What is spontaneous symmetry breaking in QUANTUM systems?

I just discovered this very interesting website through Prof Wen's homepage. Thanks Prof Wen for the very interesting question. Here is my tentative "answer":

The spontaneous symmetry breaking in the ground state of a quantum system can be defined as the long range entanglement between any two far-separated points in this system, in any ground state that preserves the global symmetries of the system.

To be more precise, denote $G$ as the symmetry group of the system and $|\Psi\rangle$ a ground state that carries a 1d representation of $G$. For an Ising ferromagnet, the ground state will be $|\Psi_\pm\rangle =\frac{1}{\sqrt{2}}\left(|\text{all up}\rangle \pm |\text{all down}\rangle\right)$. Then consider two points 1 and 2 separated by distance $R$ in the space, and two small balls around points 1 and 2 with radius $r\ll R$, denoted by $B_1$ and $B_2$. Define $\rho_1$, $\rho_2$ and $\rho_{12}$ as the reduced density matrices of the region $B_1$, $B_2$ and $B_1+B_2$, and correspondingly the entropy $S_{1}=-tr(\rho_1\log \rho_1)$ (and similarly for $2$ and $12$). The mutual information between the two regions is defined as $I_{12}=S_1+S_2-S_{12}$. If $I_{12}> 0$ in the $R\rightarrow \infty$ limit for all symmetric ground states, the system is considered as in a spontaneous symmetry breaking state.

In the example of Ising FM, $S_{12}=\log 2$ for both ground states $|\Psi_\pm\rangle$.

I am afraid it's just a rephrasing of ODLRO but it might be an alternative way to look at spontaneous symmetry breaking.

This question posted by Prof. Wen is so profound that I had hasitated to response. However motivated by Jimmy's insightful answer, I eventually decided to join the discussion, and share my immature ideas.

1) Quantum SSB is a non-linear quantum dynamics beyond the description of Schordinger's equation.

Regarding the transverse field Ising model mentioned in the comments of the question, with a small B field, the ground state is a Schordinger's cat state. Asking how does the SSB happen in the $B\to 0$ limit is the same as asking how does the cat state collapses to a definite state of live or death. Quantum decoherence plays the key role here. However quantum decoherence is an irriversible dynamics with entropy production, which, I believe, can not be described by the linear dynamics of quantum mechanics that preserves the entropy. To understand quantum SSB, we may have to understand the dynamics of quantum decoherence first.

2) Quantum SSB is a result of information renormization, which may be described by the tensor network RG.

The key of understanding quantum decoherence is to understand how entropy was produced. It had been a mystery for a long time that what is the origin of entropy? Until Shannon related entropy to information, we started to realize that entropy is produced due to the lost of information. Information is lost in the experiments inevitably because we can only collect and process finite amount of data. Because all experiements are conducted under a finite energy and information (or entropy) scale, so only the low energy and low information effective theory is meaningful to physicists. Renormalization group (RG) technique had been developed to obtained the low energy effective theory successfully. Now we need to develop the informational RG to obtain the low information effective theory. DMRG and tensor network RG developed in recent years are indeed examples of informational RG. Quantum information is lost through the truncation of density matrix, and entropy is produced at the same time, which makes quantum decoherence and quantum SSB possible. In fact, quantum SSB can been observed in both DMRG and tensor network RG as I know. Along this line of thought, quantum SSB is not a final state of time evolution under linear quantum dynamics, but a fixed point of informational RG of quantum many-body state, which is non-linear and beyond our current text-book understanding of quantum mechanics.

I'm sure Prof. Wen understands this question very well and is posting this just to inspire some discussions. So I'm just gonna go ahead and give my 2 cents.

A classical spontaneous symmetry breaking happens when the classical ground state breaks the symmetry of the Hamiltonian. For example, for a classical Ising model in 1D, spontaneous magnetization in a particular direction happens at low T, which breaks the $S\rightarrow-S$ symmetry of the Hamiltonian.

A quantum spontaneous symmetry breaking doesn't necessarily mean the quantum ground state breaks the symmetry of the Hamiltonian; instead, it's signatured by the splitting of the ground state degeneracy. Say in the case of transverse Ising model, $H=-\sum{S_i^z S_j^z}-B\sum{S_i^x}$. The ground state of the Hamiltonian for very small $B$ is the superposition of all spin up and all spin down, which still has the $S_z\rightarrow -S_z$ symmetry; but now the ground state degeneracy is lost---the ground state is now unique, rather than having a 2-fold degeneracy.

This is just a preliminary answer, so please feel free to correct me/improve the answer.