# Is the following a correct definition of spontaneous symmetry breaking?

I recall a definition of spontaneous symmetry breaking that I am not sure is fully correct as I am unable to find a reference for it.

Given a quantum system with Hamiltonian $$H$$ at $$T = 0$$, i.e. considering ground states only, let $$H$$ be invariant under a symmetry group $$G$$, meaning $$H$$ commutes with all elements of $$G$$. Then $$G$$ has a representation on the ground state space of $$H$$. If this representation is trivial the symmetry is not broken. However, if there exist states in the lowest energy eigenspace which are not invariant under action of $$G$$, i.e. the representation of $$G$$ on the ground-state space is nontrivial then the state is spontaneously symmetry broken. It would in fact be broken to the kernel of the representation of the ground-state space.

A specific example could be the transverse field Ising model with

$$H = A \sum_i Z_i Z_{i+1} + B \sum_i X_i$$

where $$G = \{ \mathbb{1}, X^{\otimes n} \}$$ forms the symmetry group. Here the lowest energy sector is given by $$|+++ \dots \rangle$$ at $$A = 0$$ on which $$G$$ acts trivially and by $$\text{span}(|111\dots\rangle, |000\dots \rangle )$$ at $$B = 0$$ on which $$G$$ is represented as trivial + alternating representations of $$\mathbb{Z}_2$$.

Considering much more sophisticated definitions of SSB in quantum systems have been put forward in this question I suspect something must be wrong with the above definition and I ask for somebody to point out what it is and whether this definition is at least close to a correct one.

• It seems that you use G to represent both the ground state and the symmetry group. Perhaps you can change it to be a bit less confusing. Dec 17, 2022 at 3:42
• $G$ denotes only the symmetry group and its representation on the lowest energy eigenspace of $H$, not the lowest energy eigenspace itself. Dec 17, 2022 at 8:03

A modern interpretation of topological order is that it can be thought of as SSB of higher-form symmetries (1-form in the case of toric code), and conventional SSB can be thought of as SSB of 0-form symmetries. In this generalized symmetry paradigm, your definition for SSB would be correct. In the toric code example, $$G$$ would be the set of loop-like symmetries which get spontaneously broken in the topologically ordered phase. Check out this review article for more on that.