QPT Concepts
Landau Theory

Breakdown of Landau Theory
Generalized Mean-Field Theory

Effective Action
Gaussian Approximation
Generalized Mean-Field Theory
RG Analysis

Quantum Ferromagnetic Transitions
in Itinerant Electron Systems^*

D. Belitz⁺
University of Oregon

Abstract: After a brief introduction to phase transitions in general, and quantum phase transitions in particular, I will discuss the quantum or zero-temperature transition in bulk (3-d) itinerant ferromagnets. Conventional theory concludes that this transition should be of second order with mean-field critical exponents. I will first show that this conclusion is incorrect due to the presence of soft particle-hole excitations that couple to the order parameter. A generalized mean-field theory that takes these modes into account predicts a first-order transition in clean systems, and a second-order one with non-mean field critical exponents in systems with quenched disorder. Finally, a full RG analysis that includes the order parameter fluctuations shows that under certain conditions the first-order transition in clean systems can become unstable with respect to a fluctuation-induced second-order transition. I will discuss these conclusions in the light of recent experiments.

^* With T.R. Kirkpatrick, T. Vojta, S.L. Sessions, M.T. Mercaldo, R. Narayanan

⁺ belitz@greatwhite.uoregon.edu

I. INTRODUCTION

1. QPT Concepts

Quantum phase transitions occur at T=0 as a function of some non-thermal control parameter.
Consider, e.g., a ferromagnet

Schematic phase diagram:

CPT triggered by thermal fluctuations

QPT triggered by quantum fluctuations

Expect different critical behavior!

Example: MnSi ( Pfleiderer et al 1997 )

Near the transition line (t -> 0), one observes critical behavior :

correlation length

relaxation time
(critical slowing down!)

order parameter

OP susceptibilty

Critical behavior at the CPT is very well understood

Question: What is the critical behavior near the QCP ?

sets an energy scale

Crossover at :

Quantum vs. classical stat. mech.:

partition fct.

classically

statics + dynamics
decouple

QM

statics + dynamics
couple!

Landau-Ginzburg-Wilson (LGW) theory:

classically

QM

Hypothesis ( Hertz 1976 ):

QPT in d-dimensions related to CPT in (d+z)-dimensions!

Caution: Hypothesis plausible, but in general NOT correct (see below)

2. Landau Theory

Order parameter: average magnetization (MFA)

(Free) energy density:

Equation of state:

Landau theory predicts continuous transition at t=0 with mean-field critical exponents

Gaussian approximation for OP fluctuations:

(Ornstein-Zernike plus electron dynamics)

Classical MFT valid for 4 < d suggests QM MFT valid for 4 < d + z = d + 3 or d > 1

d_c⁺ = 1, MF critical behavior in d = 3 ( Hertz 1976 )

Hertz actually derived the Landau theory from a microscopic model.
This theory and its generalizations ( Millis 1993 ) amounts to

It predicts

mean-field exponents

T_c ~ t^3/4 (via a DIV)

II. GENERALIZED MEAN-FIELD THEORY

1. Breakdown of Landau Theory

Indication that the above conclusion cannot be correct:

In systems with quenched disorder, z = 4
d_c⁺ = 0
= 1/2 for d > 0 according to the above argument.
This violates the Harris criterion ( > 2/d) !

The resolution of these problems turns out to affect both disordered and clean systems
( TRK & DB 1996 ; TRK, DB, et al 1997 ff )

Phase transition physics is determined by soft modes ( = massless particles) in the theory.
These are,

OP fluctuations (at t 0, any T)
particle-hole excitations (at T=0, any t)
p-h excitations contribute to f₀
FM phase broken symmetry
some p-h excitations acquire a mass
contribution to f:

Note: p-h excitations soft for all t
They are the Goldstone modes of a spontaneously broken symmetry
`` Generic Scale Invariance ''

2. Generalized Mean-Field Theory ( DB, TRK, TV 1999 )

Conclusion: Landau theory misses mode-mode coupling contribution to the equation of state:

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

first order transition !

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
m^d -> m^d/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:

III. EXPERIMENTS

MnSi ( Pfleiderer et al. 1997 , 2001 )

clean (?)
low-T transition
1st order
T_c(t) mean-field like
non-Fermi liquid
behavior in PM phase
not superconducting

UGe₂ (e.g., Saxena et al 2000 , Huxley et al 2001 , Pfleiderer & Huxley 2002 )

clean
low-T transition
1st order
non-Fermi liquid
behavior in PM phase
superconducting phase

ZrZn₂ ( Pfleiderer et al 2001 ) and URhGe ( Aoki et al 2001 )

clean
low-T transition
2nd order
Superconducting phase

Ni_xPd_1-x ( Nicklas et al 1999 )

clean (?)
low-T transition
2nd order
mean-field exponents
no superconductivity

URu_2-xRe_xSi₂ ( Bauer et al 2002 )

disordered
low-T transition
2nd order
exponents as predicted by
generalized mean-field theory
non-Fermi liquid
behavior in PM phase
no superconductivity

Summary of Experiments:

Clean systems

1st order TCP at T > 0

2nd order mean-field exponents

Coexistence of superconductivity and ferromagnetism in sufficiently clean samples
( different talk )

Disordered systems

Transition 2nd order with generalized mean-field critical behavior

Non-Fermi liquid behavior in whole regions in the PM phase

IV. SOFT-MODE FIELD THEORY (DB et al 2001a , 2001b , TRK & DB 2002 )

1. Effective Action ( = effective low-energy theory)

Keep all soft modes explicitly!
2 fields: OP fluctuations
p-h fluctuations

Action:

Local, static
LGW
functional

free fermions

coupling

2. Gaussian Approximation

paramagnon
propagator

fermion
propagator

This is Landau/Gaussian theory, cf. Sec. I.2
2nd order transition with Landau exponents

Two time scales: 1) critical
2) fermionic

3. Generalized Mean-Field Theory

MFA for OP () + Gaussian approximation for fermions:

Integrate out fermions generalized MF equation of state

Nonanalyticities due to fermion loops, , etc.

Predicts first order transition (always!)

T>0 gives fermions a mass
TCP at T>0 in agreement with exps on MnSi, UGe₂

Cannot explain experiments on ZrZn₂, NiPd

4. Renormalization-Group Analysis

Assign scale dimensions

which may be either z_M or z_q !

and look for fixed points

Discuss clean case, quote results for disordered case

a) Tree level

Look for fixed point with
[G] = [H] = 0 (Fermi liquid)
[a] = [c₁] = 0 (Landau theory)

Check stability:
[t] = 2 > 0 = 1/2
[u] = - (d - 1) < 0 for d>1
[c₂] = (4 - d - z)/2

[c₂] = - (d - 1)/2 if z = z_M
[c₂] = (3 - d)/2 if z = z_q

NB: This happens if fermion loops are involved, e.g.,

c₂ relevant with respect to Landau FP for d < 3 !
Landau FP unstable due to fermion loops

Applies to disordered case as well, only the numbers change

a) Loop level

Need to keep loops systematically

e.g., u =

u has negative correction
in perturbation theory, u(b -> ) < 0 always
fluctuation-induced first order transition
Maps on classic fluctuation-induced first order transition in superconductors
( Halperin, Lubensky, Ma 1974 )
This is the generalized mean-field theory in RG language

Go beyond perturbation theory: H(b -> ) ->
Depending on initial conditions, u(b -> ) < 0 or > 0
Transition may be first or second order !

Physical interpretation :
H ~ specific heat coefficient
Critical behavior of specific heat couples back to u
Energy scales involved: Correlation energy vs Fermi energy or bandwidth
First vs second order depends on ratio of energy scales;
strong correlation favors first order

First order transition
stable if t₁ sufficiently large
destroyed by critical fluctuations if t₁ is small

Instability of first order transition leads to non-mean field, fluctuation-induced second order
transition. Explicit calculation
Second order case in d = 3 has mean-field exponents with log corrections
(due to the marginal operator c₂ ); non-mean field exponents in d < 3
Consistent with experiments on clean systems !

Explicit calculation for disordered case
Critical behavior as described by generalized mean-field theory with log corrections
Agrees with experiment on NiPd !

V. SUMMARY AND CONCLUSION

Low-T_c itinerant ferromagnets are remarkably complex and interesting.

The T = 0 transition can be 1st order for generic reasons: A fluctuation-induced 1st order transition preempts the continuous Landau transition.

Crucial for this mechanism are soft fermionic modes and the resulting two time scales in the problem.

If the 1st order transition is too close to the second order one it is trying to preempt, it becomes unstable with respect to a fluctuation-induced 2nd order transition. This transition is in a new universality class.

Sufficiently strong non-magnetic disorder causes the transition to always be 2nd order. This constitutes yet another universality class, which is exactly solved.

These results explain, or are consistent with, all experiments to date.

Properties that are not understood (at least not completely):

Ferromagnetic superconducting phase (partially understood)
Non-Fermi liquid behavior (???)

correlation length
relaxation time (critical slowing down!)
order parameter
OP susceptibilty

partition fct.
classically		statics + dynamics decouple
QM		statics + dynamics couple!

1. QPT Concepts

Schematic phase diagram: CPT triggered by thermal fluctuations QPT triggered by quantum fluctuations Expect different critical behavior!

correlation length

relaxation time (critical slowing down!)

order parameter

OP susceptibilty

Crossover at :

partition fct.

classically

statics + dynamics decouple

QM

statics + dynamics couple!

classically

QM

(Free) energy density:

Equation of state:

(Ornstein-Zernike plus electron dynamics)

dc+ = 1, MF critical behavior in d = 3 ( Hertz 1976 )

1. Breakdown of Landau Theory

2. Generalized Mean-Field Theory ( DB, TRK, TV 1999 )

IV. SOFT-MODE FIELD THEORY (DB et al 2001a , 2001b , TRK & DB 2002 )

1. Effective Action ( = effective low-energy theory)

a) Tree level

a) Loop level

Schematic phase diagram:

CPT triggered by thermal fluctuations

QPT triggered by quantum fluctuations

Expect different critical behavior!

relaxation time
(critical slowing down!)

statics + dynamics
decouple

statics + dynamics
couple!

d_c⁺ = 1, MF critical behavior in d = 3 ( Hertz 1976 )