Emergent or concealed symmetries governing particle interactions and quantum order in materials.Hidden symmetry in quantum systems refers to symmetries that are not apparent from the system's Hamiltonian or Lagrangian but emerge dynamically, often governing particle interactions and quantum order in materials. These symmetries can be 'hidden' because they appear only at certain energy scales, in certain phases, or after a transformation that reveals a simpler underlying structure. They are crucial for explaining phenomena like topological order, where the ground state degeneracy and quasiparticle statistics are protected by symmetries that are not visible in the local degrees of freedom.
A hidden symmetry often arises when a system's low-energy effective theory has a larger symmetry than the microscopic theory. For example, in quantum spin liquids, the fractionalized excitations (spinons) obey emergent gauge symmetries that are not present in the original spin Hamiltonian. These symmetries can be uncovered by mapping the system to a dual representation, such as a gauge theory or a free fermion model, where the symmetry becomes manifest. The hidden symmetry then constrains the possible quantum phases and transitions, providing a powerful organizing principle.
Hidden symmetries are used to classify and predict topological phases of matter, which are candidates for fault-tolerant quantum computing.
| r_n | variable | Level spacing ratio for adjacent energy levels — level spacing ratio |
| s_n | variable | Spacing between consecutive energy levels — energy level spacing joule |
| min | operator | Selects the smaller of the two arguments — minimum function |
| max | operator | Selects the larger of the two arguments — maximum function |
| \Delta X | variable | Change in a physical quantity — physical quantity change |
| \theta | variable | Angle variable — angle radian |
| \theta_0 | variable | Reference angle offset — angle radian |
| \propto | operator | Indicates the left side scales with the right side — proportionality |
| \cos | operator | Computes the cosine of an angle — cosine function |
| x | variable | input point — input variable |
| x' | variable | another input point — input variable |
| σ | variable | signal amplitude — amplitude |
| ℓ | variable | length scale — length scale |
| exp | operator | raises e to the given power — exponential function |


