An introduction to spontaneous symmetry breaking

Abstract

Perhaps the most important aspect of symmetry in physics is the idea that a state does not need to have the same symmetries as the theory that describes it. This phenomenon is known as spontaneous symmetry breaking. In these lecture notes, starting from a careful definition of symmetry in physics, we introduce symmetry breaking and its consequences. Emphasis is placed on the physics of singular limits, showing the reality of symmetry breaking even in small-sized systems. Topics covered include Nambu-Goldstone modes, quantum corrections, phase transitions, topological defects and gauge fields. We provide many examples from both high energy and condensed matter physics. These notes are suitable for graduate students.

Stability and absence of a tower of states in ferrimagnets

“…The Mermin-Wagner-Hohenberg-Coleman theorem does not prohibit type-B SSB from occurring in onedimensional systems at zero temperature [5,31,32]. We can therefore confirm the generality of the analytic results of the Lieb-Mattis model in Section VI using numerical results for a one-dimensional ferrimagnetic Heisenberg chain.…”

“…Two broken symmetry generators lead to one quadratically dispersing Goldstone mode and a gapped partner mode. Finally, type-B systems do not suffer from the Coleman theorem and are stable in one dimension [5,31,32] (although both type-A and type-B systems are subject to the Mermin-Wagner-Hohenberg theorem that forbids SSB in two or lower dimensions at finite temperature [5,31]). There seems to be only one essential difference between general ("ferri") type-B SSB and the peculiar case of the ferromagnet: the latter is the same as the classical ferromagnet, whereas the ferrimagnet is a classical Néel state dressed with quantum corrections [27].…”

“…Recall that a state is defined to be unstable if there exists an extensive observable whose variance Var =  2 −  2 scales as N 2 [4,5,18]. This definition is equivalent to saying there exists a connected correlation function that does not satisfy the cluster decomposition property [5,7].…”

“…However, these unique ground states are not stable, in the sense that adding even a small symmetry-breaking perturbation will lead to a symmetry-broken ground state; in the thermodynamic limit an infinitesimal perturbation suffices to break the symmetry. Examples of this type of symmetry breaking include the breaking of Z 2 (up-down) symmetry in Ising models, of U (1) (phase rotation) symmetry in XY -models, of SU (2) (spin-rotational) symmetry in Heisenberg antiferromagnets, and of translational symmetry breaking in crystals [5].…”

“…The instability of the symmetric ground state of finitesized systems may be intuitively understood by realizing it is actually a type of 'Schrödinger cat state', namely a superposition of macroscopically distinct states which each break the symmetry differently [5,6]. There must then be some observable A with an extensive expectation value A ∼ O(N ) whose variance scales as VarA ∼ N 2 , violating the cluster decomposition property [7].…”