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.

arXiv comment149 pages; matches published version

References in corpus (55)

• Decoherence, einselection, and the quantum origins of the classical
• Quantum Spin Liquids
• Generalized Global Symmetries
• Observation of a Discrete Time Crystal
• Observation of discrete time-crystalline order in a disordered dipolar many-body system
• Quantum Time Crystals
• Zoo of quantum-topological phases of matter
• Topological classification of crystalline insulators through band structure combinatorics
• Theory of the spin bath
• Absence of Quantum Time Crystals
• A Duality Web in 2+1 Dimensions and Condensed Matter Physics
• Historical roots of gauge invariance
• Amplitude / Higgs Modes in Condensed Matter Physics
• Superconductors are topologically ordered
• Impossibility of Spontaneously Rotating Time-Crystals: A No-Go Theorem
• Spontaneously Broken Spacetime Symmetries and Goldstone's Theorem
• Unified Description of Nambu-Goldstone Bosons without Lorentz Invariance
• Particle-Vortex Duality from 3d Bosonization
• An introduction to the Ginzburg-Landau theory of phase transitions and nonequilibrium patterns
• Nonrelativistic effective Lagrangians
• Counting rule for Nambu-Goldstone modes in nonrelativistic systems
• Relativistic Fluids, Superfluids, Solids and Supersolids from a Coset Construction
• Kaon Condensation and Goldstone's Theorem
• Spontaneous Symmetry Breaking and Nambu-Goldstone Bosons in Quantum Many-Body Systems
• Why Decoherence has not Solved the Measurement Problem: A Response to P. W. Anderson
• Quantum Melting and Absence of Bose-Einstein Condensation in Two-Dimensional Vortex Matter
• On the number of Nambu-Goldstone bosons and its relation to charge densities
• Redundancies in Nambu-Goldstone Bosons
• Emergent (anomalous) higher symmetries from topological orders and from dynamical electromagnetic field in condensed matter systems
• Effective Lagrangian for Nonrelativistic Systems
• Massive Nambu-Goldstone Bosons
• A relativistic non-relativistic Goldstone theorem: gapped Goldstones at finite charge density
• More on gapped Goldstones at finite density: More gapped Goldstones
• Counting Rules of Nambu-Goldstone Modes
• Goldstone modes and photonization for higher form symmetries
• Entanglement Entropy of Systems with Spontaneously Broken Continuous Symmetry
• A short review on Noether's theorems, gauge symmetries and boundary terms
• Quasi-Nambu-Goldstone Modes in Bose-Einstein Condensates
• Nambu-Goldstone bosons with fractional-power dispersion relations
• Dispersion relations of Nambu-Goldstone modes at finite temperature and density
• On the relation between decoherence and spontaneous symmetry breaking
• Quantum Dynamics in the Thermodynamic Limit
• Remarks on nonrelativistic Goldstone bosons
• What symmetry is actually broken in the Higgs phase of a gauge-Higgs theory?
• Quantum dynamics of tunneling between ferromagnets
• The Tower of States and the Entanglement Spectrum in a Coplanar Antiferromagnet
• Broken Time Translation Symmetry as a model for Quantum State Reduction
• Conformal symmetry of trapped Bose-Einstein condensates and massive Nambu-Goldstone modes
• From physical symmetries to emergent gauge symmetries
• Deconfining the rotational Goldstone mode: the superconducting nematic liquid crystal in 2+1D
• Interpolating relativistic and non-relativistic Nambu-Goldstone and Higgs modes
• Quantum measurement driven by spontaneous symmetry breaking
• Criteria for the absence of quantum fluctuations after spontaneous symmetry breaking
• Theory of generalized Josephson effects
• Spontaneous Symmetry Breaking as the Mechanism of Quantum Measurement

Cited by in corpus (81)

• Ultra-Light Dark Matter
• Continuous Symmetry Breaking in a Two-dimensional Rydberg Array
• An Introduction to Higher-Form Symmetries
• Production and characterization of a fragmented spinor Bose-Einstein condensate
• Robust spin squeezing from the tower of states of $U(1)$-symmetric spin Hamiltonians
• Effective Field Theory for Quasicrystals and Phasons Dynamics
• Counting Nambu-Goldstone modes of higher-form global symmetries
• Pseudo-spontaneous $U(1)$ Symmetry Breaking in Hydrodynamics and Holography
• Continuous Dissipative Phase Transitions without Symmetry Breaking
• Simple views on symmetries and dualities in the theory of elasticity
• Interplay between the Lyapunov exponents and phase transitions of charged AdS black holes
• Field theories with higher-group symmetry from composite currents
• Scalable spin squeezing from spontaneous breaking of a continuous symmetry
• Exact microcanonical statistical analysis of transition behavior in Ising chains and strips
• Nonequilibrium phases of ultracold bosons with cavity-induced dynamic gauge fields
• Effective Field Theory for Spontaneously Broken Symmetry
• Phenomenology of bond and flux orders in kagome metals
• Holographic superconductivity in Einsteinian Cubic Gravity
• Effects of finite volume and magnetic fields on thermodynamic properties of quark matter and fluctuations of conserved charges
• Towards a phase diagram of the topologically frustrated XY chain
• Effective field theory of magnon: Dynamics in chiral magnets and Schwinger mechanism
• Nonequilibrium symmetry-protected topological order: emergence of semilocal Gibbs ensembles
• Unitary k-designs from random number-conserving quantum circuits
• Spontaneous breaking of U(1) symmetry at zero temperature in one dimension
• Unconventional quantum phase transitions in a one-dimensional Lieb-Schultz-Mattis system
• Quantum spin systems versus Schroedinger operators: A case study in spontaneous symmetry breaking
• Exploring energy landscapes of charge multipoles using constrained density functional theory
• An objective collapse model without state dependent stochasticity
• Quantum state reduction of general initial states through spontaneous unitarity violation
• Effects of local decoherence on quantum critical metrology
• Route to Extend the Lifetime of a Discrete Time Crystal in a Finite Spin Chain Without Disorder
• Spontaneous Symmetry Breaking, Spectral Statistics, and the Ramp
• An introduction to kinks in ${\phi }^4$-theory
• Topological modes and spectral flows in inhomogeneous PT-symmetric continuous media
• Role of conservation laws in the Density Matrix Renormalization Group
• Closure of the entanglement gap at quantum criticality: The case of the Quantum Spherical Model
• Two Nambu-Goldstone zero modes for rotating Bose-Einstein condensates
• Induced vacuum current and magnetic flux in quantum scalar matter in the background of a vortex defect with the Neumann boundary condition
• Entanglement gap, corners, and symmetry breaking
• Colored noise driven unitarity violation causing dynamical quantum state reduction
• Nature of Spinons in 1D Spin Chains
• Time-crystalline behavior in central-spin models with Heisenberg interactions
• Long-range order in the XY model on the honeycomb lattice
• Kardar-Parisi-Zhang scaling in time-crystalline matter
• Phase transitions as a manifestation of spontaneous unitarity violation
• Constants of motion characterizing continuous symmetry-broken phases
• Symmetry breaking at a topological phase transition
• Dipole symmetries from the topology of the phase space and the constraints on the low-energy spectrum
• Digital simulation of zero-temperature spontaneous symmetry breaking in a superconducting lattice processor
• Fractal dimension and the counting rule of the Goldstone modes
• Stability and Absence of a Tower of States in Ferrimagnets
• Floquet-engineered Emergent Massive Nambu-Goldstone Modes
• Tomonaga-Luttinger Liquid Behavior in a Rydberg-encoded Spin Chain
• Eccentric Catastrophes & What To Do With Them
• General theory for discrete symmetry-breaking equilibrium states
• Spontaneous symmetry breaking and Husimi Q-functions in extended Dicke model
• Extracting the symmetries of nonequilibrium quantum many-body systems
• Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement
• Entanglement gap in 1D long-range quantum spherical models
• An introduction to Goldstone boson physics and to the coset construction
• Entanglement-enabled symmetry-breaking orders
• Third law of thermodynamics and the scaling of quantum computers
• Non-commutative effective field theory of the lowest Landau level superfluid
• Lieb-Mattis states for robust entangled differential phase sensing
• Collective terahertz fluctuation modes in a polariton laser
• Continuous spontaneous localization as the white-noise limit of spontaneous unitarity violation
• The toric code under antiferromagnetic isotropic Heisenberg interactions
• Reaching Quantum Critical Point by Adding Non-magnetic Disorder in Single Crystals of Superconductor $({\text{Ca}}_x{\text{Sr}}_{1-x}{)}_3{\text{Rh}}_4{\text{Sn}}_{13}$
• Entanglement entropy as a probe of topological phase transitions
• Thermal field theories coupled to curved spacetime
• Spontaneous continuous-symmetry breaking and tower of states in a comb chain
• Spontaneous Symmetry Breaking without classical fields: a Functional Renormalization Group approach
• Symmetry-enforced minimal entanglement and correlation in quantum spin chains
• Hawking radiation as a manifestation of spontaneous symmetry breaking
• Solving Sharp Bounded-error Quantum Polynomial Time Problem by Evolution methods
• Logarithmic quantum time crystal
• Spontaneous Symmetry Breaking In Nonlinear Binary Periodic Systems
• Artificial versus Natural Atoms: The uncanny capability of the many-body Schrödinger equation to produce emergent behavior
• Undecidability of the spectral gap in rotationally symmetric Hamiltonians
• Spontaneous Symmetry Breaking and Frustrated Phases
• Signatures of three-state Potts nematicity in spin excitations of the van der Waals antiferromagnet FePSe${}_3$