I. Critical Scale Invariance Landau Theory LGW Theory Classical vs Quantum Phase Transitions II. Generic Scale Invariance Goldstone Modes Gauge Symmetry III. Interplay between GSI and Critical Behavior Superconductors / Liquid Crystals Quantum Ferromagnets IV. Conclusion

Dietrich Belitz belitz@physics.uoregon.edu University of Oregon * With T.R. Kirkpatrick and T. Vojta based on cond-mat/0403182 available at http://physics.uoregon.edu/~belitz/talks

I. CRITICAL SCALE INVARIANCE

1. Landau Theory

Landau 1937 unified mean-field theory of phase transitions

	free energy is analytic function of order parameter:
	order parameter susceptibility has Ornstein-Zernike form:

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)

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2. LGW Theory

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

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

Field-theoretic generalization of Landau theory:



with an ``action''



Free energy displays scaling:



Universality classes

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Experiment: Scaling and universality are observed

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

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

with z the dynamical critical exponent

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3 independent exponents: 2 static ones plus z

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3. Classical vs Quantum Phase Transitions

In classical mechanics, statics and dynamics decouple:

thermodynamics can be calculated independent of the dynamics

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In quantum mechanics, statics and dynamics couple:

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Must solve for thermodynamics and dynamics simultaneously

Scaling of free energy generalizes to



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LGW theory generalizes to a quantum field theory

LGW paradigm: Integrate out all degrees of freedom other than the order parameter



with an LGW action



The functional form of depends on the details of the problem

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

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Phase diagram of MnSi ( Pfleiderer et al 1997 )

Two scenarios:

• Quantum transition maps on classical transition in d + z dimensions
  
  • True only if soft-mode spectrum is the same at T = 0 and at T > 0
  • NB: If this is the case, then the quantum transition will i.g. be mean-field-like! ( Hertz 1976 )

• Quantum transition maps on classical transition with extra soft modes in d + z dimensions
  • NB: If this is the case, then the LGW paradigm will break down!

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Crucial points:

Theory needs to take in to account all of the soft modes in the problem!

LGW theory will be a local field theory only if no soft modes have been integrated out !

Check for soft modes other than the order parameter fluctuations

This hold for both quantum and classical theories!

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II. GENERIC SCALE INVARIANCE

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

Mechanical analog shows

• critical modes (transverse/radial modes)
• Goldstone mode (azimuthal mode)

More generally: Modes soft at zero wavenumber (mechanical model has no wavenumber concept)

Examples:

• spin waves in ferromagnets
• director fluctuations in nematic liquid crystals
• particle-hole excitations in metals

Theoretical realization: Classical O(2) - symmetric theory

action

• symmetric phase: 2 massive modes ( )
• broken-symmetry phase: 1 massive mode ( ) ; 1 massless Goldstone mode ( )

2. Gauge Symmetry

Local gauge invariance massless gauge fields

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Simplest realization: U(1)

Examples:

• charged particles coupled to photons (e.g., superconductor)
• nematic liquid crystals (later)

Theoretical realization: scalar gauge theory

action
symmetric phase: 2 massive modes ( ) ; 2 massless ones (2 components of A)
broken-symmetry phase: 4 massive modes (1 component of ,
3 components of A )

Gauge field eats the Goldstone mode, becomes massive in the process

Situation reversed from case without gauge symmetry ( Anderson 1963 , Higgs 1964 )

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III. INTERPLAY BETWEEN GSI AND CRITICAL BEHAVIOR

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

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1. Superconductors / Liquid Crystals / Scalar QED

Consider a superconductor

Order parameter is complex scalar field (amplitude + phase of Cooper pair)

Cooper pairs carry charge q = 2e Coupling to E&M vector potential (transverse photons)

Action given by the gauge theory given above

Extend to 3+1 dimensions A simple particle physics model: Scalar QED (scalar mesons + photons)

Treat in mean-field approximation A can be integrated out exactly

mean-field free energy

with and closely related to r and u, and

NB: F is not an analytic function of !

For later reference: In d=4 one finds

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Discussion:

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Consequences for superconductors ( Halperin, Lubensky, Ma 1974 )

• Transition is 1st order! ( "fluctuation-induced 1st order transition" , unobservable in superconductors)

• Photon becomes massive ( Meissner effect )

• Ordered phase results in loss of soft modes ( Anderson )

Consequences for particle physics ( Coleman & Weinberg 1973 )

• Massless (= critical) theory behaves like the broken-symmetry (= ordered phase) one

• Spontaneous creation of mass ( "Coleman-Weinberg mechanism" )

• Gauge bosons becomes massive

• Spontaneous symmetry breaking results in loss of massless particles ( Higgs mechanism )

Consider a nematic liquid crystal

Nematic-to-smectic-A transition maps onto superconducting one ( Halperin & Lubensky 1974 )

Same conclusions as above. 1st order transition has been observed!

Effects order-parameter fluctuations:

Quality of mean-field approximation depends on ration of length scales (type I vs type II)

Experiment 1st order in type I materials, 2nd order in type II materials

Numerics + theory 1st order transition gives way to inverted XY universality class
(incompletely understood)

2. Itinerant Quantum Ferromagnets

Soft modes:

Order parameter fluctuations

particle-hole excitations

Particle-hole excitations contribute to the free energy

in the paramagnetic phase

and in the ferromagnetic phase broken symmetry
some p-h excitations acquire a mass
contribution

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:

	MnSi	UGe2	ZrZn2
	( Pfleiderer et al 1997 )	( Pfleiderer & Huxley 2002 )	( Uhlarz et al 2004 )

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

T > 0 gives p-h exitations a mass ln m -> ln (m+T) tricritical point

G > 0 changes fermion dispersion relation md -> md/2 G > 0 changes sign of coefficient Generalized mean-field equation of state (d=3) transition second order with non-mean field (and non-classical) exponents etc.

Phase diagrams:

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Experiment: URu_2-xRe_xSi₂ shows 2nd order transition with = 3/2 ( Bauer et al 2002 )

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

At QCPs Hertz theory is valid Mean-field quantum critical behavior

Experiment: Wing structure observed in MnSi ( Pfleiderer et al 2001 ) and ZrZn₂ ( Uhlarz et al 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:

LTTs in classical fluids are an example of temporal GSI

Enhancement of LTTs at criticality Modification of critical dynamics

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