stackexchange)

The Mexican hat does give us an intuitive and pictorial understanding of spontaneous symmetry breaking in classical systems. Do we have an intuitive and pictorial understanding of spontaneous symmetry breaking in quantum systems.

Answer

Bei Zeng and I wrote a paper http://arxiv.org/abs/1406.5090 , which addresses this question:

A symmetry breaking phase for finite group G is a gLU equivalent class formed by symmetric many-body states that have GHZ entanglement.

In other words, a symmetry breaking phase is a set of

1. symmetric states $U_g \Psi = \Psi, g \in G$, and


2. those symmetric states have the same GHZ entanglement $\Psi = \sum_\alpha \Psi_\alpha ,\ \ \alpha \in G/H,\ \ H\ \subset G$, where $\Psi_\alpha$'s are locally distinguishable.

We say those symmetric states are equivalent. The set of equivalent symmetric states is a symmetry breaking phase.

So symmetry breaking = GHZ entanglement which are classified by pairs $(G , H),\ H \in G$.

More precisely:

1) A symmetric many-body state has spontaneous symmetry breaking implies that the state has a GHZ entanglement.

2) One can detect spontaneous symmetry breaking in a symmetric many-body state even without knowing the group and/or order parameter of the symmetry. One can detect spontaneous symmetry breaking in a symmetric many-body state using only probes that respect the symmetry.

3) The symmetric exact ground state of a generic symmetric Hamiltonian has spontaneous symmetry breaking iff it has GHZ entanglement.