Influence of Generic Scale Invariance on Quantum Critical Behavior:
The Case of Itinerant Ferromagnets *

I. Introduction

  1. Experimental Overview
  2. Conventional Theory

II. Generalized Mean-Field Theory

  1. Breakdown of Landau Theory
  2. Generalized Mean-Field Theory

III. Soft-Mode Field Theory

  1. Effective Action
  2. Analysis

IV. Summary and Conclusion

Dietrich Belitz +

University of Oregon

* With T.R. Kirkpatrick, M.T. Mercaldo, J. Rollbühler,
S. Sessions, T. Vojta


1. Experimental Overview

Itinerant ferromagnets whose Tc can be tuned to zero include,

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 )

T=0 1st order transition persists at nonzero magnetic field, ends at quantum critical point


This behavior of clean systems appears to be universal

URuReSi shows 2nd order transition with non-mean-field exponents (e.g., = 3/2) ( Bauer et al 2002 )

NiPd shows 2nd order transition with mean-field exponents ( Nicklas et al 1999 )

MnSi shows NFL behavior in PM phase ( Pfleiderer et al 2001 , 2004 )

2. Conventional Theory

Consider Landau theory with order parameter m = average magnetization

(Free) energy density:

Equation of state:

Landau theory predicts

Sandeman et al 2003 , Shick et al 2004 Band structure in UGe2 leads to u < 0
Origin of 1st order transition at T = 0

Hertz 1976 Mean-field theory correctly describes T = 0 transition for all d > 1 in clean systems,
and for all d > 0 in disordered ones.


1. Breakdown of Landau Theory

Hertz's conclusion is now known to be incorrect: Soft modes in addition to OP fluctuations

Breakdown of Landau theory for d 3 (clean), and d 4 (dirty), respectively
( TRK & DB 1996 ; TRK, DB, et al 1997 ff )

Phase transition physics is determined by all of the soft modes in the problem.

These are,


particle-hole excitations soft for all values of t

They are the Goldstone modes of a spontaneously broken symmetry ( Wegner 1979 )

Generic Scale Invariance

It is impossible to construct a local field theory in terms of the OP only.
(Breakdown of LGW paradigm)

This is not revolutionary. Analogous effects occur at other phase transitions,

    • Nematic-to-smectic-A in liquid crystals ( Chen, Lubensky, Nelson 1978 )

    • BCS superconductors (academic case) ( Halperin, Lubensky, Ma 1974 )

    • Critical dynamics in classical fluids ( Kawasaki 1976 , Halperin & Hohenberg 1977 )

    and in particle physics,

    • Fermi theory of weak interactions

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):

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

Effect of nonzero magnetic field (h):

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

1. Effective Action

Keep all soft modes explicitly!

two fields:

OP fluctuations

p-h fluctuations


2. Analysis

The effective action has been analyzed at various levels:


Low-Tc itinerant ferromagnets are remarkably complex and interesting.

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

Crucial for this mechanism: Fermionic modes and the resulting two time scales in the problem.

Theory explains

Sufficiently strong non-magnetic disorder drives transition 2nd order.

New universality class, which is solved exactly. Agrees with experimental results on URuReSi.

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