SciPost Phys. Lect. Notes 11 (2019) An introduction to spontaneous symmetry breaking Aron J. Beekman1, Louk Rademaker2,3 and Jasper van Wezel4? 1 Department of Physics, and Research and Education Center for Natural Sciences, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan 2 Department of Theoretical Physics, University of Geneva, 1211 Geneva, Switzerland 3 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 4 Institute for Theoretical Physics Amsterdam, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands ? [email protected] 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 conse- quences. Emphasis is placed on the physics of singular limits, showing the reality of sym- metry 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. Copyright A. J. Beekman et al. Received 05-09-2019 This work is licensed under the Creative Commons Accepted 04-11-2019 Check for Attribution 4.0 International License. Published 04-12-2019 updates Published by the SciPost Foundation. doi:10.21468/SciPostPhysLectNotes.11 Contents Preface 3 1 Symmetry 6 1.1 Definition6 1.1.1 Symmetries of states6 1.1.2 Symmetries of Hamiltonians7 1.2 Noether’s theorem7 1.2.1 Conserved quantities8 1.2.2 Continuity equations8 1.2.3 Proving Noether’s theorem9 1.2.4 Noether charge 11 1.3 Examples of Noether currents and Noether charges 11 1.3.1 Schrödinger field 11 1.3.2 Relativistic complex scalar field 12 1.3.3 Spacetime translations 13 1.4 Types of symmetry transformations 14 1 SciPost Phys. Lect. Notes 11 (2019) 1.4.1 Discrete versus continuous symmetries 14 1.4.2 Anti-unitary symmetries 15 1.4.3 Global symmetries versus local symmetries 15 1.4.4 Active versus passive, and internal versus external symmetries 16 1.5 Gauge freedom 17 1.5.1 Relabelling your measuring rod 17 1.5.2 Superfluous degrees of freedom 18 1.5.3 Distinguishing gauge freedom from symmetry 20 1.6 Symmetry groups and Lie algebras 22 1.6.1 Symmetry groups 22 1.6.2 Lie groups and algebras 24 1.6.3 Representation theory 25 2 Symmetry breaking 27 2.1 Basic notions of SSB 27 2.2 Singular limits 29 2.3 The harmonic crystal 33 2.4 The Heisenberg antiferromagnet 37 2.5 Symmetry breaking in the thermodynamic limit 41 2.5.1 Classification of broken-symmetry states 41 2.5.2 The order parameter 43 2.5.3 The classical state 46 2.5.4 Long-range order 46 2.5.5 The Josephson effect 48 2.6 The tower of states 50 2.7 Stability of states 52 3 Nambu–Goldstone modes 54 3.1 Goldstone’s theorem 55 3.2 Counting of NG modes 57 3.3 Examples of NG modes 59 3.3.1 NG-like excitations 61 3.4 Gapped partner modes 62 3.4.1 The tower of states for systems with type-B NG modes 63 4 Quantum corrections and thermal fluctuations 65 4.1 Linear spin-wave theory 65 4.2 Mermin–Wagner–Hohenberg–Coleman theorem 71 4.2.1 The variance of the order parameter 71 4.2.2 Matsubara summation 72 4.2.3 General NG mode propagator 74 4.2.4 Type-A NG modes 74 4.2.5 Type-B NG modes 75 5 Phase transitions 77 5.1 Classification of phase transitions 77 5.2 Landau theory 78 5.2.1 The Landau functional 78 5.2.2 First-order phase transitions 81 5.3 Symmetry breaking in Landau theory 81 5.3.1 The Mexican hat potential 81 2 SciPost Phys. Lect. Notes 11 (2019) 5.3.2 From Hamiltonian to Landau functional 83 5.4 Spatial fluctuations 85 5.5 Universality 88 6 Topological defects 89 6.1 Meaning of topological and defect 90 6.2 Topological melting: the D = 1 Ising model 92 6.3 Berezinskii–Kosterlitz–Thouless phase transition 93 6.4 Classification of topological defects 94 6.5 Topological defects at work 96 6.5.1 Duality mapping 96 6.5.2 Kibble–Zurek mechanism 97 6.5.3 Topological solitons and skyrmions 97 7 Gauge fields 98 7.1 Ginzburg–Landau superconductors 99 7.2 Gauge-invariant order parameter 100 7.3 The Anderson–Higgs mechanism 104 7.3.1 The Meissner effect 105 7.3.2 The Higgs boson 105 7.4 Vortices 107 7.5 Charged BKT phase transition 109 7.6 Order of the superconducting phase transition 110 A Other aspects of spontaneous symmetry breaking 112 A.1 Glasses 112 A.2 Many-body entanglement 113 A.3 Dynamics of spontaneous symmetry breaking 114 A.4 Quantum phase transitions 115 A.5 Topological order 116 A.6 Time crystals 117 A.7 Higher-form symmetry 117 A.8 Decoherence and the measurement problem 118 B Further reading 119 C Solutions to selected exercises 121 References 129 Preface Symmetry is one of the great unifying themes in physics. From cosmology to nuclear physics and from soft matter to quantum materials, symmetries determine which shapes, interactions, and evolutions occur in nature. Perhaps the most important aspect of symmetry in theories of physics, is the idea that the states of a system do not need to have the same symmetries as the theory that describes them. Such spontaneous breakdown of symmetries governs the dynamics 3 SciPost Phys. Lect. Notes 11 (2019) of phase transitions, the emergence of new particles and excitations, the rigidity of collective states of matter, and is one of the main ways classical physics emerges in a quantum world. The basic idea of spontaneous symmetry breaking is well known, and repeated in differ- ent ways throughout all fields of physics. More specific aspects of spontaneous symmetry breaking, and their physical ramifications, however, are dispersed among the specialised liter- ature of multiple subfields, and not widely known or readily transferred to other areas. These lecture notes grew out of a dissatisfaction with the resulting lack of a single general and com- prehensive introduction to spontaneous symmetry breaking in physics. Quantum field theory textbooks often focus on the mathematical structure, while only leisurely borrowing and dis- cussing physical examples and nomenclature from condensed matter physics. Most condensed matter oriented texts on the other hand, treat spontaneous symmetry breaking on a case-by- case basis, without building a more fundamental understanding of the deeper concepts. These lecture notes aim to provide a sound physical understanding of spontaneous symmetry break- ing while at the same time being sufficiently mathematically rigorous. We use mostly examples from condensed matter theory, because they are intuitive and because they naturally highlight the relation between the abstract notion of the thermodynamic limit and reality. These notes also incorporate some more modern developments that shed light on previously poorly under- stood aspects of spontaneous symmetry breaking, such as the counting and dispersion relations of Nambu–Goldstone modes (Section 3.2) and the role and importance of the Anderson tower of states (Section 2.6). A relatively new perspective taken in these notes revolves around the observation that the limit of infinite system size, often called the thermodynamic limit, is not a necessary condition for spontaneous symmetry breaking to occur in practice. Stronger, for almost all realistic ap- plications of the theory of symmetry breaking, it is a rather useless limit, in the sense that it is never strictly realised. Even in situations where the object of interest can be considered large, the coherence length of ordered phases is generically small, and a single domain cannot in good faith be considered to approximate any sort of infinite size. Fortunately, even relatively small objects can spontaneously break symmetries, and almost all of the generic physical con- sequences already appear in some form at small scales. The central message of spontaneous symmetry breaking then, is that objects of all sizes reside in thermodynamically stable states, rather than energy eigenstates. We will often switch between different types of symmetry-breaking systems. For example, most of these lecture notes concern the breakdown of continuous symmetries, but the most no- table differences with discrete symmetries are briefly discussed whenever they are arise. Even within the realm of continuous symmetries, many flavours exist, each with different physical consequences emerging from their breakdown. We emphasise such differences throughout the lecture notes, but without attempting to be encyclopaedic. The main aim of these notes is to foster physical understanding, and variations on any theme will be presented primarily through the discussion of concrete examples. Similarly, we will regularly switch between the Hamiltonian and Lagrangian paradigms. Many aspects of symmetry breaking can be formulated in either formalism, but sometimes one provides a clearer understanding than the other. Furthermore, it is often insightful to com- pare the two approaches, and we do this in several places in these lecture notes. For most of the discussion, we adopt the viewpoint that any physicist may be assumed to have a work- ing knowledge of both formalisms, and we freely jump between them, choosing whichever approach is most suitable for the aspect under consideration. These notes are based in part on lectures taught by J.v.W.