Quasiparticle anomalies near ferromagnetic instability

A. A. KataninA. P. KampfV. Yu. Irkhin

Stuttgart-Augsburg-Ekaterinburg2004

Motivation

Study of low-dimensional itinerant ferromagnetism:

• Layered systems• Thin magnetic films• Surface electronic states in 3D materials (ARPES)

The properties of layered systems are expected to be completely different from cubic systems

The T-phase diagram: 2D vs 3D

PM (Fermi liquid,

well-defined QP)Ordered phase

T

QPT

3D

How do the physical properties evolve in the vicinity of a ferromagnetic instability ?

PM (Fermi liquid,

well-defined QP)

ordered phase

T*0

C exp(T*/T)

T

QPT

RC2D

+ small interlayer coupling

TC0/ln(t/t')

0CT

Theoretical predictions for NMR

''( ) ( , ) Im |

'( ')d TQ I I i I I

2 ( 1)( )

3 zi

i I IQ

A S

MF theory: Q()

zA S QzA S Q

Q()

zA S

AFM: FM:

2 22 ( 1)( ) , ( )

3 ( ) mm

I I iA SQ

i

PM state, 2D system:

V. Yu. Irkhin and M. I. Katsnelson, Z. Phys. B 62, 201 (1986); Eur. Phys. J. B 19, 401 (2001)

Q()

ASAS

Theoretical predictions for ARPES

A(kF,)

A(kF,)

• similar to an AFM, where the suppression of the spectral weight is due to opening of a gap

*T T

*T T

Simple RPA-like calculation

Im (kF,0) is divergent at T0.

This type of divergences was discussed earliar in the AFM context (A. M. Tremblay) and for gauge field theories (P. Lee et al.)

niq

nnnn iqiiqkGTUik

,

02 ),(),(),(

3D: Im (kF,0) is only weakly (logarithmically) divergent at the magnetic phase transition temperature T = Tc

The source of potentialdivergencies

qiqqUq

q/),(1

),(),( 22

0

0

0

qiCBqAq /),( 20 [ ]

2/3Im ( ,0) (( / ) ),0

Fk T tO T tT

Bare U:

)/exp()(

)(

**1

*

TTCTT

TT

Renorm. Uef:

2D

Interpretation of the results

k

A(k,)

k

Pre-formation of the two split Fermi surfaces already in the PM phase`at low T<<T*

2 1

1

( / ) ( )Re ( , )

(ln ) /( )

T t t

Tt t

k

k

k

The spectral function depends on -k only

Qualitative physical picture

What is the nature of the anomalies found in self-energy and spectral functions ?

Formation of dynamic “domains” of electrons with certain spin projection

Formation of the two pre-split Fermi surfaces already in PM phase

Approximations

We have neglected:

o momentum- and frequency dependence of the interaction; contributions of the channels of the electronic scattering other than the ph channelo self-energy and vertex corrections beyond the RPA-like diagrams

Functional renormalization-group approach

0

]***['* ''''

T

TTTTTT VGSVdTS

.= =

...1)(

1

)(1

)(

23

12

1

izaiza

az

VGSSGV

.V

22/1

2/1

)(2

1

k

k

k

n

nT

nT

i

i

TS

iT

G

Self-energy in the fRG

Results (Hubbard model, U = 4t, t'/t = 0.45, vH band filling n = 0.47, T = 0.1t)

Self-energy and vertex corrections

Two types of corrections to the results of non-self-consistent approach:

Self-energy corrections in the internal Green functions

Vertex corrections

q,in) k,in)

Self-energy and vertex corrections

nik

nnnnnnn iiqkGikGiiiqkkTiq

,

),(),(),;,(),(

niq

nnnnnnn iqiiqkGiiiqkkTik

,

),(),(),;,(),(

)],(1)[,;,(),;,( nnnnnnn iqUiiiqkkiiiqkk

Similar to QED: no equation for !

Dyson equations

h

iii nnn

),(1),;,(

kkkWard identity:

Similar approach was applied by Edwards and Hertz to the problem of strong FM

Approximations which are used

( , ; , ) ( , ; , )n n n ni i i i k k q k k

02 2

( ,0)q

qJustification: is strongly enhanced at q=0

+1/N expansion where N is the number of spin components

1),,,(

)6

6(2

1),(

02/1

02/1

kk

k

Self-consistent without vertex corrections(analogue of FLEX):

])23(

392[50

3),,,(

)(10

3),(

202

2201

2220

40

220

422

0

202

2201

2220

kk

k

The self-consistent solutionwith vertex and self-energy corrections:

)0(0Im

Results of the solution

Other observable quantities

The density of states

)]()([2

1)'()(),()( 000 AdA

k

k

The density of states is split already in PM phase

Static magnetic susceptibility

)()()(),(),;,(

1

0220,

20

0

00

difdikGiikk

U

nn in

iknnn

ef

2)()()( zGzzf

Triplet pairing

nn i

nik

nnnnpp igdikGikGiikk

)(),(),(),(),;,( 0,

triplettriplet

)()0,();'()'()',()',( triplet zfzgzzGzzGzzzzg pp

g(i,k)

k

k

Enhancement of the triplet pairing amplitude at small , k

Quantum critical regime

What about QC regime ?1 *

1

RC: exp( / )

QC : ~

T T T

T

T*0

T

QPT

RC

There are nosolid theoretical results for the value of exponent

the quantum spin fluctuations are less important than classical, the “inverse qp lifetime” but there are no well-defined qps

the quantum spin fluctuations are more important than classical, the guess “scattering rate”requires verification vs. vertex corrections; coincides with the result by W. Metzner et. al. near Pomeranchuk instability

| Im | T

2 / 3| Im | T

Summary

Ferromagnetic fluctuations invalidate quasiparticle picture at the paramagnetic FS at low T

New quasiparticles emerge at the points of the Brillouin zone with k is the ground-state spin splitting

The density of states is pre-split at T « T* Triplet pairing amplitude is greatly enhanced at the

ferromagnetic FSs already in the paramagnetic state

Future perspectives

Non-perturbative semi-analytical tool of investigation of self-energy and vertex corrections in spin systems with strong forward scattering –

Ward identity approach + 1/N expansion

Possible extensions and applications:

Inclusion of quantum fluctuations

Long-range ferromagnetic order

Extension to vH singularity problem

More accurate description of QC regimeComparison with experimental ARPES data

Description of criteria of ferromagnetism and spectral properties of 2D and 3D ferromagnetic systems between limits of weak (Moriya theory) and strong (saturated) (Edwards-Hertz approach) ferromagnetism

Possible experimental implication

Layered manganite compound La1+xSr2-xMn2O7

Phys. Rev. Lett. 81, 192 (1998)Phys. Rev. B 62, 1039 (2000).

TC=126K

Spectral properties in mean-field theory

iG

k

1),(k

Direction of the magnetization along z-axisA(kF,)

2/

iiii ccccU

Direction of the magnetization perpendicular to the spin quantization axis:

)11

(21

),(

ii

Gkk

k

A(k,)