Non-linear sigma model

Definition

A non-linear sigma model is a type of quantum field theory where fields take values in a non-linear manifold, typically a target space like a sphere or more generally a symmetric space. This model is important for understanding spontaneous symmetry breaking and the dynamics of Nambu-Goldstone bosons, as it effectively captures the low-energy excitations that arise when a continuous symmetry is spontaneously broken.

5 Must Know Facts For Your Next Test

1. In a non-linear sigma model, the fields map to points on a target manifold, which can be curved, making them more complex than linear models.
2. These models are particularly useful in describing systems with Goldstone bosons, as they directly incorporate the effects of spontaneous symmetry breaking.
3. The action of a non-linear sigma model is often expressed in terms of a metric on the target manifold, which governs the kinetic term of the fields.
4. Non-linear sigma models are applied in various areas, including condensed matter physics, string theory, and particle physics, illustrating their versatility.
5. The presence of Nambu-Goldstone bosons in non-linear sigma models results in interesting phenomena like solitons and instantons, contributing to their rich physical implications.

Review Questions

• How does a non-linear sigma model illustrate the relationship between spontaneous symmetry breaking and Nambu-Goldstone bosons?
  • A non-linear sigma model demonstrates this relationship by providing a framework where fields take values in a non-linear manifold. When a continuous symmetry is spontaneously broken within this model, it leads to the emergence of massless Nambu-Goldstone bosons that correspond to the directions in which the potential energy remains constant. This connection highlights how low-energy excitations can be understood through geometric properties of the target space.
• Discuss how the action of a non-linear sigma model is constructed and its significance in relation to Goldstone's theorem.
  • The action of a non-linear sigma model is constructed using a metric defined on the target manifold where the fields reside. This metric determines how the kinetic terms behave and directly influences the dynamics of the Nambu-Goldstone bosons. The significance lies in its ability to demonstrate Goldstone's theorem by allowing for massless excitations corresponding to broken symmetries, showing how the structure of the target space relates to physical outcomes.
• Evaluate the implications of using non-linear sigma models in different physical contexts, such as condensed matter physics and string theory.
  • Non-linear sigma models have profound implications across various physical contexts due to their ability to capture essential features of spontaneous symmetry breaking and low-energy dynamics. In condensed matter physics, they help describe phenomena like superfluidity and magnetism by characterizing collective excitations. In string theory, non-linear sigma models are employed to describe string dynamics on curved spacetime, linking geometry with quantum mechanics. This versatility underscores their fundamental role in understanding complex systems across physics.

Related terms

Nambu-Goldstone bosons:

Particles that arise in a quantum field theory when a continuous symmetry is spontaneously broken, corresponding to the directions in field space where the potential energy is constant.

Goldstone theorem:

A theorem stating that for every continuous symmetry that is spontaneously broken, there exists a corresponding massless scalar boson known as a Nambu-Goldstone boson.

spontaneous symmetry breaking: A process where the ground state of a system does not exhibit the symmetry of the underlying laws, often leading to the emergence of Nambu-Goldstone bosons.