Department of Physics Cornell Universitypages.physics.cornell.edu/~sethna/teaching/653/GroupProjects/QuantumAndFermi10/Ghosh...Magnetic Instabilities:Stoner RG ransfoTrmation and ows

Magnetic Instabilities:StonerRG Transformation and �owsQuantum-classical crossover

Summary

Stoner Ferromagnetic instability of a Fermi Liquid

Shivam Ghosh

Statistical Physics Class Presentation

Department of Physics

Cornell University

December 2, 2010

Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid


Summary

Outline

1 Magnetic Instabilities:Stoner

Stoner Ferromagnetism

QPT at T=0

2 RG Transformation and �ows

Landau-Ginzburg-Wilson functional for interacting

Paramagnons

Carrying out RG steps

3 Quantum-classical crossover



Summary

Basic ProblemQPT at T=0

Outline



QPT at T=0



Paramagnons





Summary


Stoner instability

A magnetic phase transition of a Fermi liquid with net non

zero magnetization

There can be other magnetically ordered states with no net

magnetization (eg. AFM) . General class of magnetic

transitions is speci�ed by ordering wave vectors−→Q . Also

inludes Spin Density Waves

Will focus here on Ferromagnetic Instability. A simple

illustration of QPT

Phase transition occurs on varying a system parameter

(Coulombic repulsion U in Stoner case)



Summary


Stoner instability


zero magnetization






illustration of QPT





Summary


Stoner instability


zero magnetization






illustration of QPT





Summary


Stoner instability


zero magnetization






illustration of QPT





Summary


Hubbard Model

Hubbard model Hhubbard =−t ∑<ij>

c†i cj +U∑

ini↑ni↓

Electrons reduce U by favoring magnetic ordering

Cost: Gain in Kinetic energy



Summary


Hubbard Model


c†i cj +U∑

ini↑ni↓





Summary


Hubbard Model


c†i cj +U∑

ini↑ni↓





Summary


Phase digram and Instability Criterion

Instability occurs when UN(EF ) > 1 (Mean �eld), System has

ferromagnetic ordering

E�ect of adding �uctuations can be seen in the divergence of

the Spin response function χ0(q,ω) which diverges for

q,ω → 0 for FM



Summary


Phase digram and Instability Criterion

Instability occurs when UN(EF ) > 1 (Mean �eld), System has

ferromagnetic ordering

E�ect of adding �uctuations can be seen in the divergence of

the Spin response function χ0(q,ω) which diverges for

q,ω → 0 for FM



Summary


Outline



QPT at T=0



Paramagnons





Summary


Hertz-1976

Want to study Stoner transition both at T = 0 and extend to

�nite T

At low T (compared to EF ) Quantum �uctuations bring out

new physics

Change of e�ective dimensionality: Dynamic exponent z

Quantum to Classical crossover exponent zν



Summary


Hertz-1976


�nite T


new physics





Summary


Hertz-1976


�nite T


new physics





Summary


Hertz-1976


�nite T


new physics





Summary


Phase Diagram with Quantum, Classical Exponents

Aim to understand Phase diagram using Renormalization

Group



Summary

Landau-Ginzburg-Wilson functional for interacting ParamagnonsCarrying out RG steps

Outline



QPT at T=0



Paramagnons





Summary


Order Parameter φ(~x ,τ)

We want to �nd an order parameter which distinguishes the

para-ferro phase

Obvious choice: magnetization density φ . Also need an

'e�ective' theory forS [φ ]- describing dynamics of φ

φ = φ(~x ,τ) is a �uctuating (in imaginary time τ ∼ β ) �eld.

Quantum �uctuations arise from dependance on τ!

About rc⇒ Classical statistics in a critical fan.

Quantum statistics elsewhere



Summary




para-ferro phase









Summary




para-ferro phase









Summary




para-ferro phase









Summary


φ in mode space

In Fourier space φ = φ(q,ω). ω captures the dynamics of QF

Compare to φ4 theory where φ = φ(q). No ω dependence. No

QF!

Statics and dynamics intricately mixed up in Quantum

statistics



Summary


φ in mode space



QF!


statistics



Summary


φ in mode space



QF!


statistics



Summary


Quantum Classical←→Classical d ←→ d +1

L

Lτ

ξ

ξτ

0

β

Coarse Grain

Close analogy with a '�nite' classical system. A box with

in�nite d dimensions and a �nite ′d +1′th dimension.

Quantum←→ Classical ⇐⇒ d ←→ d +1(ξ ∼ Ld+1)



Summary


Paramagnons

Fluctuations of φ(q,ω) quasiparticles wave vector q frequency

ω . Called Paramagnons! (Recall Mihir's lecture, running

horses etc.)

Mass r0 of paramagnons gets renormalized by scattering o�

electrons, holes. We will soon �nd how r0 �ows under RG.Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid


Summary


Paramagnons

Fluctuations of φ(q,ω) quasiparticles wave vector q frequency

ω . Called Paramagnons! (Recall Mihir's lecture, running

horses etc.)

Mass r0 of paramagnons gets renormalized by scattering o�

electrons, holes. We will soon �nd how r0 �ows under RG.Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid


Summary


Building S [φ ]

Begin with Hhubbard which describes hopping fermions ψfermion

interacting through a short ranged repulsive potential U

'Integrate out' ψfermion close to Fermi surface to get an

e�ective model for φ

S [φ ] = 12 ∑q,ω

(r0 +q2 + |ω|

q

)|φ(q,ω)|2 +

u04Nβ

∑qi ,ωi

φ(1)φ(2)φ(3)φ(4)δ (1,2,3,4)



Summary


Building S [φ ]





S [φ ] = 12 ∑q,ω

(r0 +q2 + |ω|

q

)|φ(q,ω)|2 +

u04Nβ

∑qi ,ωi

φ(1)φ(2)φ(3)φ(4)δ (1,2,3,4)



Summary


Building S [φ ]





S [φ ] = 12 ∑q,ω

(r0 +q2 + |ω|

q

)|φ(q,ω)|2 +

u04Nβ

∑qi ,ωi

φ(1)φ(2)φ(3)φ(4)δ (1,2,3,4)



Summary


Outline



QPT at T=0



Paramagnons





Summary


E�ective Action S [φ ]

S [φ ] = S (2)[φ ] +S (4)[φ ](A quadratic and a quartic part)

S (2)[φ ] = 12 ∑q,ω

(r0 +q2 + |ω|

q

)|φ(q,ω)|2

S (4)[φ ] =u04Nβ

∑qi ,ωi

φ(q1,ω1)φ(q2,ω2)φ(q3,ω3)φ(−∑3i=1 qi ,−∑

3i=1ωi )



Summary




S (2)[φ ] = 12 ∑q,ω

(r0 +q2 + |ω|

q

)|φ(q,ω)|2

S (4)[φ ] =u04Nβ

∑qi ,ωi

φ(q1,ω1)φ(q2,ω2)φ(q3,ω3)φ(−∑3i=1 qi ,−∑

3i=1ωi )



Summary




S (2)[φ ] = 12 ∑q,ω

(r0 +q2 + |ω|

q

)|φ(q,ω)|2

S (4)[φ ] =u04Nβ

∑qi ,ωi

φ(q1,ω1)φ(q2,ω2)φ(q3,ω3)φ(−∑3i=1 qi ,−∑

3i=1ωi )



Summary


Interaction parameters in S [φ ]

Seen r0 and q2 before.

F [m] =∫dd r

[r0m

2 + |∇m|2 +u0m4 + ...

]r0 = 1−UN(EF )∼ [U−Uc(T )]/Uc(T ) measures distance

from criticality

Ferromagnetic instability occurs at r0 = 0⇒ UN(EF ) = 1



Summary




F [m] =∫dd r

[r0m

2 + |∇m|2 +u0m4 + ...


from criticality




Summary




F [m] =∫dd r

[r0m

2 + |∇m|2 +u0m4 + ...


from criticality




Summary


RG Transformation

Step 1: Separate out modes into 'slow' and 'fast' and

integrate out 'fast' modes

Step 2: Rescale q and ω such that they lie in the original k

space interval

Step 3: Rescale �elds φ to absorb change in rede�nition of

couplings



Summary


RG Transformation




space interval


couplings



Summary


RG Transformation




space interval


couplings



Summary


RG Transformation




space interval


couplings



Summary


Shaving o� modes

The cut o� in summations for the momentum and energy are

Λ∼ 1/a and EF

Rescale q by q/Λ and integrate out modes lying in a thin shell

[Λe−`,Λ]

Similarly, rescale ω by EF and integrate out modes lying

within [Λe−`,Λ]

e−`⇐⇒ s−1 for Kyungmin's MSRG . It parametrizes RG steps



Summary


Shaving o� modes


Λ∼ 1/a and EF


[Λe−`,Λ]


within [Λe−`,Λ]




Summary


Shaving o� modes


Λ∼ 1/a and EF


[Λe−`,Λ]


within [Λe−`,Λ]




Summary


Shaving o� modes


Λ∼ 1/a and EF


[Λe−`,Λ]


within [Λe−`,Λ]




Summary


Step 2

After RG step 1

S (2)[φ ]→ S ′(2)[φ ] = 12βN

∫ e−`

0ddqdω

(2π)d+1

(r ′0 +q2 + |ω|

q

)|φ(q,ω)|2

Note: r0→ r ′0 under step 1 of RG (postponed till later)

To restore cuto� de�ne: q′ = qe`,ω ′ = ωez`(RG Step 2)

Notice the di�erent RG step sizes along q and ω .



Summary


Step 2

After RG step 1

S (2)[φ ]→ S ′(2)[φ ] = 12βN

∫ e−`

0ddqdω

(2π)d+1

(r ′0 +q2 + |ω|

q

)|φ(q,ω)|2






Summary


Step 2

After RG step 1

S (2)[φ ]→ S ′(2)[φ ] = 12βN

∫ e−`

0ddqdω

(2π)d+1

(r ′0 +q2 + |ω|

q

)|φ(q,ω)|2






Summary


Step 2

After RG step 1

S (2)[φ ]→ S ′(2)[φ ] = 12βN

∫ e−`

0ddqdω

(2π)d+1

(r ′0 +q2 + |ω|

q

)|φ(q,ω)|2






Summary


Dynamic exponent z!

Choosing q′ = qe−`,ω ′ = ωe−z`

S ′(2)[φ ] =12βNe−(d+z)`

∫ 10 D

′(r ′0 +q′2e−2` + |ω ′|e−z`

q′e−`

)|φ(q′e−`,ω ′e−z`)|2

If we now choose z = 3 we can make q′2 and |ω ′|/q′ havesame coe�.

Also, demand now φ ′(q′,ω ′) = e−(d+z+2)`/2φ(q,ω)

S ′(2)[φ ] = 12βN

∫ 10 D

′(r ′0e

2` +q′2 + |ω ′|q′

)|φ ′(q′,ω ′)|2

Under RG r0→ r(`) = r ′0e2`



Summary


Dynamic exponent z!


S ′(2)[φ ] =12βNe−(d+z)`

∫ 10 D

′(r ′0 +q′2e−2` + |ω ′|e−z`

q′e−`

)|φ(q′e−`,ω ′e−z`)|2



S ′(2)[φ ] = 12βN

∫ 10 D

′(r ′0e

2` +q′2 + |ω ′|q′

)|φ ′(q′,ω ′)|2




Summary


Dynamic exponent z!


S ′(2)[φ ] =12βNe−(d+z)`

∫ 10 D

′(r ′0 +q′2e−2` + |ω ′|e−z`

q′e−`

)|φ(q′e−`,ω ′e−z`)|2



S ′(2)[φ ] = 12βN

∫ 10 D

′(r ′0e

2` +q′2 + |ω ′|q′

)|φ ′(q′,ω ′)|2




Summary


Dynamic exponent z!


S ′(2)[φ ] =12βNe−(d+z)`

∫ 10 D

′(r ′0 +q′2e−2` + |ω ′|e−z`

q′e−`

)|φ(q′e−`,ω ′e−z`)|2



S ′(2)[φ ] = 12βN

∫ 10 D

′(r ′0e

2` +q′2 + |ω ′|q′

)|φ ′(q′,ω ′)|2




Summary


Dynamic exponent z!


S ′(2)[φ ] =12βNe−(d+z)`

∫ 10 D

′(r ′0 +q′2e−2` + |ω ′|e−z`

q′e−`

)|φ(q′e−`,ω ′e−z`)|2



S ′(2)[φ ] = 12βN

∫ 10 D

′(r ′0e

2` +q′2 + |ω ′|q′

)|φ ′(q′,ω ′)|2




Summary


Dynamic exponent z!


S ′(2)[φ ] =12βNe−(d+z)`

∫ 10 D

′(r ′0 +q′2e−2` + |ω ′|e−z`

q′e−`

)|φ(q′e−`,ω ′e−z`)|2



S ′(2)[φ ] = 12βN

∫ 10 D

′(r ′0e

2` +q′2 + |ω ′|q′

)|φ ′(q′,ω ′)|2




Summary


S (4)[φ ] term

Carry out same steps for quartic term to �nd

u0→ u(`) = u′0e[4−(d+z)]` = u′0e

ε`

ε = 4− (d + z)

To test stability of Gaussian �xed point : du(`)d` = εu(`)

Stable ε < 0⇒ d > 4− z .Expect Mean�eld exponents for d > 1



Summary


S (4)[φ ] term


u0→ u(`) = u′0e[4−(d+z)]` = u′0e

ε`

ε = 4− (d + z)





Summary


S (4)[φ ] term


u0→ u(`) = u′0e[4−(d+z)]` = u′0e

ε`

ε = 4− (d + z)





Summary


S (4)[φ ] term


u0→ u(`) = u′0e[4−(d+z)]` = u′0e

ε`

ε = 4− (d + z)





Summary


Step 1 & Flow equations

To get the �ow equations we need to carry out Step 1 carefully

Integrate out modes in outer shell e−` < q < 1,e−z` < ω < 1

De�ne fast φ>(q) and slow φ<(q) modes with q in outer and

inner shells

S (4) = u04βN ∑

qi ,ωi

∏4i=1(φ>(q) + φ<(q))iδ (∑i qi )δ (∑i ωi )

Only few terms out of 16 contribute!



Summary






inner shells

S (4) = u04βN ∑

qi ,ωi

∏4i=1(φ>(q) + φ<(q))iδ (∑i qi )δ (∑i ωi )




Summary






inner shells

S (4) = u04βN ∑

qi ,ωi

∏4i=1(φ>(q) + φ<(q))iδ (∑i qi )δ (∑i ωi )




Summary






inner shells

S (4) = u04βN ∑

qi ,ωi

∏4i=1(φ>(q) + φ<(q))iδ (∑i qi )δ (∑i ωi )




Summary






inner shells

S (4) = u04βN ∑

qi ,ωi

∏4i=1(φ>(q) + φ<(q))iδ (∑i qi )δ (∑i ωi )




Summary


Important contributions

2 kinds of terms contribute: φ<,1φ<,2φ<,3φ<,4 and

φ<,1φ<,2φ>,3φ>,4

S (4) =u0

4Nβ∑

qi<e−l

4

∏i

φ<,i (qi ,ωi )δ

(∑i

qi

)δ

(∑i

ωi

)+

+3u02Nβ

∑qi<e−l

|φ<(q,ω)|2 ∑1>qi>e−l

|φ>(q,ω)|2



Summary


Integrating out fast modes

Z> =∫Dφ>exp

[−1

2

(r0 +q2 + |ω|

q+ 3u0

βN ∑ |φ<(q,ω)|2)|φ>(q,ω)|2

]Z> =

(r0 +q2 + |ω|

q+ 3u0

βN ∑ |φ<|2)−1



Summary


Integrating out fast modes

Z> =∫Dφ>exp

[−1

2

(r0 +q2 + |ω|

q+ 3u0

βN ∑ |φ<(q,ω)|2)|φ>(q,ω)|2

]Z> =

(r0 +q2 + |ω|

q+ 3u0

βN ∑ |φ<|2)−1



Summary


Partition Function Z

Z =∫Dφ<

(S<,quadraticS<,quarticS>

)S<,quadratic = exp

(−1

2 ∑q<e−`

(r0 +q2 + |ω|

q

)|φ<|2

)

S<,quartic = exp

(−1

4u0βN ∑

q<e−`∏

4i φ<(qi ,ωi )δ

(∑iqi

)δ

(∑iωi

))

S> = exp

(−1

2 ∑q>e−`

ln

(r0 +q2 + |ω|

q+ 3u0

βN ∑q<e−`

|φ<|2))



Summary



Z =∫Dφ<


)S<,quadratic = exp

(−1

2 ∑q<e−`

(r0 +q2 + |ω|

q

)|φ<|2

)

S<,quartic = exp

(−1

4u0βN ∑

q<e−`∏

4i φ<(qi ,ωi )δ

(∑iqi

)δ

(∑iωi

))

S> = exp

(−1

2 ∑q>e−`

ln

(r0 +q2 + |ω|

q+ 3u0

βN ∑q<e−`

|φ<|2))



Summary



Z =∫Dφ<


)S<,quadratic = exp

(−1

2 ∑q<e−`

(r0 +q2 + |ω|

q

)|φ<|2

)

S<,quartic = exp

(−1

4u0βN ∑

q<e−`∏

4i φ<(qi ,ωi )δ

(∑iqi

)δ

(∑iωi

))

S> = exp

(−1

2 ∑q>e−`

ln

(r0 +q2 + |ω|

q+ 3u0

βN ∑q<e−`

|φ<|2))



Summary



Z =∫Dφ<


)S<,quadratic = exp

(−1

2 ∑q<e−`

(r0 +q2 + |ω|

q

)|φ<|2

)

S<,quartic = exp

(−1

4u0βN ∑

q<e−`∏

4i φ<(qi ,ωi )δ

(∑iqi

)δ

(∑iωi

))

S> = exp

(−1

2 ∑q>e−`

ln

(r0 +q2 + |ω|

q+ 3u0

βN ∑q<e−`

|φ<|2))



Summary


Renormalized couplings

Expanding log in S>to 4th order in φ<will give renormalized

couplings

r0→ r ′0 = r0 + 3u0βN ∑

q>e−`∑

ω>e−zl

(r0 +q2 + |ω|

q

)−1u0→ u′0 = u0− 9u20

βN ∑q>e−`

∑ω>e−zl

(r0 +q2 + |ω|

q

)−2



Summary




couplings

r0→ r ′0 = r0 + 3u0βN ∑

q>e−`∑

ω>e−zl

(r0 +q2 + |ω|

q

)−1u0→ u′0 = u0− 9u20

βN ∑q>e−`

∑ω>e−zl

(r0 +q2 + |ω|

q

)−2



Summary




couplings

r0→ r ′0 = r0 + 3u0βN ∑

q>e−`∑

ω>e−zl

(r0 +q2 + |ω|

q

)−1u0→ u′0 = u0− 9u20

βN ∑q>e−`

∑ω>e−zl

(r0 +q2 + |ω|

q

)−2



Summary


Flow equations and comparison toφ4

Flow equations:

dr(`)

d`= 2r(`) +A

′u(`)

du(`)

d`= (4− (d + z))u(`)−B

′u(`)2

Complex scalar φ4 theory

dr(`)

d`= 2r(`) +Au(`)

du(`)

d`= (4−d)u(`)−Bu(`)2

Notice appearance of z due to quantum �uctuations!

Draw same conclusions as for φ4 but due to quantum

statistics!Shivam Ghosh Statistical Physics Class Presentation Stoner Ferromagnetic instability of a Fermi Liquid


Summary



Flow equations:

dr(`)

d`= 2r(`) +A

′u(`)

du(`)

d`= (4− (d + z))u(`)−B

′u(`)2


dr(`)

d`= 2r(`) +Au(`)

du(`)

d`= (4−d)u(`)−Bu(`)2





Summary



Flow equations:

dr(`)

d`= 2r(`) +A

′u(`)

du(`)

d`= (4− (d + z))u(`)−B

′u(`)2


dr(`)

d`= 2r(`) +Au(`)

du(`)

d`= (4−d)u(`)−Bu(`)2





Summary



Flow equations:

dr(`)

d`= 2r(`) +A

′u(`)

du(`)

d`= (4− (d + z))u(`)−B

′u(`)2


dr(`)

d`= 2r(`) +Au(`)

du(`)

d`= (4−d)u(`)−Bu(`)2





Summary

Classical d ←→ d +1 transition

In analogy with the d ←→ d +1 transition in a classical

system living in a box in�nite in ′d ′ dimensions and �nite along′d +1′th dimension

Criticality occurs when correlation length along the �nite

dimension diverges!



Summary

Classical d ←→ d +1 transition

In analogy with the d ←→ d +1 transition in a classical

system living in a box in�nite in ′d ′ dimensions and �nite along′d +1′th dimension

Criticality occurs when correlation length along the �nite

dimension diverges!



Summary

Analogy to Quantum-Classical crossover

L

Lτ

ξ

ξτ

0

β

Coarse Grain

Similarly, our system also reaches criticality (with classical

statistics existing in a 'critical fan' about the criticalpoint)

when correlation length along imaginary time τ direction

diverges ξτ ∼ β



Summary

Analogy to Quantum-Classical crossover

L

Lτ

ξ

ξτ

0

β

Coarse Grain

Similarly, our system also reaches criticality (with classical

statistics existing in a 'critical fan' about the criticalpoint)

when correlation length along imaginary time τ direction

diverges ξτ ∼ β



Summary

The cross over exponent

Begin with a small ξτ (quantum regime) and transition to

classical as ξτ ∼ β

ξτ ∼ ξ z and since ξ ∼ r−ν

0 with ν = 1/2 we get ξτ ∼ r−zν

0

Cross over condition:

ξτ = β = r−zν

0 ⇒ T = r zν0 = ((U−Uc(T ))/Uc(T ))3/2

zν is a new cross over exponent!



Summary






0


ξτ = β = r−zν

0 ⇒ T = r zν0 = ((U−Uc(T ))/Uc(T ))3/2




Summary






0


ξτ = β = r−zν

0 ⇒ T = r zν0 = ((U−Uc(T ))/Uc(T ))3/2




Summary






0


ξτ = β = r−zν

0 ⇒ T = r zν0 = ((U−Uc(T ))/Uc(T ))3/2




Summary

Crossover exponent

Boundary between classical and quantum statistics given by

T = r zν0

Classical statistics for T > r zν0 and

Quantum statistics for T < r zν0



Summary

Summary

Looked at QPT in stoner ferromagnetism

Quantum dynamics changes condition for stability of Gaussian

�xed point

z enters as a dyanmic exponent, treats spatial and time

correlations di�erently

Also see a new quantum to classical classover exponent!



Summary

Acknowledgements

All of us would like to thank Prof. Sethna for giving us an

opportunity to work on this very interesting problem and for

many helpful discussions

Shivam would like to thank Kyungmin for the wonderful

�gures!

Have a good break!


Appendix

References

Hertz J.A. (1976) Phys.Rev .B 14, 1165

Chapter 12 'Phase Transitions of Fermi Liquids' Quantum

Phase Transitions, Subir Sachdev

Millis, A. J. (1993) Phys.Rev .B 48, 7183

'Renormalization-group approach to interacting fermions' R.

Shankar, Rev. Mod. Physics, Vol. 66, No. 1, January 1994


Appendix References

For Further Reading I

Hertz J.A.(1976)

Some Press, 1990.

S. Someone.

On this and that.

Journal on This and That. 2(1):50�100, 2000.


Documents

Department of Physics Cornell Universitypages.physics.cornell.edu/~sethna/teaching/653/GroupProjects/QuantumAndFermi10/Ghosh...Magnetic Instabilities:Stoner RG ransfoTrmation and ows