1. Self-generated instability of a ferromagnetic quantum-critical point
1D physics in D >1
Andrey Chubukov
University of Maryland
Workshop on Frustrated Magnetism, Sept. 14, 2004

2. Quantum phase transitions in itinerant ferromagnets
ZrZn2
UGe2
pressure
First order transition at low T

3. Itinerant electron systems near a ferromagnetic instability
Fermi liquid
Ferromagnetic phase
What is the critical theory?
What may prevent a continuous transition to ferromagnetism ?

4. Quantum criticality Hertz-Millis-Moriya theory:
fermions are integrated out
is a quantum critical point
Z=3
Dcr = 4-Z =1
In any D >1, the system is above its upper critical dimension
(fluctuations are irrelevant?)

5. What can destroy quantum criticality?
1. Fermions are not free at QCP
ZF = 1, Dcr = 4 - ZF = 3
Below D=3, we do not have a Fermi liquid at QCP
Coupling constant
diverges at QCP

6. The replacement of a FL at QCP is “Eliashberg theory”
+ no vertex corrections
Altshuler et al
Haslinger et al
Pepin et al
fermionic self-energy (D=2) g
non –Fermi liquid at QCP
spin susceptibility
Still, second order transition
Same form as for free electrons

7. Can something happen before QCP is reached?
Khodel et al
Rice, Nozieres
Landau quasiparticle interaction function

8. Near quantum criticality
In 2D
This reasoning neglects Z-factor renormalization near QCP

9. mass renormalization
Z-factor renormalization
outside Landau theory
within Landau theory

10. Z – factor renormalization
Results:

11. In the two limits:
the two terms are cancelled out
regular piece
anomalous piece

12. Where is the crossover?
Low-energy analysis is justified only if

13. Results:

14. What else can destroy quantum criticality?
2. Superconductivity
Spin-mediated interaction is attractive in p-wave channel
first order
transition
Haslinger et al - SC

15. Dome of a pairing instability above QCP

16. At QCP
In units of

17. Superconductivity near quantum criticality
UGe2
Superconductivity affects an ordered phase, not observed in a paramagnet

18. What else can destroy quantum criticality?
3. Non-analyticity
Hertz-Millis-Moriya theory:
Always assumed

19. Why is that? Use RPA: Lindhard function in 3D Expand near Q=0
is a Lindhard function
Lindhard function in 3D
Expand near Q=0
an analytic expansion

20. Analytic expansion in momentum at QCP is related to
the analyticity of the spin susceptibility for free electrons Q:
Is this preserved when fermion-fermion
interaction is included?
(is there a protection against fractional powers of Q?)
Is there analyticity in a Fermi liquid?

21. Fermi Liquid
Self-energy
Uniform susceptibility
Specific heat

22. Corrections to the Fermi-liquid behavior
Expectations based on a general belief of analyticity:
Fermionic damping
Resistivity

23. 3D Fermi-liquid Fermionic self-energy: 50-60 th Specific heat:
(phonons, paramagnons)
Susceptibility
Carneiro, Pethick, 1977
Belitz, Kirkpatrick, Vojta, 1997
non-analytic correction

24. In D=2
Spin susceptibility
T=0, finite Q
Q=0, finite T

25. Charge susceptibility
No singularities

26. Where the singularities come from?
Singular corrections come from the universal singularities in the dynamical response functions of a Fermi liqiuid
Only U(0) and U(2pF) are relevant

27. Spin susceptibility Only U(2pF) contibutes Specific heat Q=0, finite T
T=0, finite Q
Only U(2pF) contibutes
Specific heat

28. Only two vertices are relevant:
Transferred momenta are near 0 and 2 pF
Total momentum is near 0
1D interaction in D>1 is responsible for singularities
These two vertices are parts of the scattering amplitude

29. Arbitrary D
Extra logs in D=1
Corrections are caused by Fermi liquid singularities in the effectively 1D response functions
These non-analytic corrections are the ones that destroy
a Fermi liquid in D=1

30. A very similar effect in a dirty Fermi liquid:
Das Sarma, 1986
Das Sarma and Hwang, 1999
Zala, Narozhny, Aleiner 2002
A linear in T conductivity is
a consequence of a non-analyticity
of the response function in a clean
Fermi liquid
Pudalov et al. 2002

31. Sign of the correction:
different signs
compare with the
Lindhard function
Substitute into RPA:
Instability of the static theory ?

32. One has to redo the calculations at QCP
is obtained assuming weakly interacting Fermi liquid
Near a ferromagnetic transition
|Q| singularity vanishes at QCP
implies that there is no Fermi liquid at QCP in D=2
One has to redo the calculations at QCP

33. Within the Eliashberg theory
+ no vertex corrections
fermionic self-energy g
non –Fermi liquid at QCP
spin susceptibility
Analytic momentum dependence

34. Beyond Eliashberg theory
a fully universal non-analytic correction

35. Reasoning: Non-FL Green’s functions a non-analytic Q dependence
(same as in a Fermi gas)

36. Static spin susceptibility
Internal instability of z=3 QC theory in D=2

37. What can happen? Superconductivity affects a much larger scale
a transition into a spiral state
a first order transition to a FM
Belitz, Kirkpatrick, Vojta, Sessions, Narayanan
Superconductivity affects
a much larger scale
Non-analyticity affects

38. Conclusions static spin propagator is negative at QCP up to Q~ pF
A ferromagnetic Hertz-Millis critical theory is internally unstable in D=2
(and, generally, in any D < 3)
static spin propagator is negative at QCP up to Q~ pF
either an incommensurate ordering,
or 1st order transition to a ferromagnet

39. Collaborators D. Maslov (U. of Florida) C. Pepin (Saclay)
J. Rech (Saclay)
R. Haslinger (LANL)
A. Finkelstein (Weizmann)
D. Morr (Chicago)
M. Kaganov (Boston)
THANK YOU!