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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. Quantum phase transitions in itinerant ferromagnets. ZrZn 2. UGe2. pressure.
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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
Quantum phase transitions in itinerant ferromagnets ZrZn2 UGe2 pressure First order transition at low T
Itinerant electron systems near a ferromagnetic instability Fermi liquid Ferromagnetic phase What is the critical theory? What may prevent a continuous transition to ferromagnetism ?
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?)
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
The replacement of a FL at QCP is “Eliashberg theory” • spin susceptibility + no vertex corrections Altshuler et al Haslinger et al Pepin et al • fermionic self-energy (D=2) g non –Fermi liquid at QCP Still, second order transition Same form as for free electrons
Can something happen before QCP is reached? Khodel et al Rice, Nozieres Landau quasiparticle interaction function
Near quantum criticality In 2D This reasoning neglects Z-factor renormalization near QCP
mass renormalization Z-factor renormalization outside Landau theory within Landau theory
Z – factor renormalization Results:
In the two limits: the two terms are cancelled out anomalous piece regular piece
Where is the crossover? Low-energy analysis is justified only if
What else can destroy quantum criticality? 2. Superconductivity Spin-mediated interaction is attractive in p-wave channel first order transition Haslinger et al - SC
At QCP In units of
Superconductivity near quantum criticality UGe2 Superconductivity affects an ordered phase, not observed in a paramagnet
Always assumed What else can destroy quantum criticality? 3. Non-analyticity • Hertz-Millis-Moriya theory:
Why is that? Lindhard function in 3D Use RPA: is a Lindhard function Expand near Q=0 an analytic expansion
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?
Fermi Liquid • Self-energy • Uniform susceptibility • Specific heat
Corrections to the Fermi-liquid behavior Expectations based on a general belief of analyticity: Fermionic damping Resistivity
3D Fermi-liquid 50-60 th Fermionic self-energy: Specific heat: (phonons, paramagnons) Susceptibility Carneiro, Pethick, 1977 Belitz, Kirkpatrick, Vojta, 1997 non-analytic correction
In D=2 Spin susceptibility T=0, finite Q Q=0, finite T
Charge susceptibility No singularities
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
Spin susceptibility Q=0, finite T T=0, finite Q Only U(2pF) contibutes Specific heat
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
Arbitrary D Corrections are caused by Fermi liquid singularities in the effectively 1D response functions Extra logs in D=1 These non-analytic corrections are the ones that destroy a Fermi liquid in D=1
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
Sign of the correction: different signs compare with the Lindhard function Substitute into RPA: Instability of the static theory ?
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
Within the Eliashberg theory • spin susceptibility + no vertex corrections • fermionic self-energy g non –Fermi liquid at QCP Analytic momentum dependence
Beyond Eliashberg theory a fully universal non-analytic correction
Reasoning: Non-FL Green’s functions a non-analytic Q dependence (same as in a Fermi gas)
Static spin susceptibility Internal instability of z=3 QC theory in D=2
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
Conclusions 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
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!