Self-generated instability of aSelf-generated instability of a ferromagnetic quantum-critical ferromagnetic quantum-critical point point

Andrey Chubukov

University of Maryland

Workshop on Frustrated Magnetism, Sept. 14, 2004

1D physics in D >1

Quantum phase transitions in itinerant ferromagnets

ZrZn2

UGe2

First order transition at low T

pressure

Itinerant electron systems near a ferromagnetic instability

Fermi liquidFerromagnetic phase

What is the critical theory?

What may prevent a continuous transition to ferromagnetism ?

Quantum criticality

• Hertz-Millis-Moriya theory:

fermions are integrated out

... b ]Q

|| [Q d Qd S 422-2d

Z=3 Dcr = 4-Z =1

is a quantum critical point

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

D3

1

Coupling constant diverges at QCP

The replacement of a FL at QCP is “Eliashberg theory”

• spin susceptibility

• fermionic self-energy (D=2)

-12 |)Q|/|| (Q ) (Q,

1/30

3/2 ) (i )(

non –Fermi liquid at QCP

F

2

20 E

g

16

33

g

)( ) (k, + no vertex corrections

Same form as for free electrons

Altshuler et alHaslinger et alPepin et al

Still, second order transition

Can something happen before QCP is reached?

Khodel et alRice, Nozieres

d cos ) f(cos - m

1

m

1*

) f(cos Landau quasiparticle interaction function

Fk |p| |k |

0) ,p - k( Z ) f(cos 2

p - k q , q

1

2- 2

/2sin k 2 q F

2-2F

* ) cos -(1 k 2

cos d -

m

1

m

1

0at y singularit ,

-1

m mor ), - (1

m

1

m

1 **

critical is 1

In 2D

Near quantum criticality

This reasoning neglects Z-factor renormalization near QCP

B )]k -(k v- [i Z )k -(k Z v ) (k, F*FF F

Z-factor renormalizationmass renormalization

) Z- (1 m

1

m

1

1 ZB - Z

1

*

outside Landau theory

within Landau theory

) (k,

) ,q k(G ) (q, q d T

|q| /|| q

1

) i (

qd d B

2-22*qk

*F

-2

v offunction a is B

0,

0 0,1

Results:

) (small 1 when, m

m )B(

) (small 1 when , log

m

m )B(

1*

*

on dependence no

Z – factor renormalization

In the two limits:

damping) (small 1 1.

-1

m m 1, Z

(k), ) (k,

*

) (small 1 2. 1

)k-(k

v ))k-(k v- (i v Z ) (k, F*

FF

*FF

1? when Dangerous

regular piece anomalous piecethe two terms are cancelled out

*m/m Z locality! --- )( ) (k,

Where is the crossover?

O(1) at crossover v

offunction a is ) (k,in piece Regular""*F

-2

:|q| / || dampingLandau produces also

), (k, producesn that interactio same The

2FE

g ~ 1/2

FF

2-

E

g~ or ,

E

g ~

at )( to(k) fromCroosover

smallat already occursCrossover Low-energy analysis is justified only if FE g

Results:

1 m

m ,

1

1 Z

),( ) (k,*

-1

m m 1, Z

(k), ) (k,

*

1

E

2/1

F

g

2/1

FE

g

QCP before divergenot does mass

),( ) (k, 1, When

What else can destroy quantum criticality?

2. Superconductivity

Spin-mediated interaction is attractive in p-wave channel

-2 b

Haslinger et al

SC

-0

first ordertransition

Dome of a pairing instability above QCP

0

-2 b

1/30

3/2 ) (i )(

In units of

At QCP

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:

... b ]Q

|| [Q d Qd S 422-2d

Always assumed

Why is that?

Lindhard function in 3D

|2p-Q|

p 2 Qlog

p 8

p 4

2

1

p m 0)(Q,

F

F

F

22F

2F

0 Q

Q

Expand near Q=0

2F

2

2F

0 p 8

Q - 1

p m 0) (Q,

Use RPA: , (Q) U(Q)-1

(Q) (Q)

0

0

an analytic expansion

2 2-0

0

0

Q

(Q) U(Q)-1

(Q) (Q)

Q 0 is a Lindhard function

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?

(is there a protection against fractional powers of Q?)

Q:

Is there analyticity in a Fermi liquid?

Fermi Liquid

• Self-energy

• Uniform susceptibility

• Specific heat

TC(T)

const )0Q,0( T

)(

Corrections to the Fermi-liquid behavior

Expectations based on a general belief of analyticity:

223 , , Qδ χ(Q) Tδ χ(T) Tδ C(T)

22 )(// T

Resistivity 2T (T)

Fermionic damping

3D Fermi-liquid

Carneiro, Pethick, 1977

Q log Q )( 2spin Q Belitz, Kirkpatrick, Vojta, 1997

non-analytic correction

T log T C(T) 3Specific heat:

50-60 th

Susceptibility

22 )(// T Fermionic self-energy:

2charge Q )( Q

(phonons, paramagnons)

Spin susceptibility

F

2

2spin p

|Q|

4

Um

3

4m (Q)

F

2

spin E

T

4

Um

m 2 (T)

T=0, finite Q

Q=0, finite T

In D=2

Charge susceptibility

2charge Q (Q)

2 charge T (T)

No singularities

Singular corrections come from the universal singularities in the dynamical response functions of a Fermi liqiuid

Where the singularities come from?

ysingularit / q

y singularit q- 2/ Fp

• Only U(0) and U(2pF) are relevant

p v2 -

p 2

2p-q

p 2

2p-q - 1

m ) ,2p(q

2/12

FF

2

F

F

F

FF

2F

2 q) v( - 1

m ) , 0 (q

F

2

FF223

E

T )) U(2p U(0)- )p2( U)0((U 0.03m- C(T)

Specific heat

T=0, finite Q

F

2

F2spin p

|Q|

4

) U(2pm

3

4m (Q)

F

2

Fspin E

T

4

) U(2pm

m 2 (T)

Spin susceptibility

Q=0, finite T

Only U(2pF) contibutes

Only two vertices are relevant:

0 q q,

These two vertices are parts of the scattering amplitude

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

1D interaction in D>1 is responsible for singularities

)f(

0 q ,2p q F

k -k,k - k, k- k, k - k,

Arbitrary DDDD Qδ χ(Q) Tδ χ(T) Tδ C(T) , , 1

Extra logs in D=1

These non-analytic corrections are the ones that destroy a Fermi liquid in D=1

Corrections are caused by Fermi liquid singularities in the effectively 1D response functions

T /

A very similar effect in a dirty Fermi liquid:

Das Sarma, 1986Das Sarma and Hwang, 1999Zala, Narozhny, Aleiner 2002

A linear in T conductivity isa consequence of a non-analyticity of the response function in a clean Fermi liquid

Pudalov et al. 2002

Sign of the correction:

F

2

F2spin p

|Q|

4

) U(2pm

3

4m (Q)

2F

2

2F

0 p 8

Q - 1

p m 0) (Q,

compare with theLindhard function

Substitute into RPA: |Q| -

(Q) U(Q)-1

(Q) (Q)

2-0

0

0

different signs

Instability of the static theory ?

One has to redo the calculations at QCP

implies that there is no Fermi liquid at QCP in D=2

|Q| (Q) spin is obtained assuming weakly interacting Fermi liquid

Near a ferromagnetic transition , )(

/|Q| (Q) spin

|Q| singularity vanishes at QCP

Within the Eliashberg theory

• spin susceptibility

• fermionic self-energy

-12 |)Q|/|| (Q ) (Q,

1/30

3/2 ) (i )(

non –Fermi liquid at QCP

F

2

20 E

g

16

33

g

)( ) (k, + no vertex corrections

Analytic momentum dependence

Beyond Eliashberg theory

1/2F

3/2pin p Q 0.17 - (Q) s

2charge Q ~ (Q)

a fully universal non-analytic correction

Reasoning:

-1|)Q|/|| ( ) (Q,

a non-analytic Q dependence (same as in a Fermi gas)

Non-FL Green’sfunctions

Static spin susceptibility

11/2F

3/22spin )p Q 0.17 - (Q (Q)

Internal instability of z=3 QC theory in D=2

(Q) spin1

FQ/p

What can happen?

a transition into a spiral state

Belitz, Kirkpatrick, Vojta, Sessions, Narayanan

a first order transition to a FM

(Q) spin1

FQ/p

Superconductivity affects

2/1

FF E

g p 0.01 ~ Q

Non-analyticity affects , p 0.03 ~ Q F a much larger scale

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

• D. Maslov (U. of Florida)

• C. Pepin (Saclay)

• J. Rech (Saclay)

• R. Haslinger (LANL)

• A. Finkelstein (Weizmann)

• D. Morr (Chicago)

• M. Kaganov (Boston)

Collaborators

THANK YOU!