Symmetry Breaking: Ground State vs. Hamiltonian

1. The Essence of Spontaneous Symmetry Breaking

In physics, we say a symmetry is spontaneously broken if the Lagrangian or Hamiltonian of a system is invariant under a symmetry transformation, but the ground state (or vacuum state) of the system is not.

Let's make this more precise. In the previous lessons, we saw that a continuous symmetry of a Hamiltonian $H$ implies the existence of a conserved charge operator $Q$ such that $[H,Q]=0$. In the "normal" realization of symmetry, the ground state $|0⟩$ is also symmetric, which means the charge operator annihilates it:

In contrast, spontaneous symmetry breaking occurs when the charge is still conserved, but the ground state is no longer invariant:

This second condition implies that applying the symmetry transformation to the ground state, $e^{i\alpha Q}|0⟩$, produces a new state that is physically distinct from $|0⟩$. Since $[H,Q]=0$, this new state must have the same energy as the original ground state. SSB therefore implies the existence of a set of degenerate ground states. The system settles into one of these states, thereby "breaking" the symmetry.

A simple mechanical analogy is a vertical pencil balanced on its tip. The laws of gravity are perfectly rotationally symmetric around the vertical axis. However, the balanced state is unstable. Any infinitesimal perturbation will cause the pencil to fall, and it will come to rest pointing in some specific, arbitrary direction on the table. The final, stable state is not rotationally symmetric. The rotational symmetry of the laws of physics has been spontaneously broken by the ground state of the system.

2. Discrete Symmetry Breaking: The ${\varphi }^4$ Model

Let's build a concrete model of SSB, starting with the simplest case: a discrete $Z_2$ symmetry. This is the canonical example in quantum field theory and has direct analogues in condensed matter systems like the Ising model.

Consider a real scalar field $\varphi (x)$ described by the Lagrangian density:

Instead of the usual potential, let's choose one with a "wrong-sign" mass term:

Here, ${\mu }^2$ and $\lambda$ are positive real parameters. The $\lambda {\varphi }^4$ term is crucial to ensure the potential is bounded from below, so the system is stable. This Lagrangian is clearly invariant under the $Z_2$ symmetry transformation $\varphi \to -\varphi$.

What is the ground state of this system? In classical field theory, the ground state corresponds to a field configuration that is constant in space and time and minimizes the potential energy $V(\varphi )$.

As the video shows, the potential has the shape of a double well. The stationary points are found by setting $dV/d\varphi =0$:

This gives three solutions:

1. $\varphi =0$: This is a local maximum, representing an unstable equilibrium (like the pencil balanced on its tip).
2. $\varphi =\pm \sqrt{{\mu }^2/\lambda }$: These two points are the degenerate minima of the potential.

The true ground states, or vacua, of the theory correspond to the field having a constant, non-zero vacuum expectation value (VEV):

Here lies the core of SSB. The Hamiltonian possesses the $\varphi \to -\varphi$ symmetry. However, the system must choose one of the two vacua to settle in, for example $v=+\mu /\sqrt{\lambda }$. This state is not invariant under the symmetry transformation, as it transforms into the other vacuum state, $-v$. The symmetry of the Hamiltonian is not present in the ground state.

The Role of the Thermodynamic Limit

You might ask: in single-particle quantum mechanics, a particle in a double-well potential doesn't break the symmetry. Its ground state wavefunction is a symmetric superposition of being in both wells, due to quantum tunneling. Why is a field theory different?

The crucial difference is that a field theory describes a system with an infinite number of degrees of freedom (one at each point in space). This is analogous to the thermodynamic limit ($N\to \infty$) in statistical mechanics. In this limit, the energy barrier between the two vacua becomes infinitely large, and the probability of the entire system tunneling from the $+v$ vacuum to the $-v$ vacuum becomes zero. Any infinitesimal external field or quantum fluctuation will tip the system into one of the two wells, where it will then remain.

3. The Methodology: Fluctuations and Order Parameters

Identifying SSB is the first step. The next is to understand the physics of the system in its broken-symmetry state. This is done by studying the small fluctuations of the field around the chosen vacuum.

Let's choose the vacuum $⟨\varphi ⟩=+v$. We define a new field $\eta (x)$ representing the fluctuation around this vacuum, such that $⟨\eta ⟩=0$:

The core of the methodology is to rewrite the original Lagrangian in terms of this new fluctuation field $\eta$.

The expansion reveals several critical points:

1. Constant Term: A constant term appears in the Lagrangian, corresponding to the (negative) potential energy of the vacuum itself. This is the vacuum energy and can usually be disregarded by shifting the zero point of energy (unless you are coupling to gravity).
2. Linear Term: The term linear in $\eta$ vanishes identically. This must happen, as it is the mathematical expression of the fact that we are expanding around a minimum of the potential.
3. Quadratic Term: This is the most important result. The Lagrangian for the fluctuation field $\eta$ contains the term $-{\mu }^2{\eta }^2$. This looks like a mass term, $-\frac{1}{2}{m}_{\eta }^2{\eta }^2$, implying that the fluctuation particle $\eta$ has a real, physical mass of $m_{\eta }^2=2{\mu }^2$.

The apparent "imaginary mass" ($-{\mu }^2$) in the original Lagrangian was an illusion, an artifact of trying to describe the theory from the perspective of the unstable maximum at $\varphi =0$. By expanding around the true ground state, we uncover the physically meaningful particle spectrum.

Notice that the final Lagrangian for $\eta$ contains cubic (${\eta }^3$) and quartic (${\eta }^4$) terms and does not have the simple $\eta \to -\eta$ symmetry. The original $Z_2$ symmetry is now "hidden." It is no longer obvious from looking at the excitations alone, but it is still a fundamental symmetry of the underlying theory, relating the physics of the $+v$ vacuum to that of the $-v$ vacuum.

The Order Parameter

A systematic way to diagnose SSB is to identify an order parameter. This is an observable whose expectation value is zero in the symmetric phase but becomes non-zero in the broken-symmetry phase.

For our ${\varphi }^4$ theory, the field $\varphi$ itself serves as the order parameter.

• In the symmetric phase (which would exist at high temperatures), $⟨\varphi ⟩=0$.
• In the broken-symmetry phase (at low temperatures), $⟨\varphi ⟩=v\ne 0$.

The potential $V(\varphi )$ can be thought of as a Landau free energy functional. The transition from a potential with a single minimum at $\varphi =0$ (when the ${\varphi }^2$ term is positive) to a double-well potential (when the ${\varphi }^2$ term is negative) is a classic example of a second-order phase transition.

4. Continuous Symmetry Breaking: A Preview

What happens if the broken symmetry is continuous, like a rotation? Let's consider a complex scalar field $\varphi ={\varphi }_1+i{\varphi }_2$ with the potential:

This potential is invariant under the continuous U(1) group of phase rotations: $\varphi \to e^{i\alpha }\varphi$.

The minimum of this potential is not at a set of discrete points, but along a continuous circle defined by $|\varphi {|}^2=v^2={\mu }^2/\lambda$. The system must pick one specific phase for its ground state, for instance, choosing $⟨\varphi ⟩=v$ (a real value). This choice breaks the U(1) symmetry.

As before, we can analyze the fluctuations around this vacuum. But now, we can imagine two different kinds of fluctuations:

1. Radial Fluctuations: Moving up and down the side of the "hat." This costs potential energy and corresponds to a massive particle.
2. Angular Fluctuations: Moving along the circular trough at the bottom. This costs zero potential energy.

These zero-energy excitations are a universal consequence of breaking a continuous symmetry. They correspond to massless particles known as Nambu-Goldstone bosons, or simply Goldstone modes.

Conclusion

In this lesson, we have defined spontaneous symmetry breaking and established the core methodology for its analysis.

Key Takeaways:

• Spontaneous Symmetry Breaking (SSB) occurs when the ground state of a system lacks a symmetry possessed by its Hamiltonian or Lagrangian.
• SSB is associated with a degenerate set of ground states. In the thermodynamic limit, the system chooses one of these states and remains there.
• The correct way to analyze the particle content of a theory with SSB is to expand the fields around a true vacuum (a minimum of the potential), not around an unstable point.
• An order parameter is a crucial tool for identifying SSB. It is an observable that has a zero expectation value in the symmetric phase and a non-zero expectation value in the broken-symmetry phase.
• Breaking a discrete symmetry (like $Z_2$) leads to a set of discrete, degenerate vacua and typically results in a spectrum of massive particles.
• Breaking a continuous symmetry (like U(1)) leads to a continuous manifold of degenerate vacua. This has a profound consequence that we will explore next.

In our next lesson, we will formalize the observation about zero-energy fluctuations by deriving Goldstone's Theorem. This powerful theorem states that for every spontaneously broken continuous global symmetry, a massless, spin-zero particle must appear in the theory's spectrum.