Generic Scale Invariance and Critical Behavior :
A Unifying View of Quantum and Classical Phase Transitions *

I. Critical Scale Invariance

  1. Landau Theory
  2. LGW Theory
  3. Classical vs Quantum Phase Transitions

II. Generic Scale Invariance

  1. Goldstone Modes
  2. Gauge Symmetry

III. Interplay between GSI and Critical Behavior

  1. Superconductors / Liquid Crystals
  2. Quantum Ferromagnets

IV. Conclusion

Dietrich Belitz

University of Oregon

* With T.R. Kirkpatrick and T. Vojta
based on cond-mat/0403182
available at


1. Landau Theory

Landau 1937 unified mean-field theory of phase transitions

Landau theory predicts

    continuous phase transition at (r = 0, h = 0) provided u > 0

    1st order transition if u < 0

    power-law behavior for r , h 0 , e.g.

    long-range correlations at r = 0 , e.g.

    with a correlation length

    Long range due to soft or massless modes (viz., order-parameter fluctuations)
    (cf. long-range Coulomb interaction <=> massless photons)

2. LGW Theory

Ginzburg criterion Landau theory valid only for d > dc+ (= 4 for many popular transitions)

Wilson 1970's LGW theory, and (Wilsonian) RG

Experiment: Scaling and universality are observed

Wilson & Fisher 1972 Exponents can be calculated in a dc+ - expansion

Ferrell et al 1967, Halperin & Hohenberg 1967 dynamical scaling of time correlation functions,

with z the dynamical critical exponent

3 independent exponents: 2 static ones plus z

3. Classical vs Quantum Phase Transitions

In classical mechanics, statics and dynamics decouple:

thermodynamics can be calculated independent of the dynamics

In quantum mechanics, statics and dynamics couple:

LGW theory generalizes to a quantum field theory

Imaginary time direction finite for T > 0 crossovers from quantum to classical scaling

Phase diagram of MnSi
( Pfleiderer et al 1997 )

Two scenarios:

Crucial points:


Critical soft modes occur at isolated critical points in the phase diagram

Modes can be soft in entire regions of the phase diagram Generic scale invariance

Consider two mechanisms for GSI in real space. GSI can also occur in time space long-time tails

1. Goldstone Modes

Goldstone's theorem:

Spontaneous breaking of a continuous symmetry soft modes in the entire broken symmetry phase

2. Gauge Symmetry

Local gauge invariance massless gauge fields

Simplest realization: U(1)


Part I All soft modes need to be kept

Expect GSI, if present, to influence critical behavior

Consider two examples in detail, mention a few others

1. Superconductors / Liquid Crystals / Scalar QED

Consider a superconductor

Consider a nematic liquid crystal

Effects order-parameter fluctuations:

2. Itinerant Quantum Ferromagnets

Soft modes:

Particle-hole excitations contribute to the free energy

Mean-field free energy in d = 3 same as for superconductor/liquid crystal in d = 4

Generalized mean-field equation of state (d=3):

1st order transition! ( DB, TRK, TV 1999 )

Experiments: Sufficiently clean materials all show tricritical point , with 2nd order transition at high T,
and 1st order transition at low T:




( Pfleiderer et al 1997 )

( Pfleiderer & Huxley 2002 )

( Uhlarz et al 2004 )

Effects of nonzero temperature (T) and disorder (G) :

Effect of nonzero magnetic field ( DB, TRK, J. Rollbühler 2004 ):


IV. Conclusion

Generic soft modes can strongly influence critical behavior, and even destroy it, in both classical and quantum systems

Generic soft modes are ubiquitous in metals at T = 0

Quantum phase transitions in metals are i.g. not simple

Classical example : Superconductor/nematic liquid crystal.

GSI due to gauge field fluctuation-induced 1st order transition

QM example : Itinerant quantum ferromagnet.

GSI due to Goldstone mode fluctuation-induced 1st order transition at h = 0
mean-field QCP at h 0

Another example: