Emergent Symmetry

An emergent symmetry becomes accurate as a system is viewed at lower energy or longer wavelength, even though the microscopic Hamiltonian lacks that exact symmetry. Its defining feature is an improving approximation with scale. A lattice may allow only a few discrete rotations, yet sufficiently long waves can have an isotropic dispersion. Other systems retain anisotropy at every scale: emergence must be established for the particular model and observable.

Helpful background. Exact symmetry identifies invariance of a specified Hamiltonian, approximate symmetry explains small breaking, and spontaneous symmetry breaking distinguishes the state from the laws governing it.

Rotational symmetry in a square-lattice band

Consider a single orbital on each site of a square lattice, lattice spacing $a$, and equal nearest-neighbor hopping $t>0$ along both axes. Its dispersion is

$$E(\mathbf k)=-2t[\cos(k_xa)+\cos(k_ya)].$$

The lattice has discrete rotations by $\pi/2$, rather than arbitrary rotations. Near the band minimum,

$$E(\mathbf k)-E(0) =ta^2(k_x^2+k_y^2) -\frac{ta^4}{12}(k_x^4+k_y^4)+O(t a^6|\mathbf k|^6).$$

The leading term depends only on $|\mathbf k|$, giving an isotropic effective mass $m_*=\hbar^2/(2ta^2)$. At fourth order the lattice is visible again: $k_x^4+k_y^4$ is not invariant under arbitrary rotations. At fixed small $k$, compare an axial wavevector $(k,0)$ with a diagonal one $(k/\sqrt2,k/\sqrt2)$:

$$E_{\rm axis}-E_{\rm diagonal} =-\frac{ta^4k^4}{24}+O(t a^6k^6).$$

Relative to the leading excitation energy $ta^2k^2$, the anisotropy is $-(ka)^2/24+O((ka)^4)$. Halving the wave number reduces its magnitude by about four. That is a quantitative emergence claim, with a regime $ka\ll1$ and a predicted correction.

Equal axial hoppings matter. With $t_x\ne t_y$, the leading energy is $a^2(t_xk_x^2+t_yk_y^2)$, already anisotropic. Rescaling coordinates can simplify its mathematical form but does not by itself make laboratory responses isotropic. The lattice-model introduction explains how microscopic couplings enter continuum descriptions.

Scale dependence of symmetry-breaking couplings

Near a candidate scale-invariant description, write a schematic effective theory as a symmetric part plus symmetry-breaking operators. For a dimensionless coupling $g$ with positive correction exponent $y$, its linearized running with a wave-number scale $\mu$ can take the form

$$g(\mu)\simeq g(\Lambda)(\mu/\Lambda)^y, \qquad \mu\ll\Lambda.$$

Here $\Lambda$ is a microscopic wave-number cutoff; $y$ is a correction exponent, not simply the scaling dimension of the operator. The coupling shrinks at long distance and is called irrelevant near that fixed point. An energy-scale formula uses the corresponding dynamical scaling relation. Marginal couplings can instead run logarithmically, and relevant symmetry-breaking couplings grow.

This explains how a model with no small microscopic anisotropy can approach a more symmetric description, provided it lies in the fixed point’s basin of attraction. Relevant perturbations may need tuning. A large correlation length or large sample alone does not guarantee the proposed symmetry. Cardy’s treatment develops the renormalization-group reasoning behind these statements.

An approximate symmetry can arise simply because a coefficient is small. Calling it emergent additionally identifies how the approximation improves in a limiting regime. Both descriptions can apply to the same system.

Linear dispersion and Lorentz-like descriptions

A gapless excitation with $E\simeq\hbar v|\mathbf k|$ has the dispersion expected of a massless relativistic field with speed $v$. This is a useful clue, but a dispersion alone does not establish Lorentz symmetry. The effective interactions and correlation functions must have the same symmetry, and different coupled species must have compatible limiting velocities. Anisotropic or Lorentz-breaking terms can remain important.

When a Lorentz-like low-energy theory does emerge, its velocity is set by the material, not necessarily the speed of light. Lattice corrections remain outside the long-wavelength regime. Likewise, shared critical exponents do not make every quantity universal: velocities, normalization factors, and microscopic amplitudes can remain different. See universality and Sachdev’s treatment of quantum criticality.

Emergent symmetry in an ordered phase

Emergence concerns the effective laws; spontaneous breaking concerns the state. An effective continuous symmetry can organize nearly degenerate order-parameter orientations, while a small lattice anisotropy ultimately chooses preferred directions. The resulting collective mode may be light but gapped, as discussed in Goldstone modes.

An anisotropy can also be dangerously irrelevant: it shrinks near a critical fixed point yet controls a further scale or orientation selection in the adjacent ordered phase. Thus a symmetry accurate in a critical scaling window need not govern the entire ordered phase down to the longest distances. This is why one should follow specific spectra or correlation functions as scale changes, instead of inferring an exact continuous symmetry from a nearly circular plot at one resolution.

Exercises

1. In the square-lattice example, compare the relative axial–diagonal splitting at $ka=0.2$ and $ka=0.1$ to leading order.

Solution

The magnitudes are $(0.2)^2/24\simeq1.67\times10^{-3}$ and $(0.1)^2/24\simeq4.17\times10^{-4}$, respectively. The absolute splitting scales as $k^4$ and falls by sixteen, but the leading excitation energy also falls by four; their ratio falls by four. Mixing these two comparisons would give the wrong scaling exponent.

2. A measured dimensionless anisotropy is $g(L)=0.1(L/a)^{1/2}$ within a linearized regime. Does increasing the size support restoration of the larger symmetry? When does the small-coupling estimate cease to be controlled?

Solution

The anisotropy grows with $L$, so the system moves away from the proposed symmetric description. It becomes order one at $L/a\sim100$, where the linearized small-coupling estimate can no longer predict the flow quantitatively. Small anisotropy in a modest sample can therefore describe a crossover without demonstrating long-distance restoration.

References

• J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, 1996.
• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press, 2011.
• K. G. Wilson and J. Kogut, “The renormalization group and the epsilon expansion,” Physics Reports 12, 75–199 (1974). doi:10.1016/0370-1573(74)90023-4.