Self-generated instability of a ferromagnetic quantum-critical point

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Self-generated instability of a
ferromagnetic quantum-critical point
1D physics in D gt1
Andrey Chubukov
University of Maryland
Workshop on Frustrated Magnetism, Sept. 14, 2004
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Quantum phase transitions in itinerant
ferromagnets
ZrZn2
UGe2
pressure
First order transition at low T
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Itinerant electron systems near a ferromagnetic
instability
Fermi liquid
Ferromagnetic phase
What is the critical theory?
What may prevent a continuous transition to
ferromagnetism ?
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Quantum criticality

• Hertz-Millis-Moriya theory
• fermions are integrated out

is a quantum critical point
Z3
Dcr 4-Z 1
In any D gt1, the system is above its upper
critical dimension
(fluctuations are irrelevant?)
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• What can destroy quantum criticality?

1. Fermions are not free at QCP
ZF 1, Dcr 4 - ZF 3
Below D3, we do not have a Fermi liquid at QCP
Coupling constant
diverges at QCP
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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 (D2)

g
non Fermi liquid at QCP

• spin susceptibility

Still, second order transition
Same form as for free electrons
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Can something happen before QCP is reached?
Khodel et al Rice, Nozieres
Landau quasiparticle interaction function
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Near quantum criticality
In 2D
This reasoning neglects Z-factor renormalization
near QCP
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Z-factor renormalization
mass renormalization
outside Landau theory
within Landau theory
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Z factor renormalization
Results
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In the two limits
the two terms are cancelled out
regular piece
anomalous piece
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Where is the crossover?
Low-energy analysis is justified only if
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Results
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• What else can destroy quantum criticality?

2. Superconductivity
Spin-mediated interaction is attractive in p-wave
channel
first order transition
Haslinger et al
-
SC
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Dome of a pairing instability above QCP
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At QCP
In units of
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Superconductivity near quantum criticality
UGe2
Superconductivity affects an ordered phase, not
observed in a paramagnet
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• What else can destroy quantum criticality?

3. Non-analyticity

• Hertz-Millis-Moriya theory

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Why is that?
Use RPA
is a Lindhard function

• Lindhard function in 3D

Expand near Q0
an analytic expansion
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Analytic expansion in momentum at QCP is
related to the analyticity of the spin
susceptibility for free electrons
Is this preserved when fermion-fermion
interaction is included?
Q
(is there a protection against fractional powers
of Q?)
Is there analyticity in a Fermi liquid?
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Fermi Liquid

• Self-energy
• Uniform susceptibility
• Specific heat

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Corrections to the Fermi-liquid behavior
Expectations based on a general belief of
analyticity
Fermionic damping
Resistivity
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3D Fermi-liquid
50-60 th
Fermionic self-energy
Specific heat
(phonons, paramagnons)
Susceptibility
Carneiro, Pethick, 1977
Belitz, Kirkpatrick, Vojta, 1997
non-analytic correction
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In D2
Spin susceptibility
T0, finite Q
Q0, finite T
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Charge susceptibility
No singularities
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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

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Spin susceptibility
T0, finite Q
Q0, finite T
Only U(2pF) contibutes
Specific heat
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Only two vertices are relevant

• Transferred momenta are near 0 and 2 pF
• Total momentum is near 0

1D interaction in Dgt1 is responsible for
singularities
These two vertices are parts of the scattering
amplitude
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Arbitrary D
Extra logs in D1

• 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 D1
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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
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Sign of the correction
different signs
compare with the Lindhard function
Substitute into RPA
Instability of the static theory ?
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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
D2
One has to redo the calculations at QCP
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Within the Eliashberg theory
no vertex corrections

• fermionic self-energy

g
non Fermi liquid at QCP

• spin susceptibility

Analytic momentum dependence
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Beyond Eliashberg theory
a fully universal non-analytic correction
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Reasoning
Non-FL Greens functions
a non-analytic Q dependence (same as in a
Fermi gas)
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Static spin susceptibility
Internal instability of z3 QC theory in D2
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What can happen?
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
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Conclusions
A ferromagnetic Hertz-Millis critical theory is
internally unstable in D2
(and, generally, in any D lt 3)

• static spin propagator is negative at QCP up to
  Q pF
• either an incommensurate ordering,
• or 1st order transition to a ferromagnet

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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!