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” + 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

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 regular piece anomalous piece

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

Results:

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

Dome of a pairing instability above QCP

At QCP In units of

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

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

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

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 Fermionic self-energy: 50-60 th 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 Only U(2pF) contibutes Specific heat 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 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

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 ?

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

Within the Eliashberg theory + no vertex corrections fermionic self-energy g non –Fermi liquid at QCP spin susceptibility 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? 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

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

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!