Landau–Ginzburg–Wilson Quantum Criticality

Landau–Ginzburg–Wilson (LGW) theory is appropriate when the long-distance degrees of freedom are fluctuations of a local order parameter and every other mode can be integrated out into regular, local couplings. Its quantum version adds temporal dynamics. Power counting then identifies the upper critical dimension and relevant perturbations; the construction fails when omitted gapless or topological degrees of freedom remain essential.

Required background. Quantum Phase Transitions and Competing Scales supplies $\nu$, $z$, and crossover scales; Degrees of Freedom, Symmetry, and the Local Operator Expansion supplies the EFT construction. Helpful background. Critical Surfaces, Crossover, and Corrections to Scaling supplies irrelevant-field drift and multicritical tuning.

Relativistic order-parameter dynamics

For an $N$-component real order parameter $\boldsymbol\phi$, a common Euclidean action is

$$S=\int d\tau\,d^dx\left[ \frac12(\partial_\tau\boldsymbol\phi)^2 +\frac{c^2}{2}(\nabla\boldsymbol\phi)^2 +\frac r2\boldsymbol\phi^2 +\frac u{4!}(\boldsymbol\phi^2)^2 -\mathbf h\cdot\boldsymbol\phi+\cdots\right].$$

The matched microscopic symmetries determine the components and allowed invariants. With short-range interactions and equal quadratic time and space derivatives, $z=1$. Under $x\mapsto bx$ and $\tau\mapsto b^z\tau$,

$$[\phi]=\frac{d+z-2}{2}, \qquad [r]=2, \qquad [u]=4-(d+z).$$

Thus the quartic interaction is marginal at effective dimension $d+z=4$. Below it, the Wilson–Fisher fixed point controls the transition; above it, the Gaussian fixed point controls leading exponents while $u$ can remain dangerously irrelevant. Wilson and Fisher’s expansion about four dimensions establishes the interacting fixed point Wilson and Fisher 1972, pp. 240–243.

The field dimension above is canonical. At an interacting fixed point it becomes $(d+z-2+\eta)/2$. Hyperscaling predictions further require that the singular free energy occupy correlated spacetime volumes and that no dangerous variable or extended gapless manifold changes their count.

Choosing the dynamical term

Symmetry alone rarely fixes time dependence. A particle–hole-symmetric bosonic transition may have $(\partial_\tau\phi)^2$ and $z=1$, whereas a generic density-driven transition can contain $\phi^*\partial_\tau\phi$ and have $z=2$. Coupling to a metal produces nonanalytic Landau damping. A conserved order parameter has different hydrodynamics from a nonconserved one at nonzero temperature.

The dynamical kernel must be matched before applying $d+z$ power counting. Adding $z$ after a static Landau calculation can miss which interactions and transport coefficients are relevant. Likewise, Wick rotation of a dissipative $|\omega_n|$ kernel is not obtained by replacing $\omega_n$ with $-i\omega$ term by term; its spectral representation fixes the retarded branch.

When the LGW reduction is valid

The construction requires:

1. a local order parameter that distinguishes the phases;
2. all non-order-parameter modes either gapped or retained explicitly;
3. a local derivative expansion in the relevant momentum and frequency window;
4. no anomaly, topological sector, or fractional field that obstructs the proposed symmetric phase;
5. the correct number of tuning parameters for the fixed point.

These conditions can hold remarkably well for insulating magnets, structural transitions, and some superconducting or bosonic transitions. They can fail for a Fermi surface, where integrating out gapless particle–hole pairs generates nonlocal kernels and singular vertices; for deconfined criticality, where fractional fields and gauge dynamics are central; or for transitions between topological phases without a conventional local order parameter.

Failure of a minimal LGW action does not mean symmetry is irrelevant. Symmetry still constrains the enlarged theory and its observables. Nor does a good order-parameter exponent prove that no additional sector exists: several theories can share exponents over finite ranges.

Mean-field and fluctuation checks

For uniform $\phi$ with $u>0$, minimizing $r\phi^2/2+u\phi^4/4!$ gives $\phi=0$ for $r>0$ and $\phi^2=-6r/u$ for $r<0$. The apparent dependence on $1/u$ is the simplest signal that $u$ can be dangerous above the upper critical dimension. Below it, fluctuations renormalize the exponent away from the mean-field value $1/2$.

A valid calculation checks stability of the potential, the full symmetry-allowed operator set, RG flow of anisotropies, and finite-size/crossover drift. A negative fitted quartic coefficient requires stabilizing higher powers and often signals a first-order or tricritical regime rather than the same fixed point.

Exercises

1. Find the upper critical spatial dimension for the relativistic theory.

Solution

$z=1$ and $u$ is marginal when $d+z=4$, so $d_c=3$. For $d<3$ the quartic coupling is relevant at the Gaussian fixed point.

2. What is the canonical dimension of $\phi^6$ at $d+z=3$?

Solution

$[\phi]=(3-2)/2=1/2$. The coupling $v_6$ in $\int d^3x\,v_6\phi^6$ has $[v_6]=3-6(1/2)=0$, so it is canonically marginal. Interactions decide its actual flow.

References

• Wilson, K. G., and M. E. Fisher. “Critical Exponents in 3.99 Dimensions.” Physical Review Letters 28 (1972): 240–243. DOI.