Post on 30-Dec-2015
description
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
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?
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
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